erdi+toth massaction

NONLINEAR SCIENCE THEORY AND APPLICATIONS

Numerical experiments over the last thirty years have revealed that simple nonlinear systems can have surprising and complicated behaviours. Nonlinear phenomena include waves that behave as particles, deterministic equations having irregular, unpredictable solutions, and the formation of spatial structures from an isotropic medium.

The applied mathematics of nonlinear phenomena has provided metaphors and models for a variety of physical processes: solitons have been described in biological macromolecules as well as in hydrodynamic systems; irregular activity that has been identified with chaos has been observed in continuously stirred chemical flow reactors as well as in convecting fluids: nonlinear reaction diffusion systems have been used to account for the formation of spatial patterns in homogeneous chemical systems as well as biological morphogenesis; and discrete-time and discrete-space nonlinear systems (cellular automata) provide metaphors for processes ranging from the microworld of particle physics to patterned activity in computing neural and self-replicating genetic systems.

Nonlinear Science: Theory and Applications will deal with all areas of nonlinear science - its mathematics, methods and applications in the biological, chemical, engineering and physical sciences.

Nonlinear science: theory and applications

Series editor: Arun V. Holden, Reader in General Physiology, Centre for Nonlinear Studies, The University, Leeds LS2 9NQ, UK

Editors: S. I. Amari (Tokyo), P. L. Christiansen (Lyngby), D. G. Crighton (Cambridge), R. H. G. Heileman (Houston), D. Rand (Warwick), J. C. Roux (Bordeaux)

Chaos A. V. Holden (Editor)

Control and optimization J. E. Rubio

Automata networks in computer science F. Fogelman Soulie, Y. Robert and M. Tchuente (Editors)

Oscillatory evolution processes I. Gumowski

Introduction to the theory of algebraic invariants of differential equations K. S. Sibirsky

Simulation of wave processes in excitable media V. Zykov (Edited by A. T. Winfree and P. Nandapurkar)

Almost periodic operators and related nonlinear integrable systems V. A. Chulaevsky

Other volumes are in preparation

Mathematical models of '· chemical reaction~_/

Theory and applications of deterministic and stochastic models

P. Erdi and J. T6th Central Research Institute for Physics, Hungarian Academy of Sciences and Computer and Automation Institute, Hungarian Academy of Sciences

c

Manchester University Press

Copyright© P. Erdi and J. Toth 1989

Published by Manchester University Press Oxford Road, Manchester MI3 9PL, UK

British Library cataloguing in publication data Erdi, P.

Mathematical models of chemical reactions: theory and applications of deterministic and stochastic models.- (Nonlinear science) I. Chemical reactions - Mathematical models I. Title II. Toth, J. III. Series 541.3'9'0724 QD501

ISBN 0 7190 2208 8 hardback

Typeset in Times 10/12 pt by Graphicraft Typesetters Ltd, Hong Kong

Printed in Great Britain by Biddies Ltd., Guildford and King's Lynn

Contents

Preface and acknowledgements

Symbols used in the text

1 Chemical kinetics: a prototype of nonlinear science 1.1 Mass action kinetics: macroscopic and microscopic

XI

xm

approach I 1.2 Physical models of chemical reactions 4 1.3 Deterministic and stochastic models 6 1.4 Regular and exotic behaviour II 1.5 Chemical kinetics as a metalanguage 12

2 The structure of kinetic models 14 2.1 Temporal processes 14 2.2 Properties of process-time 14

2.2.1 Discrete versus continuous 15 2.2.2 Time, thermodynamics, chemical kinetics 15

2.3 Structure of state-space 16 2.3.1 Discrete versus continuous 16 2.3.2 State and site 17

2.4 Nature of determination 18 2.5 X YZ models 19

3 Stoichiometry: the algebraic structure of complex chemical reactions 21

3.1 Conventional stoichiometry 21 3.2 Atom-free stoichiometry 26 3.3 Retrospective and prospective remarks. Suggested further

reading 28 3.4 Exercises 29

vi

3.5 Problems 3.6 Open problems

4 Mass action kinetic deterministic models 4.1 Kinetic equations: their structure and properties

4.1.1 Introduction 4.1.2 Introduction reconsidered 4.1.3 Exercises 4.1.4 Problems 4.1.5 Open problems

4.2 Verifications and falsifications of traditional beliefs 4.2.1 The zero deficiency theorem 4.2.2 Vol'pert's theorem 4.2.3 Remarks on related literature 4.2.4 Exercises 4.2.5 Problems 4.2.6 Open problems

4.3 Exotic reactions: general remarks 4.4 Multistationarity

4.4.1 Multistability 4.4.2 Multistationarity in kinetic experiments 4.4.3 Multistationarity in kinetic models of continuous

flow stirred tank reactors 4.4.4 Exercises 4.4.5 Problems 4.4.6 Open problems

4.5 Oscillatory reactions: some exact results 4.5.1 Periodicity in kinetic experiments 4.5.2 Excluding periodicity in differential equations 4.5.3 Excluding periodicity in reactions 4.5.4 Sufficient conditions of periodicity in differential

equations 4.5.5 Sufficient conditions of periodicity in reactions 4.5.6 Designing oscillatory reactions 4.5.7 Overshoot-undershoot kinetics 4.5.8 Exercises 4.5.9 Problems 4.5.10 Open problems

4.6 Chaotic phenomena in chemical kinetics 4.6.1 Chaos in general 4.6.2 Chaos in kinetic experiments 4.6.3 Chaos in kinetic models

Contents

29 32

33

33

33 35 39 39 40

40

42 45 46 47 48 48

49 49

49 50

50 51 52 52 54

54 54 55

55 56 56 57 57 58 59

59

59 60 61

Contents VII

4.6.4 On the structural characterisation of chaotic chemical reactions 62

4.6.5 Problems 62 4.6.6 Open problems 63

4.7 The inverse problems of reaction kinetics 63 4.7.1 Polynomial differential equations, kinetic differential

equations, kinetic initial value problems 64 4.7.1.1 Polynomial and kinetic differential equations 64 4.7.1.2 Further problems 65 4.7.1.3 The density of kinetic differential equations 67 4.7.1.4 Uniqueness questions 67 4.7.1.5 A sufficient condition for the existence of an

inducing reaction of deficiency zero 69 4.7.1.6 On the inverse problem of generalised

compartmental systems 72 4.7.2 The classical problem of parameter estimation 74 4.7.3 Exercises 74 4.7.4 Problems 75 4.7.5 Open problems 75

4.8 Selected addenda 75 4.8.1 Lumping 75

4.8.1.1 Lumping in general 75 4.8.1.2 Lumping in reaction kinetics 76 4.8.1.3 Possible further directions 77

4.8.2 Continuous components 78 4.8.3 Kinetic gradient systems 80 4.8.4 Structural identifiability 82 4.8.5 Parameter sensitivity 83 4.8.6 Symmetries 84 4.8.7 Principle of quasistationarity 88 4.8.8 Exercises 89 4.8.9 Problems 89 4.8.10 Open problem 90

5 Continuous time discrete state stochastic models 91 5.1 On the nature and role of fluctuations: general remarks 91

5.1.1 The logical status of stochastic reaction kinetics 91 5.1.2 Fluctuation phenomena in physics and chemistry:

an introduction 93 5.1.2.1 Stochastic thermostatics, stochastic thermo-

dynamics 93 5.1.3 Stochastic processes: concepts 96

viii Contents

5.1.3.1 Introductory remarks 96 5.1.3.2 Continuous state-space processes 97 5.1.3.3 Discrete state-space processes 99

5.1.4 Operator semigroup approach: advantages coming from the use of more sophisticated mathematics 99

5.2 Stochasticity due to internal fluctuations: alternative models 101 5.2.1 Some historical remarks I 0 I 5.2.2 Models 102

5.3 On the solutions of the CDS models I 05 5.3.1 General remarks I 05 5.3.2 Chemical reaction X !... Y I 06 5.3.3 Compartmental systems 107 5.3.4 Bicomponential reactions: general remarks 107 5.3.5 Chemical reaction X + Y ~ Z I 08

5.3.5.1 The master equation 108 5.3.5.2 Use of Laplace transformation 108 5.3.5.3 Determination of expectation I 08 5.3.5.4 The behaviour of the reaction during the

initial period of the processes 109 5.3.5.5 Determination ofstationary distribution 109

5.3.6 General equation for the generating function 109 5.3.7 Approximations 110 5.3.8 Simulation methods 112

5.4 The fluctuation-dissipation theorem of chemical kinetics 115 5.4.1 Stochastic reaction kinetics: 'nonequilibrium

thermodynamics of state-space'? 115 5.4.2 Fluctuation-dissipation theorem of linear

nonequilibrium thermodynamics 116 5.4.3 Determination of rate constants from equilibrium

fluctuations: methods of calculation 117 5.5 Small systems 119

5.5.1 Enzyme kinetics 119 5.5.2 Ligand migration in biomolecules 121 5.5.3 Membrane noise 123 5.5.4 Kinetic examinations of fast reactions 125

5.6 Fluctuations near instability points 128 5.6.1 An example of the importance of fluctuations 128 5.6.2 Stochastic Lotka-Volterra model 129 5.6.3 Stochastic Brusselator model 130 5.6.4 The Schlogl model of second-order phase transition 131 5.6.5 The Schlogl model of first-order phase transition 134 5.6.6 Stochastic theory of bistable reactions 135

Contents tx

5. 7 Stationary distributions: uni- versus multimodality 138 5.7.1 The scope and limits of the Poisson distribution in

the stochastic models of chemical reactions: motivations 138

5.7.2 Sufficient conditions of unimodality 140 5.7.3 Sufficient condition for a Poissonian stationary

distribution 142 5.7.4 Multistationarity and multimodality 143 5.7.5 Transient bimodality 144

5.8 Stochasticity due to external fluctuations 146 5.8.1 Motivations 146 5.8.2 Stochastic differential equations: some concepts and

comments 147 5.8.3 Noise-induced transition: an example for white noise

idealisation 149 5.8.4 Noise-induced transition: the effect of coloured

noise 151 5.8.5 On the effects of external noise on oscillations 153 5.8.6 Internal and external fluctuations: a unified

approach 156 5.8.7 Estimation of reaction rate constants using

stochastic differential equations 157 5.8.8 Exercises I 58 5.8.9 Problems 159 5.8.10 Open problem 159

5.9 Connections between the models 159 5.9.1 Similarities and differences: some remarks 159 5.9.2 Blowing up 159 5.9.3 Kurtz's results: consistency in the thermodynamic

limit 160 5.9.4 Exercise 160 5.9.5 Problems 161

6 Chemical reaction accompanied by diffusion 162 6.1 What kinds of models are relevant? 162 6.2 Continuous time, continuous !>tate-space deterministic

models 163 6.3 Stochastic models: difficulties and possibilities 167

6.3.1 Introductory remarks 167 6.3.2 Two-cell stochastic models 168 6.3.3 Cellular model 169 6.3.4 Other models 171

x Contents

6.4 Spatial structures 172 6.5 Pattern formation and morphogenesis 174

7 Applications 177 7.1 Introductory remarks 177 7.2 Biochemical control theory 177 7.3 Fluctuation and oscillation phenomena in neurochemistry 185 7.4 Population genetics 192 7.5 Ecodynamics 194

7.5.1 The theory of interacting populations 194 7.5.1.1 Boulding ecodynamics 194 7.5.1.2 Compartmental ecokinetics 195 7.5.1.3 Generalised Lotka-Volterra models 196 7.5.1.4 The advantages of stochastic models:

illustrations 199 7.5.2 An ecological case study 202

7.5.2.1 Arguments for a stochastic model 202 7.5.2.2 A common description of the deterministic

and stochastic models 204 7.5.2.3 Exercise 207

7.6 Aggregation, polymerisation, cluster formation 207 7.7 Chemical circuits 210 7.8 Kinetic theories of selection 213

7.8.1 Prebiological evolution 213 7.8.1.1 Introductory remarks 213 7.8.1.2 The hypercycle: The basic model 214

7.8.2 The origin of asymmetry of biomolecules 216

References 220

Index 252

Preface and acknowledgements

Chemical kinetics may be considered as a prototype of nonlinear science, since the velocity of reactions is generally a nonlinear function of the quantities of the reacting chemical components.

Both deterministic and stochastic models can be defined to describe the kinetics of chemical reactions macroscopically. (Microscopic models are out of the scope of this book.) The usual deterministic model is a subclass of systems of polynomial differential equations. Qualitative dynamic behaviour of the model can be analysed knowing the structure of the reaction network. Exotic phenomena such as oscillatory, multistationary and chaotic behaviour in chemical systems have been studied very extensively in the last fifteen years. These studies certainly have modified the attitude of chemists, and 'exotic' begins to become 'common'. Stochastic models describe both internal and external fluctuations. In general, they are a subclass of Markovian jump processes. Two main areas are particularly emphasised, which prove the importance of stochastic aspects. First, kinetic information may be extracted from noise measurements based upon the fluctuation-dissipation theorem of chemical kinetics; second, noise may change the qualitative behaviour of systems, particularly in the vicinity of instability points.

In addition to the direct utility of mathematical models in the analysis of\ complex chemical systems a unified conceptual framework is offered to the \ mathematical treatment of problems of chemical kinetics and related areas in !

biomathematics. Biochemical control processes, oscillation and fluctuation / p~enomena in neurochemical systems, coexistence and extinction in popul- j at10ns, prebiological evolution and certain ecological problems of Lake 1 Balaton can be treated in terms of this framework. Though the main body of/ the book deals with spatially homogeneous systems, spatial structures in chemical systems, pattern formation and morphogenesis related to reaction-diffusion models are also mentioned briefly.

The material that the book contains has until now been scattered in journals and proceedings. In spite of the undoubted penetration of modern mathematical techniques into the kingdom of chemical kinetics, the gap between the practical necessity of the chemist and the theorems of the

xii Preface and acknowledgements

mathematician is obvious. This book is the result of a ·quasicontinuous discussion between a chemist (P.E.) and a mathematician (J.T.) about mathematics and the natural sciences, and we certainly do not want it to be either a 'user's manual' for kineticists or a zoo of 'pseudo-applications' of

, mathematical problems. What we hope to emphasise is that chemical

l1 kinetics, a beautiful discipline in its own right, is really a pro,totype of nonlinear science.

We are deeply indebted to many Hungarian scientists working in Buda and Pest, Debrecen and Veszprem, with whom we have had many discussions, informal and formal, about mathematical techniques and theoretical concepts of chemistry.

We would like to express particular gratitude to our former colleague, Vera Hars with whom we have discussed many topics studied and not studied in this book.

Many problems on stochastic kinetics have been debated with Professor Michel Moreau (Paris). One of us (P.E.) enjoyed his hospitality in June 1985; the main part of Chapter 5 was written during this time. Some short visits to Bordeaux were useful for both of us.

Very special thanks to Dr Holden for inviting us to contribute this monograph to the series.

A large part of the figures are reprinted with permission of the authors (as cited at the appropriate place) and of the copyright holders, as follows: Acta Biochimica, Elsevier Scientific Publishers Ireland Ltd, North-Holland Physics Publishing, Pergamon Journals Ltd, Plenum Publishing Corp., Publishing House of the Hungarian Academy of Sciences.

Budapest, February 1986

Peter Erdi, Janos T6th


Chapter 1

A

A( solid)

A

a

a •.•

8

B(solid)

b

c

c

notation for an abstract chemical component (or species) in the elementary reaction aA + bB -+ cC + dD reactant component in the solid phase reaction A(solid) -+ B(solid) + C(gas)

matrix of infinitesimal transition probabilities (infinitesimal operator) of the stochastic model of reactions stoichiometric coefficient (or molecularity) of the chemical component A in the elementary reaction aA + bB -+ cC + dD element of the matrix of infinitesimal transition probabilities (infinitesimal operator) of the stochastic model of reactions element of the matrix of infinitesimal transition probabilities (infinitesimal operator) of the stochastic model of reactions notation for an abstract chemical component (or species) in the elementary reaction aA + bB -+ cC + dD product component in the solid phase reaction A(solid) -+ B(solid) + C(gas)

stoichiometric coefficient (or molecularity) of the chemical component B in the elementary reaction aA + bB -+ cC + dD notation for an abstract chemical component (or species) in the elementary reaction aA + bB -+ cC + dD product component in lhe solid phase reaction A(solid) -+ B(solid) + C(gas)

stoichiometric coefficient (or molecularity) of the chemical component C in the elementary reaction aA + bB -+ cC + dD initial vector of concentrations

xiv

c(t)

c(t)

c1 (t), c2 (t), c3 (t) D

d

f

/;,jj ks M

N

NM n(t) n, n'

~M

r(t)

-S-ST t v


vector of concentrations of chemical components at time t (the unit of a component of the concentration vector is to be understood as kmol/m3 here and everywhere below) time derivative of the concentration vector versus time function at time t (the unit of a component of the time derivative of the concentration vector is to be understood as kmol/m3 s) here and everywhere below) concentration of the component A at time m the elementary reaction aA + bB-+ cC + dD concentration of the component B at time in the elementary reaction aA + bB-+ cC + dD concentration of the component C at time in the elementary reaction aA + bB-+ cC + dD concentration of the component D at time m the elementary reaction aA + bB -+ cC + dD components of the vector of concentrations at time t notation for an abstract chemical component (or species) in the elementary reaction aA + bB-+ cC + dD stoichiometric coefficient (or molecularity) of the chemical component D m the elementary reaction aA + bB-+ cC + dD right-hand side of the kinetic differential equation scalar valued functions of the state variables in eqn ( 1.6) Boltzmann's constant (1.38 x 10- 23 J/K) number of chemical components in the investigated complex chemical reaction (perhaps after lumping) number of chemical components of a chemical system (if this number is very large) space of M-dimensional vectors of nonnegative integers number of components at time t possible number of components in the stochastic model probability of the event written in parenthesis is the absolute distribution of the number of components at time t space of M-dimensional vectors of real numbers reaction rate of the elementary reaction aA + bB-+ cC + dD (kmol/(m 3 s)) covalent bond between sulphuric atoms absolute temperature (K) time (s) volume of the chemical system (m- 3)

scalar valued functions of time in eqn (1.6) a vector valued stochastic process describing the time evolution of the number of components at time t

Symbols used in the text XV

Chapter 2

A g(t) Jf

X

Xo

<f>x. <p(t, . )

!l(t) co(g( t), t)

Chapter 3

A .s:1 (a) Ac-CoA ADP ATP c E r(r) M M+

this sign is used throughout to mean 'is defined by' or 'is by definition'

state-space of a dynamic system state of a thermodynamic system at time t function giving the state of a thermodynamic system at time t as a function of site and time site of a thermodynamic system at time t history of the site of a thermodynamic system up to time t the set of positive integers the set of real numbers euclidean space of M-dimensional real vectors number of constitutive quantities point of the time domain T of a dynamic system time domain of a dynamic system point of the time domain T of a dynamic system state of a dynamic system initial state of a dynamic system a function from T x A into A describing the time evolution of a dynamic system for different initial states the set of integers the set (or lattice) of ordered M-tuples of integers motion of a dynamic system starting from state x0

automorphism from the state-space A onto the state-space itself of a dynamic system constitutive quantities of a thermodynamic system at time t value of the constitutive functional of a thermodynamic system at timet this sign is used throughout to mean 'is defined by' or 'is by definition'

(external) chemical component atoms (a= I, 2 .. . A) acetyl coenzyme A adenosine 5 '-diphosphate adenosine 5'-triphosphate enzyme-substrate complex in the Michaelis-Menten reaction enzyme in the Michaelis-Menten reaction length of the reactant complex vector ex(·, r) number of the chemical components rank deficiency of the stoichiometric matrix

xvi

NA f\\liAxR

p IR+ IRM R rank(·) s

T

X X(n) y z Z(·, m) z !X(m, r) !X(., r) p(m, r) Ph r) y y(., r) p(m) p

0

Chapter 4

A a b c d E


space of A-dimensional vectors of nonnegative integers the set of A x R matrices of integers product in the Michaelis-Menten reaction rank deficiency of the atomic matrix space of M-dimensional vectors of real numbers number of the elementary reactions rank of the matrix in parenthesis substrate in the Michaelis-Menten reaction; dimension of the stoichiometric space notation for the transpose chemical component chemical component (n = 1, 2 ... M) chemical component chemical component formula vector of the mth component the atomic matrix stoichiometric coefficient in the reactant complex the rth reactant complex vector stoichiometric coefficient in the product complex the rth product complex vector the stoichiometric matrix the rth elementary reaction vector formal mass of the mth component vector of formal masses of the chemical components (may be considered as having the dimension kg) empty or zero complex containing no chemical component this sign is used throughout to mean 'is defined by' or 'is by definition'

Riemannian manifold parameter of the Lorenz equation parameter of the Lorenz-, and of the Rossler equation parameter of the Rossler equation parameter of the Rossler equation set of positive equilibrium points

~(IR X A) f

the set of functions with domain in IR and range in A right-hand side of the induced kinetic differential equation

f GL(M) Hi, ... , HN-l h

parameter of the FKN (Field-K~ros-Noyes)-model group of (invertible) general linear transformations first integral length of a short time interval

Symbols used in the text xvii

k k 1 ,k2 , ••• ,k(r) k+ (r), k- (r)

L M N o(h) q R IR R, R2 R3 r §

s s T T T, T, t v w X(m) x, x 1(t), x 2 (t), ... x(t) Xo (x*, y*) y y, y' z a.(m, r) a.(·,r) P(m, r) P( ·, r) y

y(.' r) 0 0/ A. j.l

v P(m) p

vector of rate constants rate constants rate constants of a reversible complex chemical reaction number of linkage classes lumping matrix number of complexes a quantity of smaller order than h parameter of the FKN-model number of the elementary reactions the set of real numbers the number of first-order endpoints the number of second-order endpoints the number of entry points parameter of the Lorenz equation stoichiometric space dimension of the stoichiometric space parameter of the FKN-model time transformation N-dimensional domain transformation in a dynamical system time potential (to a gradient system) parameter of the FKN-model chemical component (m = I, 2, ... , M) concentrations of chemical components vector of concentrations at time t vector of initial concentrations equilibrium concentration concentrations of chemical components complex vector concentrations of chemical components stoichiometric coefficient in the reactant complex the rth reactant complex vector stoichiometric coefficient in the product complex the rth product complex vector the stoichiometric matrix the rth elementary reaction vector deficiency deficiency of the /th linkage class measure in the continuous component model measure in the continuous component model matrix defining the generalised Lotka-Volterra model formal mass of the mth component vector of formal masses of the chemical components

xviii

0"

0

Chapter 5

A A A A(x) Am a, a1 , a 2

B B B B(x) Bm b b(t) b(X,) b lib II c C, C 1 , C2

C(·) C1(t) C2 (t) c c(t) c. D D D Dj DZ d(t) q)

DNA E E e(t) F F, f


parameter of the Lorenz equation empty or zero complex containing no chemical component

chemical component infinitesimal generator constant coefficient function constant constants chemical component cofactor matrix constant coefficient function constant constant quantity of component B diffusion function eigenvector determinant chemical component constants correlation function time-dependent function time-dependent function concentration quantity of component C coefficient chemical component domain of operators initial number of components velocity of the conditional moments variance of a random variable quantity of component D differential operator deoxyribonucleic acid enzyme expectation of a random variable quantity of enzyme E generating function marginal generating function conditional probability function

Symhols used in the text xix

f function in the domain of A

f forcing function G(x, A) auxiliary function g(x, t) absolute probability density function g(jil) transition factor gK conductivity of potassium channel g(x) auxiliary function H integration constant H magnetic field H Hamiltonian-like function h function in the Banach space Z h auxiliary function /(ro) spectrum line I, dichotomous noise

state of the system J Jacobian of likelihood function J(t) measured current p current through a single open channel j state of the system K normalisation constant k constant k state of the system k, kl rate constants k estimate of rate constant ks Boltzmann constant L linear hull L well Lmax number of wells /I' /2 rate constants M dimension of the state-space M constant M magnetisation M number of channels M KL phenomenological coefficient m expectation of the Poissonian distribution No set of nonnegative integers Nexp experimental value of molecule numbers NI (t) number of channels N"o. Neb Ne2 number of channels in equilibrium .AI normalisation constant n number of particles in volume A V P product Po(s, t) transition probability Pk(t) absolute distribution

XX

Pt(t) ~

p pjk(t) pj~(t) qj qn IRI IR+ IR R ial(x) r s s S(ro), S( v) S1( ro) // s s s(t) T T T, 11 t u u, ill u v v v AV w wn w, W(xJx') w(x/ VJx' / V) W;(t) X X X, X x(t)


the probability that one channel is open probability pressure infinitesimal transition probability Kurtz-type infinitesimal transition probability parameter of the exponential distribution constant set of real numbers set of positive numbers set of reactions Raoult constant polynomial reaction vector substrate entropy spectral density function current spectrum set of components point of the time domain 11 variable in the Laplace transform quantity of substrate temperature upper limit of time interval continuous semigroup time parameter set point of time 'free-energy-like' function external parameter transformed variable potential system volume transformed variable volume of a 'small' subsystem Brownian motion a Markov process Wiener process infinitesimal transition rate infinitesimal transition rate time-dependent functions chemical component state of the system stochastic process Lmax-dimensional vector state of the system


Xm y y y(t) y z z z

(l

(l

& (l( t)

OtE IR" (l

p p p ~ ~ r. r. f y y y(ro) y 0 0 OKD

E

s, s, TJ, e 3 I(

A A AA., A8

A., A.,, A.2 A; A. 1.1 1.1(.)

extrema of stationary probability density function chemical component Poisson process expectation state of the system chemical component Banach space complex number

stoichiometric coefficient transformed constant rate constant 'distance' from the equilibrium solution of a linear equation constant stoichiometric coefficient transformed variable constant rate constant transformed constant constants to be determined transformed variable auxiliary function Fourier transform of D1

rate constant auxiliary function ball radius in the Lindeberg condition Dirac delta measure of 'deviation' from the white noise idealisation fluctuating parameter Ornstein-Uhlenbeck process stochastic process time interval zero complex parameter of noise strength parameter of the process Y parameter in Subsection 5.5.2 eigenvalues stationary stochastic process nontrivial eigenvalues constants rate constant rate constant death rate function

xxi

XXJI

J.l J.l J.lo JlK v

~I ~ crJ t

t

X ljl ljl(.) (l)

Chapter 6

A A A a a IX a (IX', y')

8 B b b c c r

death rate constant bifurcation parameter bifurcation point eigenvalue angular frequency stochastic process stochastic process variance dimensionless time shift of time relaxation time potential function auxiliary functions stochastic dynamic system birth rate constant birth rate function frequency


this sign is used throughout to mean 'is defined by' or 'is by definition'

chemical component parameter of the Gierer-Meinhardt model generator of the cellular stochastic model of Kurtz lower bound of the (real, physical, one-dimensional) space parameter of a biochemical system homogeneous stable steady state concentration of the activator in the Gierer-Meinhardt model argument of the maximum of the stationary distribution in a two-cell model chemical component parameter of the Gierer-Meinhardt model upper bound of the (real, physical, one-dimensional) space parameter of a biochemical system chemical component parameter of the Gierer-Meinhardt model parameter of a biochemical system parameter of a biochemical system diffusion matrix parameter of the Gierer-Meinhardt model parameter of the Gierer-Meinhardt model parameter of the Gierer-Meinhardt model component formation vector

Symhols used in the text XXlll

f G g y (y', cr')

h i L; n1 nz P(n 1 , n2 ; t) P*(x, y; t)

p

QT Rl Rz 1!\11"

r

sn ~

s T T t

u U(r, t) u

v v X;, X

(x) (y) L\ L\ v(x) P;(X, t) p(x) <p(r, t) Q

component formation function group of transformations component formation function homogeneous stable steady state argument of the maximum of the stationary distribution in a two-cell model concentration of the inhibitor in the Gierer-Meinhardt model number of component (uniformly) elliptic differential operator number of molecules in cell I in a two-cell model number of molecules in cell 2 in a two-cell model value of the absolute distribution function probability of finding cell I in region (x) and cell 2 in region (y) concentration of a biochemical component = Q X (Q, T) reaction operator acting on n1

reaction operator acting on n2

set of n-dimensional real vectors cell number a characteristics of the Gaussian noise <p(r, t) concentration of a biochemical component the first neighbour of cell r an element of the group of transformations upper bound of the time domain point of the time domain chemical component function of two variables, periodic in the second variable concentration of chemical component U chemical component concentration of chemical component V number of particles of species i in cell r point of the (real, physical, one-dimensional) space region around state x in a two-cell model region around state y in a two-cell model Laplacian operator discrete Laplacian operator unit outward normal at place x mass density of component i at place x and at time t initial mass density at place x Gaussian noise domain in m-dimensional space this sign is used throughout to mean 'is defined by' or 'is by definition'

xxiv Symbols used in the text

Chapter 7

c c Co

E

s So

v

chemical component enzyme-substrate complex concentration of the enzyme-substrate complex initial concentration of the enzyme-substrate complex chemical component enzyme concentration of the enzyme initial concentration of the enzyme Michaelis constant reaction rate constant in the Michaelis-Menten reaction reaction rate constant in the Michaelis-Menten reaction reaction rate constant in the Michaelis-Menten reaction chemical component product in the Michaelis-Menten reaction concentration of the product in the Michaelis-Menten reaction initial concentration of the product in the Michaelis-Men ten reaction chemical component substrate concentration of the substrate initial concentration of the substrate (initial) velocity of the Michaelis-Men ten reaction maximal (initial) velocity of the Michaelis-Menten reaction this sign is used throughout to mean 'is defined by' or 'is by definition'

1 Chemical kinetics: a prototype of nonlinear science

1.1 Mass action kinetics: macroscopic and microscopic approach

The intention of the theory of chemical reactions is to describe the interactions among the components (or species) of a chemical system. The composition of a chemical system can generally be characterised by a finitedimensional vector. The dimension of the vector is the number of interacting qualities (i.e. components); the value of the vector describes the quantities of the qualities. Reactions take place among the components causing the change of the quantities of the components. A chemical reaction is traditionally conceived of as a process during which chemical components are transformed into other chemical components. Stoichiometry investigates the static, algebraic structure of the network of reactions. It might be considered as the calculus of changes in composition that take place due to the reaction. In other words, stoichiometry provides the framework within which chemical 'motion' must take place. ~/-

Reaction kinetics (it is important to emphasise that it is:kinetics, and not dynamics, since no 'force law' exists, at least not according to any traditional interpretation) governs the velocity of the composition changes. Kinetics assigns to each reaction of the network a velocity function. The most common model can be associated with the mass action law. The velocity of the reaction

aA + bB--+ cC + dD

is given by this law as

r(t) = - 1/a(dcAfdt) = - l/b(dc8 /dt) = 1/c(dccfdt) = l/d(dc0 /dt) = k(cA(t)t(c8 (t))b,

(l.l)

(1.2)

2 Mathematical models of chemical reactions

where the scalar k is the rate constant characterising the velocity of the process.

If the participating molecules of a chemical reaction go through states that may be identified as separate chemical components, then the reaction is called complex; otherwise it is simple, or elementary in the kinetic sense. Reaction ( 1.1) is an elementary reaction. By an elementary reaction we mean a chemical reaction with some mechanistic significance between reaction participants (e.g. collisions).

The molecularity of an elementary reaction is the number of molecules that are to collide in order that the elementary reaction take place. In a bimolecular reaction the transformation is the result of the collision of two molecules. The collision of three or more molecules is highly improbable, a seemingly trimolecular reaction usually is the resultant of mono- and bimolecular elementary steps. In our example the molecularities of the chemical species A, B, C and D are a, b, c and d.

The order of a reaction with respect to a component is the exponent to which the concentration of the components influencing the rate of the reaction are to be raised in order to get the rate expression. The order of the reaction is the sum of the orders with respect to all the components. The basic assumption of mass action kinetics is that the orders of an elementary reaction with respect to the components are given by the stoichiometric coefficients or molecularities as in ( 1.2).

These above are by no means intended to be exact definitions; formal definitions will be given in later chapters. An excellent analysis of the physicochemical background has been given by Horn & Jackson (1972).

Equation ( 1.2) is a system of ordinary differential equations. The number of differential equations is equal to the dimension of the component vector, since the temporal change of each component is described by a differential

/equation. The kinetic behaviour of a chemical reaction is described tradition( ally by a system of (generally nonlinear) differential equations: I ! c(t) = f(c(t)); c(O) = c0 , (1.3)

where c(t) E IRM denotes the quantities of the chemical components at time t, \ f is determined by the stoichiometry, and c0 is the initial value of the \components.

\ A number of questions arise immediately. How can we derive and interpret kinetic equations? Are they sufficiently general for describing chemical reactions taking place in different materials and in different circumstances (e.g. at low and high temperature and pressure)? What is the mathematical structure of the kinetic equations? What can we tell about the properties of the solutions? Can we define or derive other kinds of models?

The mass action type kinetic equation was first derived by Wilhelmy ( 1850), who measured the velocity of mutarotation of simple sugars. Wilhelmy knew that the equation he derived has general character: It is also a

Chemical kinetics 3

problem of interest to establish in what way the chemical action depends on general conditions, at least in this special case. However, this is certainly only one member of greater series of phenomena which all follow general laws of nature. (Wilhelmy, 1850, but see Leicester & Klickstein, 1952). The kinetic mass action law was suggested by Guldberg & Waage (1867; see also Bastiansen, 1964), who assumed the reversibility of each elementary reaction. Deriving two-term rate laws they identified the terms as 'forward' and 'reverse' rates. As a matter of fact, we adopt a rather pragmatic approach and do not restrict ourselves in this book to investigating systems of 'reversible' reactions. Concepts such as 'microreversibility' and 'detailed balance' are strongly interconnected with the notion of reversible reaction and will be analysed tangentially later.

To be more definite, the mass action law is a postulate in the phenomenological theory of chemical reaction kinetics. In the golden age of the quantum, chemistry seemed to be reducible to (micro)physics: 'The underlying physical laws for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that exact application of these laws leads to equations much too complicated to be soluble' (Dirac, 1929). As was clearly shown by Golden (1969) the treatment of chemical reactions needs additional requirements, even at the level of quantum statistical mechanics. The broad-minded book of Primas (1983), in which the author deeply analyses why chemistry cannot be reduced to quantum mechanics is strongly recommended.

The derivation of mass action kinetic laws from elementary principles (at least for reversible bimolecular gas-phase reactions) was attempted in an already classical paper of Ross & Mazur (1961 ). Starting from a Boltzmanntype equation (supplemented by a term due to reactive collisions) they derived mass action kinetic equations using the Chapman-Enskog approximation method. In this special case phenomenological kinetic equations can be considered as the 'zeroth order approach' of equations of nonequilibrium statistical mechanics (see also, perhaps, the somewhat arbitrarily given references: Garcia-Colin eta/., 1973; Eu, 1975; Moreau 1975a, 1975b).

Macroscopic theories of chemical reactions cannot take into consideration the spatial course of reaction, though it is obviously relevant from a microscopic point of view. Therefore a chemical reaction is handled as interactions among components being present at the 'same place'. (The 'same place' is considered as the region of space that is sufficiently large to make possible the underlying mechanism of reaction, but sufficiently small to be able to assume that it is a 'point'.) A chemical reaction has no spatial 'cause', and it can be considered as the rearrangement of a 'point having internal structure', Since a chemical reaction is not a 'spatial' phenomenon, it is difficult to give its descriptive picture, i.e. to create its (macro)physical model. Macroscopic spatial motion of the constituents of a chemical system belongs to the topics of the transport theory (defined in the narrow sense, i.e.


neglecting chemical reactions). However, chemical reactions and transport processes substantially interpenetrate, and they can be separated only by abstraction and/or neglect. The inherent contacts between chemical reactions and transport processes are reflected in the kinetic theories of special classes of materials (dilute gases, electrolytes, enzymes, polymers, plasmas, solids etc.). In the next section these particular systems will be briefly analysed.

1.2 Physical models of chemical reactions

According to the basic assumption of the reaction kinetic theory of dilute gases a chemical system consists of a finite number of discrete qualities (molecules, atoms, radicals; i.e. chemical components). Chemical reaction can result from the collision between the individuals of some components. The celebrated microscopic theory of chemical reaction (namely the transition state theory of Glasstone et a/., 1941) based on this approach is theoretically not well-founded (for a critical survey see Simonyi & Mayer 1975) in spite of its great practical success. According to one assumption of the theory the equilibrium (Maxwell-Boltzmann) distribution is not essentially influenced by the reaction. This assumption is only reasonable for the case when the activation energy of the reaction is much larger than the k8 T thermal energy (k8 is the Boltzmann constant).

A theory appropriate for describing reactions in dilute gases cannot be easily extended to reactions taking place in condensed phase. The spirit of transition state theory can be applied to ion reactions in electrolytes, if the concentrations are replaced by the practically useful, but theoretically not well-founded notion of 'activity'. Since the activity coefficients of each ionic species is a function of the quantities of all ionic species present, the picture of collisions among the individuals of finite discrete qualities cannot be maintained rigorously.

The velocity of migration of individual ions is characterised by mobility. The mobility of ions directly influences the diffusion constant, and indirectly the reaction rate, through its effect on diffusion. For the connection between the theory of ion reactions and the Debye-Hiickel theory of electrolytes see, for example, Moelwigh-Hughes (1971).

The overwhelming majority of chemical reactions taking place in living organisms are enzyme reactions. An enzyme reaction is a particular kind of chemical reaction, since the size of the components can be considerable different. A reaction of an enzyme with a substrate molecule cannot be interpreted as collision between spheres of near-equal diameter. From a formal point of view the enzyme reaction can be qualified as a heterogeneous or, more precisely, surface reaction, since the individual velocity of one component is very small, and can even be zero. Another difficulty is the identification and enumeration of the chemical components. The bulk of enzyme molecules are proteins. The primary structure of the proteins, i.e. the

Chemical kinetics 5

sequence of amino acids, does not uniquely determine the chemical component. A protein with a given primary structure may have different spatial arrangements (conformations) which could be different from an energetic point of view. Conformational changes can be sometimes associated with changes of covalent chemical bonds (e.g. formation or destruction of -SS- bonds). It is not obvious, whether the chemical components can be identified by their primary or perhaps by their secondary structure. Additionally, there is a question of interpretation: whether the changes of conformation can be considered as chemical reactions. Perhaps we may refer here to a remark of Primas (1983, p. 149):

The quest for the referent of modern scientific theories is nontrivial and the danger of making category mistakes is serious. For example, it is not at all clear, that collision theory (as used for the description of molecular beam experiments) and phenomenological kinetics (as used to describe enzyme kinetics in biochemistry) have reference categories of the same logical type.

Very often cells contain a small number of individual molecules of a given species. In this case the quantity of these enzymes can be given by an integer number and so the notion of concentration has no sense, even as an approximation.

Plasmachemistry has recently become very important, both from the theoretical and from the practical point of view (see, for example, Venugopalan, 1971). In 'cold' plasmas the temperature of neutral components is different from that of heavy ions and electrons (temperature is defined as the average kinetic energy divided by the Boltzmann constant). The statement, 'the reaction takes place at T' is ambiguous, and so the classical definitions of 'reaction rate' and of 'temperature dependence of reaction rate' have to be modified.

In another class of plasmas ('hot plasmas') the activation energy has the same order of magnitude as k8 T, therefore one assumption of transition state theory does not hold. As in the case of enzyme systems, it is not too easy to define the chemical components constituting plasmas. The chemical (kinetic) properties of the components can be fundamentally modified by the electric field, therefore it is difficult to know after which changes a chemical component may be considered as identical to itself.

A statistical mechanical treatment of plasmachemical kinetic phenomena based on the original Boltzmann equation is not possible, since the probability of three-particle collisions is not negligible. (However, the modern theory of nonequilibrium statistical mechanics tries to extend the range of validity of the Boltzmann equation to describe collisions among three or more particles.)

Plasmachemical reactions can be considered as macroscopic stochastic processes described by Kolmogorov equations, which roughly speaking refer to the temporal evolution of probabilities. In a plasmachemical context these equations are weakly related to the Pauli equations describing the change of


the occupation numbers of 'internal states'. This approach forms an intermediate level between strict statistical mechanics and the phenomenological description.

The number of components in polymers is generally 'superfluously' large. A chemical system having a very large number of components (denoted by N) can be formally simplified:

(I) by lumping into a system having M ~ N number of quasicomponents; (2) if the state space is characterised by a function and not an N-dimensional

vector.

The applicability of the second technique is quite obvious, since the state is often characterised in polymers by parameters, such as the molecular weight distribution, that can be characterised by a function.

A number of thermomechanical phenomena taking place in polymers can be explained by the term 'memory': the change of state not only depends on the present but on past values of the state variables. According to the theory of 'fading memory' (Coleman, 1964) the influence of the near past is greater than that of the far past.

Chemical reactions in solids cannot be interpreted using the theory of gas phase reactions, since a solid phase can be considered as a macroscopic continuum and not a system of independent, discrete entities. Changes in such systems imply changes in the internal structures and symmetries of the whole macroscopic system. Chemical reactions are intimately interconnected with diffusion in condensed matter (Goselle, 1984). From a theoretical point of view it is hard to make a distinction between phase transitions and chemical reactions. Transformations of solids can be classified into two groups (Budnikov & Gisling, 1965): with or without change in composition of the phases.

The decomposition of salts of oxyacids are typical examples of the reaction A<sotidl-+ B(sotid) + C(gas)· The two characteristic steps of the transformation, i.e. the formation and the growth of clusters can be described by deterministic (Riickenstein & Vavanellos, 1975), or stochastic (Kitahara et a/., 1975) models. Since the role of fluctuations is very important in the vicinity of cri.tical points, the kinetic theory of the phase transitions is based on the theory of stochastic processes (Hohenberg & Halperin, 1977; Wilson & Kogut, 1974).

Table 1.1 summarises the results of the present analysis in connection with the properties of special kinetic systems.

1.3 Deterministic and stochastic models

In spite of the diversity of chemical reactions taking place in different material systems, the mathematical structure of reaction kinetics is remark-

Chemical kinetics 7

ably general. In the models of 'pure' reaction kinetics the reaction rate is the only dependent variable (constitutive quantity) and the composition is the only independent variable. The spatial distribution of the components are considered homogeneous.

As was mentioned earlier, deterministic models of chemical reactions might be identified with eqn (1.3). However, not all kinds of systems of differential equations, not even all those with a polynomial right-hand side can be considered as reaction kinetics equations. Trivially, the term -kc2 (t)c 3 (t) cannot occur in a rate equation referring to the velocity of c1 ,

since the quantity of a component cannot be reduced in a reaction in which the component in question does not take place. Putting it another way, the negative cross-effect is excluded. A necessary and sufficient condition is required to restrict eqn (1.3) to be able to be a kinetic equation.

The clearing up of this question is important from the point of view of practical chemistry as well as of mathematics. In the first place, if a polynomial differential equation has been fitted to experimental data, then it is a question, whether this equation can be considered as a model of reactions. In the second place, utilising the special structure of the kinetic differential equations, surprisingly strong theorems exist for the qualitative properties of the solutions. More precisely, certain systems of differential equations can be studied more efficiently if they can be models of chemical reactions.

According to the assumptions on which classical kinetics is based, deterministic models are adequate as long as deviations from the 'macroscopic average values' remain negligible. A number of situations can be listed to argue for relevance of fluctuations in chemical sytems:

(I) The size of the chemical system is small. In this case we cannot apply continuous state models even as an approximation. (For the case of a finite particle number, a precise continuous model cannot be defined, since in the strict sense the notion of concentration can be used only for infinite systems.) Discrete state space, but deterministic, models are also out of question, since fluctuations cannot be neglected even in the 'zeroth approximation', because they are not superimposed upon the phenomenon, but they represent the phenomenon itself.

(2) The system operates near an instability point of a deterministic model. In this case small fluctuations may be amplified and produce observable, even macroscopic effects. It may also happen that the deterministic model of a system is structurally stable while the stochastic model is not, or vice versa.

(3) Fluctuations can be a source of information. According to the so-called fluctuation-dissipation theorem the dissipative processes leading to equilibrium are interconnected with the fluctuations around the equilibrium point. Using the spirit of this theorem the kinetic rate constants can

Table 1.1

Characteristic Characterisation of the Memory effects Character of the Relationship between properties/material matter evolution equation chemical reaction and

classes transport processes

dilute gases finite number of - nonequilibrium equilibrium discrete qualities statistical established is not

mechanics Oth influenced approximation essentially by the

t chemical reaction

electrolytes discrete qualities- - deterministic kinetic mobility (but activity equations ~ coefficient of each i diffusion constant ion is the function of stochastic mesoscopic t a/lion equations reaction rate concentrations)

enzyme systems the size of the conformation change small number of conformation change components is can exhibit time molecules can be considered practically delay; complex

h J,. d . . as diffusion,

incommensurable enzyme systems stoc astlc escnpt10n chemical reaction J, as phase transition

biological memory

plasma

polymers

solid state

finite number of discrete qualities (the temperature of different components can be different)

the number of components can be superfluously great; global state variables (functions)

macroscopic continuum

the actual value of stress tensor depends on the history of deformation tensor

pretreatment influences the behaviour ( topochemistry)

extended Boltzmann electric stimulus: equation chemical properties

change stochastic mesoscopic

description

deterministic model: functional differential equations (in two senses!)

fluctuations are significant near instability points

j, stochastic description

thermomechanical effects

the effect of phase transition and chemical reaction cannot be separated

diffusion and reaction occur together


be calculated from equilibrium measurements. (4) So far we have argued for the role of internal fluctuations. These

fluctuations are generated spontaneously by the system during its normal operation. However, the parameters of a macroscopic system can be externally controlled, and therefore they are subject to environmental fluctuations. Chemical system perturbed by 'external noise' also may result in radical change in the qualitative behaviour of the system.

To describe fluctuation phenomena a continuous time, discrete state stochastic model has to be defined. To use discrete variables instead of concentrations, the functions of the latter have to be transformed into the functions of the number of components:

c(t) -+ v-'n(t),

where Vis the volume ofthe system, c(t) is a concentration vector and n(t) is the number of components at a fixed time t.

Introducing a stochastic description let~ be a stochastic vector process, the dimension of which is equal to the dimension M of the concentration vector.

P.(t) = &(~(t) = n)

is the probability that the vector of the numbers of components is n (n of course also is vector), P.(t) is the distribution function (ne N<r). The temporal evolution of the distribution is determined by the assumption that the chemical reaction is considered as a Markov process. Here we have to refer to van Kampen's remark (1981, p. 80):

When a physicist talks about 'process' he normally refers to a certain phenomenon involving time. Concerning a process defined in this way it is meaningless to ask whether or not it is Markovian, unless one specifies the variables to be used for its description. The art of physicist is to find these variables that are needed to make the description (approximately) Markovian.

A Markovian description can be naturally introduced by generalising deterministic systems modelled by ordinary differential equations. In other words, the stochastic version of a deterministic process without 'after-effect' is a Markov process.

However, the Markov character of the chemical process represented by the component vector has not been derived from microscopic models. Therefore Markovicity is not more (and of course not less) than a plausible assumption.

Discrete state space stochastic models of chemical reactions can be identified with the Markovian jump process. In this case the temporal evolution can be described by the master equation:

(dP./dt) = A(P.(t)). (1.4)

Equation (1.4) is a linear differential-difference equation, the special struc-

Chemical kinetics II

turc of A is determined by the stoichiometry. Introducing a .... as the infinitesimal transition probability, which gives the probability (per unit time) of the jump from n' to n, the master equation can be interpreted as a gain-loss equation for the probability of each state n:

dP.fdt = ~)aDD.P.(t) - a •.• P.(t)). (1.5) a'

The first term of the right-hand side is the gain due to transition from all the other states n', and the second term is the loss due to the jump to other states.

In spite of the fact that (1.4) and (1.5) are linear, they can govern situations qualified 'experimentally' as 'nonlinear phenomena'. The terms 'linear' and 'nonlinear' are mathematical concepts, which refer to equations. The equations (1.4) and (1.5) can be converted into nonlinear equations for a macroscopic variable, for example by taking the expectation.

I.4 Regular and exotic behaviour

Traditional reaction kinetics has dealt with the large class of chemical reactions that are characterised by having a unique and stable stationary point (i.e. all reactions tend to the 'equilibrium'). The complementary class of reactions is characterised either by the existence of more than one stationary point, or by an unstable stationary point (which could possibly bifurcate to periodic solutions). Other 'extraordinarities' such as chaotic solutions are also contained in the second class. The term exotic kinetics refers to different types of qualitative behaviour (in terms of deterministic models): to sustained oscillation, multistationarity and chaotic effects. Other irregular effects, e.g. hyperchaos (Rossler, 1979) can be expected in higher dimensions.

Exotic chemical systems, mostly oscillatory reactions, but also systems exhibiting multistationarity and chaotic effects, have extensively been investigated. Phenomena in chemical, biological and industrial chemical systems are the experimental basis of the theoretical studies.

For almost a century chemical kinetics used deterministic models only, which completely neglects the existence of fluctuations. However, at the same time the early thermodynamic fluctuation theory was already examining the quantitative aspects of the fluctuations.

Since classical thermodynamics, including fluctuation theory, adopts continuous macroscopic variables, their fluctuations are described generally by Gaussian distributions. . It is often mentioned that the stationary distribution of chemical reactions IS. generally the Poisson distribution (Prigogine, 1978). However, the sigmficance of the Poisson distribution is rather limited; other unimodal distributions can also reflect the regular behaviour of the fluctuating chemical sytems. Multimodality of the stationary distribution might be associated, at least loosely speaking, with multistationarity of deterministic


models. The maxima correspond to the stable, and the minima to the unstable stationary points. In certain situations this qualitatively clear picture is slightly modified (:Erdi et a/., 1981; Ebeling & Schimansky-Geier, 1979).

The qualitative theory describes the long-term behaviour of stochastic systems without solving the equations. Recurrence, stationarity and ergodic properties are the most important concepts which characterise the stochastic process and/or the state-space.

Every network of chemical reactions can be classified by its qualitative properties. To identify stoichiometric and kinetic conditions for exhibiting different kinds of exotic behaviour is an important problem of theoretical reaction kinetics.

I. 5 Chemical kinetics as a metalanguage

The structure and models describing chemical reactions are almost trivial. Chemical kinetics generally takes into consideration binary and, rarely, ternary interactions among the molecules. It is a natural tendency to decompose complex phenomena into binary, or perhaps ternary interactions. Therefore the formal theory of chemical kinetics can be extended to describe transformation phenomena (using the term in a broad sense) in populations whose basic components are not molecules.

A simple class of chemical reactions, namely compartmental systems (when a complex is identified precisely with one component) are appropriate to model- real or abstract- discrete spatial transport phenomena. Isotope exchange, charge transfer between energy levels, absorption of drugs and multistage reactors can be arbitrarily mentioned as capable of being described by formal compartmental models. Compartment analysis is also a conceptual tool of ecological system modelling.

Chemical kinetics describes, in fashionable terminology, competition, cooperation and selection among the constituents. A number of biomathematical models at different hierarchical levels, such as prebiotic chemical models (Eigen, 1971; Eigen & Schuster, 1979), genetic models (Wright, 1931), ecological models (Volterra, 1931) and sociobiological models (Maynard Smith, 1982), lead to the same type of differential equations (Schuster & Sigmund, 1983):

.X;(t) = x;(t)(_t;(x(t))- jtl xj(t)./j(x(t))} n

L xj(t) = I; xj(t) ~ 0. j= I

(1.6)

There is little chance of giving a general qualitative analysis of (1.6) except for low-dimensional cases. Limit cycle and chaotic behaviour might occur in certain situations, as has been demonstrated.

Chemical kinetics 13

The conceptual and mathematical framework of competitive and cooperative phenomena has been extended to socioec?nomic situations (see, for example, Peschel & Mende, 1985; Peschel & Bre1tenecker, 1984). Many problems of epide~iology ~nd. of mic_robiology ~a~ also be converted into 'chemical language . Quant1tat1ve soc1ology (We1dhch & Haag, 1983) also uses models having quite similar structure, as chemical kinetics has.

Over the past years enormous progress has been made in the study of macroscopic structures formed in chemical systems. Chemical reaction was considered as an analogue of phase transition (Schlogl, 1972), at least from a certain formal point of view. Gradually chemical reaction has become the prototype of nonlinear systems, capable of exhibiting instabilities, bifurcation phenomena etc., and therefore its formal aspects have been investigated extensively, by synergetics (Haken, 1977, 1983), and by the theory of dissipative structures (Nicolis & Prigogine, 1977).

2

The structure of kinetic models

2.1 Temporal processes

Mathematical models of chemical reactions are based on the assumption of spatial homogeneity, i.e. diffusion and other transport processes can be neglected. From the formal point of view chemical reaction is handled as a temporal process. To investigate the structure of kinetic models we adopt the approach of dynamic systems theory in this chapter, while historical motivations are left more or Jess out of consideration.

A dynamic system is an ordered pair: (A, cp), where A is the state space, and cp: lr x A --+A, is a function which assigns to an arbitrary point x 0 E AI the point x E AI, characterising the state at t, assuming that the system was in x0 at t = 0. A fundamental property of cp is the validity of the identity

cp((t + s), x0 ) = cp(s, cp(t, x0 )). (2.1)

The motion of a dynamic system is the one variable function:

cpx.: T--+A cpx0 = cp( ·, Xo),

(2.2)

where lr c IR and A c IRM (MEN) or A consists of random variables taking their values from !RM. (For every tE lr cp(t, ·):A--+ A is an automorphism.)

The process to be described can be classified (perhaps rather arbitrarily) either by the properties of the 'process-time', or by structure of the state space, or by the nature of determination.

2.2 Properties of process-time

Structure of kinetic models 15

2.2.1 Discrete versus continuous

• a notion of time is required for dealing with the phenomena of change; ;~~ there are many diff~ren~ kinds ~f change and, depen~ing ?n the particular kind on which attentiOn IS focusmg, several alternative time concepts are arrived at' (Denbigh, 1981, p. 4). The time can be chosen as continuous (l c IR) or discrete (lf c Z) variable.

The arguments generally adopted for choosing a continuous time variable are that:

(I) time is really continuous; (2) calculations with continuous time models are easier, or at least they have

greater tradition.

Arguments for selecting a discrete time variable are that:

(I) time is really discrete; The idea that time has no objective existence but depends on events led some scientists to abandon the assumption that it . is a continuous variable. Following the establishment of the Planck-Einstein quantum theory, it was suggested by Poincare and others that time is quantized; one calculation gave the value 10- 24 s to the 'chronon'.

Perhaps the best argument for time quantization (at least in the empirical spirit of quantum mechanics) is that we perceive temporal intervals of finite duration rather than durationless instants; and it is dangerous to assume that the world has properties that can never be observed. But this psychological argument, as developed for example by Bergson, provides no quantitative justification for a division of physical time into chronons of the order of 10- 24 s. (Brush, 1982)

(2) calculations with discrete-time models are easier since they are really simpler;

(3) the notion of 'immediate next time' can be interpreted, and this is rather difficult in the case of continuous time;

(4) digital computers calculate at discrete points of time only; (5) experimentalists measure at discrete points only.

Weaker arguments could have been made for adopting continuous time: analogue computers operate in continuous time, and certain physicochemical quantities can be transduced continuously. However, it is important to note that in most applications the 'real' time is transformed to 'state-time' (e.g. number of generations).

2·2.2 Time, thermodynamics, chemical kinetics

We have to resist the temptation to expatiate on the question of the 'chemical


time arrow'. The notion of time has mysterious elements, perhaps due to the fact that our ideas about temporal processes come from different sources. Research on the nature of time has been extensive (e.g. Fraser, 1972, 1978; Fraser & Lawrence, 1975; Zeman, 1971; Denbigh, 1981). History, micro- and macrophysics, biology and cosmology have all adopted different notions of time. Perhaps thermodynamics (since it ought to be the 'science of time' (Prigogine, 1980; Prigogine & Stengler, 1984) could offer a unified framework to interpret irreversible and cyclic phenomena, but little has been achieved as yet. In spite of the fact that chemical reactions play an important role in Prigogine's arguments, the relationship between chemical kinetics and thermodynamics is not clear. The development of chemical kinetics was practically independent of the formation of nonequilibrium thermodynamics. This is surprising to a certain extent, since chemical reaction is included in the topics of thermodynamics. Thermodynamics investigates processes and equilibrium due to energetic interactions. Processes and equilibrium due to chemical reactions can and should be investigated by the concepts of nonequilibrium thermodynamics. Physical transport processes are spatia-temporal phenomena, but the macroscopic theory of chemical reaction, as was mentioned earlier, cannot take into consideration the spatial course of the reaction. According to the formalism of the conventional theory of irreversible processes the reaction rate is a thermodynamic flow, just as the rate of diffusion or of thermal conduction, and so chemical kinetics is a particular chapter of thermodynamics. The authors of this book feel attracted by general theories, and, in addition, they know that chemical reactions produce entropy. Unfortunatley they cannot see a better solution for the time being than to investigate models of reaction kinetics without giving a detailed thermodynamic treatment.

2.3 Structure of state-space

2.3.1 Discrete versus continuous

The state-space (or 'site space' according to the terminology soon to be introduced) can be chosen either continuous (A\ c IRM) or discrete (A\ c z_M). To emphasise the existence of 'elementary' particles of a population (as, for example, sometimes in reaction kinetics and in population dynamics), a discrete state-space formalism can be used. Continuum mechanics is an illustration for preferring a continuous state-space, since the mass points can arbitrarily occupy space.

The notion of state was derived from the theory of mechanics and of thermodynamics and generalised by mathematical system theory. Following the tradition of the field theory of continua (Truesdell & Noll, 1965) quantities of a theory can be classified into two categories: state variables and constitutive quantities. State variables are functions such that their values


ecify the state of the system. The constitutive quantities are functions of the :fate. in the sense that their value is univocally determined (at least in principle) once the state of the system has been assigned:

Q(t) = co(g(t), t), (2.3)

where Q is a constitutive quantity, g denotes the state of the system and co: A x v -+ !R' is the constitutive functional mapping the state into a constitutive quantity (the 'functional' assigns a number to a function, here the term refers to every mapping having the function as argument); re N. The case r = J means that the value of the constitutive quantity is a scalar.

The state of an M-component chemical system is described by a vector c,

c: lr-+ IRM, t >-+ c(t) E IRM. (2.4)

As was mentioned earlier, sometimes the state is described by a function (e.g. in polymers and oils) and not by a vector, and so the concept of a 'continuously changing component' can be introduced. For the sake of the general mathematical treatment we mention that a finite-dimensional vector can also be interpreted as a function: IRM can be considered as an abbreviation:

IRM: = IR:u ..... Ml = {f;f: {I, 2, ... , M}-+ ~}.

Generally xr = {f, f: Y-+ X}; the cardinality of the set is IXI1 r1•

The state of the system with continuously changing components is described at a fixed point of time by a (not necessarily scalar valued) function, f: IRM-+ !Rm.

Adopting the notation introduced just above the state of the system is c, where c: lr-+ (!Rm)RM or it is an element of the set:

MT M [(!Rm)R ]' = {f; j: l[" -+ (!Rm)R }.

However, the state of polymers can be represented by the molecular weight distribution, which is a scalar-scalar function.

According to the conventional treatment of pure homogeneous reaction kinetics the state is a finite-dimensional vector and the only constitutive quantities are the reaction rates.

2·3.2 State and site

The theory of thermomechanics adopts the concept of 'materials with ~emory'. According to this concept the constitutive quantities depend on the t~ory of the independent variables, and not only on their actual value. In

~t er words, it is not definite that the instantaneous value of state variables 1.~ the _state) completely determines the state.

1 he stte function is A: lr -+ IRM. Since the state is determined by earlier va ues of the site function, therefore the state g is interpreted as


g: lf--+ G, t t-> A' (2.5)

where the definition of the 'history' A' is

A'(s) = A(t - s), s > 0. (2.6)

Knowing the history the state can be set up:

.tf(A,.) = g, (2.7)

i.e.

.tf(A, t) = g(t) = %(A')= A'(O). (2.8)

.tf is a mapping assigning a function to a function and to a number. It is a possible but, without doubt, a strange construction. If we assume that the history of the site does not influence the state, then the constitutive functional reduces to a function. Furthermore, if we also assume the invertibility of this function then the differences between state variables and constitutive quantities are not significant. These assumptions are tacitly assumed in the usual theories of chemical thermodynamics.

The stochastic version of a memory-free deterministic process is a Markov process- more precisely, a first-order Markov process. It is interesting to remark that in the theory of stochastic processes the concept of 'historydependent' processes, had been adopted by the time the theory was established (i.e. in the mid-thirties).

2.4 Nature Clf determination

An (A, q>) dynamic system is deterministic if knowing the state of the system at one time means that the system is uniquely specified for all telf. In many cases, the state of a system can be assigned to a set of values with a certain probability distribution, therefore the future behaviour of the system can be determined stochastically. Discrete time, discrete state-space (first order) Markov processes (i.e. Markov chains) are defined by the formula

&'(~,+ 1 = al~o = ao, ~~ = al, . .. , ~~=a,)=&'(~,+ 1 = al~, =a,). (2.9)

Knowing the total history of the process we can extrapolate its future behaviour with the same probability as if we knew only the actual current state.

The Markovian property of vectorial processes can be lost by reducing the dimensionality of the process. 'One cannot expect that the knowledge of only a few components is again sufficient to predict the future probability' (van Kampen, 1981, p. 80).

Deterministic dynamic systems generated by autonomous ordinary differential equations,


dx(t)fdt = f(x(t))

, n be associated with time homogeneous Markov processes. Time homogen~~y means that the transition probabilities are stationary, i.e.

,gil(~ .• = xl ~~ = y) = P"_..(s - t). (2.1 0)

In this case we can say that the 'evolution law' does not depend explicitly on the time. Consequently, the time origin can be defined arbitrarily.

2.5 XYZ models

At least eight different kinetic models can be defined, depending on the specification of time (X), state-space ( Y) and nature of determination (Z). As was explained earlier, time can be discrete (D) or continuous (C), the statespace can be also discrete (D) or continuous (C), and the nature of determination is deterministic (D) or stochastic (S).

Mass-action-type kinetic differential equations can be identified with the CCD model, while the more often used stochastic model is the CDS model. These two models will be the object of detailed investigations in Chapters 4 and 5.

DCD models have achieved a particular significance in the last decade in connection with chaotic phenomena. There are at least two distinct methods of relating DCD models to CCD models. The easier, but less rigorous, way is by the discretisation of time. An autonomous differential equation

dx(t)fdt = f(x(t), t), x(t0 ) = x 0

can be transformed as

x(t + h) = x(t) + f(x(t), t)h + o(h).

The other method can be applied if the differential equation has a periodic solution, i.e. when a Poincare map can be constructed.

Take a hyperplane of dimension n - I transverse to the curve t--+ x(t) ~hrough Xo. A map F: U--+ IR"- 1 is induced by associating with t0 the nearest mtersection of the trajectory (with initial condition x(O) = x 0 ) with the given hyperplane. If the first such intersection occurs at x 1 we define F(x0 ) = x 1 .

The form of F is independent of the index of the series and also of the coordinates, therefore

x.+ 1 = F(x.) (2.11)

c~n be specified. Thus a difference equation has been obtained from a differential system. W~at is important for us is that certain qualitative properties of multidi

;~nsional differential equations can be studied by a one-dimensional 1 erence equation, or an iterated map, in modern terminology. Iterated


maps on the interval were studied by Collet & Eckmann (1980). Poincare maps are extensively used to investigate the possibilities of chaos in chemical systems- see, for example, Swinney & Roux (1984).

Other, less important, XYZ models will be mentioned in the text as they arise.

3 Stoichiometry: the algebraic structure of complex chemical reactions

From now on we shall concentrate on reactions taking place in a well-mixed vessel (test-tube, cell, reactor etc.) of constant volume at constant pressure and temperature. More generally speaking, it is assumed that all the transport processes in the thermodynamic sense (including volume changes due to the reaction) are negligible. Further - at the present tacit -assumptions will be made clear below. The present chapter sets up a framework for the reactions: it deals with stoichiometry.

The skeleton-like character of stoichiometry should be emphasised: it does not depend on the type of the model (discrete or continuous, deterministic or stochastic, see Chapter 2) that is used afterwards when describing the time evolution.

Stoichiometry will here be treated in two sections: following history, in the ~rst section the atomic structure of chemical components or species is taken mto consideration, while in the second one it is not. Disregarding the atomic structure allows more freedom: more models can be drawn into the purview 0~ enquiry and treated with methods that were originally developed for the atms of reaction kinetics.

3·1 Conventional stoichiometry

let us consider the complex chemical reaction

2CO + 3H2 = C2H2 + 2H20 (3.1) CO+ H20 = C02 + H 2 . (3.2)

In this reaction the following chemical components or species are involved:

X (I) =: CO X(2) =: H 2 X(3) =: C2 H2 X(4) =: H2 0 X(S) =: C02,


thus the number of components is five. The atomic components may be described by the atomic matrix:

CO H 2 C2 H 2 H 2 0 C02

Z= r: ~ ~ r n H 0 c

structure of the

where the first column expresses that the first component (CO) contains none of the H atoms, one 0 atom and one C atom, the second column expresses the atomic structure of the second component and so on. The structure of the elementary reactions (this name will be used in spite of possible criticism from the point of view of physical chemistry) may be described by the matrix of elementary reaction vectors, i.e. by the stoichiometric matrix:

I 2 -2 -I co -3 I Hz

y= I 0 CzHz, 2 -I H 2 0 0 l C02

where the negative numbers of the first column express that two molecules of CO and three molecules of H 2 are needed in order that the first elementary reaction take place and if it does take place l molecule of C2 H 2 and 2 molecules of H 2 0 are formed.

An elementary requirement is the law of atomic balance: the number of atoms should be the same on both sides of each of the reactions. Calculating the number of atoms of each kind on both sides of both of the reactions is equivalent to multiplying the matrices Z and 1 according to the rule of ordinary matrix multiplication and setting the product equal to zero:

z,~ [~ n Two questions arise:

( l) What is the number of independent components in this complex chemical reaction?

(2) What is the number of independent elementary reactions in this complex chemical reaction?

In order to make the above questions meaningful independence should be defined in a sensible manner. For example, in the first case it is intuitively clear that C 2 H 2 and C6 H6 or CO, H 2 0 and HCOOH are dependent.

In the same way one would say that the sum of the elementary reactions (3.1) and (3.2) (or in other words the overall reaction)

Stoidwmetry 23

3CO + 2H2 = C2H2 + H 20 + C02

is linearly dependent on (3:1) and (3.2). Following this introductory example we give a general treatment of the

notions and statements used above. At first, a tacit assumption used until now was that all the elementary reactions are reversible. From now on it is allowed (it is unnecessary to exclude from the investigations) that some of the elementary reactions be irreversible. Therefore, if one of them is not irreversible, then one has to write it down twice, namely, in both directions.

The (complex chemical) reaction under study consists of the elementary

reactions M M

I <X(m, r)X(m)-+ I P(m, r)X(m)(r = I, 2, ... , R) (3.3) m~ I m~ I

where X( I), ... , X(M) are chemical components (molecules, radicals, ions), and the nonnegative integers

<X(m, r) and P(m, r)

are the stoichiometric coefficients or molecularities. The elementary reaction vector expressing the change caused by the rth elementary reaction of (3.3) is the difference of complex vectors, the product complex vector, and the reactant complex vector:

y(·, r) = P(·, r) -<X(·, r) = (p(l, r) -<X(I, r), ... , P(M, r) -<X(M, r)).

The matrix y withy(·, r) as its rth column is the stoichiometric matrix. It may be assumed (but only very rarely used) that the orders of all the elementary reactions, i.e. the numbers

M

I <X(m, r) = l"(r) m~ I

(the length of the reactant complex vector <X(·, r)) are less than or equal to three because of the improbability of collisions between more than three molecules (Benson, 1960; Laidler, 1965). The first order reactions, i.e. those for which l~(r) ::;:; I (r = I, 2, ... , R) are singled out for distinction both from the theoretical and the practical viewpoints.

A complex is called short, if it is not longer than two. A mechanism is a se~ond order mechanism, if all the reactant complexes are short and if there ~XIsts at least one of length two. A set of elementary reactions is said to be Independent if there is no way of expressing any of the elementary reaction vectors as a linear combination of the others. In the opposite case the elementary reactions are said to be dependent. From this definition it is clear ~hat the number of independent elementary reactions is the number of ~~dependent columns of y. But this number is called in linear algebra the rank ~- y: rank(y). This number is usually denoted by Sand is considered as the

Imension of the stoichiometric space, i.e. the dimension of the linear


subspace of IRM spanned by the columns of y, i.e. by the elementary reaction vectors.

It may be worth emphasising that if one only considers different independent elementary reactions then an elementary reaction and its multiples are considered to be the same.

In the present section it is assumed that the components have an atomic structure, i.e. they can be conceived of as formal linear combinations of the atoms (which may in fact be atoms or ions or other particles as electrons etc.) d(l), d(2), ... , d(A):

A

X(m) = L Z(a, m)d(a). a=l

The nonnegative integers Z(a, m) make up the atomic matrix Z with the columns called formula vectors. A (complex chemical) reaction among atomic components as above is an atomic reaction.

The law of atomic balance means for the rth elementary reaction that

Zy( ·, r) = 0 (ENA),

and when it holds for all elementary re2.ctions then one has

Zy = 0 (ENA X R).

Now let us return to the independence of elementary reactions in this general setting. A set of elementary reactions is said to be independent if there is no way of expressing any of the formula vectors as a linear combination of the others. In the opposite case the components are said to be dependent. From this definition it is clear that the number of independent components is the number of independent columns of Z. But this number is the rank of Z: rank(Z). In our example (3.1 )-(3.2) this number is equal to 3.

The following question arises. Given a set of atomic components how many independent elementary reactions are possible? Using the notations introduced above the question becomes: how many linearly independent solutions g are there to the equation

Zg = 0? (3.4)

As is well known, eqn (3.4) admits R+ = M - rank(Z) linearly independent solutions and these can be determined by standard methods (see, for example, Korn & Korn, 1968, Sec. 1.9).

Let us consider as an example the matrix Z of reaction (3.1 )-(3.2), then as rank(Z) = 3, R+ = 5 - 3 = 2. Equation Zg = 0 is in this case

[ ~I ~ ~ ~ ~] 0 2 0 1

Stoichometry 25

A there is a 3 x 3 submatrix with a nonvanishing determinant in the upper s . left corner of Z we may wnte

2g2 + 2g3 = - 2g4 -g4- 2gS

2g3 = - gs.

The general solution is therefore

g = (-g4- 2gs, -(3g4 + gS)j2, {g4 + gS)j2, g4, gS).

or g = (-2g4 _ 4gs, -3g4 _ gs, g4 + gs, 2g4, 2gs).

Taking g4 = I, g 5 = 0 and then g4 = 0, g 5 = I we get

g1 = (- 2, - 3, I, 2, 0) and g2 = (- 4, - I, I, 0, 2) (3.5)

and the general solution is the linear combination of these vectors with parameters as coefficients. The elementary reaction corresponding to g1 is just that in (3.1) and the representation of y( ·, 2) in terms of g1 and g2 is

y (., 2) = -0.5gl + 0.5g2.

An alternative approach to this question has been given by Aris & Mah (1963) who provided a method by which the number of independent reactions may be determined from experimental observations.

The natural sequel to the above question is: given a set of elementary reactions how many independent components are possible? Using the notations introduced above the question becomes: how many linearly independent solutions are there to the equation

zTy = OT or to yTz = 0? (3.6)

As it is known (3.6) admits M+ = M- rank(y) linearly independent solutions and these can be determined by standard methods.

Let us consider as an example the matrix y of reaction (3.1 )-(3.2) then as rank (y) = 2, M+ = 5 - 2 = 3. Equation yTz = 0 is in this case

[ -2 -I

-3 I I 0

r ~~~ = 0. L zs

As there is a 2 x 2 submatrix with nonvanishing determinant in the upper left corner of Z we may write

-2z 1 - 3z2 = -z3 - 2z4 -zl + z2 = z4- zs.


The general solution is therefore

z = ((z3 - z4 + 3z5 )J5, (z3 + 4z4 - 2z5 )/5, z3 , z4 , z5 ).

Taking z3 = 1, z4 = 0, z 5 = 0; z3 = 0, z4 = 1, z 5 = 0; and z3 = 0, z4 = O, z 5 = 1 we get

z1 = (l/5, 1/5, 1, 0, 0), z2 = ( -1/5, 4/5, 0, 1, 0), and Z3 = (3/5, -2/5, 0, 0, 1)

(3.7)

and the general solution is the linear combination of these vectors with parameters as coefficients. The representation of the components in the reaction (3.1 )-(3.2) in terms of Z~o z2 and z3 is

Z(·, 1) = z2 + 2z3 Z(·, 2) = 2z4 + z3 .

3.2 Atom-free stoichiometry

Let us consider the reaction

A + Y-+ X A + X-+ 2X + Z Z-+ Y

where

A = Br03, X= HBr02 , Y = Br-, Z = 2Ce4 +.

(3.8)

(3.9)

It is obvious that none of the elementary reactions obey the law of atomic balance. Still, the Oregonator model of the Belousov-Zhabotinskii reaction, of which (3.9) is a subset, has served well for a long period of time when carefully interpreted.

As a second example let us consider the well-known Michaelis-Menten reaction

E+ S¢-C¢-E+ P. (3.10)

It is clear that in general the atomic structures of the components enzyme (E), enzyme-substrate complex (C) and product (P) need not be known in order to get an idea of the time evolution of the process. What is more, their structures are generally not known.

The moral to be drawn from these examples is either that one has to abandon the law of atomic balance, or that the atomic structure of the chemical components should be disregarded.

Let us even go a step further. In reaction (3.9) A is considered to be an external component, i.e. a component that is held at a constant concentration. This can experimentally be realised by constant supply from a reservoir. Any of the models in Chapters 4 and 5 below may be formulated in the same way as (3.9) and can be written in shorthand as

Y-+ X X-+ 2X + Z Z-+ Y. (3.11)

Stoidwmetr}' 27

(If it is to be emphasised that a component is not external, it is called . terna/.) Now the following controversy appears: the second elementary Ill h ' f ' ' reaction of (3. II) expresses t e creatiOn o mass, I.e. reactiOn (3. II) does not obey the law of mass balance. As models like the Oregonator or the Lotka-Volterra model,

X-+ 2X X+ Y-+ 2Y Y-+ 0, (3.12)

that have an elementary reaction expressing the destruction of the chemical component Y have turned out to be really useful, and as this shorthand description of the reactions expresses the essence better, from now on it is allowed (it is unnecessary to exclude from the investigations) that some of the elementary reactions (including the possibility of all as well) do not have to be mass-conserving. The exact definition of (mass) conservativity is this: the (complex chemical) reaction (3.3) is conservative if there exist positive numbers p( I), ... , p(M) (being possibly conceived of as masses of the components) such that for all elementary reactions

M M L a.(m, r)p(m) = L P(m, r)p(m) (3. 13)

nr = I m= I

holds. Equation (3. 13) means that the masses on both sides of the elementary reactions are the same if calculated with the components of the vector p = (p(l), ... , p(M)). The requirement may be written down in concise form as

pa. = PP or p(a.- P) = 0.

The components of the vector p (especially if nonnegative values are allowed too) may be conceived of as other conservative quantities, e.g. the number of subunits (such as radicals or electrons) or charges, too.

As an example let us consider reaction (3.10). Here the usual quantities that are supposed to be conserved are the masses of the enzyme molecules and substrate molecules, therefore the matrix of ps, if denoted by Z here too, is

s c 0 I I I

p

~] ~-It is straightforward to generalise properties of the second elementary

reaction of(3.11) and the third elementary reaction of(3.12) respectively. If there exists a vector p with positive co-ordinates such that

M M L et(m, r)p(m):::; L ~(m, r)p(m) (3.14)

nr=l m=l

~o:ds ~in~quality is to be understood for all of the co-ordinates here and e ow tf It relates to vectors, therefore (3.14) is equivalent to stating that


ptl ~ pp)

then the reaction is superconservative while if M M

L a.(m, r)p(m) ;;:: L ~(m, r)p(m) m= I m~ I

holds (or, equivalently,

(3.15)

it is subconservative. It is clear that (3.14) expresses construction, whereas (3.15) expresses destruction of material.

A pair of questions arises here. Given a (possibly huge) complex chemical reaction, how can it be decided whether it is conservative, sub- or superconservative or perhaps none of them (as reaction (3.12) is, see Exercise 5, Sec. 3.4)? On the other hand, if a reaction is conservative, sub- or superconservative, what conclusions can be inferred from this fact? Unfortunately, only partial answers can be given to these questions and they are relegated to the 'Open problems' section (Sec. 3.6) at the end of the chapter.

3.3 Retrospective and prospective remarks. Suggested further reading

The first important contribution to atomic stoichiometry in this century seems to be provided by Brinkley (1946). He has shown the importance ofthe rank of the atomic matrix and presented a proof of the phase rule of Gibbs ( 1876). A systematic outline of stoichiometry was presented by Petho (sometimes Petheo) and Schay (Petheo & Schay, 1954; Schay, Petho, 1962). They gave a necessary and sufficient condition for the possibility of calculating an unknown reaction heat from known ones based upon the rank of the stoichiometric matrix. They introduced the notion of independence of components and of elementary reactions, the completeness of a complex chemical reaction (see the Exercises and Problems) and gave a method to generate a complete set of independent elementary reactions with as many zeros in the stoichiometric matrix as possible (see Petho, 1964).

This line was taken up again by Noyes (1984). A starting point of atom-free stoichiometry is the paper by Horn &

Jackson (1972) where the notion of conservativity was defined and the paper by Vol'pert (1972) that introduced subconservativity. Significant contributions were provided later by Horn and Feinberg (Horn, 1973c; Feinberg & Horn, 1977), and others (Deak (in T6th & Erdi, 1978, pp. 253-5); Willamowski and Rossler, 1980).

A major part of the literature that deals with stoichiometry has not been summarised and cited here but will be treated later, mainly in Chapter 4: this includes Dalton's laws of definite and multiple proportions, and the important works by Aris ( 1965, 1968) and Bowen ( 1968). There exist notions

Stoidwmetr_\' 29

( weak reversibility, or the value of deficiency, or connectedness of the s~~~~-space) that also belong to the ~lgebraic ~escription of a reaction: but these are exclusively used tog~ther with a specific model, and so they will be onsidered in the correspondmg chapters (e.g. Chapters 4 or 5).

c It may be useful to note that questions of stoichiometry do arise in connection with the stochastic model: see Chapter 5.

An important area of stoichiometry that lies outside the framework of this book is experimental and relates to the determination of the atomic and the stoichiometric matrices. An example of this literature is the paper by Aris & Mah (1963).

3.4 Exercises

1. An atomic reaction is said to be complete if any of the conceivable elementary reactions fulfilling the law of atomic balance can be written as linear combinations of the elementary reactions of the given reaction.

(a) Check that reaction (3.1)-(3.2) is complete. (b) Check that the reaction 2C + H 2 + 2C2 H 2 = C 6 H 6 is not complete.

2. Is the reaction

Br2 ~2Br H 2 + Br~HBr + H H + Br2 -+ HBr + Br

complete? 3. Show that an atomic reaction obeying the law of atomic balance is

necessarily conservative as well. 4. Show that any reaction can be made conservative by introducing a single

new internal component. 5. Show that the Lotka-Volterra reaction is neither sub- nor

superconservative.

3. 5 Problems

In Problems 1-3 it is supposed that the reaction is atomic.

I. Show that completeness of an atomic reaction is equivalent to the properties:

2.

(i) Zy = 0 (ii) rank(Z) + rank(y) = M.

Hint. Cf. Schay & Petho (1962). (i) Give a formal proof of the intuitively obvious fact that elementary reactions cannot take place among a set of atomic components unless any of the atoms is contained in at least two components. (ii) Show that the condition is not sufficient.


Hint. Consider the components CH4 , CO, H 2 0. 3. Show that

(i) the law of atomic balance implies conservativity; (ii) conservativity implies the existence of formal 'atoms', i.e. the components in a conservative reaction may be conceived of as formal linear combinations of (possibly hypothetic) atoms.

Hint. Start from the definitions. 4. (i) A compartmental system is a special case of the first order reaction; it

is a reaction in which the length of all the complexes is not more than one. A compartmental system is closed if it does not contain the zero complex. Show that a compartmental system is conservative iff it is closed. (ii) A generalised compartmental system (cf. T6th, 1978; Othmer, 1981) is a reaction in which all the complexes contain a single component, and all the components are contained in a single complex. It is closed if it does not contain the zero complex. Show that a generalised compartment system is conservative iff it is closed.

Hint. Could you find a vector to prove (i)? Statement (ii) is easy to reduce to statement (i).

5. The Vol'pert graph of the elementary reaction consists of two sets of points, one for the components and one for the elementary reactions. The number of directed edges (arrows) from the point corresponding to component X(m) to the point corresponding to the rth elementary reaction is r:x(m, r), and the number of directed edges from the point corresponding to the rth elementary reaction to the point representing the component X(m) is ~(m, r).

Fig. 3.1 shows the Vol' pert graph of the Lotka-Volterra reaction as an example.

It may be illuminating that graphs used by the biochemist are either Vol'pert graphs as in the case of the Lohmann reaction shown in Fig. 3.2 or are the duals of the Vol'pert graph (and this is the more usual case) as

X

2

y

3

Fig. 3.1 Vol'pert graph of the Lotka-Volterra mechanism.

Stoidwmetry 31

Creatine Creati nephosphate

ATP ADP

Fig. 3.2 Vol'pert graph of the Lohmann mechanism.

Citrate

Oxalacetate

H.~ I ~ Ac-CoA

L-Malate -<( Ac-CoA \

Aconitate

) lsocitrate

Glyoxylate -------

Fumarate Succinate ...._________.. Fig. 3.3 Vol'pert graph of the glyoxylate cycle.

in the case of the glyoxylate cycle shown in Fig. 3.3. (By the dual of a graph we mean a graph obtained from the original one by interchanging the vertices and edges but leaving the incidence relation unchanged.)

6· Show that acyclicity of (absence of directed cycles in) the Vol' pert graph of a reaction implies subconservativity.


Hint. Cf. Vol' pert ( 1972). 7. Show that conservativity is equivalent to the boundedness of the sets

x 0 + S, where x 0 is an arbitrary element of (IR + )M and S is the stoichiometric space.

Hint. Use the Kuhn-Tucker theorem. Cf. Horn & Jackson (1972). 8. Show that

(i) there exist 43 essentially different mechanisms with three short complexes, and (ii) 22 of them are conservative while 21 are not.

Hint. The solution of (i) has been given by Horn ( 1973a, b, c), while (ii) is a simple consequence of the definition of conservativity.

3.6 Open problems

I. Give general characterisations of the properties of conservativity, suband superconservativity. The equivalent conditions should have chemical meanings, properties that are easier to check than the original ones.

4 Mass action kinetic deterministic models

4.1 Kinetic equations: their structure and properties

4.1.1 Introduction

What are the simplest requirements if one wants to have a model for the time evolution of the concentration of chemical components or constituents in a reaction such as

2CO + 3H2 --. C2H2 + 2H20 2CO + 3H2 +- C2H2 + 2H20 (4 I) CO+ H20--. C02 + H 2 CO+ H20 +- C02 + H2, .

(similar to the first example of Chapter 3) and if one wants to rely on the methods of classical mathematical analysis? It is natural to choose a continuous time, continuous state model, in which the concentration of the chemical component CO at time te IRis described by x 1 (t), the concentration of H2 by x 2 (t) and so on.

Using these more abstract notations the reaction in question takes the form

2X(l) + 3X(2)--. X(3) + 2X(4) 2X(I) + 3X(2) +- X(3) + 2X(4)

X(l) + X(4)--. X(2) + X(5) X(l) + X(4) +- X(2) + X(5).

(4.2)

Let us consider a small time interval (t, t + h) and let us make the simplest Possible assumptions on the change of the components. These clearly are that .\'1(t +h)- x 1(t) depends on the quantities of the reacting components of the elementary reactions in which X(l) takes part. This dependence can be taken as linear, at least if the stoichiometric coefficient of the component in question is one. If one takes the dependence on h to be linear too (at least up


to an error o(h), a quantity tending to zero if divided by h when h tends to zero) then the change of x 1 (t) as a consequence of the third elementary reaction is

X 1 (t + h) - X 1 (t) = k 3 X 1 (t)X4 (t)h + o(h).

Furthermore, let us suppose that the effect of the single elementary reactions on the change do not interact, there is no cross-effect among them: they simply sum up.

How do we handle the problem when the stoichiometric coefficients are different from one? The effect of the second elementary reaction on the change of x 3 (t) would be

x 3 (t +h)- x 3 (t) = -k2 x 3 (t)x41 (t)x42 (t)h + o(h)

if we had two different components X(41) and X(42) instead of X(4). Therefore it is natural to assume that lumping these two components into X(4) one has

The minus sign above clearly expresses that the quantity of the component X(3) decreases in this elementary reaction, and it does so by one. Hence the change of x 1 due to the first elementary reaction is

x 1(t +h)- x 1 (t) = -2k1 (x1 (t))2 (x2 (t))3h + o(h),

where the coefficient - 2 expresses the fact that in this elementary reaction X (I) changes by two.

Using these principles an equation of the kind

x(t + h) - x(t) = f(x(t))h + o(h) (4.3)

will arise for the vector of the concentrations

x(t) = (x1 (t), x 2 (t), x 3 (t), x 4 (t), x 5 (t)).

Dividing eqn (4.3) by h and letting h tend to zero an equation of the form

x(t) = f(x(t))

is obtained that in our particular case is

.l- 1 = -2k 1(xd2 (x2 )3 + 2k2 x 3 (x4 ) 2 - k 3 x 1x 4 + k 4 x 2 x 5

.l-2 = -3k 1 (x 1 ) 2 (x2 )3 + 3k2 x 3 (x4 ) 2 + k 3 x 1 x4 - k 4 x 2 x 5

.l-3 = +kl(xd2(x2)3 - k 2 x 3 (x4 ) 2 (4.4)

.l-4 = 2k 1 (x 1 ) 2 (x2 ) 3 - 2k2 x 3 (x4 ) 2 - k 3 x 1 x 4 + k 4 x 2 x 5

.l-5 = +k3 x 1 x4 - k4 x 2 x 5

What we have arrived at in eqn (4.4) is the usual mass action kinetic deterministic model of reaction (4.1), well known to every student of introductory physical chemistry. It may also be said that (4.4) is the induced

Deterministic models 35

kinetic differential e~uation. o.f re~ction (4.1). This is the model we shall eneralise and study m detail m this chapter.

g The mass action kinetic deterministic model of the reaction

M k(r) M L fX(m, r)X(m)-+ L P(m, r)X(m) (r = l, 2, ... , R) (4.5) m=l m=l

is the autonomous polynomial differential equation

R M

"""' = L (p(m, r)- fX(m, r))k(r) n xm"(m,r) (m = l, 2, ... 'M) (4.6) r= I m= I

or, if the following notations are introduced: M

x"< "' = f1 x:!"'·'' "f = p - fX k = (k(l ), ... , k(R)) m= I

zi 0 0 0 z2 0

diag(z) = 0 ZM-1 0 0 0 ZM

then eqn (4.6) is transformed into

x = y diag(x")k (4.7)

or, introducing

A(x) = y diag(x"), x = A(x)k.

(4.8)

In order to get accustomed to this abstract formulation it may be instructive to consider special cases. Before doing so we remark that in (4.7) y only depends on stoichiometry, i.e. on the matrices fX and p, while diag(x")k expresses the dependence on the kinetics, on the rates of the elementary reactions (although in the case of mass action type kinetics it partly depends on stoichiometry through fX). This idea of separation goes back to Othmer (1981).

4.1.2 Introduction reconsidered

What we arrived at as a model to be investigated is an autonomous polynomial system of differential equations. The main questions studied in connection with differential equations are:

(I) What is a differential equation (of a certain type; no useful general definition can be given)?

(2) What is a solution?


The answers to these question can be found in introductory textbooks on differential equations (see, for example, Arnold, 1975; Bogdanov & Syroid 1983; Coddington & Levinson, 1955; Kamke, 1959). Assuming a certai~ amount of knowledge of the basic notions we only mention that our main equation (4.6) (or (4.7) or (4.8)) is explicit, because the derivatives of the highest (first!) order are expressed by other quantities. It is ordinary, because the unknown functions depend on a single variable (interpreted here as time). The system only contains the first derivatives of the unknown functions therefore it is a first order system. The right-hand side does not explicit); depend on time and so it is autonomous, and the dependence is given by (multivariate) polynomials, thus it is polynomial.

(3) How do you formulate a problem in the language of differential equations, i.e. how do you model a phenomenon using these tools?

It is only for the special area of homogeneous reaction kinetics that this question has been treated in this chapter so far. This question is usually treated in textbooks of the corresponding fields of application, and not in textbooks on differential equations; the only exception seems to have been written by Ponomarev (1962).

(4) Does there exist a solution to a given differential equation? (5) Is it unique, or, if it is not, how much is it non unique?

The existence and uniqueness of the solution of mass-action-type kinetic differential equations (or, more precisely, initial value or Cauchy problems for this type of differential equations) are ensured by general theorems, such as the Picard-Lindelof theorem (see the textbooks cited above).

(6) How can the solutions be obtained by analytical or numerical methods?

An analytical method provides the solution in 'closed form', i.e. a formula can be given for the time evolution of the concentrations. This possibility only exists for a rather limited class of differential equations, even in the special subclass of kinetic differential equations. Methods of how to derive solutions can be found in the book by Kamke ( 1959) or in problems books like those by Filippov (1979), Matveev (1983), or Krasnov et a/. ( 1978). The special case of kinetic differential equations is treated by Rodiguin & Rodiguina ( 1964) who published the explicit solution for many first order reactions, and by Szabo ( 1969), who collected results for second order reactions too (together with realistic chemical examples).

As analytical solutions can be obtained rarely, numerical methods are widely used in treating problems of chemical kinetics. Special methods of numerical analysis had to be developed to overcome the problems raised by


the stiff differential equati~ns t~at ~re so co~m~n in chemical kinetics: A tern of differential equatiOns ts satd to be sttff (m vague nonmathemattcal

sysms) if there are differences of several orders of magnitude between the ~:~es of changes of the different components. A new class of methods has been initiated by Gear (see the later book by Gear (1971)) and some of the

rograms using his or other methods (as, for example, that by Rosenbrock ~nd Storey (1966)) and. tailored specially to solve kinetic differential equations have been provtded by Allara & Edelson (1975), Deuflhard eta/.

( 1981) and others. The paper by Nowak & Deuflhard (1985) may serve both as an introduc

tion and as a survey of recent developments for realistic large systems that have dozens of components and hundreds of elementary reactions.

It may happen that it is hard to get an exact or approximate solution either by analytical or by numerical methods. In these cases one still wants to obtain at least qualitative information about the nature of the solutions. To obtain qualitative information about the solutions without solving the equation is the aim of a branch of the theory of differential equations: of the qualitative theory of differential equations.

This theory can also be used to give a mathematically strict verification of properties shown by numerical calculations. Some of the problems of this theory (that are relevant in our field) are formulated in the next questions, recent developments of the theory can be traced using the books edited by Farkas (1981), by Sz-Nagy & Hatvani (1987) and by Farkas & Hatvani (1975).

(7) Are the solutions of (4.6) nonnegative (or positive) if the initial condition contains a vector of nonnegative (or positive) concentrations?

The affirmative answer to both of these questions has been given by Vol'pert (1972). The meaning is that our model is plausible as it does not allow negative concentrations. This is a fact that should be proved as it is not selfevident from the form of (4.6) . . More refined theorems by the same author connected to this area will be

ctted below as problems.

(8) What can be said about the extrema of the solutions? Are the solutions monotonous, or do they have a specified number of maxima or minima?

Partial answers to these (rather hard) questions can be found among the Problems and in Section 4.5.

Another set of problems relates the long time behaviour of solutions.

(9) Is there a unique, asymptotically stable equilibrium point of the system (4.6)?


The simplest feature or pattern that a reaction can show is that after an initial transient period all the concentrations tend to a limiting value that is called the steady state concentration, stationary concentration, equilibrium concentration (this may be criticised from the thermodynamic point of view, but is often used in the theory of ordinary differential equations), or singular point. This equilibrium is independent of(or hardly dependent on) the initial vector of concentrations, i.e. it is asymptotically stable.

It will be the task of Section 4.2 to examine this widely accepted view. Section 4.4 deals with the possibility of the existence of more than one equilibrium points.

(10) Do the equations admit periodic solutions?

It may also happen that the concentration versus time curves tend to a periodic trajectory after an initial transient. This behaviour of differential equations is used when modelling the extremely important phenomena of oscillatory reactions. These reactions and their models will be treated in Section 4.5.

(II) Does a certain kinetic differential equation of the form (4.6) admit chaotic solutions?

A relatively new phenomenon both in theory and chemical experiments is the appearance of chaos. This behaviour can be initially characterised because, although the trajectories remain in a bounded set, they tend neither to a given equilibrium point nor to a periodic trajectory. It seems as if they have a stochastic nature in spite of the fact that there is no intrinsic or external stochasticity present.

The possibility of chaos in chemical kinetics will be touched on in Section 4.6.

The questions so far (after necessary preliminaries included in (I), (2) and (3)) have dealt with direct problems. These problems have the general form: given a differential equation what can we say about its solutions? Another set of problems called inverse problems are even more important both in chemical kinetics and in general. It may even be said that solutions of direct problems only have practical importance when used to solve an inverse problem.

( 12) Given solutions, or properties of solutions, which differential equations have the prescribed functions as solutions, or have solutions with the prescribed properties?

Section 4.7 treats an inverse problem whose solution can be formulated. This is the problem of the characterisation of kinetic differential equations among the polynomial differential equations. In other words: how can one decide if a


. polynomial differential equation may, or may not, be considered as the given · d · · · d I f h · I continuous time, contmuous state etermm1st1c mo e o a c em1ca

. ? reactwn. . , , . The last section of the chapter 1s called Selected addenda , expressmg the

fact that here very brief _re_m~rks have been made on_ diff~ren~ topics nnected with the determimst1c models of formal reactwn kmetlcs. The

~:ngth of a subsection here is by no means proportional to the importance of

the individual topics.

4.1.3 Exercises

1. Give a complete qualitative characterisation of the dynamic behaviour of the reactions

(i) (ii)

2X~Y

2X--+ Y--+ 3Z

using the explicit solution of the induced kinetic differential equation. 2. Using the result mentioned in Problem 6 below, or by any other method,

show that in the irreversible Michaelis-Menten reaction all the concentrations are strictly positive, if the initial concentrations of the enzyme and the substrate are positive.

3. Show that the differential equation

.X = s(y - xy - x - qx2 )

J = S- 1(/z - }' - X}')

z = w(x - z) (4.9)

is the induced kinetic differential equation of the Field-Koros-Noyes model of the Belousov-Zhabotinskii reaction.

Hint. Consult Field et a!. (1972) and Field & Noyes (1974).

4.1.4 Problems

I. Formulating the requirements at the beginning of Section 4.1 in mathematical terms, show that they necessarily lead to mass action kinetics.

Hint. Consult T6th & Erdi (1978) (but it is in Hungarian) or Aczel (1966).

2· Show that the domain of all the solutions of the induced kinetic differential equation of (i) first-order reactions (ii) reactions with a subconservative mechanism may be extended to the whole positive half-line of real numbers.

f:lint. The induced kinetic differential equation is linear in case (i), wh•le the trajectories in case (ii) remain in a bounded set.


3. Show the same holds for reactions with a single internal component for which the largest stoichiometric coefficient occurs on the /eft-hand side of an elementary reaction, in other words, for which the longest complex is a reactant complex.

Hint. The trajectories are either wandering in a bounded set or tend to a finite limit point.

4. Construct a reaction which has an induced kinetic differential equation that has solutions defined on the whole positive half-line of real numbers and which still does not belong to any of the reaction classes above.

5. Show that all the co-ordinates of a solution of an induced kinetic differential equation are either strictly positive for all positive times, or everywhere zero.

Hint. Consider the induced kinetic differential equation as a linear differential equation with variable coefficients.

6. Let the subset of those components given which have strictly positive initial concentrations. Give an algorithm to generate the set of components which are strictly positive for all subsequent (positive) times.

Hint. Consult Vol'pert (1972).

4.1.5 Open problems

1. Analysing Exercise 1 (ii) try to characterise a large set of generalised compartmental systems for which the induced kinetic differential equation can be solved in closed form.

Hint. It is advantageous, although neither necessary nor sufficient if the Vol'pert graph of the reaction has a tree structure.

4.2 Verifications and falsifications of traditional beliefs

The trajectory of a general autonomous system of differential equations can wander anywhere in the state-space. What kind of restrictions are obtained if one considers the trajectories of a kinetic differential equation? It was mentioned earlier (Subsection 4.1.2) that the solutions of a kinetic differential equation remain in the first orthant if they started there. More refined statements regarding positivity and nonnegativity have also been stated as Problem 6 of Subsection 4.1.4. Now let us try to delineate an as narrow as possible set in the state-space for the trajectories. As a next step towards this goal let us write the kinetic differential equation (4.6) in the form

x = (p- (X)dgx11k

and let us take the integral of both sides from 0 tot, assuming that x(O) == xo: I

x(t) = x 0 + (p - (X) f dg x(s)11k ds.

0


This formula expresses the fact that x(t) - x0 lies in the linear space of the

column vectors of

y = p - at, in S.

Let us consider an example. The kinetic differential equation of the reaction

k

2X(l)~X(2)

is:

and now

therefore

k,

.X1 = - 2kt (xt )2 + 2k2X2 x2 = kl(x1)2- k2x2.

y= [-2

I

dimS= I.

(4.10)

The meaning of the result above can be seen in Fig. 4.1, which shows the set, E, of positive equilibrium points that is now a parabola. This parabola intersects at a unique point the part of

x 0 + S

which lies in the first quadrant. Explicit solution of (4. 10) shows that the solutions (remaining all the time in the corresponding set

X 0 = S)

tend to this intersection point, i.e. the equilibrium point is (locally relatively) asymptotically stable. What we have given here is a more explicit explanation for a special case of what we mean (or what the traditional chemist means!) by regular kinetic behaviour, that is generally expected from reactions. General criteria would be needed in order to be able to decide whether large cl_asses of reactions do show regular behaviour. (The fact that not all the k~netic equations behave in this regular way is shown by simple kinetic differential equations with periodic solutions or multiple equilibrium points; s~e Sections 4.3-4.6). To put it in more technical terms, regular behaviour is ~ e same as the quasithermodynamic behaviour introduced by Horn & ~ckson ( 1972). This behaviour can be found in a reaction if there exists a

sm_gle equilibrium point in all positive reaction simplexes; all the equilibrium P~IDts are relatively asymptotically stable and no nontrivial periodic solUtions exist.

th It would be desirable if a general sufficient condition ensuring quasiermodynamic behaviour was formulated and related to the mechanism,


x.

x,

s X0 + S

Fig_ 4.1 The trajectories of a simple mechanism.

and was independent from the actual value of the reaction rate constants. He "Te we present two such general criteria. The first is the Zero Deficiency The!orem by Feinberg (1972b), Horn & Jackson (1972) and the second is a the orem by Vol'pert also from 1972.

C Before formulating these theorems we mention in passing that the aim of Kr .am beck (1984) also was to give narrow limits for the wandering of tra_jectories in the case of first order reactions.)

4.2 .I The zero deficiency theorem


X( I)--. X(2) X( I)+ X(3)--. X(4)

\ / X(2) + X(5)

an<t the same reaction with a reversible first step:

(4.11)

Deterministic models

X(l)~X(2)

X(l) + X(3)-+ X(4). '\ /

X(2) + X(5)

43

(4.12)

H e as usual all the complexes are written down exactly once. The directed er l · d · h l · aphs above with the comp exes as vert1ces an w1t e ementary reactiOns as

;~ges are the FHJ (Fein_berg-Ho~n-Jackson)-graphs of the reaction. The FHJ-graph of both reactiOns cons1sts of two connected parts called linkage classes. Let us denote the number of linkage classes by Lin the general case.

Now the deficiency of the mechanism is defined as

o = N- L- S,

where N is the number of different complexes and S is the dimension of the stoichiometric subspace.

In the case of our examples N = 5, L = 2, S = 3, thus o = 0. It might be pointed out that the value of the deficiency is independent of the value of the reaction rate constants; thus it is a characteristic of a mechanism rather than of a reaction. It is also true that the value of the deficiency remains untouched by making an elementary reaction reversible (i.e. adjoining the antireaction pair of an elementary reaction) or by making a reversible elementary reaction irreversible.

The FHJ-graph may or may not have an important property of being weakly reversible. This means that going along a directed route from one complex into another one implies the existence of another directed route from the second complex into the first one. If this is not fulfilled the graph (or the reaction) is not weakly reversible.

Example ( 4.11) is not weakly reversible as there is a directed route (of length I) from X( I) to X(2) whereas there is no route in the backward direction. . Example ( 4.12) is weakly reversible, although it is not reversible. The mverted implication does hold: obviously, all reversible reactions (reactions for which all the elementary reactions have their antireaction pair) are weakly reversible as well. ~y now all the notions are at hand to state the zero deficiency theorem by

Femberg, Horn and Jackson. Let us consider a mechanism with deficiency zero. Then

(I) if the mechanism is weakly reversible then with any choice of positive rate constants there exists in each stoichiometric compatibility class

x 0 + Sn(IR+)M

exactly one equilibrium concentration vector; every equilibrium point is rhela~ively asymptotically stable and no nontrivial periodic solutions to t e mduced kinetic differential equation exist;


(2) if the mechanism is not weakly reversible then an equilibrium con. centration cannot be positive, i.e. if it exists then at least one of its co. ordinates is zero.

The first part of the theorem means that within the class of zero deficiency mechanisms weak reversibility is enough to ensure regular behaviour.

One of the simplest examples with nonzero deficiency and with exotic behaviour is the Lotka-Volterra reaction:

X-+ y X+ Y-+2Y

y-+ 0,

where N = 6, L = 3, S = 2, thus o = I, therefore the zero deficiency theorem cannot be applied even in the reversible case.

The ecological interpretation of part (2) of the theorem lies in the fact that it may give an answer to questions like this: can a reaction modelling an ecological system evolve towards an equilibrium population in which all species coexist, or, to put it another way, none of them become extinct? Statements on coexistence have been formulated by several authors recently, as, for example, by Epstein (1979), by Goh (1975), or by Freedman & So (1985).

Finally, let us make a remark that may enlighten the significance of relative asymptotic stability. As is known (from Exercise I of 4.1.3) the solutions of the induced kinetic differential equation of the reaction

2X~Y

remain in the set

A = ((xo. Yo) + S) n (~+ )2

if the initial condition is

x(O) = x 0 , y(O) =Yo·

The theorem only asserts that solutions starting from a point close enough to the equilibrium point (x*, y*) in the set A tend to (x*, y*).

The question may arise how broad is the class of deficiency zero mechanisms. Investigations of textbooks on classical chemistry shows that a large part of the mechanisms possess this property. A systematic investigation of mechanisms with three short complexes by Horn ( l973a, b, c) has shown that nearly all these mechanisms are of zero deficiency.

Another answer to the same question is that all the compartmental systems so important in chemical and biological modelling (see, for example. Jacquez, 1972) are of deficiency zero (Horn, 1971). A generalisation ofthe~e mechanisms, the generalised compartmental system, also belongs to thiS class; see Problem 2.


From the viewpoint of classi_cal kinetics it is the ~lass of detai!ed ~alanced

h. nisms that is the most Important. A reverstble mechamsm ts called mec a h . . . . * " h" h

1 ·led balanced if t ere extsts a postttve concentratiOn vector x 10r w tc aetal e (r)X*"(·,r) = k- (r)x*P(·,r) (4. J3)

if the elemen~ary reactions and the corresponding rate constants are denoted in the followmg way:

M k+(r) M

L ~(m, r)X(m) ~ L '(m, r)X(m). m ~ 1 k (r)m ~ 1

It is obvious that a concentration x* for which ( 4.13) holds is an equilibrium concentration. It is not so obvious that mechanisms with the above property are of deficiency zero. This fact is an important by-product of the Feinberg-Horn-Jackson theory and shows that the zero deficiency theorem expresses a generalisation of classical beliefs.

Quasithermodynamic behaviour of detailed balanced reactions has also been proved in full generality by Shear ( 1967) (he even thought that he had proved a more general statement than he actually did; see the paper by Higgins (1968)), and by Vol'pert and Khudyaev (1975).

Finally, let us mention that the question as to whether a directed graph has the property of weak reversibility or not is well known in graph theory. A paper providing a quick algorithm utilising the specific structure of the graph is that of Ebert ( 1981 ).

4.2.2 Vo/'pert's theorem

Weakly reversible reactions represent an extreme case in a certain sense: all the complexes are contained in at least one of the cycles of the FHJ-graph, and so there are many cycles in the graph. Now let us consider the other extreme case - an acyclic graph.

Let us consider a mechanism with an acyclic Vol'pert graph, and let us suppose that the zero complex is not a reactant complex in any of the elementary reactions. Then

(I) the solution of (4.6) is defined on the whole positive half-line; (2) the solutions tend to a limit as time tends to infinity, and this limit is an (J equilibriu_m point of (4.6); ~

) the reactiOn rates of all the elementary reactions vanish at all the nonnegative (!) equilibrium points of (4.6) (or to use the FHJ

(4) terminol~g7, the reaction is detailed balanced); (S) no nont~tvtal periodic solution exists;

~here ~xtsts a positive constant K such that for all solutions to (4.6) the ollowmg inequality holds:

46 Mathematical models l~l chemical reactions

X

Jlxm(s)lds < K (m =I, 2, ... , M). (4.14)

0

Several remarks may help in understanding the significance of the theorem.

As acyclic reactions are subconservative (Problem 6 of Section 3.6) statement (I) is a consequence of Problem 2 of Subsection 4.1.4.

Statement (3) together with the zero deficiency theorem implies all the other statements, as the deficiency of detailed balanced reactions is zero.

The meaning of statement (5) is that the solutions are not too wavy, they do not go up and down too often. (A periodic function does not fulfil the inequality in this statement, therefore statement (5) implies statement (4).)

As the V-graph of a weakly reversible mechanism is cyclic (Exercise 4 below), the regular behaviour of a mechanism cannot be inferred from both zero deficiency theorem and by Vol'pert's theorem at the same time: at most one of them can be used.

As the FHJ-graph and the V -graph of a compartmental system may be considered to be identical, for compartmental systems it is literally true that regular behaviour can be assured when there are either no or many cycles.

4.2.3 Remarks on related literature

Some of the recent papers that have not been adequately covered here, although their reading is highly recommended, will be mentioned briefly. To orient the reader, we should like to make some comments on these works in the order of their publication date.

It was the paper by Wei (1962) and later the papers by Aris (1965, 1968) that had perhaps the greatest influence on the development of today's formal reaction kinetics.

Gavalas (1968) was an early pioneer in the treatment of the deterministic models of chemical reaction kinetics. His book deals with homogeneous systems and systems with diffusion as well. Basing himself upon recent results in nonlinear functional analysis he treats such fundamental questions as stoichiometry, existence and uniqueness of solutions and the number and stability of equilibrium states. Up to that time this treatise might be considered the best (although brief and concise) summary of the topic.

The finely written manuscript by Horn (1973e) is an introductory textbook to the Feinberg-Horn-Jackson theory. Certainly more will be (or, has been) included in the lecture notes by Feinberg (in preparation).

Vol'pert et a/. (1974) investigate the co-limit sets and determine the Lyapunov exponent for large classes of reactions.

Yablonskii and Bykov (1977) summarise results of the Novosibirsk group in comparison to modern development elsewhere. Their main interest is in


the general treatment of catalytic processes. The results concern both transient and equilibrium behaviour (including multistationarity in the second case).

West (1977) outlines a general theory of the deterministic model including stoichiometry in detail. Stability of the equilibrium points is also treated (although in a less detailed way than by Stucki (1978)). Relations with thermodynamics are elaborated. The authors of the present book are not aware of a bound copy edition of this manuscript rich with information.

Stucki ( 1978) mainly deals with the stability of the steady states of kinetic differential equations. He gives an introduction to stability theory that is more detailed than those found in the usual textbooks on ordinary differential equations. He also shows how to apply the different methods to problems of biochemical kinetics. His paper also includes some of Clarke's results up to that time.

Clarke (l974a, b, 1980) gives a detailed analysis of the stability of the steady state using the structure of the Vol'pert graph and several derived graphs to check the Routh-Hurwitz criterion.

Kinetic logic (Thomas, 1979; King, 1986) is also a tool that provides a structural approach to classifying dynamical systems' behaviour. As these authors are mainly interested in exotic behaviour we shall say a few words more about this topic in Section 4.3.

Orlov & Rozonoer (l984a, b) present a general phenomenological approach to the macroscopic description of the dynamics of open systems. They prove theorems on existence, uniqueness and stability of the stationary states. Singularly perturbed equations are also considered. The variational principle for the studied equations is formulated as well. As an application of the general results, nonisothermal kinetic differential equations are considered, detailed balanced and balanced systems are studied and the method of quasistationary concentrations is discussed. For open isothermal systems a theorem on the existence of a positive stationary state is proved providing a solution to an old basic problem.

4.2.4 Exercises

I. Show that the Vol'pert graph of a weakly reversible mechanism is necessarily cyclic.

2. Calculate the explicit solution of the kinetic differential equation of the reaction 2X(l) ~ X(2) and show that they have all the properties mentioned above. (Think about the impracticality of carrying through the same procedure for a reaction with, say, five chemical components!)

3. Which reactions are of zero deficiency in the following two classes of reactions: (i) reactions with two complexes,

(ii) reactions with one component?


4.2.5 Problems

1. Show that all the reactions with three short complexes and with a single linkage class show regular behaviour, either by proving the equality of the deficiency to zero, or by directly verifying the properties needed for quasithermodynamic behaviour.

Hint. Consult Horn (l973a, b, c). 2. Show that the deficiency of a (generalised) compartmental system is zero.

How can this statement be generalised? Hint. Consult Horn & Jackson (1972, p. 94).

3. Show that independence of the complex vectors is assured by the strict but chemically relevant requirement:

yry' = 0 for all complex vectors y, y'.

(The requirement above expresses exclusion of a special kind of autocatalysis and autoinhibition.)

(The statements of the last two problems together imply the fact that if no elementary reaction contains the same component as both a reactant and a product and if all the complex vectors are orthogonal to each other then the deficiency of the mechanism is zero.)

4. Show that the kinetic differential equation of a reversible reaction has at least one positive equilibrium point.

Hint. Consult the paper by Orlov & Rozonoer (l984b). 5. A mu/tice/1 reaction system consists of a finite number of cells of the same

volume and temperature and having the same reaction within them. Transport between the cells can be described by first order (formal) reactions. Show that, if the (common) mechanisms within the cells are of zero deficiency, and are weakly reversible, then the whole multicell reaction system is quasithermodynamic.

Hint. Consult Shapiro & Horn (1979). The results are worth comparing with those by Othmer (1985) on the dependence of oscillations in networks on the coupling strength, because his concept of network is a slight generalisation of the concept of multicell reaction system.

4.2.6 Open problems

1. How is it possible to gain more detailed qualitative information about the behaviour of the trajectories relating questions such as (i) domain of the solution,

(ii) exponential stability etc. without solving the kinetic differential equation of a reaction?


4.3 Exotic reactions: general remarks

The most regular behaviour of a solution of a differential equation is if it tends to a constant in the state-space. It is a bit more complicated if, after an initial transient period, the solution tends to a periodic orbit. In more than two dimensions it may also happen that the trajectories remain in a bounded set, but they neither tend to an equilibrium point, nor to an oscillatory solution. This is a loose definition of chaos.

These three essentially different possibilities can be seen in Fig. 4.2.

X X X

Fig. 4.2 Regular and exotic solutions of differential equations.

Investigating all possible solutions of a differential equation it may happen that, although the solutions tend to an equilibrium point, the equilibrium point depends on the initial conditions. Here we do not think of the necessity to remain in the same stoichiometric compatibility class, we think of the non uniqueness within a single class. This is the case of multistationarity. The same thing may happen with multiple periodic solutions and with multiple chaotic attractors, or even with several attractors with all the three types.

In the previous section regular behaviour has been dealt with; now we turn to multistationarity, oscillation and chaos. The other possible combinations seem· to be more complicated, and they have not been regularly studied so far. (See, however, Rossler (1983a).)

In these sections we remain within the area of homogeneous reaction kinetics. Questions regarding phenomena connected with spatial effects, like propagating fronts, will be touched on in Chapter 6.

4.4 Multistationarity

4.4.1 Multistability

Mu/tistability is the more general notion we start with. It means that the system of differential equations has multiple attractors. Here we only deal with multistationarity, the case in which all the attractors are equilibrium points.


4.4.2 Multistationarity in kinetic experiments

The first experimental results on multistationarity seem to be those by Liljenroth ( 1918).

Transitions between multiple steady states has been reported by Creel & Ross ( 1976) who performed closed system studies under laser illumination, of the photochemically controlled equilibrium between N0 2 and its dimer.

Ganapathisubramian & Showalter (1984b) measured the steady state iodide concentration in the iodate-arsenous acid system as a function of reciprocal residence time and found multistationarity.

4.4.3 Multistationarity in kinetic models of continuous flow stirred tank reactors

In this field, as in any other part of reaction kinetics, there are two approaches: investigation of simple, but possibly realistic models; see, for example, Caram & Scriven (1976), Othmer (1976), Luss (1980, 1981) Gray & Scott (1983a, b) or the example of Horn & Jackson (1972, cited here as Exercise I) or a search for general criteria that ensure or exclude multistationarity in large classes of mechanisms. This second approach was initiated by Rumschitzky & Feinberg (in preparation; Rumschitzky, 1984). Here we present results of this second approach, and some of the Problems contain results on specific models derived using the first approach.

The theorems by Feinberg, Horn, Jackson and Vol'pert provide sufficient conditions to exclude multistationarity. These theorems can be applied in the case of homogeneous systems, and in the case of inhomogeneous systems, if the system can be modelled by formal elementary reactions as shown several times above. An especially important case of an inhomogeneous systems is the isothermal continuous (flow) stirred tank reactor (CSTR). By a CSTR we mean one in which there is perfect mixing and in which, at each instant, every component within the reaction vessel is also contained in the effiuent stream.

Investigation of several examples shows that the deficiency of the models of these reactors is usually more than zero, and their Vol'pert graph is usually cyclic. The authors mentioned above started another approach to avoid these problems. Their results belong to two different classes: some of them come from the direct investigation of formally homogeneous models of CSTRs and others of them come from the application of the 'deficiency one' theory. Before applying this theory it must (and can) be shown that chemically relevant assumptions define a class of models with deficiency one.

Now let us turn to the simplest results in the first class: to the theory specifically devised to treat reactions in CSTRs. It turns out that it is only the structure of the mechanism in the reactor that counts.

A critical species (or component) is a component that appears in two or more distinct complexes.

Suppose that the reaction under consideration is such that:


(I) there exists no species that appears in two or more complexes within the same linkage class;

(2) there exists at most one linkage class in which two or more critical species appear.

Then the induced kinetic differential equation for the corresponding CSTR cannot admit multiple positive steady states, no matter what the feed composition may be and no matter what (positive) values the residence time and rate constants take.

From the theoretical point of view it is an easy consequence of this statement that if there are no critical species in the reaction then multiple positive steady states cannot occur. The practical information provided by this consequence, however, is hardly trivial.

Another corollary of the first statement is: if there exists precisely one critical species, and if within each linkage class there exists at most one complex in which that species appears, then multiple positive steady states cannot occur.

More refined statements can be found in Rumschitzky (1984) and in Rumschitzky and Feinberg (in preparation).

A statement of the second class will also be cited from Feinberg ( 1980). We emphasise that now we will not deal any more with CSTRs.

Consider a weakly reversible reaction with L linkage classes. Let 3 denote the deficiency of the reaction and let 31 (I= I, 2, ... , L) denote the deficiency of the /th linkage class and suppose that

(I) (i) 31 :::;; I (/=I, 2, ... , L), (4.15)

(2) (ii) (4.16)

Then the induced kinetic differential equation admit precisely one positive equilibrium point in each stoichiometric compatibility class.

Finally, let us mention that it is a wide spread belief that any reasonable model of an oscillatory chemical reaction is expected to show bistability for another region of the parameter values. This belief formed the bases of a criticism on the validity of the explodator. Literature on this topic can be found in Farkas et a/. (submitted).

4.4.4 Exercises

I. Show that the weakly reversible conservative mechanism of Horn & Jackson (1972)

3X~X+ 2Y lj t !I 2X+ Y~3Y


has three positive equilibrium points if 0 < E < 1/6, and it has a single positive equilibrium point if E is larger then 1/6.

2. Show that the mechanism

A(l) + A(2)-+ A(3)~2A(4)

" / A(S)

in CSTR admits precisely one positive equilibrium point. 3. Show that the mechanism

A(3) + A(4)~A(l)~2A(2) A(l) + A(5)~A(6)-+ A(7)~2A(8)

in CSTR admits precisely one positive equilibrium point. 4. Show that the mechanism

2A(l)~A(2)~2A(3)~A(4)

~ ~ A(l) + A(3)

A(l) + A(4)-+ A(S) A(3) + A(5)~2A(7) ~ / A(6)

admits precisely one positive equilibrium point. 5. Show that a weakly reversible reaction which is of deficiency zero or one

and which consists of a single linkage class admits precisely one positive equilibrium point.

6. Show that while the Edelstein mechanism (Edelstein, 1970)

X~2X

X+Y~Z~X

which is known to have the capacity to generate multistationarity, does not fulfil the condition given in Exercise 3 above, while its 'perturbed' version

X~2X

~ X+Y~Z~X

admits precisely one positive equilibrium point.

4.4.5 Problems

I. Prove (by analytical or numerical methods) that the (logarithm of the) steady state iodide concentration as a function of the (logarithm of) k + k' is as shown in Fig. 4.3, if the kinetic differential equations of the iodate-arsenous acid reaction is to be taken


72 .. 70

-6.8

~6.6~ • 64 0 0 -52

6.0

S.~ .6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 s.o· + log 11r + k'l

(a)

6.8,-----------,

6.4 sn

...... •"sp

2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 5.0 + log (lr + lr')

(C)

6.8r----:::--:==--------, ·y; 6.4

sp\ /sp sn

4.4

4.01.__,..,~~~~:c-:':c-:'-;--;!"::--:"::--:':" 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.8 3.8 4.0

5.0 + log (k + k')

(b)

6.8r-------------,

6.4

~::v + 5.2 0

~4.8

4.4

sn

4•0 ':':2.2-:-:'2.-o-4 -:2':'".6 --:2:':".8-3;!-:.0,..--:-.3.2,.--:-3.4-:--=3.-:-6 --:3':'".8-.~.0 5.0 + log (lr + lr')

(d)

53

Fig. 4.3 Multistability, mushrooms and isolas. (a) k' = 0 s- 1• (b) k' = 1.20 x 10- 3

S- 1• (C) k' = J.30 X J0- 3 S- 1• (d) k' = 1.42 X 10- 3 S- 1•

x = R(x, y) + kx(O) - (k + k')x y = - R(x, y) + kx(O) - (k + k')y,

where the following notations, abbreviations and data are used:

x = Wl y = [103] a= [H+] R(x, y) = (k1 + k 2 x)xya2

x(O) = 8.40 X 10-5 moldm- 3 y(O) = 1.01 X 10- 3 moldm- 3

a= 7.59 X 10-3 moldm- 3 k2 = 4.5 X 108 (moldm- 3)-4s-l

k1 = 4.5 x 103(moldm- 3)- 3s- 1

and it is known that

x + y = k(x(O) + y(O))!(k + k').

Hint. Consult Ganapathisubramanian & Showalter (1984a) where chemical details and previous models may be found.

2. Show that condition (4.16) is equivalent to the requirement that the stoichiometric space be the direct sum of the stoichiometric spaces generated by the linkage classes separately.

Hint. Use the definition. 3. Show that the absence of weak reversibility from the assumptions of the

last on (4.15) and (4.16) might preclude both the existence and uniqueness


of positive equilibrium points. Hint. Cf. Feinberg, 1980.

4.5 Oscillatory reactions: some exact results

In the last two decades there has emerged great interest in reactions, in which at least one of the concentrations is a periodic function of time. There are two possible application areas of these reactions: they may form the chemical basis of periodical phenomena in biology, much as biological clocks; and in chemical engineering it may happen that the yield may be enhanced when working under periodic instead of steady state conditions.

Here we intend to give a very brief introduction to the wide area of oscillatory reactions. The interested reader should consult the collection of papers edited by Field and Burger (Field & Burger, 1984). The review papers there have been written by the foremost experts in this field, who have already been cited so often here, and will continue to be so. The book also contains a scientiometric assay by Burger and Bujdos6, that may be interesting for an even larger audience.

4.5.1 Periodicity in kinetic experiments

The first example of oscillation in a homogeneous chemical reaction was discovered by Bray (1921 ). This is an acidic hydrogen peroxide and iodate ion system.

Frank-Kamenetskii (1947) has given a chemical interpretation to the Lotka-Volterra model: he has shown that oscillations in the cold flame can be described using the model.

Winfree (1984a) provided a detailed account of the pioneering work of Belousov, Zhabotinskii and others in the USSR.

4.5.2 Excluding periodicity in differential equations

The first results on excluding periodicity in differential equations are due to Poincare and Bendixson. Their most widely used theorem states that if the divergence of the right-hand side of a two-dimensional system of differential equations is of constant sign in a domain then the differential equation cannot have periodic solutions in that domain. This statement has been generalised by Dulac (1937) in such a way that the divergence can be calculated after having multiplied the right-hand side with an arbitrary, sufficiently smooth function.

The problem with all these types of theorems is that they only relate to two-dimensional systems because they essentially utilise the fact that the plane is divided into two parts by a closed, sufficiently smooth curve (Jordan theorem). Therefore it would be desirable to have theorems relating to three-


or multidimensional systems. An attempt to formulate such a theorem was made by Demidowitsch (1966), but to his conditions a further one had to be added in order that the statement remain true, and this was done by Schneider (1969). Other attempts in this direction can be found in T6th (1987). a theorem from that paper will be cited.

Let T c IRN (N > 1) be a domain for which the set

ff(x) =: {yEIRN-I; (.X, y)ET}

is convex for all xE R Let us suppose that Pis a continuously differentiable function from T to IRN, and let us suppose that the differential equation

.X= Pox (4.17)

has N - I continuously differentiable first integrals H 1 , ••• , H N _ 1 for which

a2H1(x, y 1 )

a2HN-I(x, yN-I>

Then the differential equation ( 4.17) has no periodic solution.

4.5.3 Excluding periodicity in reactions

Based upon the results ofBendixson and Dulac the investigations ofHanusse ( 1972), Tyson & Light (1973) and P6ta (1983) have shown that in twocomponent bimolecular systems there is only one oscillator: the Lotka-Volterra model.

Studying three-dimensional systems P6ta has shown (P6ta, 1985), again using Dulac-type theorems, that there only exists a single bimolecular reaction which may exhibit oscillatory behaviour, and this is the Ivanova reaction:

X + Y-+ 2 Y Y + Z-+ 2Z Z + X-+ 2X.

He has also shown that in a continuous-flow stirred tank reactor no bimolecular reaction can show sustained oscillations.

It might be repeated here that the statements in Section 4.2 also preclude periodicity in several classes of reactions.

4.5.4 Sufficient conditions of periodicity in differential equations

It is a complicated problem proving that a given differential equation has periodic solutions. Although there exist more than one method, especially in the two-dimensional case, all the methods involve more complicated calculations than the methods of excluding periodicity.

The most natural method in the two-dimensional case is the one that proves that the trajectories remain in a bounded domain that contains an


isolated unstable stationary point. This condition is enough to ensure that there exists a periodic solution (see, for example, Coddington & Levinson, 1955).

The second class of methods is based upon fixed point theorems (see, for example, Andronov eta/., 1967) while the third, possibly most popular, one is the one that proves that an Andronov-Hopf bifurcation occurs. Without giving details we recapitulate here what it means. One follows how the eigenvalues of the linearised part of the right-hand side change while a parameter (called the bifurcation parameter) changes. It may happen that there exists a value of this parameter when a pair of conjugated complex eigenvalues appear with zero real part and the derivative of the real part with respect to the parameter is different from zero. This - together with regularity conditions (see, for example, Schneider (1980) or the literature cited there)- is enough to ensure the appearance of a periodic solution in the neighbourhood of the stationary point. Problems 2 and 4 of Subsection 4.5.9 show kinetic examples to illustrate the application of this argument.

4.5.5 Sufficient conditions of periodicity in reactions

The theory of oscillating reactions started with the papers by Latka. In Latka (1920) he provided the Lotka-Volterra model and has shown that it leads to sustained oscillations. Since then many models have been shown to lead to oscillations; some papers in this area are: Hsii (1976), Wolfe (1978), Kertesz (1984), Schneider et a/. (1987) etc.

It is interesting to note that several authors were able to provide sufficient conditions for classes of reactions, e.g. Schnakenberg ( 1979), Csaszar et a/. (1981) and experts in kinetic logic (see the papers by Thomas (1979, 1981), King (1980, 1982a,b, 1983, 1986) and Glass (1977)). The problem is that all these statements are mere indications from the strictly mathematical point of view.

4.5.6 Designing oscillatory reactions

After the investigations mentioned in Subsection 4.5.3 another question has arisen. Is it also true that the Lotka-Volterra model is the simplest unique one among all those models having the same linearised form (around their own stationary state)? From the chemical and the mathematical point of view this is a very different question.

From the mathematical point of view the answer (Toth, Hars 1986b) is a new proof that the Lotka-Volterra model is the only possible twocomponent bimolecular oscillator - under a different set of assumptions than those made in Subsection 4.5.3. The method of the proof can also be used to prove that the two-dimensional explodator model by Farkas et a/. (1985) is almost as unique in its own class.


However, the chemical approach seems to be more useful. The investigations by T6th & Hars (1986b) bear on the area of designing periodic reactions. Experimental work in this field has been reviewed by Epstein . (1983), while Escher (1981), Schnakenberg (1979) and Csaszar eta/. (1981,) 1983) have discussed the theory. The results may be obtained as results of a mixed-integer programming problem too, and this formulation may be the first step towards a method for the design of chemical models with prescribed irregularities, for example with periodic or exploding solutions, witJ:t multistationarity, or with local controllability.

4.5. 7 Overshoot-undershoot kinetics

A more natural phenomenon seems to be the oligo-oscillation, or the overshoot-undershoot phenomenon. These expressions denote the case when there is only a finite number of local extrema on the concentration versus time functions. Natural as it is, it has rarely been studied in a well-controlled experiment (see, however, Rabai eta/., 1979). It has also rarely been studied from the theoretical point of view. This situation can be explained by the fact that the qualitative theory of differential equations usually makes statements on long-range behaviour and much less on transient behaviour. The only exception seems to be that P6ta ( 1981) has given a complete proof of the statement called Jost's theorem which says: in a closed reversible compartmental system of M components none of the concentrations can have more than M - 2 strict extrema. The methods used by P6ta makes it possible to extend this result (see Problem 6 below). Another result of this type, relating nonlinear kinetic differential equations, can also be found among the Problems.

4.5.8 Exercises

I. Show that the induced kinetic differential equation of the simple consecutive reaction

X--+Y--+Z

under the usual initial condition

x(O) = I y(O) = z(O) = 0

can only have solutions for which y has exactly one strict local maximum on the interval [0, + oo ). Is the special form of the initial condition essential?

2. Applying the last statement of Subsection 4.5.2, show that the induced kinetic differential equation of the reaction

H 2 0 + H02 ~ 30H


has no periodic solution if all the reaction rate constants are taken to be I. Can you find other methods useful in this special case?

4.5.9 Problems

I. Applying the last statement of Subsection 4.5.2 show that, if the number of atoms is one less than that of the components in an atomic reaction obeying the law of conservation of atomic numbers, and if the stoichio metric matrix is of full rank, then the induced kinetic differential equation of the reaction has no periodic solution.

Hint. Consult T6th 1987. 2. Show that the differential equation (4.9) (the induced kinetic differential

equation of the Field-K6r6s-Noyes mechanism model of the Belousov-Zhabotinskii reaction) does have periodic solutions at certain values of the parameters.

Hint. Consult Hsii (1976). 3. Show that the generalised Volterra equation

xm = xm( f ampxp +am.) (m = I, 2, ... , M) p=!

(4.18)

does have a stable limit cycle for the case M = 3 (in contrast to the M = 2 case).

Hint. Consult Hofbauer (1981 ). 4. Show that the induced kinetic differential equation of the reaction

k, x~z

k,\ fk, y

k. 2X+ Y~3X

shows stable oscillations for some regions of the rate constants and certain intervals of total mass.

Hint. Consult Schneider et a/., (1987). From that paper, or from previous papers by Erdi eta/. (e.g. Erdi & T6th, 1981) it turns out that this reaction can be considered as a model for synaptic transmission based upon the transmitter recycling hypothesis.

5. Show that the reaction

x~2x, X+ Y~z. z~2Y, Y~o

called explodator shows oscillatory behaviour with growing amplitude, while it has periodic solutions if the elementary reaction (limiting reaction) 0 ~ X is added.

Hint. Consult Farkas & Noszticzius (1985) and Kertesz (1984). 6. Show that a first order reaction among M components with real

eigenvalues of the coefficient matrix of the induced kinetic differential


equation is such that none of the concentrations can have more than M- 2 strict local extrema. Give sufficient conditions for having real eigenvalues.

Hint. Cf. P6ta (1981), Hearon (1963) and West (1981). 7. Show that the induced kintic differential equation of the reaction

I I I 7 Y-+ 0 ~X-+ X + Y-+ 2X

4

under the initial conditions

x(O) = 0 and y(O) ~ I

can only have solutions for which x has exactly one strict local extremum in [0, + oo).

Hint. Consult Pinter & Hatvani (1977-80).

4.5.10 Open problem

I. Give general sufficient conditions of periodicity for large classes of reactions.

4.6 Chaotic phenomena in chemical kinetics

'The chemical community today views chemical chaos much as it did chemical oscillation 20 to 30 years ago. There are a number of "enlightened" students of and believers in the phenomenon, but the vast majority of chemists are either ignorant of or sceptical about the possibility of genuine chaos in a well-controlled chemical system.' This generally accepted characterisation of the situation has been given by Epstein (1983, p. 47). Being aware of the truth of these words we are obliged to give a short introduction to the mathematical theory of chaos and strange attractors. Then we turn to chemical kinetic experiments (mostly done on different variations of the Belousov-Zhabotinskii system) where apparently chaotic behaviour has been measured. Then we consider the answers given by different authors to the question: is it possible to produce chaotic behaviour in as realistic as possible kinetic models, i.e. induced kinetic differential equations of complex chemical reactions? Finally, attempts made towards the structural characterisation of chaotic kinetic models will be reviewed.

4.6.1 Chaos in general

Numerical investigations of the solutions of the equation

.X= cr(y- x) y = rx- y- xz z = -bz + xy

- nowadays called the Lorenz equation - have shown (Lorenz, 1963) that


they may tend neither to a stationary point nor to a periodic orbit: still they remain in a bounded set. This is a qualitatively new phenomenon in the possible behaviour of solutions of differential equations. It only appears when the dimension is more than or equal to three.

This kind of behaviour - called chaotic behaviour - has no generally accepted, strict in the mathematical sense definition, so we mention some of the other characteristics.

In the theory of differential equations the Peano inequality ensures that the solutions starting near each other cannot diverge from each other at a rate that is above exponential. Chaotic solutions, however, are examples where the divergence is not less than exponential. To put it another way, no matter how exact the measurement of the initial conditions was, one cannot well estimate the value of the solution at a later time.

Another characteristic of the solutions of such an equation is that they not only look like realisations of stochastic processes, they can be proved to have the same properties if one uses statistical methods.

It is a promising fact that the routes to chaos are qualitatively universal: there exist sequences of events that occur when a parameter of the right-hand side is changed until the chaotic regime is reached.

The reader can obtain a detailed, and more rigorous and vivid, introduction to the topic when he turns to the recent collection of papers edited by Holden ( 1986) (see also Berge et a/., 1983).

4.6.2 Chaos in kinetic experiments

Recent reviews on chaos in chemical kinetics have been given by Swinney & Roux (1984) and by Epstein (1983). Here we shall rely heavily on these papers.

Olsen & Degn (1977) first observed nonperiodic behaviour in an enzyme system. Soon after observations of nonperiodic behaviour in the Belousov-Zhabotinskii (BZ) reaction were reported by Schmitz eta/. (1977), Rossler & Wegmann (1978), Vidal eta/. (1977), Hudson eta/. (1979) and others.

By the time of the 1981 Bordeaux conference observations of nonperiodic behaviour in CSTRs had begun to be analysed by different methods. These methods are reviewed by Swinney & Roux (1984) and newer methods have been proposed and shown to work in the chemical applications.

However, at that time there still existed a healthy scepticism regarding the existence of nonperiodic behaviour in well-controlled reactions. After all, nonperiodic behaviour can arise from fluctuations in stirring rate or flow rate, evolution of gas bubbles from the reaction, spatial inhomogeneities due to incomplete mixing, vibrations in the stirring motor, fluctuations in the amount of chemical components (such as bromide and dissolved oxygen) in the feed, and so on. Any experimental data, no matter how well a system is


controlled, will contain some noise arising from fluctuations in the control parameters; therefore it is reasonable to ask: 'Will noise, always present in experiments, inevitably mask deterministic nonperiodic behaviour (chaos)?'

Based it upon a summary of a large body of experience by groups in Bordeaux, Virginia and Texas, Swinney and Roux state that the answer is no: nonperiodic behaviour observed in chemical reactions is, at least in some cases, chaos, not noise. It is even sometimes possible to separate the experimental noise from the deterministic dynamics. The main argument for this is that alternating periodic-chaotic sequences, period doubling, intermittency, wrinkles on a torus- all these routes to chaos- have been found in kinetic experiments.

4.6.3 Chaos in kinetic models

Two lines of research have been taken. Some people strive to construct ever more realistic (from the chemical point of view) models, starting from simple formal models, while others try to show that refined models of the Belousov-Zhabotinskii reaction are able to reproduce the apparently chaotic behaviour found in experiments.

As to the first line, Rossler ( 1976) was the first to provide a chemical model of chaos. It was IlOt a mass-action-type model, but a three-variable system with Michaelis-Men ten-type kinetics. Next Schulmeister ( 1978) presented a three-variable Lotka-type mechanism with depot. This is a mass-action-type model. In the same year Rossler ( 1978) presented a combination of a LotkaVolterra oscillator and a switch he calls the Gause switch showing chaos. This model was constructed upon the principles outlines by Rossler ( 1976a) and is a three-variable nonconservative model. Next Gilpin ( 1979) gave a complicated Lotka-Volterra-type example. Arneodo and his coworkers ( 1980, 1982) were able to construct simple Lotka-Volterra models in three as well as in four variables having a strange attractor.

The most realistic reaction to provide chaos so far was presented by Willamowski & Rossler ( 1980) at the 1979 Aachen meeting. This model may be considered as an approximation (via the introduction of external components) of a conservative, unconditionally detailed balanced reaction. The reaction is worth depicting here:

A + X=::; 2X X + Y =::; 2 Y E + Y =::; B X + Z =::; C D + Z =::; 2Z. (4.19)

The next step should lead to the meeting point with the other line of research and should provide chemically realistic models of an actual reaction that shows irregular behaviour. Chaos in the model should be proved and understood theoretically, not just shown numerically.

At the time of writing there is no sufficient reason to be more resolute than Ganapathisubramanian & Noyes ( 1982) who said: ' ... uncertainty must


remain as to whether the experimental observations are more susceptible to random fluctuations than the experimentalists themselves believe or whether the computational model has neglected a mechanistic feature that is responsible for the observed chaos. The jury is still out.'

4.6.4 On the structural characterisation of chaotic chemical reactions

The examples given above were mostly constructed upon the basis of deep qualitative knowledge of the behaviour of the trajectories of the individual differential equations. Some papers suggest that, following their methods, the reader is able to construct an unlimited number of chaotic kinetic models. Still, there is a desire to obtain information more easily. This means that statements like the zero deficiency theorem are needed that assure or exclude chaotic behaviour using only knowledge on the algebraic structure of the complex chemical reaction. So far only small steps have been taken in this direction.

King ( 1983) has given a method that is absolutely convincing from the intuitive point of view, but it seems not to have been founded strictly in the mathematical sense. Using similar but more complicated reasoning that he used in the case of the Lotka-Volterra reaction, he shows that the discrete state model obtained by his lumping method in ten (actually fewer, as some of the models are equivalent, if viewed strictly) models result in two interlocking cycles. From our present point of view, all the models that he considered contain negative cross-effect and so are not kinetic. Still, the method seems worth applying to the chemical models considered above. This kinetic logic approach (see Thomas, 1979) provides tools to construct differential equations with prescribed qualitative properties, but the equations constructed are generally nonpolynomial ones. The formal kineticist would prefer developments leading to polynomial models.

Another kind of approach has been initiated in the recent investigations by Toth and Hars (1986a). They studied linear transforms of the Lorenz equation (1968) and of the Rossler model (1976) in order to obtain kinetic models. The failure of their efforts underline the importance of negative cross-effects. Based upon these results the following conjecture can be formulated: if a nonkinetic polynomial differential equation shows chaotic behaviour then it cannot be transformed into a kinetic one.

4.6.5 Problems

I. Show that none of the proper and improper orthogonal transforms of the Lorenz equation are kinetic equations.

Hint to solution I. The general form of the proper orthogonal transformations can be found in Section 14.10 of Korn & Korn (1968). The proof based on this is contained in Toth & Hars ( I986a).


Hint to solution 2. Apply the theory of algebraic invariants of differential equations as outlined, for example, in the book by Sibirskii (1982, p. 91 et seq.).

2. Show that none of the orthogonal transformations transform the model

.X = x - xy - z y = x - ay z = bx - cy + d

by Rossler (1976a) into kinetic equations for all a, b, c, de IR+. Hint. The hints to Problem should be supplemented by the following

argument. Start with a= b = c = d = 0 and use continuity arguments.

4.6.6 Open problems

I. Show for reaction ( 4.19) (or for any of the chaotic kinetic models) that its influence diagram defined by King contains two interlocking cycles.

Hint. Consult King ( 1983) in order to obtain state diagrams of complicated models.

2. Prove or refute the conjecture formulated in the last sentence of Subsection 4.6.4.

4. 7 The inverse problems of reaction kinetics

The majority of problems of reaction kinetics can be formulated using the general scheme below. Let .Y be the set of (complex chemical) reactions and let A and B be nonempty sets. Let us consider the diagram:

... ,. "Y--+g;(IR X A)--+B,

where g;(!R x A) is the set of functions with domain in IR and range in A. Here v e .Y expresses the structure of the reaction, ell( v) is the dynamics of the process, and 'I' gives the quantities derived from the process.

A problem is a direct problem, if v e .Y, A, B, ell and 'I' are given and «D( v) or 'I'(«D(v)) is looked for.

A problem is an inverse problem, if A, B, ell and either «D(v) or 'l'(ell(v)) is given and either v is looked for, or a subset of .Y containing v.

A direct problem is attacked if, for example, the solution of a kinetic differential equation is determined either analytically or numerically, or when the qualitative properties of the solutions are investigated (in this case a subset of.~ (IR x A) is determined in which «D( v) may belong), or when the stochastic model of a reaction is simulated.

Solutions of all the steps of direct problems serve as tools to solve inverse problems: the theory of formal reaction kinetics is applied in such a way that it helps to draw conclusions on the mechanisms of (not necessarily chemical) processes from measurements of a component of the process.

Here we concentrate upon a special part of direct and inverse problems:


upon the characterisation of kinetic differential equations within the set of polynomial differential equations. Many problems will be posed, but only some will be solved, and later the results given here will be applied.

Finally, let us mention that the general definition of the inverse problem given here is in accordance with that used in reaction kinetics (Emanuel & Knorre, 1975, p. 154; Jacquez, 1972, pp. I, 132, 152, 187, 211; T6th & Erdi, 1978, pp. 303-7; Yablonskii & Bykov, 1977) and that used in other sciences and mathematics (Anger, 1979, p. 107; Keller, 1976; Peil, 1979; Sverdlove, 1977; Anikonov, 1978). The importance of the notion is reflected in the fact that a new journal entitled Inverse Problems has been started in 1985.

4.7.1 Polynomial differential equations, kinetic differential equations, kinetic initial value problems

4.7.1.1 Polynomial and kinetic differential equations

A function P: ~M -+ ~Q (M, Q e N) is an (M, Q)-polynomial, if all its coordinate functions are univariate polynomials of all of its variables. In the present book we shall encounter (M, M)-polynomials and (M, I )-polynomials. As an example let us consider the (3, 3)-polynomial:

[ -crx +cry] P(x, y, z):= rx- y- xz

xy- bz

The differential equation

is the well-known Lorenz equation (Lorenz, 1963). One of the specialities of this Lorenz equation is that it contains a term - xz expressing the decrease of y in a process in which y does not take part. Such a term cannot be present in a kinetic differential equation. We formulate this important statement in a more detailed way and present its proof as well.

The right-hand side of the induced kinetic differential equation of reaction (4.5) is such an (M, M)-polynomial the mth co-ordinate of which does not contain a negative term without xm among its factors. In other words, it is a polynomial without negative cross-effects.

The proof of this theorem uses the form of the kinetic differential equation (4.6):

R M

Xnr = L (JJ(m, r) - (X(m, r))k(r) n (xn,.)<l(m',r)

r =I m' =I

(m = I, 2, ... , M). (4.20)


Let us suppose that a term, say,

M

(p(m, r) - CX(m, r))k(r) n (Xm·)m(m'.r)

m'= I

has a negative coefficient. As

k(r) > 0,

therefore

p(m, r) < cx(m, r)

should hold. But, because ofO ~ p(m, r) and cx(m, r) ~ I, the product above surely does contain xm among its factors.

The consequence of the theorem as applied to the Lorenz equation is that no reaction can induce the Lorenz equation, and so this equation cannot be considered as the kinetic differential equation of a complex chemical reaction.

It is an astonishing fact that the converse of the theorem holds as well. If the right-hand side of a differential equation is an (M, M)-polynomial without negative cross-effects then it may be considered as the induced kinetic differential equation of a reaction, or, in other words, if there is no negative cross-effect in the right-hand side then there exists a reaction with the given equation as its deterministic model.

A constructive proof of this theorem is stated as a problem: one of the inducing reactions can easily be constructed.

Polynomial differential equations without negative cross-effects will usually be called kinetic differential equations from now on.

4. 7 .1.2 Further problems

Two examples will be shown here proving that even the mechanism corresponding to a given kinetic differential equation is not unique.

Any reaction can be enlarged by adding the reaction

I I 2X~X+ Y--.2Y (4.21)

without any change of the original kinetic differential equation. (It is interesting, however, that the simpler reaction

I I 2X ~X--. 0,

where Y is considered to be an external component, is easy to distinguish from the empty reaction, the complex chemical reaction without elementary reactions and components, if one considers the usual stochastic model. In this case the variance of the quantity of X tends to infinity, see Chapter 5.)

The canonic reaction corresponding to a kinetic differential equation


constructed as described by Hius & T6th (1981) is:

2X+ 2V

~ 2X+ V-+X+ V

~ 2X+ Y+ V

2P/3Y + V

/ 3P 3Y+ 2V-+2Y+ 2V

""'- 4P

"-..'3Y+ Z + 2V

1 /'4Z + V

/2y 4Z-+X+ 4Z.

~3Z On the other hand, the same equation is the induced kinetic differential equation of the reaction

3

2X + V-+ 3 Y + 4 V

' ,/

1 4Z p

as well. This reaction is weakly reversible, conservative, and of deficiency zero. This example shows why the two statements above are so important for a pure mathematician: no general theorem other than the zero deficiency theorem is known to provide statements so simply about the qualitative behaviour of the solutions of the (complicated nonlinear) kinetic differential equation induced by the above reactions.

This example also shows that the canonic mechanism is not the simplest one and it is not minimal in any sense (although it is unique). Its major advantage is that it can quickly be constructed and by an algorithm.

Several groups of problems arise here:

(I) Is the lack of negative cross-effects not too strong a restriction in the sense that a 'randomly selected' polynomial differential equation is usually nonkinetic. How 'dense' is the set of kinetic differential equations within the set of polynomial ones?

(2) For the sake of easy manipulation it would be useful to find an inducing reaction with the minimal number of complexes, elementary reactions, linkage classes etc. What kind of reasonable assumptions assure the


existence or the uniqueness of such a mechanism? (3) The reaction may be looked for within a class of reactions with a given,

chemically relevant property. Such a property may be conservativity or subconservativity (the canonic reaction is never conservative!), reversibility, weak reversibility or acyclicity, small (zero or one) deficiency. When does an inducing reaction exist within a given class to a given differential equation, when is it unique, or if it is not unique then how far is it nonunique?

The chemically relevant property may also relate qualitative properties of solutions of induced kinetic differential equations, such as, presence or absence of multistationarity, oscillation and chaos. Representatives of the 'kinetic logic' approach such as Thomas, King and Glass have given a solution to these kinds of problems. Thomas (1981) has shown by an example how to derive a set of differential equations with a given type of solutions, such as, for example, having three stable steady states. The only problem of this method at present seems to be that the emerging equations are not of the polynomial type. Therefore the ideas are to be tailored according to the needs of formal reaction kinetics.

( 4) Similar but more complex problems will be obtained if only the 'essential' part of a differential equation is considered.

(5) Is it possible to obtain a kinetic differential equation from a nonkinetic one with transformation of a given type?

Let us formulate some of the questions more precisely in order either to answer them or to set them as a target. Before doing so we mention that question (5) will not be treated here separately, as a question of this type has been treated in Subsection 4.6.4, and other similar ones will be treated in Subsection 4.8.6 (on symmetries).

4.7.1.3 The density of kinetic differential equations

According to several different real situations different definitions have been given of the random event that a polynomial differential equation is kinetic (Toth, 1981 b, pp. 44-8). The results can be summarised as follows. If one selects a polynomial differential equation with fixed coefficients and the random selection only concerns the exponents than the probability of getting a kinetic differential equation is I. If the exponents are fixed and the coefficients are randomly chosen then the probability of getting a kinetic differential equation is 0. Finally, as a consequence of the statements above, if both the coefficients and the exponents are randomly selected then the probability of getting a kinetic differential equation is again 0.

4.7.1.4 Uniqueness questions

A general theorem can be stated saying that for any given kinetic differential

68 Mathematical models of' chemical reactions

equation there exists an inducing reaction with the minimal number of complexes and elementary reactions. If a certain inequality holds for the coefficients then it is also true that the number of complexes equals the different exponent vectors on the right-hand side of the equation. This theorem is rather awkward to formulate, and is given as Problem I of Subsection 4.7.5.

The right-hand side of the induced kinetic differential equation of reaction (4.21) is the (2, 2)-polynomial 0. Such a case cannot occur with reversible, or even with weakly reversible reactions. Neither can it occur with acyclic reactions.

The right-hand side of the induced kinetic differential equation

c =Poe

of a weakly reversible reaction cannot be the polynomial 0. The proof of this statement is based on the fact that in this case

span(9Pp) = §,

(Feinberg & Horn, 1977, p. 90), where§ is the stoichiometric subspace. But dimS~ I, thus span(9Pp) cannot be the single 0 vector.

On the other hand, it is easy to construct two essentially different conservative, reversible reactions with the same induced kinetic differential equation (see Exercise 4). Such an example shows that it is not true that a kinetic differential equation is induced by a unique reaction even within the class of reversible and conservative reactions. The addition of other, chemically relevant properties may be enough to ensure uniqueness.

The right-hand side of the induced kinetic differential equation of an acyclic reaction cannot be the polynomial 0. As the reaction is acyclic it should contain at least one point without arrows pointing towards it. If one of these points corresponds to a component then the derivative of this component surely contains a (negative) term that cannot be counterbalanced as a result of other elementary reactions as no other point points toward this point. If all these points correspond to elementary reactions, than the components formed in this elementary reactions have a constant inflow that cannot be counterbalanced.

Summarising the result of the previous two paragraphs: the right-hand side of an induced kinetic differential equation can only be zero, if the reaction is cyclic but not weakly reversible. This class of reactions has again proved to be the most complicated and interesting.

The design of periodic reactions may also be treated as a uniqueness problem. In Subsection 4.5.6 we sketched how is it possible to use the methods of mathematical programming to design periodic reactions, or, to use the present terminology, to investigate the uniqueness of reactions with a prescribed linearised part.


4.7.1.5 A sufficient condition for the existence of an inducing reaction of deficiency zero

As is known from an earlier exercise (Problem 4(ii) of Section 3.5) a generalised compartmental system is a reaction consisting of elementary reactions of three types

k,m . y"' X(m) ~ y'X(i) y"' X(m) ;;o

0 ~ y"' X(m),

where

MeN; m, ieJt' = {1, 2, ... , M}; y"', /eN

(4.22a) (4.22b) (4.22c)

for all i, meJt', k;mE ~R; for all i, meJI = Jt' u {0} (i ¥- m). In other words, all the components (here: compartments) are contained in a single complex and all the complexes, except the empty one, contain a single component.

The example

2X~O~X

shows that it is not true that a reaction only cons1stmg of elementary reactions of the type above is a generalised compartmental system, because in the example component X is contained in both the complexes X and 2X.

Let us repeat that a generalised compartmental system is closed if it only contains elementary reactions of the type (4.22a), while it is strictly half open or strictly open according to whether it contains elementary reactions of the type (4.22b) or (4.22c) too.

These definitions express that a generalised compartmental system is closed if and only if no matter enters it from and leaves it for the outside world. It is strictly half open if and only if no matter enters it but matter does leave it. Finally, it is strictly open if and only if matter does enter it and may leave it.

These formulations are in accord with the results of Problem, Section 3.5. As we know from Problem 2 of Subsection 4.2.5, the deficiency of a

generalised compartmental system is zero. It is also known that there exists a directed route from one component into the another in the V-graph of a generalised compartmental system if and only if there is a directed route from the first given component into the second one in the FHJ-graph of the reaction.

Earlier it was shown that if a reaction is acyclic then it cannot be weakly reversible (Exercise I of Subsection 4.2.4). This statement shows that in the case of generalised compartmental systems the connection between the two graphs is even stronger: there exists not just a one-to-one correspondence between cycles and closed directed routes; the two graphs are essentially identical. (Here the empty complex may not be excluded.)


These statements together with the results of Feinberg, Horn & Jackson, and Vol'pert imply that the dynamic behaviour of a generalised compartmental system is completely characterised in two extreme cases: if the graph (either the V- or the FHJ-graph) is weakly reversible, or, if the graph is acyclic then the reaction behaves regularly in the classical sense. The zero deficiency theorem throws light on intermediate cases too: in the cyclic but not weakly reversible case, and in the acyclic and not weakly reversible case, some of the co-ordinates of the equilibrium point of the kinetic differential equation must be zero.

The induced kinetic differential equations of generalised compartmental systems of the three types are as follows:

(I) in the case of a closed generalised compartmental system

X;= (- ~kj)/(xY! + /~k;;(xj)·< (2) in the case of a strictly half-open generalised compartmental system

x; = (- ~kji)Y<xY! + /~kij(xj)Y1 (3i: k 0 ;E IR+);

(3) in the case of a strictly open generalised compartmental system

-~; = (- . L kji) /(x;)-'! + / L k;;(x)-'.1 + k;0 (3i: k;0 E IR + ), ,~o '

where throughout

ie{l, 2, ... , M}, te~'"' kije!R;, yjeN (i,je{O, I, ... , M}).

An easy-to-formulate (but still hard to work through, see Exercise I (ii) of Subsection 4.1.3 and Open Problem I of Subsection 4.1.5) generalisation of some of the results collected by Rodiguin & Rodiguina (1964) is: if the graph of a generalised compartmental system is a tree (i.e. it is acyclic) then the solutions of the induced kinetic differential equation may often happen to be explicitly determined (but not in all cases of tree graph, as one familiar with compartmental systems would expect).

Now let us formulate a statement on the existence of a deficiency zero inducing mechanism. It will be shown that if the right-hand side of a kinetic differential equation is the sum of univariate monomials and if all the variables have the same exponent in all the rows (this assumption is necessary as well!) then- if an additional condition is met and only then- there exists an inducing generalised compartmental system to the system of differential equations. Let us formulate the statement more precisely.

There exists an inducing generalised compartmental system of M compartments to the system of differential equations


(where for all

i, je{l, 2, ... , M}, Y;. yje N, Y; # yj, i #j, aij, b;E IR)

which is

( 1) closed, if and only if b; = 0, -a;;, aij, d; e ~R;; a;; = d; yi, (2) strictly half-open, if and only if b; = 0, -a;;, aij, d; e ~R;;

71

(4.23)

(4.24a) a;;:::;;; d;yi,

3ia;; < d;/, (3) strictly open, if and only if b;,

(4.24b) -a;;, aij, d;E ~R;; a;;:::;;; d;yi, 3ib;EIR+,

(4.24c)

where throughout

i,je{l, 2, ... , M}, i #j, M

d; = - L ajijyi. j~l

The necessity of the conditions is trivial. Sufficiency follows from the construction below. Let us consider the reaction

. a;j/./ . y'ej --+ y'e;

. dj

y'ej --+ 0 h-!..'

0 ~ yie; (i, je{l, 2, ... , M}, i # j).

(4.25)

The reaction rates are placed above the corresponding arrow expressing an elementary reaction. Here, and only here the elementary reactions are understood in such a way that a reaction rate may be zero as well: in this case the corresponding elementary reaction is excluded.

Reaction (4.25) above clearly induces (4.22a), (4.22b) or (4.22c), and is closed, strictly half-open or strictly open corresponding to the conditions. In case (l) one even gets a conservative inducing reaction as a generalised compartmental system is conservative if and only if it is closed. Let us consider the example

Here

thus

c1 = -c 1 + 3c2 c2 = 3cl - c2.


a;;> d;yi (i = 1, 2).

Therefore there is no generalised compartmental system that induces this kinetic differential equation. The question is if there exists a deficiency zero mechanism that induces this kinetic differential equation?

4.7.1.6 On the inverse problem of generalised compartmental systems

A necessary and sufficient condition has been given for the existence of an inducing generalised compartmental system to the system of differential equations (4.23). However, certain generalised compartmental systems (and differential equations) are to be considered as identical, as they are essentially not different. This will be done below and thus a problem of the type (3) formulated in Subsection 4. 7.1.2 will be solved here.

(The reader should be cautioned that a more technical part follows, although almost all the details are relegated to the Problems section.)

A component X(m) of a generalised compartmental system for which

(I) k;m = 0 (i e {0, I, ... , M}) is a first-order endpoint, the number of such points is denoted by R 1 ;

(2) kom e ~ + is a second-order endpoint, the number of such points is denoted by R2 ;

(3) kmoe ~+ is an entry point, the number of such points is denoted by R 3 •

R 1 obviously equals the number of zero column vectors of the matrix K, R 2 equals the number of positive elements in the zeroth row of the matrix K, while R 3 equals the number of positive elements in the zeroth column of the matrix K.

Let us consider the following strictly open generalised compartmental system:

k k k k k 4X(l) ~ 3X(2) ~ 2X(3) ~ X(4) _:; 2X(5) ~ 3X(6)

~ ~k., k.. J ~ 4X(7).

Here the matrix K of the reaction rate constants is:

0 ko1 ko2 0 0 0 0 0 0 0 kl2 0 0 0 0 0 0 0 0 k23 0 0 0 0

K= 0 0 0 0 k34 0 0 0 k40 0 0 0 0 0 0 0 0 0 0 0 ks4 0 0 0 0 0 0 0 0 k6s 0 0 0 0 0 0 0 k? s 0 0


and R1 , the number of first-order endpoints, is 2, R2 , the number of secondorder endpoints is 2, R3, the number of entry points, is I, and they can be calculated from the structure of K as said before.

The core of a generalised compartmental system is obtained through substituting all first-order endpoints by the zero complex. Thus the core of our example is:

k k k k k 4X(l) ~ 3X(2) ~ 2X(3)::. X(4) _:; 2X(5) ~ 0. I k., ~k·~ k.. T ~.. I

) 0 ~t=-======___:_ _ ____]_ The core of a system of differential equations of the form (4.23) is the

differential equation where the variables having the same index as the zero column vectors of the matrix (a!i) have been deleted. The induced kinetic differential equation of our generalised compartmental system is:

it= -4k01 (x1 ) 4 + 4k12(x2 ) 3 x2 = - 3(ko2 + kl2)(x2)3 + 3k23(x3)2

i 3 = - 2k23 (x3)2 + 2k34x4 X4 = -(k34 + ks4)X4 + k4o Xs = -2(kos + k7sHxs)2 + 2ks4X4 Xo = 3kos(Xs)2

X7 = 4k7 5 (x5 ) 2

The core of this differential equation consists of the first five equations. Clearly, these make up the induced kinetic differential equation of the core of the given generalised compartmental system. This is not by chance, because in general it is true that the induced kinetic differential equation of the core of a generalised compartmental system is the core of the induced kinetic differential equation of the generalised compartmental system.

Let us call two differential equations of the form (4.23) equivalent, if their core is the same. Two generalised compartmental systems will also be called equivalent, if their core is the same.

All the necessary preparations have been done to answer the question: what is the number of generalised compartmental systems that 'essentially' induce eqn (4.23)? Part of the answer will be formulated here, without proof. The full answer can be found in the Problems, Section 4.7.4.

Let us suppose that the coefficients of the right-hand side of the differential equation fulfil (4.24a). Then there exists a generalised compartmental system and 2R, - I other strictly half-open generalised compartmental systems consisting of M- K1 (K1 e {1, 2, ... , R1 }) components inducing a differential equation equivalent to the original differential equation.

Another related problem that seems worth investigation is, how the present result changes in the case of compartmental systems if one considers, instead of the· core defined here, the controllable and observable part as defined in linear system theory, see, for example, Brockett (1970).


In Section 4.8.6 we shall turn to the problem of how is it possible or not possible to obtain a kinetic differential equation from a polynomial nonkinetic one.

4.7.2 The classical problem of parameter estimation

The most popular and most important inverse problem is the estimation of reaction rate constants, see, for example, Deufthard et a/. (1981) Hasten (1979), or Vajda et a/. 1987). Using the terminology introduced above CJ) is the function that gives the solution of the kinetic differential equation as a function of the reaction, while 'I' o CJ) provides the values of the solution at discrete time points together with a certain error. In this case a subset of "Y with the same mechanism is delineated and the aim is to select a reaction from this set in such a way that the solution of the kinetic differential equation be as close to the measurements as possible by a prescribed, usually quadratic, norm. As the solution is a nonlinear function of the parameters, therefore a final solution to the general problem seems to be unobtainable both because a global optimum usually cannot be determined and because the estimates cannot be well-characterised from the statistical point of view. In addition to these problems, reaction rate consants only have a physicochemical meaning if they are universal, i.e. the reaction rate constant of a concrete elementary reaction must be the same whenever it is estimated from any complex chemical reaction.

4.7.3 Exercises

I. Verify that a polynomial differential equation without negative crosseffect can be induced by a reaction (called canonic reaction) that is constructed in the following way: (i) for each term kx~ on the right-hand side of the mth equation, where k

is positive, let us define the elementary reaction k

L(X(m')X(m')--+ L(X(m')X(m') + X(m),

and (ii) for each term - k' x~ on the right-hand side of the mth equation,

where k' is positive, let us define the elementary reaction k

L (X(m')X(m')--+ L (X(m')X(m') - X(m).

Hint. Cf. Hars & Toth (1981). 2. Show that

(i) the generalised Volterra equation


is a kinetic differential equation. How can this statement be generalised?

(ii) this equation and (4.18) are essentially the same, thus both the statements here and the statement of Problem 3 of Subsection 4.5.9 are true for both forms.

4. 7.4 Problems

I. Show that the generalised Volterra equation (4.26) in the case M = 3 is conservative if and only if the matrix v is of the form

where a, b, c > 0 (only if

a 0

-c

abc :F 0

has previously been assumed) and the vector in the definition of conservativity may be chosen p = (lfa, lfb, lfc).

Hint. Start from the definitions. 2. The property that the vector on the right-hand side of a differential

equation points into the interior of the first orthant is obviously a necessary condition for negative cross-effects to be absent. Give a polynomial example showing that it is not sufficient.

4.7.5 Open problems

1. Try to formulate and give a (constructive) proof of the uniqueness theorem on the inducing reaction to a given polynomial differential equation mentioned in Subsection 4. 7.1.4.

2. Does there exist a kinetic differential equation for which all the conditions (4.24) hold except the inequalities and which is still induced by a deficiency zero mechanism (although surely not by a generalised compartmental system)?

Hint. Cf. the example at the end of Subsection 4. 7.1.5.

4.8 Selected Addenda

4.8.1 Lumping

4.8.1.1 Lumping in general

Let us start from outside chemical reaction kinetics, from the theory and practice of modelling in general. Let us suppose that the phenomenon to be described can bt: characterised by a vector-valued function x of N coordi-

7~ Mathematical models of chemical reactions

n~es, where N is an enormously large number. One should like to reduce the niiillWr of coordinates using, for example, a linear transformation, that is by 1i1ear lumping:

i=Mx, (4.27)

Vl\ere M is an M x N matrix, and M is much smaller than N. Such a lumping is only meaningful if the resultant process behaves

s~ilarily, with respect to some of its properties. For example, if the original poce~s obeys a differential equation

i=Kx

tkn ;1 requirement may be that the resultant process obeys a differential ~11ation of the same form:

i= tx \lith ;tn M x M matrix t. In this case the lumping is said to be exact.

[t jS an easy exercise (Exercise 1 of Subsection 4.8.9) to show that the lllatriX M in eqn (4.27) realises an exact lumping only if there exists an },f x M matrix K for which

KM=Mi. (4.28)

In this case K may even be determined from (4.28), although not necessarily ~niq11ely. It may also be interesting whether the lumping is proper or not. Linear

lumping can be written in the following way:

M

Xm = L mmjxj (m = l, 2, ... , M). j=l

(4.29)

lhe tum ping (4.29) is said to be proper, if no xj is contained in two different .r s ~sa summand with a nonzero coefficient mmj· This is equivalent to saying th"atno column of M contains more than one nonzero element, or, to put it another way, all the columns of M are either zero vectors, or lie in the direition of the coordinate axes.

4.8.1.2 Lumping in reaction kinetics

A n~e statement related the application oflumping in reaction kinetics is due to russ & Hutchinson (1971):

A cpsed compartmental system is transformed into another closed compartmental systc111 by proper exact lumping.

Thii statement means that the transformed equation may be considered as theinduced kinetic differential equation of another closed compartmental sysr-m with newly introduced quasicomponents.


In the same paper cited above, Luss & Hutchinson consider the reaction

MA(i)-+B (i=1,2, ... ,N)

where M is a nonnegative real number. This system of equations may be lumped exactly into a scalar equation only if the trivial case M = I holds. In the case when M may be different from I upper and lower bounds may be obtained for x. The authors succeeded in giving these bounds in such a way that they only depend on the initial concentration of the lumped component. Based upon the difference between the two bounds they suggest how to lump the components, which are to be lumped into a new quasicomponent.

Another interesting result of theirs is that they are able to estimate the sensitivity of the reaction rate of the lumped component with respect to changes in the original reaction rates and concentrations without knowing original reaction rate constants and initial concentrations, using nothing other than easily measurable data.

Finally, let us mention that lumping is especially important in the presence (in the model) of continuous components (Aris, 1969; Bailey, 1972). It may be worth citing Aris (1969) here: 'When infinitely many components are involved, a first order system may be lumped into virtually any kinetic law, and probably an approximate form of such lumping accounts for the success of some of the less enlightening empirical rate expressions.'

4.8.1.3 Possible further directions

The number of open questions is much larger than the number of solved ones in this area. We cannot even formulate them in a form suitable for the 'Open problems' subsections.

(I) It is obvious that one can also define nonlinear lumping. (This is the case e.g. if one replaces mole numbers by mole fractions or weighted mole fractions in a nonconservative system.)

(2) The same, or almost the same definitions apply to other types of models as well (inside and outside reaction kinetics). The mathematical problems arising are quite different. This may be the appropriate place to mention that lumping is widely used in econometrics (see, for example the basic reference text by Malinvaud (1969), or the book edited by Los & Los (1974). In this context lumping is called aggregation.

Another name for the same notion is collapsing the states, and this expression is used in the theory of stochastic processes. In this area some of the questions are as follows: - Is the lumped process Markovian, if the original one was? -Does it fulfil at least the Chapman-Kolmogorov equations? (This is a

less severe requirement than the previous one.) It is very hard even to formulate questions in this field. Starting points for the interested


reader may be the works by Erickson (1970), Heller (1965) and Rosenblatt (1971 ).

(3) The basic requirement may be other than conservation of the form of the constitutive equation; other meaningful requirements in reaction kinetics include conservation of conservativity, deficiency, acyclicity, and controllability.

(4) From the point of view of practice, by far the most important question is the following one. What can be said about the original system when some of the properties of the lumped system are known? This area seems to be unjustifiably neglected. Still, two recent works may worth mentioning, the one by Balakotaoiah and Luss (1982) connecting this area with multistationarity using the method of singularity theory, and the papers by Pismen (1984, 1985) on the dynamics of lumped reactions near singular bifurcation points.

It may be the case that reducing the number of components does not help to solve a kinetic problem. In this case increasing the number of components may be a solution. How, and why, is the topic of the next subsection.

4.8.2 Continuous components

The state of the system we investigate in the present book is almost exclusively characterised by a finite dimensional vector. Sometimes it may prove insufficient. The usual stochastic model can be conceived of as one where the state is a random variable, i.e. an element of an infinite dimensional linear space. If one wants to describe spatial effects (in a deterministic model) then a possible way to do so is to characterise the state by a function (by the mass density) and to write down a differential equation for the time versus mass density function. This will be a differential equation for a function with values in an infinite dimensional state space. (Memory effects can also be taken into consideration using an infinite dimensional state space. Cf. Atlan & Weisbuch, 1973.)

In the present section we should like to present a model in which the infinite dimensional state space arises in the most natural way: in the case of continuous components.

There are two problems leading to the notion of continuous components. At first, the number of species or chemical components may reach a huge number in such areas as oil chemistry, polymerisation or biochemistry. In these cases the number of the variables in the induced kinetic differential equation is so large that this system is difficult to treat; it may prove more promising to have a continuous manifold of chemical components.

Theoretical, as opposed to practical, considerations may also lead to the introduction of the notion of continuous components. Let us consider the case of the ethane molecule. This molecule can in the first approximation


H H

(a) H

(b)

Fig. 4.4 Eclipsed (a) and staggered (b) conformations of ethane.

79

H

have two basic conformations: the eclipsed one and the staggered one depending on the angle between the two methyl groups that can take the value ofO" or 180". Actually, this angle may take any value between o· and 180" (see Fig. 4.4), and the C-C bond energy of these species is different. Therefore a more subtle analysis should take into consideration ethane species characterised by an angle anywhere between o· and 180". The same kind of problem arises when considering transition states in the transition state theory.

Let us consider the isomerisation of ethane in a more detailed way. Let the species characterised by the angle m be X(m), then one has the following elementary reactions:

k(p.m) X(m) -+ X(p) (p, me .A)

where .A is the set of components. If .A is a finite set then one has the induced kinetic differential equation

Xm = -xP L k(p, m) + L k(m, p)xp (me .A). (4.30) peJI peJI

If .A is an infinite set, say the interval [0, 180), then it is natural to replace the sums in (4.30) with integrals. In order to do so a measure Jl is to be introduced on .A to express the importance of the component X(m). Thus the generalised form of eqn (4.30) is

xnr = -xP f k(p, m)dJ.l(p) + f k(m, p)xpdJl(p) (me .A). (4.30)

./{ ./{

In the more general case the basic equation R M

Xm = L (~(m, r) - cx(m, r))k(r) fl r= I m'= I

x~<.m".r) m (4.32)

is to be substituted by an equation for functions with values in an infinite dimensional vector space. To this generalisation another measure is needed, and this time on qt, the set of elementary reactions. Then (4.32) is generalised to give


xm = I(~(m, r)- c:t.(m, r))k(r)exp I (c:t.(m, r)logxmd!l(m)dA. (4.33)

.If

This is the approach proposed and investigated in detail by Herman (1978). It is different and seemingly more natural than that chosen by Aris & Gavalas (1966). The main difference is that Aris & Gavalas wrote down equations for special, mainly linear, cases that seem not to have a common generalisation. It is an as yet unsolved problem whether their models can be derived as special cases from the model by Herman.

The existence and uniqueness of the solution to (4.33) under appropriate mild conditions have been shown by Herman (1978). It has also been shown how to transform (4.33) into an equivalent partial differential equation more amenable to numerical solution and theoretical investigations.

The mathematical problems coming from the investigation of (4.33) are obviously more difficult than those coming from the usual discrete component model. Still, the greatest hurdle is not of pure mathematical nature: it is, how to interpret and how to measure and estimate the measures 11 and A.. This is the main obstacle delaying the breakthrough of continuous component models.

4.8.3 Kinetic gradient systems

As an application of the statement on the inverse problem treated in the previous section the question will be investigated here (based upon T6th, 1979): what is the role of gradient systems in chemical reaction kinetics?

This question has a certain mathematical interest, as gradient systems are relatively easy to deal with from the point of view of exotic phenomena (or, to put it in a more mathematical way, from the point of view of the qualitative theory of differential equations) in general (see, for example, Hirsch & Smale, 1974) and from the point of view of catastrophe theory in particular (Thorn, 1975, p. 55).

The significance of the question is from the point of view of thermodynamics: according to several schools only systems described by gradient systems are worth being studied. The question is, therefore, which reactions belong to the class of gradient systems?

A differential equation,

x =Pox,

is called a gradient system, if there exists a continuously differentiable scalar valued function V such that

P = V'(= grad V)

Such a function V is usually called a potential.


Further preparatory definitions are needed. A mechanism is called cross-catalytic, if in each elementary reaction either

(I) none of the components decreases, or (2) the reactant complex contains at most one component.

The meaning of this definition is that no chemical component causes the decrease of another one. Nothing has been supposed about the effect of components on themselves. Our definition is in concord with the definitions used in classical chemical kinetics ( cf. Bazsa & Beck, 1971 ). A mechanism is called canonically cross-catalytic, if for all the sets of reaction rate constants the canonic complex chemical reaction corresponding to the induced kinetic differential equation of the complex chemical reaction is cross-catalytic.

A cross-catalytic mechanism is obviously canonically cross-catalytic as well, but not vice versa.

A mechanism is called weakly realistic, if in each elementary reaction either

(I) the reactant complex contains at most one component, (2) all the stoichiometric coefficients of the reactant complex vector are less

than two.

Having introduced all the necessary notions the following theorem may be stated.

If the induced kinetic differential equation of a complex chemical reaction with a weakly realistic mechanism is a gradient system, then the mechanism itself is canonically cross-catalytic.

The proof is left to the reader as a problem (of applying the elementary theorem of calculus on the existence of a potential).

Let us try to interpret the meaning of the theorem. If elementary reactions of type (l) are present among the elementary reactions of a weakly realistic mechanism then it is not conservative. Therefore the result above throws light upon the observation of Tuljapurkar & Semura (1979, p. 31) according to which few gradient systems are known that are relevant from the point of view of applications in chemistry or related subjects, because if the induced kinetic differential equation of a conservative reaction is a gradient system, then the underlying mechanism can only belong to one of the classes below:

(I) to the class of not weakly realistic and not canonically cross-catalytic mechanisms, or

(2) to the class of canonically cross-catalytic mechanisms without elementary reactions of the type (I).

Closed compartmental systems with a symmetric matrix of reaction rate constants belong to the second class. This statement contradicts the state-


ment by Bataille eta/. ( 1978, p. 165), as a consequence of differences between our definitions and theirs. Their second example, however, is a reaction with a conservative, not weakly realistic, and not cross-catalytic mechanism, the induced kinetic differential equation of which is not a gradient system in the sense of our definitions, and so their second example is in accord with our statements.

The relations between classes of reactions are shown in Fig. 4.5.

Conservative

Weakly realistic

Fig. 4.5

Canonically cross-catalytic

___ Cross-catalytic

Canonically gradient

Fig. 4.5 Relevant classes of reactions related to gradient systems.

4.8.4 Structural identifiability


k, X-+Z

and let us suppose that the concentration of X is measured. It is well known, but it is obvious too, that this measurement does not uniquely determine the values of the parameters k 1 and k 2 separately (only their sum). The question arises how can we decide before the experiments knowing only the mechanism of the reaction which of the rate constants can be calculated from the planned measurements. Answers to this question based upon linear system theory can be found in, for example Cobelli (1976) that contains many


(mainly biological) examples. He only treats the simple case of compartmental systems. Critical review of the literature and further conditions with examples have been provided by Vajda (1982). General results on more complicated cases can be found in Vajda (1983).

4.8.5 Parameter sensitivity

The value of the reaction rate constants is usually uncertain. The object of sensitivity analysis is to systematically determine the effect of uncertain parameters on system solutions. A recent review paper has been written by Rabitz et a/. (1983). The methods of sensitivity analysis can be divided into global and local methods.

Global methods are aimed at providing information on the change of solutions in a large domain of parameters, i.e. rate constants. As in this case the induced kinetic differential equation should be solved tens of thousands of times in the case of larger systems these methods are only of rather limited use.

Let us consider the solution of the induced kinetic differential equation at a fixed time point as the function of the rate constants and let us consider the Taylor series of the solution with respect to the parameters around the nominal value of the parameters. The coefficients of this Taylor series are the local sensitivity coefficients. In general only the first order coefficients are considered.

Several methods are used to calculate these coefficients, such as the finite difference approximation, the direct method and its modifications, the Green-function method, and the method of polynomial approximations.

The (local) sensitivity analysis can be applied in several areas.

(I) The question, how uncertain is the value of the solutions as a consequence of uncertainty in the rate constants, can be answered by this method. The most exact approach to this question is how the probability density function of the parameters are transformed into the probability density function of the solutions at any desired time point (Costanza & Seinfeld, 1981 ).

(2) Sensitivity analysis can be used to judge whether a certain rate constant can be determined from a certain measurement or not. One of the very successful methods of answering this question is the principal component analysis (Vajda et a/. 1985, Vajda & Tuninyi, 1986).

(3) The sensitivity coefficients may help us to fit the parameters to measurements. In this case the initial estimates of the parameters are used to

calculate the sensitivity coefficients and these calculations are repeated with newer values of the parameters, if necessary.

(4) The investigated complex chemical reaction may have a characteristic feature such as the period length in an oscillatory reaction, the argument


of the local maximum of one of the concentrations etc. Such are the basic notions of feature sensitivity analysis (Skumanich & Rabitz, 1982, Beumee et a/. 1985). Using the sensitivity coefficients other types of sensitivity coefficients can be derived in such a way that they reflect the change of the interesting properties as a consequence of changes in the parameters.

(5) Sensitivity analysis can be used to reduce the number of elementary reactions based upon their relative importance. The method and its convincing application to the Belousov-Zhabotinskii reaction can be found in Vajda eta/. (1985) and in Vajda & Tuninyi (1986).

(6) Strength of dependence between the parameters can also be studied by this method.

4.8.6 Symmetries

Mathematical models are intended to describe real phenomena, and in many cases different phenomena can be explained by a common model. It is a natural question to set up criteria, according to which two phenomena can be qualified as identical, similar, or diverse. The question is in close connection with the invariance of natural laws and with the fundamental symmetries being established as a consequence of these invariances.

The term invariant is the adjective of a function, number, property etc., which remains unchanged under a certain mapping, transformation, or operation. For instance the numbers 0 and I are invariants under the operation of squaring, or parallel elements remain parallel during an affine transformation, i.e. in this case being parallel is an invariant property.

A mapping, transformation, or operation is symmetric, if after their application a certain function, number or property remains invariant.

In mechanics, time and space are primary concepts. Not only Einsteinian but also Newtonian mechanics is relativistic, as the latter complies with Galileo's principle of relativity. Galilean transformation, which introduced the relativity of space but retained absolute time, does not change the Newtonian equation. Using the canonic formalism it was shown about 1850 that the invariances under displacement in time, position and angle give rise to conservation of energy, linear momentum, and angular momentum respectively.

Invariance principles are closely connected with transformation groups, since Felix Klein suggested classifying geometries based on the invariance of theorems under application of transformation groups. An algebraic structure is a transformation group, if the elements of the structure are transformations satisfying the group axioms.

Emmy Noether showed in 1918 the way in which invariance under a continuous group gives rise to constraints of motion and proved that the ten first integrals of Newtonian mechanics follow from the in variance properties


of the Lagrangean under the infinitesimal transformations of the tenparameter Galileo group.

However, the Maxwell equations of classical electrodynamics are not invariant under the Galileo transformation. H. A. Lorentz introduced another transformation which plays a similar role in electromagnetism as the Galileo transformation in mechanics.

A question open for discussion is whether the equations of motion or the invariance principles are the more basic 'first principles'? It was a choice of Einstein that he gave the Lorentz transformation primary importance, and so the equations of motion had to be modified.

The theory of dynamic systems has been strongly motivated by the problems and formalisms of mechanics, and the investigation of the principles of certain invariant transformations is an important subject of modern theory.

Let us consider the differential equation

i(t) = f(x(t)),

where x(t) may either be an element of ann-dimensional Euclidean space or, more generally, be taken from a Riemannian manifold A. Let us suppose that for all initial conditions the solution to the equation is unique and it is defined on the whole real line. Then the mapping

T: ~-+ ~n

is a transformation if it is defined by

T,x(O) = x(t).

(The group property here means that T,+s = T,T,.) T, is a characteristic example of continuous parameter dynamics calledflow. The induced kinetic differential equation of a reaction may also generate a flow if it is known that all the solutions of the corresponding initial value problems are defined on the whole real line. A flow is denoted by (A, T), while if a measure ~invariant under the transformations Tis also given then the notation is (A, T, ~). The invariance of a measure ~ means precisely that

~(T,B) =~(B)

holds for all B c A and for all t e R Differential equations of pure mechanical systems generate transformation

groups for which the Lebesgue measure is invariant: this statement is called the Liouville theorem. Major results of the modern theory of dynamic systems are connected with physical sciences, mostly with mechanics. Differential equations of physics may refer to particle or planetary motions described by ordinary differential equations, or to wave motion described by partial differential equations. Dissipative effects are neglected in all these systems, and so the emphasis is on 'conservative' or 'Hamiltonian' systems.


More specifically, the evolution equations of conservative systems can be derived from a time-independent Hamiltonian, but there are systems originating from a time-dependent Hamilton function where the Liouville theorem is valid. Another important viewpoint of classification may be whether a dynamic system is integrable or non-integrable.

In spite of the fact that analogies have been sought between Hamiltonian mechanics and chemical reaction kinetics (Yourgrau & Row, 1957; Kerner, 1964; Oster & Perelson, 1974; Engel, 1979, 1980, 1982; Reti & Ropolyi, 1984) a narrow area of overlap has been found between the two topics. We may ask what are the necessary and sufficient conditions for the induced kinetic differential equation of a chemical reaction to be Hamiltonian, and we might be glad that the application of these conditions will not be the empty set. The overwhelming majority of chemical systems (and it is interesting to note that not all) are dissipative systems. (At the moment we adopt the following definition: a system, which cannot be derived from a Hamiltonian is called dissipative.) The picture is even more confusing: in the physical literature one can find arguments for the existence of so-called dissipative Hamiltonian systems (Kaufman, 1984).

Though the structure of mechanics and chemical kinetics is highly different, and the search for analogies between them might be misleading, there are some significant questions to be answered. In which cases can models of chemical reactions be derived from 'Hamiltonians'? What are the assumptions for the existence of (time-independent) first integrals? Is it possible to find chemical reactions for which the divergence of the right-hand side of the induced kinetic differential equation is zero? (As is well-known, for two-dimensional systems Hamiltonian systems are equivalent to 'zero divergent' systems, whereas in higher dimensions the first property implies the second one.) Instead of a detailed analysis some illustrative examples are given in Table 4.1 to show the connections among the three properties just

Table 4.1 Connections between the existence of a Hamiltonian, the existence of a first integral, and zero divergence.

Differential equation

.x- = x, _v = -y

.X"= c (c > 0)

.X"= 0 i =X, _i· = J' .X"= X

Hamiltonian

xy

First integral div f= 0?

xy + +

X + xjy

mentioned. However the examples describe nonconservative reactions, and so the analogy is rather strained.

To avoid the disturbing effects of analogies perhaps the best procedure


would be to abandon their advantages as well. The precisely defined induced kinetic differential equations of chemical reactions can be investigated by the aid of pure mathematical concepts. The structure and role of 'concentration space' is distinct from those of real space, and so there is not too much to be done (if anything) with co-ordinate transformations leading to 'relativistic reaction kinetics'. However, other, not-too-sophisticated transformations may be relevant from the practical point of view of the chemist.

Let us consider the differential equation

i =fox (4.34)

with an everywhere-defined, continuously differentiable right-hand side. A (state) transformation is an everywhere-defined, continuously differentiable function «p. Such a transformation transforms (4.34) into

:X= «p'o«p-•ox-fo«p-•ox = («p'·f)o«p-•ox.

The transformation of eqn (4.34) is called linear, if

«pEGL(M)

(4.35)

i.e. if «pis an invertible linear transformation. The transformation is said to be a change of scale, if

pr10«pOpr1 «p=

prMO«pOprM

and

[}l~m C IR+,

where

If a transformation D is linear and it is also a change of scale, then D is a change of the unit of measure. This is equivalent to saying that Dis a positive definite diagonal transformation. (Time) transformation of equation (4.34) is defined by a diffeomorphism

by which the new function

X:=xoT

is introduced. The transformed equation of (4.34) is:

X=fOX·t.

Change of time scale and change of time unit can be defined analogously to the previous case.

Mathematical models of chemical reactions

1he class of differential equations for which (~ + )M is an invariant set will \e interesting for us. (~+)M is an invariant set if and only if the vector f(x) shoWs inward from every point x of (~+)M. If (~+)M is an invariant set then l!l~)M has the same property. According to a theorem due to Vol'pert (~+)M is an invariant set of the induced kinetic differential equations of reactions. Of course, polynomial and nonkinetic differential equations, or even nonpolynomial ones may have this property.

l3ased upon the definitions above the following statements can be made.

II) Every change of scale (including every change of unit of measure) preserves the character of polynomial equations in the sense that kinetic differential equations remain kinetic, while nonkinetic ones remain nonkinetic.

(2) Every change of time scale (including every change of time unit) preserves the character of polynomial differential equations in the sense above.

(3) If(~+)M is an invariant set of a differential equation then the change of scale and the change of time scale preserves this property.

Group analysis is the general mathematical method for searching symmetries of differential equations. Group analysis of differential equations was initiated by the Norwegian mathematician Sophus Lie at the end of the ni11eteenth century. The first problem is to determine the general form of differential equations that admit a given group as a symmetry group. Loosely speaking G is a symmetry group for a differential equation if for all g EGg o p is a solution to the differential equation if p itself was a solution. The inverse to this problem is, how to find the Lie-group of symmetries of a given differential equation. (The fundamental books on the subject are: OvsjaJlnikov (1962, 1982), Bluman & Cole (1974) and Sibirskii (1982).) In a series of papers Steeb (1978, 1979; Schwarz & Steeb, 1983) pointed out (cf. w~lfman, 1979) that if an autonomous differential equation (for which all the solutions are defined on the whole real line) has a limit cycle solution, then the limit cycle trajectory is an invariant function of a one-parameter group admitted by the equation.

4.8-7 Principle of quasistationarity

frOm time to time a paper emerges in the literature aimed at emphasising the n~essity of the foundation of the principle of quasistationarity, pseudosteady state hypothesis, or Bodenstein(-Semenov) method. The essence of the method seems to be an absolutely crazy idea- from the mathematical point ofview. In a system of differential equations let us consider the variables that ta~e on 'small' values to be constant. So: if a function is small, so is its derivative! It turns out that among the conditions that occur in chemical re~ction kinetics it does work well.


According to our knowledge, after Bodenstein (1913) it was Semenov ( 1939), Frank-Kamenetskii ( 1940) and Hirschfelder ( 1957) who treated the problem from the chemists' side. Sayasov & Vasil'eva (1955) and Heineken eta/. (1967) were the first to cite and use the seminal paper by Tikhonov ( 1952) on singular perturbation theory that is the adequate tool to treat such problems. Relevant contributions have since been made by Aris and coworkers (Aris, 1972; Georgakis & Aris, 1975; Viswanathan & Aris, 1975) and by Vasil'ev eta/. (1973). It was Noyes (1978) who from the chemical side was to reconsider things again. And now we have the (by the way, very useful) paper by Klonowski (1983) aimed at the mathematical foundation of simplifying principles for reaction kinetics. The paper is based upon the mathematical results of the Russian school, but seems to acknowledge nobody from among the other authors mentioned above. In spite of this, we would recommend just this paper the reader as it is self-contained; and this is very important as Tikhonov's theory is not an easy one for the nonmathematician.

4.8.8 Exercises

I. Show the sufficiency of condition (4.28) for exact lumping. 2. Investigate exact lumping in the case when M = I. 3. Formulate a continuous component model for xylene isomerisation.

(Xylene is known to have three distinct isomers: ortho-, meta- and paraxylene and the transition between them is usually described by the triangle reaction, cf. Aly eta/. (1965).)

4. Verify that (i) a generalised compartmental system is weakly realistic, (ii) a second order reaction (in which all the reactant complex vectors

have a length smaller than three) is weakly realistic, (iii) the reaction rates of all the elementary reactions of type (ii) in a

weakly realistic mechanism are linear functions of all the concentrations separately.

4.8.9 Problems

I. Show that condition (4.28) for exact lumping is necessary as well. Hint. Start from the definition.

2. Give a necessary and sufficient condition for the solvability of ( 4.28) inK. What about the case when M is of full rank?

3. State and prove existence and uniqueness theorems on the continuous component model.

Hint. The Picard-Lindelof theorem for differential equations in Banach spaces may be used; see, for example, Lang ( 1962).

4. Reformulate the abstract differential equation (4.33) into a partial

cJO Mathematical models of chemical reactions

differential equation in the case when ..It = [0, I), q{ = [0, I) and both J.1 and A. are the Lebesgue measure.

?· Prove the statement of Subsection 4.8.3. Hint. Cf. Rudin, 1978, Remark 10.35.

6. Solve the induced kinetic differential equation of the reaction 1 2

X+-0+-X+Y 1! 1! 1j 2X2. Y.!.-2Y

Hint. The induced kinetic differential equation for x and y is a Cauchy-Riemann- (or Erugin-) system, therefore for z = x + iy an easily solvable (separable) differential equation can be written down.

4.8.10 Open problem

I. Find a physicochemical meaning of the measures J.1 and A. and devise a method (theoretically) to determine them.

5 Continuous time discrete state stochastic models

5.1 On the nature and role of fluctuations: general remarks

5.1.1 The logical status of stochastic reaction kinetics

A new approach has been adopted in the last two decades in the theory of chemical reactions that considers the chemical reaction as a stochastic process. In this chapter we deal with stochastic models of chemical reaction. More specifically, our intention is

(I) to discuss the theoretical foundation, (2) to set up the model, (3) to investigate its properties, (4) to give some hints on the applications.

Arguments for the application of stochastic models for chemical reactions might come at least from three directions, since the model

(a) takes into consideration - the discrete character of the quantity of components - the inherently random character of the phenomena·

(b) is in accordance (more or less) with the theories of - thermodynamics - stochastic processes

(c) is appropriate to describe - 'small systems' - instability phenomena.

We have to make clear that the formulation of the theory of stochastic


kinetics does not reduce the importance of deterministic kinetics, since for great classes of phenomena the stochastic model is only slightly 'better' than the deterministic approach, while the mathematics of the stochastic model is much more complicated. Practically speaking, the set of stochastic models for which we can find an analytical solution is much more restricted than that of deterministic models.

The subject of this chapter is organised as shown in Table 5.1.

Table 5.1

The logical status and mathematical framework of stochastic models of macroscopic physicochemical systems

Theoretical background

5.1

Stochastic models of chemical reactions

5.2, 5.8

The fluctuation-dissipation theorem

Examination '- of chemical kinetics: of properties 1 theory and applications

,(;) QJ,..~, .

.s- /c 'I J ~.s'f. V~

~--------~~v~-------, &~ I'"

Methods of solutions: ;so

exact results approximations qualitative properties simulation methods

5.3

Stationary probability distributions

5.7

Small systems 5.5

Unstable systems 5.6

5.4

Stochastic models 93

5.1.2 Fluctuation phenomena in physics and chemistry: an introduction

More than !50 years ago the Scottish botanist Robert Brown discovered the existence of fluctuations when he had studied microscopic living phenomena. However, the physical nature of the motion, which was named after its discoverer, was not known for a long time. As Darwin wrote in 1876: 'I called on him [Brown] two or three times before the voyage of the Beagle (1831 ), and on one occasion he asked me to look through a microscope and describe what I saw. This I did, and believe now that it was the marvelous currents of protoplasm in some vegetable cell. I then asked him what I had seen; but he answered me, "That is my little secret".'

The knowledge that Brownian motion could be detected particularly well in colloidal solutions dates from the eighteen-seventies. The mass of the -literally microscopic - Brownian particle is much greater than the mass of the solvent molecules, and the observable motion is the result of the individual motions of the small molecules. (For the early history of Brownian motion up to 1900 see Kerker (1974), and particularly Brush (1976, Chapter 15). A rather sophisticated explanation of Brownian motion was given by Einstein (1905) and Smoluchowski (1916), who calculated the temporal change of the expectation of the square of the displacement of the Brownian particle, and the connection between the mobility of the particle and the -macroscopic - diffusion constant. The experimental verification of the heterogeneous nature of colloid solutions, and of the EinsteinSmoluchowski theory resulted in three Nobel prizes for colloid chemists in 1925-26 (Zsigmondy, Svedberg, Perrin). A time-dependent theory of fluctuations has been formulated in connection with Brownian motion (Uhlenbeck & Ornstein, 1930; Wang & Uhlenbeck; 1945). However, the theory of Brownian motion is not closed, and it is still open for physical and mathematical researches (Levy, 1948; Kac, 1959; Ito & McKean, 1965).

The other class of fluctuation phenomena, well-known since the work of Gibbs (done in 1902, see Gibbs (1948)) and Einstein (1910), is equilibrium fluctuation. The theory of equilibrium (thermostatic) fluctuations considers the equilibrium state as a stationary stochastic process (see, for example, Tisza & Quay (1963) and Tisza (1966)). By thermostatic fluctuation theory the statistical character (e.g. the distribution functions and moments derived from it) can be computed.

5.1.2.1 Stochastic thermostatics, stochastic thermodynamics

It was ·logical to extend the results of the thermostatic theory to temporal changes. Stochastic thermostatics adopts an intermediate level between statistical and phenomenological thermodynamics. Analogously, in principle the stochastic treatment of thermodynamics processes has an intermediate character between nonequilibrium statistical mechanics and phenomenolog-


ical (deterministic) thermodynamics. While statistical thermodynamics deals with the motion of microparticles, conventional thermodynamics describes the temporal change of macroscopic quantities which used to be interpreted as expectations. Taking into consideration the variance and the higher moments a more detailed description of irreversible processes might be given. Since the stochastic treatment of thermodynamics adopts an intermediate level between the strict microscopic and strict macroscopic levels, it might be called mesoscopic, according to the terminology of van Kampen (1976).

A typical question of thermostatic fluctuation theory is the following: a system with volume V consists of n particles. What can we tell about the number of particles in a volume ~V <iii V? For the case of ideal gases, in thermodynamic limit (N-+ oo, V-+ oo, N/V =constant) the number of molecules in~ V can be described by a Poissonian random variable. Denoting the random variable with n,

E(n) = D 2 (n) = N~VJV, (5.1)

since for the Poisson distribution the expectation E is equal to the variance D. Fluctuations can be characterised sometimes by the relative variance - the quotient of the variance and expectation:

D(n) _ I _ I E(n)- JE(;i}- jN~VJV (5.2)

Density fluctuations of the atmosphere are not significant from the mesoscopic point of view. In the range associated with the wavelength of light, the order of magnitude of the fluctuation is 10- 3• On a fine scale the refractory index and the dipole moment of the atmosphere is represented by random variables. The greater the variance of these random variables is, the more significant the light scattering. The first precise determination of the A vogadro number was based on the intensity of scattering of light due to density fluctuation of the atmosphere.

Though the fluctuations considered above are 'useful', at least in the sense that the measure of fluctuation provided valuable information on the discrete nature of matter, fluctuations in an electric circuit resulting from thermal motion of electrons are generally considered as 'undesired' noise from the experimental point of view, since they reduce the 'signal/noise' ratio. The terms 'fluctuation' and 'noise' are used interchangeably, though the second used to have some pejorative connotations. However, these kinds of fluctuations can be useful (Johnson, 1928; Nyquist, 1928). As Nyquist noted, the spectral density of thermal noise and certain characteristics of the circuits, i.e. admittance, are interconnected, and he gave the form of this function, at least in a certain frequency regime. Callen and his coworkers recognised that the Nyquist formula can be generalised to include nonelectrical cases. The fluctuation-dissipation theorem obtained by the generalisation connects the fluctuations around the equilibrium with the


dissipative processes leading to equilibrium. Transport coefficients can be calculated based on this theory (Kubo, 1957). The fluctuation-dissipation theorem can be extended to chemical reactions, too, and it also has important biological applications.

Investigating the behaviour of 'small systems', i.e. the case when the number of particles is 'small', fluctuation phenomena are very important. More precisely they cannot be interpreted as the superposition of certain deterministic behaviours, as they are the very phenomena under observation. A thermodynamic description of small systems was attempted by Hill ( 1969). Under the traditional conditions of chemists, 'small systems' are rather rare: e.g. I 06 molecules (which are considerably less than Avogadro's number) are contained in the volume of 10- 12 ml of a 10- 2 mol dm -I solution. Such small volumes are availabe to experimentalists using optical or sometimes electrical methods.

The volume of a 'small system' can be larger if the size of some component is much greater as usual. This condition is often fulfilled in enzyme systems. In this case results corr.ing from a deterministic model can be completely misleading.

Fluctuation phenomena occurring in consequence of transport processes through biological membranes can be readily studied. Membranes are twodimensional, and 106 molecules are contained in a 500 jlm 2 membrane surface, which is available to experimental techniques (Neher & Stevens, 1977). (Membrane noise can be analysed by using the language of stochastic kinetics, as will be shown in Subsection 5.5.3, as well as in Chapter 7.)

Fluctuation phenomena are particularly significant in connection with 'critical' phenomena. Critical phenomena are highly interconnected with instabilities. Intuitively a system exhibits instability if small changes of the 'control' parameter of the system implies the change of the macroscopic structure of the system. The second order phase transition has been studied extensively from the physical point of view, as a typical (equilibrium) critical phenomenon. The enormous development of synergetics (see, for example, Haken, 1983) has been initiated by searching for phase transition-like, analogous (non-equilibrium) critical phenomena. Fluctuations have very significant effects in the vicinity of 'critical points'. The 'square rule' expressed by (5.2) is not even approximately valid, since fluctuations are amplified to cause drastic macroscopic effects. Nonequilibrium transitions can also be interpreted in terms of thermodynamics (Glansdorff & Prigogine, 1971; Nicolis & Prigogine, 1977). Accordingly, a nonequilibrium stationary regime might be unstable as a consequence of fluctuations due to the discrete character of the matter, and a new stationary regime will be stabilised after decay of the transient phenomena. Newer results of the stochastic aspects of nonequilibrium transitions in chemical systems have been recently reviewed by Nicolis (1984). Internal and external fluctuations represent two disjoint classes: the former is due to the inherently random character of the system,


the latter term expresses the approach by which the (generally, but not necessarily) deterministic system is influenced by external (environmental) noise. While internal fluctuations are generally small (except for instability phenomena), external noise can be arbitrarily large, since they are controlled from 'outside'.

The theory of noise-induced transition (Horsthemke & Lefever, 1984a) emphasise that fluctuations (noise) superimposed on deterministic motions might have a crucial role in forming 'ordered structures'; i.e. they may operate as 'organising forces'. Noise may destroy the stability of the deterministic attractors and stochastic models might exhibit completely different qualitative properties from those of the deterministic model.

5.1.3 Stochastic processes: concepts

5.1.3.1 Introductory remarks

A stochastic process is a collection of random variables {~,}. tElr. Two important cases are the discrete parameter process, when lr c ~ 0 = {0, I, 2, ... } and the continuous parameter process, when lr c IR 1. We shall mostly consider the second case, and, in addition, restrict ourselves to Markov processes (processes without after-effect).

Systems under investigation are characterised at a fixed time, t, by a finitedimensional vector random variable ~ 1 (t), ~2 (t) ... ~M(t). The state of the system can be characterised (not completely) by the absolute density function x(t)-+ g(x, t) where x(t) is the value of ~(t). The temporal evolution of the system is given by a partial differential equation for g. Very often we have to be satisfied with deriving and solving differential equations for the first two moments only. Since we are interested in Markov processes, we have to know the conditional probability density function f; f( y, s; x, t) gives the probability density that a system will be at x at a time t, if at the times it was at y. The connection between the absolute g( ·) and conditional probability function f( ·) is given by

X

g(x, t) = I f( y, 0; x, t)g( y, 0) dy (5.3)

-:x:

Additionally, the Chapman-Kolmogorov equation is valid:

X

f( y, s; x, t) = I f( y, s; z, u)f(z, u; x, t) dz; s < u < t (5.4)

-x

To study the different classes of processes the state-space and the character of the motion have to be specified:

Stochastic models

State space:

Character of the motion:

5.1.3.2 Continuous state-space processes

97

Jump

(1) Loosely speaking, continuous motion means that during a small time interval the system undergoes only a small change. This kind of motion is reminiscent of diffusion.

More precisely, for diffusion processes the following conditions are fulfilled:

lim lf!lt I f(y, t; X, t + !lt)dy = 0, (5.5)

lim lf!lt I (x - y)f( y, t; x, t + !lt) dy = a(x, t), (5.6)

lx- .1'1 < 6

lim lf!lt I (x; - y;) (xk - Yk)f( y, t; x, t + !lt) dy = b;k(x, t). (5.7) At-0

The verbal interpretation of (5.5) is that the process is continuous- this is the Lindeberg condition. The function a(x, t) is the velocity of conditional expectation ('drift vector'), and bij(x, t) is the matrix of the velocity of conditional covariance ('diffusion matrix'). The latter is positive semidefinite and symmetric as a result of its definition (5.7).

Using the mild conditions of existence of the derivatives

iJj(y, s; X, t) d 02j(y, s; X, t) -"-.o...::....:,.-:---'~ an

oy; oy; Yk (5.8)

the conditional probability density function satisfies both the Kolmogorov I ('inverse', 'backward') and II ('direct', 'forward') equations:


oj(y, s; X, t) + ~ ·( )oj(y, s; X, t) a ~ ~ x, t a s i= I Y; (5.9)

+! I bij(x, t)i)2j(y, s; x, t) = 0, 2 i.j= 1 oy;oyj

of( y, s; X, t) + f Oa;(X, t)j( y, s; X, t) ar i=l ax;

_! f 02b;;(X, t)j(y, s; X, t). 2 i.j= I OX;OXj

(5.10)

The forward Kolmogorov equation (5.10) is referred to in physical literature as the Fokker-Pianck equation. For the absolute density function g(x, t) (which contains less information than/) the following 'Kolmogorovlike' equation holds:

og~, t) + i~l Oa;(X, ;~~(X, t) - ;~ i)~~Xj [bix, t)g(x, t)] = 0. (5.11)

We adopt the adjective 'Kolmogorov-like', since the Kolmogorov equation is used to refer only to (5.9) and (5.10) for conditional (or 'transitional) quantities.

Equation (5.11) is a generalisation of the diffusion or heat conduction equation.

(2) Noncontinuous processes in continuous state-space occur when the condition (5.5) is not fulfilled. In this case we need more general equations than the Kolmogorov equations. The main point is that analogously to (5.6) and (5. 7) the jth velocity of conditional moments Dj can be defined:

Dj(x, t) = lim 1/~t E[(~(t + ~t)- ~(tWI~(t) = x]. (5.12) Ar-o

For the absolute density function g the Kramers-Moyai-Stratonovich equation (Kramers, 1940; Moyal, 1949; Stratonovich, 1963) can be derived:

og(x, t) x 1 ( a )j a = L 1 -!1 [Dj(x, t)g(x, t)]. t j=lj. uX

(5.13)

The terminology is nonstandard, and in physical literature the Kramers-Moyal expansion is given as a (nonsystematic) procedure to approximate discrete state-space processes by continuous processes. The point that we want to emphasise here is the clear fact that, even in the case of a continuous state-space, the process itself can be noncontinuous, when the Lindeberg condition is not fulfilled. The functions for the higher coefficients do not necessarily have to vanish.


5.1.3.3 Discrete state-space processes

We will only consider state-spaces with a countable number of states. First the Pij(s, t) transition probability and the Pk(t) absolute distributions are defined, then the evolution equations are derived:

Py(s, t) = &'(~(t) = jl~(s) = i); (i,je NM). (5.14)

The absolute distribution Pk(t) = &'(~(t) = k) can be related to the initial distribution Pk(O) = &'(~(0) = k) and the transition probabilities:

Pk(t) = L P;(O)P;k(O, t); (k E NM) (5.15) iei'\IM

Under mild conditions the following limits (infinitesimal probabilities) exist:

I. I - Pnn(t - /:it, t)_. ( ) O 1m A -.en t > Ll.t~O t

The infinitesimal transition probabilities pjk(t) have

Pit)= 0 and LPjk(t) = I, je NM.

Using the Chapman-Kolmogorov equation we can derive

and

(5.16)

(5.17)

(5.18)

(5.19)

(5.20)

which are Kolmogorov equations with the P;k(t, s) = O;k boundary condition, and

(5.21)

which is (Pk(O) = okD) Kolmogorov-like equation for a given or deterministic initial condition, i.e. &'(~(0) = D) = I.

These kinds of equations have to be specified to obtain continuous time discrete state space models of chemical reactions.

5.1.4 Semigroup operator approach: advantages coming from the use of more sophisticated mathematics

Although the main body of our treatment of stochastic models does not


adopt the more abstract approach offered by the modern theory of stochastic processes, we may not go further without mentioning its main concepts.

At least two kinds of advantages come from the use of the semigroup operator approach. First, a mathematically justified approximation of discontinuous processes by continuous processes can be obtained. Second, not only models for purely temporal processes (e.g. 'pure' chemical reactions) can be given, but stochastic models of spatio-temporal phenomena (e.g. chemical reactions with diffusion) can be defined in a mathematically concise manner. The celebrated monograph on the modern approach to Markov processes is still Dynkin's book (1965), and for chemical applications see Arnold & Kotelenez (1981).

Some definitions:

A strongly continuous semigroup T, on a Banach space Z is a map T,: IR+ -+ L(Z), with the conditions

T,+s = T,T., Vt, sE IR+ (semigroup property) T0 =I

lim II T,h- hll-+ 0, VhEZ ,_o+

In consequence of the definitions

IIT,II :!:';;Me"''

for some constant ro and M ~ 0. A closed, linear, densely defined operator A on Z is defined by T,:

Af =lim T,J- f fi-o+ 0

(5.22) (5.23) (5.24)

(5.25)

(5.26)

for all f such that T,J is differentiable. A is called as the infinitesimal generator ofT,. From the definition of/the evolution equation immediately follows:

d dt(T,/) =AT,/= T,Af V/ED(A) (5.27)

where D(A) denotes the domain of A. What we have seen that continuous semigroups can define linear operators

and evolution equations for motions in rather abstract spaces. A diffusion process might be defined by the infinitesimal generator:

M a I iJ2 A = L a- + - L b.-- (5.28)

;~ 1 'ox; 2i.j~l ''ox;oxj

where the function a; and bij are the same as in (5.6) and (5.7). A relatively simple example will be given for approximating a Poisson

process (a very particular discrete process) by a Brownian motion (Kurtz, 1981).

Stochastic models

Let the generator of the Poisson process Y given by

Af(k) = A(f(k + 1)- f(k));

where A is the parameter of the process Y. Additionally, let

Wn(t) = n- 112(Y(nt)- An(t))

be a Markov process with generator

AJ(x) = nA(f(x + n- 112)- f(x))- n 112A~~x.

Under the mild conditions:

lim A f(x) = l/2A d2f(x) n dx2 '

n-ao

101

(5.29)

(5.30)

(5.31)

(5.32)

and convergence is uniform in x, then according to a theorem of Kurtz:

lim supn-'(Y(nt)- Ant)= 0 (almost sure) (5.33)

for all T > 0; and Wn::. W, where W is a Brownian motion defined with generator

- d2J Af = l/2Adx2

(for A = 1 this is the generator of the Wiener process that we shall use in the definition of the stochastic integral).

Equation (5.27) is an ordinary differential equation in function space. Giving an appropriate interpretation of the state-space it can describe 'spatiotemporal phenomena'. Thinking of chemical applications we might set the state-space as the unification of the 'real three-dimensional' space and the m-dimensional 'component' space. However, the formulation of stochastic models of chemical reactions accompanied by diffusion is not easy, and practically all of the applications treat both the component space and the real space by discrete methods. Although it is quite natural to apply the notion of discrete state-space for chemical reactions, at least from the mesoscopic point of view, it might be better if diffusion were described in terms of continuous models. In Chapter 6 we return to this question.

5.2 Stochasticity due to internal fluctuations: alternative models

5.2.1 Some historical remarks

Perhaps it is not well known that Leontovich (1935) was the first to investigate a stochastic model of chemical reaction. From the beginning and from the end of the nineteen-forties the works of Delbriick ( 1940) and Siegert ( 1949) might be mentioned. While the former studied fluctuations in the


autocatalytic formation of trypsin (EC 3.4.21.4) from trypsinogen, the latter did not adopt a direct chemical language when studying discrete transport among the internal energy levels of molecules. The first complete treatment of a se~ond-order reaction was given by the Hungarian mathematician A. Renyi (1953). This paper give evidence that the differential equation for the expectation of the stochastic model cannot be identified with the differential equation associated with the usual deterministic model. Prekopa (1953) introduced a nontrivial application dealing with the decomposition of longchained molecules. For the early literature of stochastic kinetics, consult McQuarrie (1967, 1968, 1969) and van Kampen (1981 ); Gardiner ( 1983) reviews the newer developments.

5.2.2 Models

The chemical reaction (!/, 9f, (ex, ~). k) is studied. Assumptions:

(I) time is considered as continuous, the quantities of components are discrete variables;

(2) chemical reaction is associated with a ;(t) discrete state space Markov process;

(3) o(L\t) is the probability that during L\t more than one reaction takes place, i.e.

I. o(L\t)_ O Im---

<1r-o L\t

(4) the infinitesimal transition probability matrix is specified by

Pjl = L k(r)ja.<·.r) (j, leN~, j # /). re R1_ i

(5.34)

and the other elements are 0. (This assumption corresponds directly to the mass action kinetics.)

(4)* the infinitesimal transition probability matrix is specified by

pft = L k(r)[j]~<·.rl (j, /eN, j # /) (5.35) reRI-i

and the other elements are 0. The definition of symbol [j]~< .r) is:

M

[j]~<·.r) = 0 jm( jm- I) ... Um- cx(m, r) + I). m= I

(This assumption corresponds to the Kurtz-type or combinatorial model.)


Remark: As a practical example, the assumption (4)* means the following for the reaction 2X--+ Y. The deterministic model is

i: = kc2 = - 1-kp V2

(Vis the volume, j is the number of the component X). The combinatorial model gives the expression

pff = k(r)j(j- l)

Additionally, it can be seen that for the case of second order reaction the connection between the deterministic rate constant k and the discrete stochastic rate constant k(r) is given by

k(r) = kj/V2.

The two models are equivalent when the infinitesimal transition probabilities are linear in every co-ordinate of the indices.

Illustration: let us consider the reaction

Then

& X+ Y --+2X

a X+ Y--+2Y

'i 3X + 2 Y--+ 2X + 3 Y

!/={X, Y}; ~ = {1, 2, 3}; k(l) = &; k(2) = ~; k(3) = y; ex.(·, l) = (1, If, ex.(·, 2) = (1, If, ex.(", 3) = (3, 2)r, ~( ·, l) = (2, O)r, ~( ·, 2) = (0, 2f, ~( ·, 3) = (2, 3f, ~(1.-1{ ={I}, ~(-I. I{= {2, 3} Pi.i+<I.-1{ = lij1h Pi.i+<-1.1{ = ~jlj2 + .YRH. Pi1 = 0 otherwise

Furthermore k - •••

Pj.j+(l.-l)T- CX.Jih PL+(-I.I)T = ~jij2 + yjl(h- l)(jl- 2)j2(j2- l)

=.YRR- 'YRj2- 3.YRH + 3yjfj~ + 2yj!H + (~- 2y)jlj2, pjl = 0 otherwise.

Using assumption (4) and adopting the condition that the process is homogeneous in time (i.e transition probability function depends only on t - s and not specifically on t and s) the Kolmogorov equations (5.9) and (5.1 0) can be reduced to (linear) ordinary differential-difference equation ('differential' in time, and 'difference' in state and in accordance with (5.27) we get, for a deterministic initial condition


Pit)= L., L k(r)f(·,r) Pkt(t) keNO reAk-J

- Pit) L L k(r)j"(·.r>, keN: re.Jik- 1

Pt(t) = ~ Pk(t) ~ k(r)ka.(·,r) 1 L-., 1 L..

keN(, reJI1- k

(5.36)

- Pit) L k(r) /"(·.r>,

The Kolmogorov-like equations for the absolute probability distribution function can be derived by using assumption (4):

Pj(t) = 4 P1(t) L k(r)f"(·.r) /eN(, re.Jfj-/ (5.37) -Pit) L k(r)j"(·.r>(jeN~)

reA,_ j

Equations similar to (5.36) and (5.37) can be derived using assumption (4)* instead of (4): in this case we might call the equations the combinatorial model. Generally these equations are called the master equations. The structure of (5.37) is clear. Two types of elementary reactions are taken into consideration. The effect of the first class of reaction is that the state j is available from I (I can denote different possible states). The second class describes all of the possible transitions from the state j. Therefore we can write:

Pj(t) = L transition to state j from /- transition from state j to /. I

Sometimes the expressions 'gain term' and 'loss term' are used. Unfortunately the stochastic model appears more 'complex' than the associated deterministic model. Even in the case of the reaction X~ Y for Pit) we get

Pj(t) = &.(j + l)Pj+ l(t)- &Pit) Pj(O) = oii.• (5.38)

where j 0 is the quantity of X at the time 0. The dimension of the system of differential equation is j 0 + I.

It is often remarked that stochastic models of chemical reactions can be easily extended to 'birth and death' type phenomena that take place in other populations of entities. Although we agree with this approach in principle, we have to remark that from the mathematical point of view the relationship between the three categories, namely stochastic models of reactions, simple birth and death processes (Karlin & MacGregor, 1957) and Markov population processes (Kingman, 1969) is not simple and is illustrated in Figure 5.1.

Stochastic models

Stochastic models of reactions

Simple birth and death processes

105

Population Markov processes

Fig. 5.1 Logical relationships among population Markov processes, stochastic models of chemical reactions and simple birth and death processes.

5.3 On the solutions of the CDS models

5.3.1 General remarks

The solution of the master equation (5.37) contains all the information about the system that is required in practice. Unfortunately, closed form solutions cannot be obtained even for the large class of reactions that are not important in practice.

We just mention some methods that have been used in simple special cases:

- successive integration and induction (Sole 1972), - determination of eigenvalues and eigenvectors (in cases when the state-

space is finite and the system size is small), application of Laplace transformation (for the case X+ Y~Z (see Renyi, 1953).

The most important technique is the generating function method that transforms the system of(ordinary) differential-difference equations into one partial differential equation. Examples will be given for illustrating the scope (and limit) of this method.

Not being able to solve the master equation in the more general cases we are often satisfied by the determination of the first and second moments. Furthermore, different techniques can be applied to approximate the 'jump processes' by continuous processes, which are more easily solvable. The clear structure of the stochastic model of chemical reactions allows the possibility of simulating the reaction. By simulation procedures realisations of the processes can. be obtained. The methods for obtaining solutions will be illustrated by discussing particular examples.


5.3.2 Chemical reaction X~ Y

The reaction consists of one elementary reaction step with the rate constant k. For the absolute distribution Pj(t) = &'(l;(t) = j) the form of the master equation is:

Here j 0 is the quantity of X at t = 0. (Again the initial condition is deterministic.) Equation (5.39) is a system of linear differential equations with constant coefficients and, in principle, its solution can be given by the eigenvalues and eigenvectors of the coefficient matrix. However, because of the large number of the unknown functions this method cannot be used in practice.

To solve the equation the generating function can be introduced:

00

F(z, t) = L Pj(t)zj (zeC; lzl ~ I) j=O

(5.40)

Without giving the details of the calculations the partial differential equation for F is:

oF(z, t)fot = k(l - z)oF(z, t)foz; F(z, 0) = Zj' (5.41)

(5.41) is a first-order linear partial differential equation with the known solution:

F(z, t) =[I+ (z- l)exp(-kt}F•. (5.42)

Pit) can be obtained by using the formula

Pj(t) = (1/j!)o{F(O, t) (5.43)

and so

Pj(t) = ~J) exp(- jkt)(l - exp(- kt))j-j•. (5.44)

The moments are also directly obtained from the generating function:

E[l;(t)] = o 1 F(l, t) = Dexp( -kt) (5.45)

D 2 [!;(t)] = oyF(l, t) + o1 F(l, t)- [F(l, t)F = j 0 exp(-kt)(l- exp(-kt)). (5.46)

In words, at every time point t -F 0 the quantity of component X can be given by a binomial distribution with expectation j 0 exp(- kt) and variance j 0 exp(- kt) (I - exp(- kt) ). As can be seen, the expectation agrees with the well-known deterministic solution. The two models are said to be 'consistent in the mean' (Bartholomay, 1957).


5.3.3 Compartmental systems

Compartmental systems might be considered -from a formal point of view - to be the analogues of isomerisation reactions. They are also quite often used in the 'biomathematical' literature.

It is very important that stochastic models of compartmental systems are completely solved, since by the aid of the method of generating functions the absolute probability distribution can be expressed for every time instant as a function of the rate constants and the initial conditions. In particular, the state of the system is characterised by the sum of random variables having independent multinomial distributions, which can always be determined exactly. General compartmental systems have been solved by Siegert (1949) and simplified by Krieger & Gans (1960) and Gans (1960). Their model was interpreted in terms of transport among levels of internal energy. Darvey & Staff ( 1966) interpreted the earlier results in terms of chemical reactions. The master equation for the probability distribution function is

n n

dP · · (t)fdt = ~ ~ k1 (J"1 + I)P. . . (t) l1h···ln i...J L. m l1h···ln

I= lm= I n n (5.47)

- ~ ~ k, }·,p . (t) ~ ~ m l1h···ln • I= lm= I

The multivariate generating function is defined by :c oo :c m

F(z, t) = F(ziz2 ... z., t) = L L ... L Pj,j, ... j.(t) n z{'. (5.48) j, = 0 j, = 0 j 0 = 0 I= I

The differential equation for F(z, t) is n n

aF(z, t)jat = L L klm(ZI - zm)aF(z, t)jaz (5.49) I= II= I

(with F(z, 0) = Zmj'). Equation (5.49) can be solved, and it is possible to show that F(z, t) is the

generating function of the multinomial distribution:

(5.50)

where w;(t) are time-dependent functions (0:::; w;(t) :::; 1). The consistency in the mean can also be proved.

5.3.4 Bicomponential reactions: general remarks

The mathematical treatment of stochastic models of bicomponential reactions is rather difficult. The reactions X + Y-+ Z and X + Y ¢ Z were investigated by Renyi ( 1953) using Laplace transformation. The method of the generating function does not operate very well in the general case, since it leads to higher-order partial differential equations. In principle chemical


reactions can be classified according to whether the equation for the generating function can be solved exactly or not. Methods of solution have been developed in the last decade, but not too much can be expected from the practical point of view in the near future. Information obtainable by elementary methods and approximations will be illustrated by the example of the reaction X + Y ¢ Z.

5.3.5 Chemical reaction X+ Y¢Z

5.3.5.1 The master equation

The master equation will be given. Let ~(t) be a random variable denoting the quantity of X; D1 , and D3 are the (deterministic) initial conditions for X, Y and Z; and 1.1 and A. are the 'forward' and 'backward' rate constants. Then for Pj(t) = &'(~(t) = j)

dPif)/dt = A.[(j + l)(D2 - D1 + j + l)Pj+ 1(t)- j(D2 - D1 + j)Pit)] + Jl[(D3 + D1 - j + l)Pj_ 1(t)- (D3 + D 1 - j)Pit))

Pj(O) = Ojo,· (5.51)

5.3.5.2 Use of Laplace transformation

Renyi (1953) gave a recursive formula for the Laplace transform of the distribution,

00

~(s) = f e-" Pj(t) dt, (5.52)

0

but he did not calculate the distribution Pj(t) itself.

5.3.5.3 Determination of expectation

It is sometimes sufficient in practice just to determine the expectation and variance. Multiplying eqn (5.51) by j and summing over all possible j the equation for the expectation can be obtained:

d/dt E(~(t)) = A.E[(D2 - ~(t)]E[D3 - ~(t)] - 1.1E(~(t)) + A.D 2 (~(t)). (5.53)

We see that in the equation there is a term containing the variance

D2 [~(t)] = L (j- E[~(t)])2 Pj(t) j

= L F Pit) - {E[~(t)]}2

= E[~(t)2 )- {E[~(t)]}2.

(5.54)

In general the treatment of bicomponential reactions is rather difficult


because the expression for the derivatives of the moments contain the higher moments, and so a closed system of equations cannot be set up.

Sometimes the approximation E[!;(t)2 ] = (E[!;(t)])l is adopted. This assumption is equivalent to the condition that D 2 [!;(t)] = 0 and so it corresponds to the omission of the stochastic character.

If D 1 -+ oo (i.e. for large systems), then A.D 2 [!;(t)]-+ 0 and with TJ(t) = D 2 - !;(t) and l;(t) = D3 - l;(t), (ll(t) is the quantity of Y, l;(t) is that of Z) we obtain

dfdt E[!;(t)] = A.E[TJ(t)]E[!;(t)] - ~E[I;(t)]. (5.55)

This equation is analogous to that for the deterministic model. Similar kinds of equations can be derived for more general cases.

5.3.5.4 The behaviour of the reaction during the initial period of the processes

Pj(t) can be approximated using Maclaurin series expansion:

<1J t" Pj(t) = L Pj"l(O)nt.

• =0 •

Truncating the expansion at the first term

Pj(t) = PiO) + Pj(O)t + o(t).

Using the formula (5.57) for the initial value problem (5.51) we get

Po,-2(t) = 0 + o(t) Po,-t(t) = o + t~D3 + o(t)

P0 ,(t) = I - t~ + o(t) Po,+ t(t) = 0 + o(t).

(5.56)

(5.57)

(5.58)

The procedure can also be applied in the case of nondeterministic initial conditions. Truncating the expansion at a higher power, the distribution can be approximated for longer time.

5.3.5.5 Determination of stationary distribution

The stationary solution can also be calculated:

where Pb' is a normalizing constant.

5.3.6 General equation for the generating function

(5.59)

From the structure of the equation (5.37) for the absolute probability


function P1 it follows immediately that the general evolution equation for the generating function FK(z, t) is

a ( a)"(·,r) at r(z, t) = L k(r)(z~< .r) - z"< .r)) az FK(z, t)

rE.!I

(5.60)

Here the superscript K refers again to the combinatorial model. Although this equation is a general model, it is the main tool for obtaining exact solutions, and that is why it is mentioned in this section on solutions.

5.3. 7 Approximations

A great amount of 'stochastic physics' investigates the approximation of jump processes by diffusion processes, i.e. of the master equation by a Fokker-Planck equation, since the latter is easier to solve. The rationale behind this procedure is the fact that the usual deterministic (CCD) and stochastic (CDS) models differ from each other in two aspects. The CDS model offers a stochastic description with a discrete state space. In most applications, where the number of particles is large and may approach Avogadro's number, the discreteness should be of minor importance. Since the ceo model adopts a continuous state-space, it is quite natural to adopt CCS model as an approximation for fluctuations.

One of the most extensively discussed topics of the theory of stochastic physics is whether the evolution equations of the discrete state-space stochastic processes, i.e. the master equations of the jump processes, can be approximated asymptotically by Fokker-Planck equations when the volume of the system increases. We certainly do not want to deal with the details of this problem, since the literature is comprehensive. Many opinions about this question have been expressed in a discussion (published in Nicolis et a/., 1984). However, some comments have to be made.

The question can be studied only in those models that contain the volume explicitly. The original master equation works with numbers of components, i.e. discrete valued extensive quantities. To investigate our- physically and not mathematically motivated - problem these quantities have to be transformed to quasicontinuous variables by dividing with the volume V. 'Quasicontinuity' means that it becomes continuous when Vtends to infinity.

The first modern treatment of the connection between the master equation and the Fokker-Planck equation was given by van Kampen (1961) who introduced a power series technique. He explored a systematic expansion method in a series of papers (see van Kampen, 1981 ).

As a result of the expansion in a 'small parameter', that is generally v- 1, a Fokker-Planck equation can be obtained approximately. From this approximation the average behaviour of the system and a Gaussian approach to the fluctuations can be calculated. The intensive quantity (random variable) ~(t) = ~(t)/V can be set up in the form

Stochastic models Ill

~(r) = y(t) + v- 1' 2TJ(t), (5.61)

where y(t) is the expectation of the process ~(t), and T)(t) is the stochastic process describing the fluctuation. Motivated by equilibrium statistical mechanics Kubo et a/. (1973) postulated that the form of the probability distribution function P,(t) is

Px(t)= Cexp[V- 1<1>(x/V, t)]. (5.62)

This is known as Kubo Ansatz. It means that the probability distribution function can be expressed as an exponential of some free-energy-like potential functions.

Remarks and formulae

I. The integral form of the master equation:

d/dt Pj(t) = f{W(jil)Pit)- W(/lj)Pit)}d/ (5.63)

could be obtained in principle from microscopic models by averaging; W(jl/) is the infinitesimal transition rate.

2. Turning from mathematics to physics, the extensivity postulate can be applied:

W(xix') = Vw(x/Vix'JV),

i.e. the transition probability W is an extensive quantity. 3. The Kramers-Moyal-Stratonovich equation is:

_ oo (-I)" (a)· dP,(t)/dt- -~~ n! v·- 1 ay D.(y, t)Px(t), (5.64)

where D. is the velocity of the nth conditional moment, and the notation x = jjv is used.

4. The Fokker-Planck equation

(5.65)

may be derived by truncating the Kramers-Moyai-Stratonovich expansion aftt;r the second term.

5. Other kinds of Fokker-Pianck equations can be also derived. The continuous state-space stochastic model of a chemical reaction, which considers the reaction as a 'diffusion process', neglects the essential discreteness of the mesoscopic events. However, some shortcomings of (5.65) have been eliminated by using a direct Fokker-Pianck equation obtained by means of nonlinear transport theory (Grabert et a/., 1983).

6. Another approximation method for solving the discrete master equa-


tions have been suggested by Haag ( 1978). First he introduced a 'transition factor' g(jil) defined by Pif) = g(jil)(t)P1(t), then the master equation has been transformed to describe the slowly varying change of these transition factors. The main point of the procedure is again (similar to the Kubo Ansatz) an assumption that there is a slowly varying variable (f)(j) defined by the relation g(jil) = exp((f)(j) - (f)(l)).

The continued fraction representation of the transition factor has been applied for solving one-variable chemical master equations (Haag & Hiinggi, 1979, 1980). For simple birth and death processes with birth and death rate functions \j1 and J.1 the nearest neighbour transition gj' is:

st _ psr;pst _ \jlj- I gj = j j- I = ('I' + ) _ st •

'l'j Jlj Jlj+ lgj+ I (5.66)

This relation generates the continued fraction that is, in general, infinite. Explicit expressions were derived for the continued recursion coefficients in terms of the elementary reaction rates. The method has also been applied for cases when two-step jumps occur.

5.3.8 Simulation methods

As was mentioned earlier the master equation (5.37) generally cannot be solved. To get some experience of the behaviour of chemical systems we might do stochastic simulation experiments using Monte-Carlo techniques (Introductions to Monte-Carlo methods are given in Hammersby & Handscomb 1964, and Srejder 1965. Their applications in chemical physics are discussed in Binder (1979.)

We shall consider two algorithms. (l) A DDS Markov process (i.e. Markov chain) can be associated with a

CDS model, if we change the assumption (3) of Subsection 5.2.2 'by substituting I instead of A.t, and 0 instead of o(L\t). The Markov chain obtained by this approximation is simulated. The probability that a reaction will proceed in a time instant is proportional to the k(r) rate constant and to the quantities of the reactant components. Having selected a reaction, the value of the process ~(t) is modified according to the stoichiometry. Iterating the procedure generates a realisation of the process. (An introduction to the use of stochastic simulation methods for chemical reactions is the paper of Rabinovitch (1969). The idea of the algorithm just mentioned was suggested by Lindblad & Degn (1967). We have used this algorithm: see Erdi et a/. 1973) and Sipos eta/. (l974a, 1974b).)

(2) From the theoretical point of view, it is better founded to treat the chemical reaction as a jump process (Hars, 1976; Gillespie, 1976, 1977, 1978, 1979; Hanusse, 1973, 1980).

Based on a theorem of Doob (1953, pp. 244) we can assume that the 'waiting time' of the system in state j can be considered as an exponentially


distributed random variable, where the value of the (naturally statedependent) parameter of the exponential distribution is qj = "f.1pj1• i.e. we take into consideration all the possible reactions, and sum the probabilities of the reactions taking place in the interval At.

Example

For the reaction

kl kl A+ B¢. C + D A+ C¢. B + D

k, k.

q;(t) = k 1a(t)b(t) + k 2 c(t)d(t) + k 3 a(t)c(t) + k 4 b(t)d(t) where a(t), b(t), c(t) and d(t) are the numbers of components A, B, C and D respectively at the timet.

Simulation experiments are performed by determining the duration while the system does not change its state, and then the reaction what occurs at the end of the interval.

The steps of the algorithm

0. Set the time variable t = 0. Prescribe the initial numbers of molecules (a deterministic set of initial conditions is assumed). Specify the elementary reactions and their rate constants. Specify the 'stopping time'.

I. Generate two (pseudo )random numbers to select a time interval At, and a reaction occurring at time t + ll.t.

2. Advance t by ll.t, and modify the value of the quantities of those components which are involved in the reaction just selected. (For example, if the reaction A + B-+ Cis selected, then a(t), b(t) and c(t) will be replaced by a(t + ll.t) = a(t) - I, b(t + ll.t) = b(t) - I, and c(t + At) = c(t) + 1.

3. If the simulation time did not reach the stopping time, recalculate the p~ infinitesimal probability, otherwise the simulation has been finished. Furthermore, even if the time is less than the stopping time, but aiiJ't = 0, the calculation is terminated. Fort< stopping time and pj1 '# 0 return to step I.

Simulation experiments with the Lotka-Volterra reaction are given (Fig. 5.2). It can be seen that even the qualitative results of the simulation can depend primarily on the 'distance' from the equilibrium, and even on the operation of the pseudo-random number generator. The set of realisations can be classified:

(a) the oscillator can be 'regular', with a constant amplitude, as in the deterministic case;

(b) fluctuations give rise to an increase of the amplitude (an 'explosion'); (c) fluctuations give rise to a damping of the oscillation;


(a) Damping oscillation (b) Sustained oscillation

Y(l( j'v-Lb-____J

(c) Explosion (e) No oscillatory behaviour

(d) Oscillation with alternating amplitude

Fig. 5.2 The character of the behaviour of the Lotka-Volterra system depends on the value of rate constants, initial values, and even on the starting point of the pseudorandom number generator.

Stochastic models

(d) amplitudes may alternate; (e) oscillation does not occur.

115

The analysis of the behaviour of the Lotka-Volterra model leads us to investigate fluctuations near an instability point, and we return to this point in Section 5.6.

5.4 The fluctuation-dissipation theorem of chemical kinetics

5.4.1 Stochastic reaction kinetics: 'nonequilibrium thermodynamics of statespace'?

Our starting point is the Kramers-Moyal-Stratonovich equation (5.13). Neglecting the higher-than-first-order conditional moments the 'drift' equation is obtained:

M

og(x, t)fot + L ofox;D) (x, t)g(x, t) = 0. (5.67) i= I

Since g is a probability density function, and a deterministic initial condition has been assumed, the following conditions hold:

"' g(x, t) ~ 0; f g(x, t)dx = I; g(x, 0) = ox,(x), (5.68)

-x

where ox0 is the Dirac distribution. (Of course the term 'distribution' is used in the sense of 'generalised function' and has nothing to do with 'probability distribution'.) Considering (5.67) as a partial differential equation for the distribution g(x, t) it can be proved:

The unique solution of (5.67) satisfying the condition (5.68) is the ox(t)

distribution, where x(t) is the solution of the initial value problem

x(t) = Dl(x(t)); x(O) = Xo. (5.69)

The statement can be proved by solving (5.67) for the function g( · ), and taking into consideration the condition (5.68). The theoretical basis of the procedure can be found in Hormander (1964), and the statement may be interpreted in four different ways.

(I) The deterministic motion (i.e. the solution of the initial value problem) can be considered as a particular stochastic process satisfying just (5.67) and (5.68); (Lax 1966a).

(2) Starting from a stochastic model and neglecting the velocity of higherthan-first-order moments, the (5.69) solution is obtained, which is the deterministic motion.

(3) From a formal point of view (5.13) is analogous to the continuity


equation describing nonlinear conduction and neglecting the source term. The motion in the state-space constituted by the components can be interpreted as the resultant of the 'convective' and 'conductive' motion. The deterministic model of chemical kinetics describes just the 'convective' motion in the state-space.

(4) In other words, the 'convective' motion does not modify the shape of the density function (which characterises the state of the system), it only shifts the 'density cloud'. The 'spreading' of the cloud is caused by the 'conductive' motion. However, in real three-dimensional space the convective motion is not a dissipative process, i.e. it does not change the entropy of the system, and the 'convective' motion of chemical reactions is dissipative. The state-space of the chemical components, is not isotropic, unlike mechanical state-space.

5.4.2 Fluctuation-dissipation theorem of linear nonequilibrium thermodynamics

Since chemical reaction is considered as a stochastic process, and furthermore as a 'thermodynamic' process, it is a natural question to ask what are the counterparts of the statements of the fluctuation theory of nonequilibrium thermodynamics. In the theory of thermodynamics the fluctuation-dissipation theorem is associated with the observation that the dissipative process leading to equilibrium is connected with fluctuations around that equilibrium. This fact was pointed out in a particular case (related to Brownian movement) by Einstein. Different representations of the theorem exist for linear thermodynamic processes (Callen & Welton, 1951; Greene & Callen, 1951; Kubo 1957; Lax, 1960; van Vliet & Fasset, 1965; van Kampen, 1965.)

The time representation gives the relationship among the velocities of conditional moments. For linear thermodynamic processes (which are associated with diffusion processes, i.e. Dn(x, t) is nonzero for n = I, 2 only) we get

aD aD 2D2 = oa.1 E[a.(O)a.r(t)) + E[a.(O)a.(tf] oa.1 , (5.70)

where a.(t) denotes the 'distance' of the physical quantities from equilibrium. The time representation can be converted to a spectral representation by

the Wiener-Khintchine relation. A fundamental property of stochastic processes a.(t) that are stationary in the broad sense is that they can be characterised by the time correlation function:

0

C(•) = lim 219 f a.(t)a.(t + •) dt.

o-x (5.71)

-0


The time correlation function expresses the effect of the value of the process at t for values at t + 't. Characteristic properties of C when E[cx] = 0:

lim,C(•) = £[~2 ]; lim C(•) = 0. (5.72)

The correlation is maximal, if 't = 0, i.e. if cx(t) determines uniquely the value cx(t + 't), and it is minimal when 't is much larger than the average duration of one fluctuation. Within the validity range of linear thermodynamics

C(•) = exp[- A.-]E[cx2 ]. (5.73)

The spectral density function of the fluctuation can be calculated from the autocorrelation function by the Wiener-Khintchine relation (Wiener, 1930; Khintchine, 1934). The original formulation of the theorem refers to stationary stochastic processes; for a possible generalisation see, for example, Lampard, 1954. The relationship connects the autocorrelation function to the spectrum:

00

S(ro) = _!_ f C(.-)cosond.- = _!_(cx2 )'q y(ro) . (5.74) 27t 27t y(ro) + ro2

-oo

y(ro) is the Fourier transform of D1 (x, t) (dissipation coefficient), while (cx2 )'q characterises the measure of equilibrium fluctuation. Applying the Wiener-Khintchine theorem we get, from (5.74),

oc

S2 .(ro) = 2~ f E[cx(O)cx(•)rcos.-ro.-d.- = 217tE[cx*, oo]y(;)<~ ro2 • (5.75)

0

5.4.3 Determination of rate constants from equilibrium fluctuations: methods of calculation

Let us consider the following problem: having measured the equilibrium concentrations in a complex chemical reaction is it possible to determine all the reaction rate constants of the reaction? The answer will depend on the model used to describe the reaction, and it will be 'no' in the case of the ceo model, and generally will be 'yes' in case of the CDS model.

Some of the possible ways to solve the problem will be shown on the following example:

k. kJ A+ B¢C A+ C¢B+ D. (5. 76)

kz k.

Kinetic information can be obtained from equilibrium data in CCD models; and in CDS models from the time representation of the fluctuations or the frequency representation of fluctuations (spectrum).


The CCD model of (5. 76) is the following system of differential equations:

a= -k.ab + k2c- k3ac + k4bd b = -k1ah + k 2c + k 3ac- k4bd

i: = k 1ab- k 2c- k 3ac + k4bd d = k 3 ac - k 4 bd.

(5.77)

Knowing the equilibrium value (a*, b*, c*, d*) of the concentrations, k 1 /k2 and k3 /k4 can be determined:

(5.78)

Not more than M (the number of components) reaction rate constants can be expressed by the equilibrium concentrations and by the other reaction rate constants, even in the most favourable case. (The maximal number of reaction rate constants in a chemical reaction is N(N + I), where N is the number of different reactant and product complexes.)

The CDS model considers the equilibrium as a stationary stochastic process. Information on stationary stochastic processes can be obtained from the equilibrium distribution or from the realisations.

The equilibrium distribution characterises (but not uniquely) the equilibrium fluctuations. In a closed compartmental system, M- I functions of the reaction rate constants can be determined. It may be guessed that not more than M functions of the rate constants can be determined also in the general case.

It is also possible to analyse the realisations using the method of waiting times. This method is based on the theorem of Doob mentioned earlier. The time spent in the statej is an exponentially distributed random variable with parameter qj, where I - q/lt + o(At) is the probability of the event that no reaction will take place in the interval (t, t + At). In the case of the reaction (5.76):

(5.79)

As the expected value of an exponentially distributed random variable with parameter A. is I /A., and the arithmetic mean is the unbiased estimator of the expected value, therefore 1/qj may be estimated for all such j that appear among the concentration vectors measured in equilibrium. Making the supposition - that is not well-founded statistically - that when 1/qj is estimated qj may be estimated, a satisfactory number of independent functions of the reaction rate constants can be determined. Let us suppose the equilibrium valuesj; = (a;. b;, c;, d;) have been measured (i = I, 2, 3, 4) in reaction (5.76). Then the following functions of the reaction rate constants can be estimated:

k•a•h• + k2c1 + k3a1c1 + k 4 h1d1, k 1a 3h3 + k 2c3 + k 3a 3c3 + k4 b3d3, k1a2h2 + k2c2 + k3a2c2 + k4b2d2, k1 a4 b4 + k 2c4 + k3 a4 c4 + k4 b4 d4 .

(5.80)

As these functions are linearly independent (in the ks) when the j;s are


appropriate, therefore all the four reaction rate constants can be determined. The method of jumps makes possible a statistically well-founded analysis

of the realisations (Billingsley, 1974). The procedure is as follows: examine how many transitions have taken place from state j to states j + (- I, - I, I, O),j+(l,l, -I,O),j+(-1,1, -l,l)andj+(l, -1,1, -1). Relative frequencies of the transitions give an estimation of the probabilities

(5.81)

It is not clear for the time being- from the point of view of statisticshow to extend the methods of waiting times and jumps when errors of measurement are also taken into account.

The spectrum of the fluctuations also contains useful information on the kinetics of the reaction. Quantities of some of 'the components and some functions of' the reaction rate constants can be calculated from the spectrum.

k For the reaction X ::i: Y the spectrum is

k,

S( ) = 4(k1 + k 2 )- 1j*(l - j*/N) v I + [21tv(k1 + k2 ) 1)2

(5.82)

Here N is the total quantity of the components and j* is the equilibrium quantity of the component X (Feher & Weismann, 1973).

Noise analysis, in particular the evaluation of experimental spectra, has been extensively used for interpreting the kinetics of transport processes in membranes. (See, for example Frehland (1982) and Section 5.5 ofthis book.) In Chapter 7 an example will be given of how to use stochastic kinetics to obtain stoichiometric and kinetic information referring to the transmitterreceptor interaction at the surface of the postsynaptic membrane of nerve and muscle cells.

Experimental methods that may be used in applications of the fluctuation-dissipation theorem of chemical kinetics will be mentioned in Subsection 5.5.4.

5.5 Small systems

It was mentioned earlier, in Subsection 5.1.2, that in small systems the continuous state deterministic model is not relevant, since the notion of concentration is meaningless. Furthermore, stochastic phenomena are not superimposed on certain deterministic processes, but they constitute the very phenomena themselves.

5.5.1 Enzyme kinetics

Aninyi & T6th (1977) have investigated the Michaelis-Menten reaction

<I y

E+S"¢-C-+E+P p

(5.83)


in small systems. They considered a cell compartment of sufficiently small volume that contains only one enzyme molecule and several substrate molecules. The CDS model of (5.83) was studied with the initial condition e(O) = I, where e is the quantity of the E enzyme molecule.

The master equation is

dfdt P.,(t) = ~(2- e)P.- 1.,- 1(t) + y(2- e)P.- 1.,(t) + ex(e + l)(s + l)P.+ l.s+ 1(t)- [exes+(~+ y)(l - e)]P.,(t)

(5.84)

withe= 0, I; P.,(t) = 0, if e < 0, e > I, s < 0 or s > s0 - I + e, where s0 is the initial number of substrate molecules S: s(O) = s0 .

The evolution equation for the generating function would be of the second order. However, introducing the marginal generating function

s0 - I+ e

F.(z, t) = L z' P.,(t) (e = 0, I) s=O

we get

0 0 ofot F.(z, t) = ex(e + I) ot F.+ I (z, t) - exez ot F.(z, t)

- (~ + y)(l - e)F.(z, t) + (~ + y)z(2- e)F._ 1(z, t) (e = 0, 1).

The solution of this equation is (Aninyi, 1976):

F0 (z, t) = rexp[-~/ex(z- l)exp(-yt)

+ r :z: ~exp(-(ex + y)t)

+ f. f r<•l [y - (y + A.l"l)z]q. exp(J..\•l) L., L., I _ J..(•J I

i= In =0 1

F 1 (z, t) = r<-•> - r exp[- ~/ex(z- I) exp(- yt)

-f. f r<•>[Y- (y + A.\"l)z]q•+texp(J..\•>t) L.. L., i _ J..(•J I '

i= In= 0 1

nwhere A.; =I= Y; and

_ (A.\">) 2 + (ex+ ~ + y)A.\">) + exy q. = ex(y + J..~•l) (i = I, 2).

r, rand r can be determined from the

F.(l, t) + F2(l, t) =I; F0 (z, 0) = 0; F1(z, 0) = z0 (Az)

boundary and initial conditions.

(5.85)

(5.86)

(5.87)

(5.87)

(5.88)

Assuming that the solutions F.(z, t) are true generating functions, i.e. they are polynomials of finite degree in z, it can be shown that the summation, contains a finite number of terms only, and r = r = 0. The q.s are integers,


0 ~ q. < S0 - 1, (q. = n), and the A.s are the roots of the equations

1..2 + [~ + cx(n + 1) + y]A. + cxy(n + 1) = 0; (n = 0, 1, ... S0 - 1).

All the A.s are different negative real numbers. The absolute probabilities can be calculated from the generating function,

and they appear as finite sums of exponential functions. The CDS and CCD models of Subsection 4.1.1 have been compared. The

deterministic value always evolves above the expectation, and their differences can be 20-30% of the former (see Fig. 5.3).

E, S

1.0

(a)

0123456789 Time

E, S

1.0

0.5

0

-0.2~

\ \ \ ·,

' ·,·,E(~) ......

(b)

--·-1'-.2345678

~ E(t;)- D(s)

Fig. 5.3 Time course of the Michaelis-Menten reaction. k 1 = k_r= k 2 = I. Deterministic solution ([£], [S]) and stochastic expected values (£@, E(t;>) are compared (a). The stochastic expected value £(~) is plotted together with its variance (b).

5.5.2 Ligand migration in hiomolecules

The binding of ligands to proteins can be typically treated by CDS models, see, for example, Alberding eta/. ( 1978). In particular consider the binding of carbon monoxide to myoglobin where the ligand CO may encounter four potential barriers. The random variable ~L(t) denotes the number of ligands


in well L in a given biomolecule with L = I, 2, ... Lmax· (in this example Lmax = 4). The transition from stage L to K is proportional to the number XL. (XL is the value of ~L(t)). The binding is so tight that well I acts as a trap, and so transitions from well I can be neglected. The first ligand to occupy well I blocks further transitions.

Adopting a Markovian approach to ligand migration the system is characterised by a distribution function P(~L(t) =XL; A) (L = I, ... , Lmax).

where A denotes such parameters as temperature, pressure, pH and ligand concentration.

Using the abbreviation X= (Xi, ... , XLm.J the stochastic evolution equation is

iJP(X,t) _ Lmax Lmax ,

a - I I (XL+ I)yKLP(XL + 1, xK- 1, x) f L=2K=2

Lm •• + L (XL+ l)ysLP(XL + I, X')

L=2 L#K Lmax Lmax Lmax

+ L PLP(XL- I, X')- L L XLyKLP(X) L=2 L=2K=2

L#K Lmax Lm.u

- L XLYsLP(X)- L PLP(X) L=2 L=2 Lmax

+ I (XL+ I)y,Lox,.,P(XL + 1, xi- 1, X') L=2 Lmax

- L XLYlLOx,.oP(X). L=2

(5.89)

Here X' is the reduced vector derived from X by removal of the explicitly written components. YKv YsL and PL are rate parameters, S refers to the solvent, t and A are omitted for the sake of brevity. Using again the multivariate generating function (5.48), F is in the form:

I oo oo Lmu

F(z,, ...• ZLmax; t, A):= L L . . . L P(X; t, A) n z[L. (5.90) X1 =OX2 =0 XLmax""o L= I

The evolution equation for F is

Lmax Lmax oF iJF(z, t; A)jot = L PL(zL- l)F + L YsL(l - zL)"JZ

L=2 L=2 L Lmax Lmax oF

+ L"J;2K"J;2 YKL(zK- zL) iJzL

Lmax

+ L ... L L y,d(XL + l)Ox,.lP(XL + I, xi - I, X') X 1 XLmax L = 2

Lmax

- XLOx,.oP(X)] n z{1• (5.91) I= I


Before solving the equation we have to mention an experimental observation done by optical absorption measurements, which measures an average over a very large number of biomolecules:

(5.92)

From (5.90) and (5.92)

N •• p(t, A) = F(O, I, ... , I; t, A), (5.93)

To solve (5.90) analytically for F(O, I, ... , I; t, A) with the initial distribution P(X, t = 0, A) we need the 'phenomenological coefficient' matrix:

Lmu

MKK = YsK- L YLK((~L(t))), L,.l L,.K

(5.94)

(5.95)

where the (~K(t)) bracket denotes the average occupation number in well K at time t. The calculated value of N.,P(t, A) is

Ncxp(l, A)

= exp { -llbll- 1 :~: ~L(A>[:~>K 1 BKL(exp llKt- l)bK1 + B 1Lb11 r]} :X: :X:

* L . .. L P(O, x2, ... , XLmax; t = 0, A) X.z = I XLmax = 1

(5.96)

)I {I - llbll- 1 :~~ BKLbK1 expll~rL Here bK = (bK1, ••• , bKL ) are the eigenvectors of M with corresponding eigenvalue IlK• B is the c~"factor matrix of eigenvectors, and II b II denotes the determinant of b.

Experiments suggested that the deterministic and stochastic approaches give different results when the probability of finding more than one ligand bound to a given biomolecule cannot be neglected, even if we accept that continuous deterministic models might be defined for small systems.

5.5.3 Membrane noise

Electrical noise in biological membrane systems is explained in terms of opening and closing of ionic channels (for two useful review see De Felice ( 1981) and Frehland (1982)). The qualitative behaviour of membrane activity strongly depends on the density of membrane channels (Holden, 1981; Holden & Yoda 1981). Integrating numerically the celebrated HodgkinHuxley equations (Hodgkin & Huxley, 1952) it was demonstrated that the number of channels is a bifurcation parameter. The fact that channel


numbers can be less than 104 and channel density can be as low as I J.lm- 2

implies the adequacy of using stochastic methods for describing membrane kinetics. The ionic transport system is thought to consist of a number NP of identical channels. According to the basic assumption the measured current is proportional to the number of channels being in conducting 'open' state.

The measured current J(t) as a random variable is given:

J(t) := Npj5P 1 (t) = j 5 Nt(t) (5.97)

where }5 is the current through a single open channel, P 1 (t) is the probability that one channel is in the open state. Current fluctuations occur because of the fluctuation in the number of open channels N 1 (t)

The variance crf is given by

crf = (}5 ) 2 NpPf(I - Pf), (5.98)

where Pf = lim P 1 (t). r~x

The kinetic model adopted by Hill & Chen ( 1972), (but see Chen, 1978), assumed that the channels are independent and that each one consists of x independent subunits. Each subunit can be in either one of two configurations. The state of the channel is given by the number of subunits in a particular configuration. A channel is thought to be nonconducting unless all x subunits are in the appropriate configuration. For the particular case of x = 2; the kinetic scheme is

rm~rn~rn P0 P 1 P2

(5.99)

where [1] denotes a channel in which j of the subunits are in adequate configuration, pj is the fraction of M such channels in that state. At equilibrium

N]. = Mp~ = M[cx/(cx + ~)]2 = Mn~ N0 = Mp"• = M[~/(cx + ~)]2 = M(l - nx)2 (5.100)

Nr = Mp; = M[2cx~/(cx + ~)2] = M2nx(l - nx).

Often the current spectrum is measured. Theoretically, the spectrum of 'component' fluctuation is

S(ro) = 8Mn~(l- nx>[l +n~2 ,2 + 4 1+-ro~~2l (5.101)

The current spectrum S1(w) can be associated to S(ro) by the relation

(5.102)

where gK is the conductivity of an open (potassium) channel, Vis the actual potential, EK is the equilibrium potential for K+.

More complex models exist to describe different types of membrane noise.


Noise analysis is a method of differentiating between concurrent transport mechanisms (Fig. 5.4). A particular problem, transmitter-induced membrane noise, will be studied in Chapter 7.

Assumed transport mechanism

no

Noise analysis cannot provide sufficient information to reject the mechanism assumed; if you have another assumption. repeat the calculation

yes

Formal chemical reaction

Kolmogorov equations

Correlation function

Calculated noise spectrum

Fig. 5.4 Scheme of an algorithm to differentiate among mechanisms of membrane transport processes.

5.5.4 Kinetic examinations of fast reactions

It was seen earlier (Subsection 5.4.3) that at least the 'spirit' of the fluctuation-dissipation theorem can be utilised in chemical kinetics, since rate constants can be evaluated from equilibrium fluctuation measurements.

According to the classical approach of experimental reaction kinetics, the reagents have to be mixed, then the temporal changes of the components have to be followed. The classical methods are not appropriate for the study of fast reactions, since the reaction is much more rapid than the mixing.


The relaxation method is an important technique for obtaining rapid kinetic measurements in solution (e.g. Eigen & De Mayer, 1963). According to this technique the chemical and physical equilibrium is disturbed by perturbing the system with a jump in an intensive variable (such as temperature, pressure or concentration). Certain functions of the rate constants can be determined by the process relaxing to a new equilibrium. The relaxation technique is very useful, though its theoretical foundations are not completely well-established. First, the perturbation disturbs the physical as well as the chemical equilibrium; the common treatment of physical and chemical relaxation processes - mostly with deterministic models- is difficult (see Czerlinski, 1966; Pecht & Riegler, 1977). Though it is ingenious, the method offered by Hayman (1971), according to which the volume and the entropy are interpreted as 'chemical components', and their change is considered as a formal chemical reaction, it cannot qualify as the final solution of the problems. Second, there are opposing requirements for the size of the concentration change due to the perturbation: it has to be small enough to allow the use of the linear equation, but it also has to be large enough to justify the neglect of fluctuations.

From the theoretical point of view measurements based on fluctuation phenomena are better, since the equilibrium fluctuation may be interpreted as spontaneous perturbation and relaxation. Consequently, kinetic information is available without perturbing the system externally. In practice, chemical fluctuation measurements refer to small systems, therefore the problem is discussed in this section. Fluctuation measurements can be classified as follows (Romine 1976): indirect measurements, and direct measurements, which may be based on the time representation of fluctuations, or the frequency representations of fluctuations.

The indirect measurements are derived from static not kinetic measurements, where the effect of chemical fluctuations appears as a disturbance. It is well-known (e.g. Gordon, 1969) that the connection between the function describing the shape of the spectrum line and the associated physical quantity is

X

/(ro) = l/27t I exp(iro•)C('t)d't, (5.1 03)

-x

where the physical quantity is considered as a stationary stochastic process ex(t) with a correlation function c('t). ex can be the magnetic momentum of the spin, or the dipole moment of a molecule. ex might be a function of temperature T, pressure, P, and chemical composition, c; (the function is denoted by &). The effect of fluctuation of these intensive variables and of time:

dex = (iJ&jiJp) dp + (iJ&jiJT) dT + ;~1 (~~)de; + (iJ&jiJt) dt. (5.1 04)


& is considered as a stationary stochastic process, and so the derivatives are taken in some stochastic sense. It is a problem to separate the effect of the component fluctuation from those resulting from temperature and pressure fluctuations. Another, finer problem is the decomposition of the effect of component fluctuation on 'diffusion' and 'reaction' terms. Extensively used measurement procedures are based on the line broadening of nuclear magnetic resonance peaks (e.g. Bradley, 1975); however, the area of application is limited.

Another indirect method is based on the light-scattering analysis of chemically reactive solutions (when Cl is taken as the relative permittivity). The fluctuation of the relative permittivity reflects the fluctuation of the quantities of the components (at least when the polarisability of the reagent and product molecules differs). Sometimes a rough approach is adopted, according to which temperature and pressure fluctuations can be neglected (Berne, eta/., 1968). The broadening of the Rayleigh component of scattered light has been analysed by the methods of conventional linear nonequilibrium thermodynamics (Blum & Salsburg, 1968, 1969). They pointed out that in the limit of zero scattering angle the (spatial gradient-dependent) transport processes do not contribute to the line broadening. They considered at how small an angle the measurements have to be obtained. Other applications are by Yeh & Keeler (1969, 1970), who determined certain functions of rate constants of dissociation reactions taking place in 0.1-1 M electrolytes and of conformation reactions of macromolecules. The conformation change of DNA has been investigated by Simon (1971). The results of these measurements sometimes gave relaxation time with a 10-8 s order of magnitude. The modern theory· of light scattering, and its kinetic and biological applications are surveyed in the book of Berne & Pecora (1976).

The direct measurement of fluctuations seems to be feasible, given current and anticipated measurement techniques. The measurement of optically active systems is particularly favourable, since laser-based absorbance measurements, fluorescence spectroscopy, circular dichroism, rotary dispersion and Raman spectroscopy are all avariable.

Conductance measurements have also been used in electrolytes for determination of composition fluctuation. Most of the techniques mentioned above are appropriate to investigate both the time, and spectral, representation of fluctuations. The scope, and limits of, the experimental techniques for determining the composition fluctuation was analysed by Magde ( 1977).

The representation of fluctuations connected with fluorescence correlation spectroscopy has been adopted for the study of the reversible binding of ethidium bromide to DNA. The reaction has biological significance in connection with the discovery of the transcriptive mechanism of the genetic code, since ethidium bromide inhibits nucleic acid synthesis. The DNAethidium bromide complex is strongly fluorescent, its fluorescent quantum


yield is 20 times larger than that of pure ethidium bromide. A single-step reversible bimolecular reaction scheme can be associated with a lumped model of the reaction. The rate constants have been determined from the time correlation function and the expectation of the quantity of DNA.

An example of direct measurements based on the frequency representation of fluctuation is the study of the association-dissociation reaction of BeS04

in a 0.03 M solution, with conductance measurements (Feher & Weismann, 1973). It is particularly interesting that they could increase the ratio of the reaction noise to Johnson noise of the circuit, since the former is a quadratic function of the applied direct voltage, while the latter is independent of the voltage.

Though modern experimental techniques are appropriate for fluctuation measurements, the specific examination of the fluctuation phenomena due to chemical reaction is still rather difficult.

5.6 Fluctuations near instability points

5.6.1 An example of the importance of fluctuations


).

A+ X -+2X 11

X-+0, (5.105)

where A is the external and X the internal component, and 0 is the zero complex. This reaction can be associated with a simple birth-death process. The deterministic model is the following:

dx(t)/dt = (A.- jl}x(t); x(O) = x 0 •

(Here A.= A.[A].) The solution is

x(t) = x 0 exp(A. - jl}t.

(5.106)

(5.107)

If A. > jl, i.e. the birth rate constant is greater than the death rate constant, x is an exponentially increasing function of the time. If A. < jl, x decreases exponentially. For the case of A.= ll

x(t) = x 0 . (5.108)

The stochastic model of the reaction is

dPk(t)/dt = -k(A. + !l)Pk(t) + A.(k- 1)Pk_ 1(t) + !l(k + I)Pk+ 1(1) Pk(O) = okx.; k = I, 2, ... N. (5.109)

There are two consequences of the model:

(1) the expectation coincides with the process coming from the deterministic theory, i.e.

Stochastic models

E[~(t)] = x 0 exp(A. - ll)t,

which in case of A. = ll reduces to the form

E[~(t)] = X 0 .

(2) the variance of the process is

D2 [~(t)] = (A.+ ll)t.

For the case of A.= ll

D2 [~(t)] = 2A.t,

129

(5.ll0)

(5.lll)

(5.ll2)

(5.113)

i.e. progressing with time, larger and larger fluctuations around the expectation occur (Fig. 5.5).

E[l;(t)] ± D[l;(t)]:_. ____ _

,..,..... .,.,.

E[l;(t)]:------

--·-· ...-· -·..--·---·

Fig. 5.5 Amplification of fluctuations might imply instability.

It is quite obvious that in this situation it is very important to take the fluctuations into consideration. Such kinds of formal reactions are used to describe the chain reactions in nuclear reactors (e.g. Williams, 1974). In this context it is clear that the fluctuations have to be limited, since they could imply undesirable instability phenomena.

5.6.2 Stochastic Lotka- Volterra model

Let us consider the (irreversible) Lotka-Volterra reaction:


k, A+ X-+2X

k, X+ Y-+2Y (5.114)

k, y -+0.

Here X, Yare internal and A external components, and 0 is the zero complex. The deterministic kinetic equation is

dx(t)/dt = k 1ax(t)- k 2 x(t)y(t) dy(t)fdt = k 2 x(t)y(t) - k3 y(t). (5.ll5)

The equation (5.115) has one trivial (x~1 = y~1 = 0) and one nontrivial stationary solution:

(5.116)

According to linear stability analysis the trivial stationary point is an unstable saddle point, while the nontrivial stationary point is a marginally stable centre.

The Lotka-Volterra system exhibits undamped oscillation, and the amplitude of the oscillation is determined by the initial values (and not by the structure of the system). The equation of the trajectory is

x + y - x 0 In x - y 0 In y - H = 0, (5.117)

where H is the integration constant that is determined by the initial value. The stochastic version of the Lotka-Volterra model leads to qualitatively

different results from the deterministic approach. The master equation is

dP,,.(t)/dt = - (k1 xa + k 2 xy + k3 y)Pxy(t) + k 1 (x - I )aPx _ 1.,.(t) + k 2 (x + l)(y- l)Pn 1.y- 1(t) + k 3 (y + l)P,_,.+ 1(t). (5.118)

It can be shown (Reddy, 1975) that

P~~.:= &'(~ 1 = 0, ~2 = 0) = I, (5.119)

i.e. the absorbing state x = 0, y = 0 is the only stationary state. This result is different from that obtained from the deterministic model.

Similar results were given by Keizer (1976). However, he also demonstrated that the fluctuations might be bounded, if fluctuations for the external components (including the zero complex) are also allowed. Possible behaviours of the realisations were demonstrated by simulation methods (Fig. 5.2).

5.6.3 Stochastic Brusselator model

The Brusselator model (Prigogine & Lefever, 1968) is a formal reaction scheme:

Stochastic models

k, A-+X

k, B+X-+Y+C

k, 2X+ Y-+3X

k. X-+E,

that has a deterministic kinetic equation:

dx(t)fdt = k 1a- k 2 hx(t) + k 3 x 2 (t)y(t)- k 4 x(t) dy(t)/dt = k 2 hx(t) - k 3 x 2 (t)y(t).

Introducing the dimension-free variables

- {k; y- {k; X= yk;.x, = yk;.y,

and the parameters

A= (kta)/k4 jk3 /k4 , B = (k 2 /k4 )h,

the following system of differential equations is obtained:

dXfd• =A- BX+ X 2 Y- X dYfd• = BX- X 2 Y.

Equation (5.122) has one stationary solution:

X 0 = A, Y0 = B/A.

131

(5.120)

(5.121)

(5.122)

(5.123)

The necessary and sufficient condition of the stability of the stationary point is B < I + A 2 • It can be shown for this system that an unstable stationary point implies a limit cycle.

·stationary and time-dependent solutions of the master equation of the Brusselator have been approximately established by Turner ( 1979). The main point of his procedure is that the stationary solution of the master equation can be seen as a time average over a period of the limit cycle of Pn(t), the time-dependent solution. Some illustrative results of the symmetry-breaking bifurcations in the stochastic Brusselator model are shown in Fig. 5.6 (after Nicolis, 1984).

5.6.4 The Sch/ogl model of second-order phase transition

Chemical reaction models of 'phase transition-like' phenomena have extensively been studied. Schlogl's model (1972) of the second-order phase transition is

k~ A+ X~2X

k~ (5.124)


log P (b) c) (a) y

X y

X "'(X,, Y,)

Fig. 5.6 Transition to a limit cycle in a two-variables stochastic model: (a) For A. < A.,, there is only one stationary distribution of probability P, peaking on the stable steady state (x,, y,); For A. > A.c, there is a family of time-periodic distributions (differing by a phase), peaking on the stable limit cycle solution (x,(t), y,(t)).

Here A, B and C are external components, and X is the single internal component. The deterministic model is

dx(t)Jdt = - x 2 + (l - ~)x + y,

setting knA1 = I; kl = I; ~ = k;[B]; y = k2[C]. The stationary value of x will be denoted by X 00 •

The two 'phases' are represented by the cases whether x differs from 0 or not. y > 0 implies X 00 > 0, therefore 'phase transition' does not occur. For y=O

_ {I - ~. if ~ < I Xoo - 0, if~> 1.

It means that for ~ < I the stationary concentration is finite. A change in ~ could lead to a 'phase-transition'. The analogy with the most frequently investigated second-order phase transition phenomena, namely to the behaviour of ferromagnetic materials, is trivially demonstrated by making the following correspondences:

Xx +-+M; y+-+ H; ~+-+ TJTc

where M is the magnetisation, H is the magnetic field, T is the absolute temperature and T,. is the critical temperature.

What follows from the stochastic analysis of the system? In the case of y = 0 the only stationary state, analogously to the Lotka-Volterra case, is the absorbing state 0, i.e.

P,(oo) =&I(~= 0) = I. (5.125)

This result is different from that obtained based on the deterministic model. Oppenheim et a/. (1977) attempted to resolve this contradiction (but see Horsthemke & Brenig, 1971 ). They restricted the state-space by making


y

0.75

0.5

0.25

0.25 0.5 0.75 n

P=1~T=Tc

Fig. 5.7 Second-order transition in the deterministic (Schlogl) model.

x = 1 a reflecting state and introduced the concept of quasistationary state. The starting points of their analysis are:

(a) in the case ofmultistationarity in a deterministic model it depends on the initial value which stationary state will be realised;

(b) the stationary distribution of the master equation does not depend on the initial distribution;

(c) the temporal change of the distribution depends on the initial distribution;

(d) the time course of the reaction can be characterised by two separable time scales: the quasistationary states which are associated with the deterministic solution in the thermodynamic limit are realised much more rapidly than the states of the stationary distribution.

Though (a)-( d) are not generally fulfilled, e.g. the range of validity of(b) is definitely narrow ('ergodicity') in the case of y = 0 they hold. The temporal


change of the shape of the distribution and the relaxation times are visualised in Fig. 5.8.

Initial distribution T = O(N") Quasistationary

t = 0 distribution

T = O(e•) Equilibrium

distribution

P.:.:U_ p:.:.u p:lL 0 (1 - ~) 0 (1 - ~) 0 (1- ~)

n n n

Fig. 5.8 Relaxation of distribution function Pn(t) from initial state to final state via quasistationary state.

5.6.5 The Schlogl model of first-order phase transition

The Schlogl model of the first-order phase transition is given by the reaction

k: A+ 2X~3X

(5.126)

A, Band Care external components, X is the only internal component. With the notation

a= (kt/k!)[A]; k = (k;/k!)[B]; b = (k2/k!)[C],

the deterministic model is

dx(t)/dt = x3 - ax2 + kx - b = R(x).

Under the circumstances a > 0, k > 0 and b > 0 and the polynomial R(x) has three real, positive roots if

(a - (a 2 - 3k)112) (a - (a 2 - 3k)112) R 3 ~ 0 and R 3 ~ 0

R has a triple root, if k =am and b = a 3/27. The phases are represented by the stationary points. The triple root might be associated with the 'critical point'.

Since the constitutive equation of the ':'an der Waals gases is also a third order polynomial, R(x) can be associated with the equation

p _ RT _ a 1 + a2 -y V2 V3


by making the following correspondences:

x+-+ v-•; k+-+RT; b+-+p,

where V is the volume, p is the pressure and T is the temperature. Stochastic analysis of this model has been reported many times in different

contexts. Janssen (1974) calculated the stationary distribution:

_ I b Tin (i- l)(i- 2)a + b pn- p 0 fifki= 2 (i- l)(i- 2) + k (5.127)

This is not the only example dealing with the role of fluctuations in general, and in the vicinity of bifurcation points, specifically.

Based on the approximative calculations of Nitzan et a/. (l974a) it was suggested that the stationary distribution can be multimodal, and the location of the maxima can be associated with the stable stationary points of the deterministic model, and the minimum corresponds to the unstable stationary point. This result does not hold for small systems; and the question ofmultimodality will be discussed in Subsection 5.7.4. Matheson et a/. (1975) obtained a qualitatively similar result, they also estimated the relaxation time of the process from the quasistationary state. Nicolis & Turner (1977) calculated the variance at the critical point and in its vicinity. Under and beyond the critical point the variance is a linear function of the volume, but at the critical point it shows a stronger volume-dependence:

lim cr2 ( oo) = const. V312 + o( V). v-oo N-oo

N/V-consl.

This result shows that the formulae (5.1) and (5.2), which suggest that the fluctuations are insignificant for large systems, are not always valid.

Further remarks

Gillespie (1979, 1980) has emphasised the possibility of using stochastic simulation methods for identifying the quasistable stationary states and calculating the associated mean transition times, mean fluctuation periods, and effective fluctuation ranges.

It was argued (Horsthemke & Brenig, 1971; Blomberg, 1981; Haenggi et a/., 1984) that the nonlinear Fokker-Planck equations (derived in a slightly different way) also operate correctly in the critical point.

Related questions in connection with stochastic bistable systems are discussed in Subsection 5.6.6.

5.6.6 Stochastic theory of bistable reactions

I. The motivation for studying bistable reaction systems came from two different directions. First, a model of a Brownian particle moving in a


potential well was adopted by Kramers (1940) to reformulate the diffusion model of the chemical reactions at the microscopic level. Microscopic models of chemical reactions are beyond the scope of this book. From a more general point of view it is interesting that Fokker-Pianck equations in a double well potential are widely used to describe phase transition phenomena (see van Kampen, 1981, VIII. 7 and X1.6). Exact solutions for diffusion in bistable potentials in the case of particular potential functions have been given (Hongler, 1979a, 1982; Hongler & Zheng, 1982, 1983). Second, multistationarity has been demonstrated experimentally in continuous stirred tank reactors, as was mentioned earlier in Subsection 4.4.2.

2. The estimation of relaxation times from the Fokker-Planck equations was discussed by Matsuno eta/. (1978). A Fokker-Pianck equation can be written as

a a az atP(x, t) = - ax[A(x)P(x, t)] + (I/2V) axz [B(x)P(x, t)]. (5.128)

The stationary distribution is

P 51(x) = (K/B(x))exp[- VU(x)],

where K is a normalisation constant and X

U(x) = -2 f[A(x')/B(x')]dx',

0

In the physical literature, adopting the Landau-Ginzburg picture, U is often referred to as the 'free energy'. In the chemical context it is better to use the term 'free-energy-like'. The stationary states of the system occur at the extrema of U(x). Metastable states can be identified with local minima. The relaxation time of the process leading from metastable to stable state is estimated as

.\'11 X

'"' = 2 V f exp[VU(x)] dx f 1/B( y){exp[- VU( y)]} dy (5.129)

-"" where x. and xm are neighbouring unstable and metastable states. It can be seen that the relaxation time is exponentially dependent on the volume. By evaluating the integrals the relaxation time is estimated as

'tm ~ const.exp[V(U(x.)- U(xm))].

This formula shows not only the exponential dependence of the relaxation time on the system size, but also the effect of the height of the potential barrier ( U(x.) - U(xm)).

3. Estimation of relaxation times from the master equation using spectral analysis of the coefficients matrix was used by Dambrine & Moreau (1981a, b) for a chemical system having a finite state-space. The reaction


X+ Y~2X, X~Y (5.130)

obeys the master equation with the following transition probabilities:

~ = o, if IJ- II ;?; 2,

Pli+ 1 = ct)(N- J) k - •

Pj-l.j- C2)

p'ff = (c 1 j(N- j) + c2j).

Here, c1 and c2 are constants determined by the rate constants, N is the total number of molecules. Exact bounds for the first two nontrivial eigenvalues of the coefficient matrix were given as:

IA2I- 2 = 't2 4i IAtl-' = 'tt In thermodynamic limit 't 1 "' (D(exp(CN)) and (D(l) < 't2 < (D(ln N).

Remark

The method has also been extended to infinite state-space models by Borgis & Moreau (l984a) who studied the reaction A -+ X, A + X-+ 2X, A + 2X-+ 3X. The coefficient matrix of the master equation is a tridiagonal infinite matrix. The eigenvalues and eigenvectors of this matrix are the limits of eigenvalues and eigenvectors of truncated matrices obtained by introducing a reflexing state for X= N where N is the number of molecules at which the system is truncated.

4. A scaling theory of transient phenomena near an instability point has been formulated by Suzuki (1976a, b, 1980, 1981, 1984). The theory is based on a generalised scale transformation of time and equivalently on a nonlinear transformation of stochastic variables. The whole range of time is divided into three regions, namely the initial, scaling and final regimes (Fig. 5.9). He

Fig. 5.9 Illustration of macroscopic enhancement of fluctuation from the initially microscopic one. Fluctuations in the initial and final regime can be well described by Gaussian approximation. In the transient regime fluctuation enhances macroscopically, as can be calculated based on a generalised scale transformation of time. (a) Initial regime. (b) Scaling regime. (c) Final regime.


o•[J(t)J -,, /' /(t)

t

Fig. 5.10 Anomalous fluctuation near the instability point: J(t) denotes the most probable path or deterministic motion y(t) and D 2 [y(t)) is the variance of the order parameter.

calculated that the fluctuation becomes anomalously large in the scaling regime (Fig. 5.10).

5. Stochastic bifurcation problems in chemical systems have been analysed by Lemarchand ( 1980). He proposed a systematic expansion of the freeenergy-like quantity ('stochastic potential') in powers of v- 1:

(5.131)

to obtain the stationary distribution as well as the time-dependent evolution near the maximum of the probability distribution. (Some further readings are: Lemarchand & Fraikin, 1984; Fraikin, 1984; Walgraef eta/., 1982).

The expansion of <I> around an extremum is expressed in the representation provided by the eigenfunctions of the linear stability operator. General expressions of the successive derivatives are obtained independently of the nature of the bifurcation. The method has been extended to describe the effect of inhomogeneous fluctuations as well as in reaction diffusion systems (Fraikin & Lemarchand, 1985).

5. 7 Stationary distributions: uni- versus bimodality

5. 7.1 The scope and limits of the Poisson distribution in the stochastic models of chemical reactions: motivations

I. In the literature of stochastic reaction kinetics it was often assumed that the stationary distributions of chemical reactions were generally Poissonian (Prigogine, 1978). The statement is really true for systems containing only first-order elementary reactions, even when inflow and outflow are taken into account (i.e. for open compartmental systems; see Gans, 1960, p. 692). If the model of open compartmental systems is considered as an approximation of an arbitrary chemical reaction 'near' equilibrium, then in this approximation the statement is true.


2. Thermodynamic fluctuation theory characterises equilibrium fluctuation by the so-called Einstein relation connecting the probability density function g with the (appropriately defined) entropy function S:

g'1(x) = C exp(- S(x)fk8 ), (5.132)

where k8 is the Boltzmann constant, C is a normalisation constant. Expanding S around x*, which is considered a thermodynamic equilibrium point, i.e. a stationary point of S, and stopping at the second term in the expansion, the density function is approximated by

g'1(x) = [ det ;;~x*) JM12 exp[(- I /2k8)(x - x*)T S" (x*)(x - x*)]. (5.133)

From a mathematical point of view the approximation by a Gaussian process is appropriate, in the sense that for every stochastic process having first and second moment there exists a Gaussian process with the same first and second moment.

Fluctuations might also be expressed without utilising the entropy function:

(5. 134) g'1(x) = [21t mt x! JM/2 exp( -~(x- x*)T(dgx*)- 1(x- x*))

(x* is ~ector variable, dg x* is the diagonal matrix formed from x*). In small systems, i.e. for relatively large fluctuations (5. 134), approxima

tion can be by a Poisson distribution:

* N -N Il (x!)Nmexp(- I Nm) P(N) = (x ) e = m= 1 m= 1

N! M

0Nm! m= I

This relationship can be interpreted as an inverse procedure of the usual approximation by a normal distribution.

3. Some historical remarks. The physical assumption adopted by van Kampen (1976) is that the grand-canonical distribution of the particle number of an ideal mixture is Poissonian. Based on this - strongly restrictive - assumption, and utilising the conservation of the total number of atoms the stationary distribution can be obtained. This stationary distribution can be identified with the stationary solution of the master equation, and it is not Poissonian in general, even for large systems.

Hanusse (1976, pp. 86-8) claimed that in the case when not only monomolecular reactions are in the system, the stationary distribution can be different from the Poissonian. Zurek & Schieve (1980) have simulated a particular cheq~ical reaction exhibiting a non-Poisson distribution.

The Poisson distribution seems to have particular role, since it is sometimes considered as a reference distribution. For nonlinear systems the


cr2 variance of the stationary distribution is expressed as the function of the expectation m, and 11:

cr2 = m(l - f.l)

where ll (the parameter of the Poisson distribution) is qualified as a parameter characterising the reaction (e.g. Malek-Mansour & Nicolis, 1975).

A sufficient condition for having Poissonian stationary distribution in a certain class of birth and death processes was given by Whittle (1968). However, his assumptions are slightly different from those of chemical reactions, therefore the search for precisely defined assumptions or for certain classes of reactions is necessary.

The fundamental works of P. Medgyessy on the decomposition of superpositions of density functions and discrete distributions (Medgyessy 1977) has been applied to set up conditions for determining the modality (i.e. the number of maxima) of density functions and distributions.

5.7.2 Sufficient conditions of unimodality

The following definition is applied: the {Pj:j E N0 } distribution is unimodal, if in the series Pi - P0 , P 2 - Pi, P 3 - P 2 , ••• there is precisely one change of sign.

Let us consider the reaction having one internal component:

k(r) cx(r)X-+ ~(r)X (5.136)

For the generating function F of the stationary distribution we can write the following ordinary differential equation:

R L k(r) (z~(r) - z"(r))~"(r) F(z) = 0 (5.137) r= I

where ~denotes the differential operator. A theorem of Medgyessy (1977): Let us assume that the generating

function F of the {P.} (P0 > 0) discrete distribution satisfies the differential equation

M

L (Amz'" + Bmz'"+ 1 )~mF(z) = 0, (0 < lzl ~ 1), (5.138) m=O

where MEN, Am, Bm E IR are constants. Furthermore let

m~/m(n:l)m!#O, ifnEN (5.139)

and

(5.140)


and we assume that in the series

(5.141)

there is at most one change of sign. Then {Pn} is unimodal. Applying this theorem for the reaction (5.136) we get the existence of m

such that ~(r) = CI(r) = m + I. Additionally k(r) = Am = - Bm. Therefore

M (n + )) R (n + 1) m~O Am m m! = r~l k(r) ~(r) (~(r))!

and

m~o Bm(~)m! = ,t. k(r)(~~r))(~(r))!. Equations (5.139) and (5.140) hold precisely if r exists such that ~(r) = 0 (and then CI(r) = 1 ). From the chemical point of view it means that the reaction (5.136) contains an elementary reaction X-+ 0 (first-order decay or outflow).

Based on another theorem we may state that a birth and death-type chemical reaction with one internal component leads to unimodal stationary distribution. More precisely, let us assume that a chemical reaction can be identified with a birth and death-type process with the birth \j1 and death J.1 rate:

I

\j!(j) = \j!o, and JJ(j) = L J.li i=l

(jei\1 0 ; \j!0 EIR+; /eN)

Then the stationary distribution of the stochastic model of the reaction exists, and it is unimodal.

The proof of this proposition is based on a slight generalisation of a theorem of Medgyessy.

Let us assume that

are functions such that

y(j) + o(j) ~ I (jE 1\1 0 )

holds, and suppose that for the distribution {Pi}

pj ~ y(j)Pj+ I+ O(j)Pj-1

holds. Then Pi is unimodal. Going back to the statement on birth and death-type reactions, existence

and uniqueness of the stationary distribution follows from the KarlinMcGregor (1957) condition. For the stationary distribution of the reaction we can write

142 Mathematical models of' chemical reactions

(J.t{j) + \jl(j))Pj = J.l{j + l)Pj+ I + \jl(j- l)Pj- I (5.142)

Utilising the particular form of \jl, the monotonicity of J.l can be written as:

- J.l{j+ l) \j/(j-1) pj = pj +I J.l{j) + \jl(j) + pj- I J.l{j) + \jl(j)

>-: P Jl{j) + P o/o ...- j+ I J.l{j) + \jl(j) j-1 J.l{j) + \jl(j)

Applying Medgyessy's above mentioned theorem with

Y(1.) _ J.l{j) . ~( ·) _ o/o ( · ~~., ) - J.l{j) + \jl(j), u 1 - J.l{j) + \jl(j) 1 E '~o

the statement is proven. From the chemical point of view these kinds of result suggest that the

stationary distribution of the reaction

kM k 2 k 1

MX-+(M- l)X-+ ... -+X~O (5.143) k_,

(M > 0, k_ 1 > 0, kM > 0; km ~ 0, m = l, 2 ... M- l) is unimodal.

5. 7.3 Sufficient condition for a Poisson ian stationary distribution

Theorem. The necessary and sufficient condition for a simple birth and deathtype process to have a Poissonian stationary distribution (if it has a nondegenerate stationary distribution at all) is that it be a linear one (Erdi & T6th, 1979).

Sketch of the proof If a simple birth and death-type process has a unique stationary distribution then its form is:

rzi Pi = const. -M,-------- (jE 1\J~) (5.144)

TI <pm( l) · · · <pm(jm) nr =I

where rx E IRM is the solution of a certain system of linear equations. It can be seen by induction that this distribution has a form

Pi= exp(- I Am)~,i m= I J.

(jEN~; AE(IR+)M; Am= prmA)

if <pm is a homogeneous linear function. If the stochastic model of a chemical reaction can be identified with a

simple birth and death process, and the process has stationary distribution, then it is precisely Poissonian, if the reaction is an open compartmental system.


Remarks

(I) The reaction a b

0-+X kX-+(k- I)X (a, heiR+, keN)

(5.145)

can be described by a simple birth and death-type process, but it generally does not lead to a Poissonian distribution (only for the case k = 1).

(2) Particular results can be derived for certain classes of the Markov population process (Kingman, 1969). Necessary and sufficient criteria were given for polynomial simple Markov population processes to have Poissonian stationary distributions when the detailed balance condition holds. Special relations among the coefficients are necessary to get Poissonian distribution (Toth & Torok, 1980; Toth, 1981; Toth eta/., 1983).

(3) Connections among stochastic models of reactions, simple birth and death-type processes, compartmental systems and population Markov processes are illustrated in Fig. 5.11.

Compartmental Systems

Fig. 5.11 Interrelations between different types of processes (MPP, Markov population process; CCR, complex chemical reaction; SBD, simple birth and death process; S, V, simple Markovian jump processes and models of reactions in the scalar and vector case respectively).

5.7.4 Multistationarity and multimodality

It was mentioned (in Section 1.4) that multistationarity in deterministic models might be associated (however, only approximately) with multimodality of the stationary distribution. It is generally assumed that

(I) the number of equilibrium points in the deterministic model coincides with the number of maxima of the stationary distribution;


(2) equilibrium points can be associated with the location of maxima of the stationary distribution;

(3) stable equilibrium points coincide with maxima, unstable equilibrium points coincide with minima of the stationary distribution.

However, Cobb (1978) illustrated- with the aid of a nonreaction kinetic model - that there is no one-to-one correspondence between the location of equilibrium points and of extrema of stationary distributions. He applied a stochastic version of catastrophe theory, and stochastic catastrophe theory was also applied by Ebeling (1978).

The Schlogl reaction has been the workhorse for studying the occurrence of exotic behaviour in deterministic and stochastic models (Erdi et a/., 1981; Ebeling & Schimansky-Geier 1979). The results of both papers suggest that multistationarity and multimodality correspond to each other asymptotically with increasing volume. (The presentation in the latter paper is much more detailed than our arguments.) ceo models refer to infinite and continuous systems, while CDS models describe finite and discrete systems. Both the volume-dependence and the continuous approximation of the stationary distribution of the master equation illustrate that for large systems multistationarity and multimodality coincide. However, in small systems all of the four combinations might occur, i.e. uni- and multistationarity can be associated as well to uni- and multimodality for certain parameter values. Illustrations of the 'volume decrease' induced transition is given (Fig. 5.12).

Remark

The possibility of the occurrence of unistationarity and multimodality is demonstrated by the following rather artificial but formally correct model.

The reaction

(5.146)

is unistationary for fixed ex, ~ e IR +, and for certain parameter values it shows multimodality. The example is not completely correct, since the stationary distribution is a mixture of a number of unimodal distributions, where the weights of the mixture are the members of the initial distribution of the process. The components of the mixture, i.e. the

{Po. Pq, P2q ... }, {P~> Pq+l• P2q+l ... }, { ... , Pq-1• P2q-l ... }

are unimodal. It is obvious that for example taking the first and last distribution with sufficiently large weights the mixture will be unimodal.

5.7.5 Transient bimodality

The investigation of stochastic bistable reactions showed that processes leading to the stationary state might be decomposed in certain situations by


s s P(N) P(N)

EM

(a) v = 10

10 n 10 n s s P(N) P(N)

J\ EM

(b) v =50

50 n 50 n s s

P(N) P(N) EM

(c) v = 100

100 n 100 n s s

P(N) P(N) EM

(d) v = 200

200 n 200 n

Fig. 5.12 The volume-dependence of the exact stationary probability distribution function of system I and comparison with its Euler-McLaurin approximation (after Ebeling and Schimansky-Geier).

two different time scales. Many systems, e.g. thermal and chemical explosions, can be characterised by a fast (explosive) transient leading to a stationary state. It was demonstrated by a simple model calculation (Frankowicz & Nicolis, 1983; Frankowicz eta/., 1984) that during the explosion the probability distribution develops a long tail and may even split into two parts, thus exhibiting transient bimodality. The visualisation of the temporal change of the shape of the probability distribution can be seen in Fig. 5.13.

In particular, a stochastic modei of thermal explosion in a chemical system Was studied by Baras & Frankowicz (1984). Their main results are: (i) during the ignition regime the probability distribution function displays transient bimodality; (ii) the distribution of ignition times displays a long time tail; (iii) the mean ignition time calculated from the stochastic model is significantly shorter than that obtained from the deterministic model.


T = 0.25 T = 4.0

T = 6.25 T = 7.25

T = 99

Fig. 5.13 Time evolution of the probability distribution: V = 1000, X 0 = 1500, 0 = -0.38, ()' = -1.5.

5.8 Stochasticity due to external fluctuations

5.8.1 Motivations

While internal fluctuations are self-generated in the system, and they can occur in 'closed' and 'open' systems as well, external fluctuations (or noise) are determined by the environment of the system. A characteristic property of internal fluctuations is that they scale with the system size, i.e. they tend to vanish in the thermodynamic limit (except in the vicinity of critical points). Since external noise reflects the random character of the environment, the measure of fluctuations is completely independent of the system size. A natural way to introduce external noise is to assume that the control parameters (i.e. the rate constants for isothermal 'pure' chemical systems) are not strictly constants, but that it is better to consider them as stationary stochastic processes.

Chemical systems, on which the effects of external noise have been observed, have been studied both experimentally and theoretically in the last few years. De Kepper & Horsthemke (1978) studied the effect of fluctuating illumination on photochemical reactions (see also Micheau eta/., 1984). The effect of nonperfect mixing in continuous stirred tank reactor (CSTR) was


analysed (e.g. Boissonade et a/., 1984; Horsthemke & Hanson, 1984). Theoretical studies have been done for an enzyme- kinetic system (Hahn et a/., 1974), and for certain chemical models (Arnold et a/., 1978). The effect of external noise was investigated in a number of nonlinear physical systems (electrical circuits: Stratonovich (1967); a hydrodynamical system: Moss & Weiland (1982); an optical system: Bulsan et a/. ( 1978); plasmas: Rise (1982); nuclear reactors: Williams, 1974). The concept of growth in a random environment has been adopted successfully in the biological context (May, 1972; Nobile & Ricciardi, 1984a, b). External noise ('developmental noise' as it was termed by the celebrated biologist Waddington (1962)) might have a crucial role in the formation of ordered biological structures. External 'noiseinduced ordering' was introduced to model the ontogenetic development and plastic behaviour of certain neural structures (Erdi & Barna, 1984, 1985).

According to the traditional point of view fluctuations are averaged out. It was clearly demonstrated that noise can support the transition of a system from a stable state to another stable regime. Since stochastic models might exhibit qualitatively different behaviour than their deterministic counterparts, external noise can support transitions to states which are not available (or even do not exist) in a deterministic framework. (The theory of noiseinduced transition, as well as its applications are discussed in the book of Horsthemke & Lefever (1984b).

In Subsection 5.8.2 we give a short introduction to the mathematical formalism of external noise. In Subsection 5.8.3 specific models are given for illustrating the existence of noise-induced transitions in chemical systems. Further remarks in connection with the development of the theory will be given in Subsection 5.8.4.

5.8.2 Stochastic differential equations: some concepts and comments

Let us consider a stochastic dynamic system:

X= (f, X,~. A, x0 ), (5.147)

where f is the forcing function, X is the state-space, ~ is the vector-valued stochastic noise process, A is the external parameter, x 0 is the initial condition (it can be random). The stochastic dynamic system reduces to a deterministic dynamic system when random effects are neglected. Assuming furthermore that the system can be described by a system of autonomous ordinary differential equations, the deterministic model is:

dx(t)/dt = f 0 (x(t), A). (5.148)

In many applications of the theory of noise-induced transition it is assumed thatf0 is linear in A, i.e.

dx(t)/dt = h(x) + G(x, A) = h(x) + Ag(x) (5.149)

is valid.


Stochastic dynamic systems can be classified according to the very nature off Arnold & Kliemann (1981) summarised the qualitative behaviour of X both for linear and nonlinear systems (for a condensed survey see Arnold ( 1981 ). The term 'linear' is not specific here, since f can be linear either in state or in noise, even in both. In applications it is assumed very often that the forcing function has a 'systematic' or 'deterministic' part, and a term due to the 'rapidly varying, highly irregular' random effects:

'rate of change of state' = 'deterministic rate' + 'random rate'.

This is a verbal version of the old Langevin equation (Langevin, 1908). According to a standard further assumption the random term is a linear function of a 'white noise'. White noise is considered as a stationary Gaussian process with £[!;,] = 0 and £[1;,1;,·1 = 011·1 t - t' I, where o is the Dirac delta function.

It is weB-known today that the heuristic derivation of the Langevin equation is mathematicaBy not weB-founded. A stochastic process X, obeys an Ito stochastic differential equation (SDE) which has the form (in the autonomous case).

dX, = a(X,)dt + b(X,)dW,; t0 ~ t ~ T~ oo,

or in the form of an integral equation

X,= Xo + f a(Xs)ds + f b(Xs)dWs

(5.150)

(5.151)

X, can be considered as an !Rn valued stochastic process and W, is an rdimensional Wiener process. The introduction of the Wiener process was motivated by its connection with white noise. Accordingly,

dW, = l;,dt or equivalently W, = f l;sds. (5.152)

HeuristicaBy it means that the second term in the right-hand side of (5.149) is subject to the sum of white noise terms.

In the literature of application of stochastic processes there was extensive discussion of the interpretation of stochastic integrals. The key example is the particular case b(X,) = W,, and so the integral J:. Ws d Ws has to be calculated. Using the Ito calculus (1951) we set

( f WsdW, = l/2[W,2 - Wtij- (t- t 0 )]

'• Ito

and by adopting the Stratonovich integral (Stratonovich, 1963)

f WsdWs = l/2[W,2- Wtij].

Str

(5.153)

(5.154)


The unpleasant fact is that for arbitrary function b there is no clear connection between the two integrals. What is more or less true is that the Ito integral cannot be interpreted as an ordinary Riemann-Stieltjes integral. Stratonovich was motivated by extending the applicability of classical integration rules. The main advantage of the Ito integral is the fact that, as a function of the upper limit, it is a martingale, and this notion is useful tool in the theory of stochastic processes (Doob, 1953).

What is very important from both the theoretical and practical points of view is the connection between the SDE and the Fokker-Pianck equation. If X0 is independent of W., the solution of (5.150) is a (strong) Markov process, in particular a diffusion process specified by the infinitesimal generator:

a a2

A= ~a~axi + lf2~~(bb')iiaxiax/

The practical importance of the possibility of transforming an SDE to a Fokker-Pianck equation comes from the fact that the shape of the probability density function which can be unimodal or multimodal is appropriate to characterise the system qualitatively.

There are two possibilities of associating an Ito-type SDE (5.149) with the deterministic equation (5.148), depending on the nature of the noise. Heuristically, the noise might be introduced

(I) assuming that the parameter A is not a constant, but might be replaced by a A, = A + cr~, stationary stochastic process, and g(x) is constant;

(2) assuming that the fluctuations depend explicitly on the state of the system, which means that g(x) is not constant.

5.8.3 Noise-induced transition: an example for white noise idealisation

The role of external white noise in a mass action kinetic model was clearly demonstrated by Arnold et a/. ( 1978). Let us consider the reaction network

k, '· /2 X¢ Y A+X+ Y-+2Y+A B+X+ Y-+2X+B. (5.155)

k,

A and B are external, X and Y are internal components. Since the reaction is mass-conserving, the system can move under the constraint C1 (t) + C2 (t) = N = const. for all t, where C1 (t) and C2 (t) are the quantities of X and Yat timet. Introducing X(t) = (I/N)C1(t), and

= k /(k + k ) d (.1 = (12B- IIA)/N ex - 2 I 2 an P - k I + k 2

and making the special choice k 1 = k 2 ; the deterministic model is

dx(t)/dt = f(x(t)) = h(x) + Ag(x) = 1/2- x + ~x(l - x). (5.156)


For the stationary case

- x; + <a - l)x, + 1/2 = 0 (5.157)

holds. The x, versus a function is illustrated in Fig. 5.14. It can be shown that x, ~ 0 is stable, i.e. the deterministic model of the

system can be characterised by one, stable stationary point.

~----------------

--------------#---------------1-

0 ________________ l _______________ _ Fig. 5.14 No instability is displayed in the deterministic model.

Assuming that A and Bare rapidly fluctuating quantities, the idealisation of white noise can be adopted, and a might be considered as a stationary stochastic process, in particular Gaussian white noise with expectation 0 and variance cr 2 • For the associated stochastic process X, the following Ito SDE can be derived:

dX, = {1/2- X,+ ~X,(I - X,)}dt + crX,(I - X,)dW,. (5.158)

The fluctuation is state-dependent, i.e. the noise is multiplicative. Equation (5.158) can be associated with the Fokker-Pianck equation:

o,g(x, t) = -o.{l/2- X+ ~X(I- x)}g(x, t) + (l/2)cr 2 oxxX2 (1- x)2g(x, t). (5.159)

Under the boundary condition, referring to g as a probability density function, the stationary probability density function is

g,(x) = X[x- 1(1- x)- 1]exp{2/cr 2 (- 2x(ll- x)- ~ln[(l- x)/x])} (5.160)

The extrema xnr of g,(x) can be calculated from the relation

(5.161)

We get


l/2- Xm + J3xm(l - Xm)- <:r 2 Xm(l - Xm) 2 (l - 2xm) = 0. (5.162)

This is a third-order equation for xm, and it might occur that it has three positive roots. For the special case J3 = 0 (5.162) yields

Xm = l/2 and x± =(I + Jt - 2/cr 2)/2. (5.163)

For cr 2 = 2 xm is a triple root. The situations for cr 2 < 2 and cr2 > 2 are given by Fig. 5.15.

Fig. 15.5 illustrates that transition may occur which has no deterministic counterpart.

x.

0

Fig. 5.15 Extrema of the probability density function as a function of 13 and for three values of the variance.

5.8.4 Noise-induced transition: the effect of coloured noise

The white noise idealisation is motivated by the assumption that the environmental state varies on a much faster time than the macroscopic state of the system. In consequence of this assumption the notion of white noise, i.e. a 'completely random' process characterised by zero correlation time has been adopted.

Since white noise is qualified by a single parameter, namely by its intensity, the characterisation of real noise requires at least another parameter, its correlation time. The theoretical treatment of nonlinear systems subject to real, i.e. 'coloured', external noise has two newer difficulties. First, only the white noise idealisation leads to a Markov process. Second, from the practical point of view the Gaussian and Poissonian distributions are relevant only to describe white noise, and real noise can have a richer description. The disadvantage due to the loss of Markov property is partially compensated by the fact that non-Markovian processes have smoother realisations than Markov processes. Therefore no particular (Ito or Strato-


novich) interpretation of SDE has to be used. The integral form of a SDE is I I

X,= X 0 +I a(X,)ds + cr I b(XJI;;sds. (5.164)

Here both integrals can be interpreted as ordinary Riemann integrals (/;;, is the nonspecified fluctuating parameter).

An often-used coloured noise is the Ornstein-Uhlenbeck process: its expectation £[/;;,] = 0 vanishes, and its correlation function is exponential:

E[/;;,/;;5 ] = ( cr 2 /2y) exp(- 'Y It - sl) (5.165)

To investigate the robustness of white-noise-induced transitions a nonzero but short correlation time has been used (see, for example, Chapter 8 of Horsthemke & Lefever (l984b)).

dX~ = [h(X~) + Ag(X~) dt + (/;;,/E)g(X,) dt = /(X~) dt + (/;;,/E)g(X~)dt.

d/;;, = -(/;;,/E2 )dt + (cr/E)dW,.

(5.166)

(5.167)

The correlation time is E2 , where E is the single parameter to measure the 'deviation' from the white noise situation. The robustness of white-noiseinduced phenomena, at least for small correlation time, are verified by this perturbation expansion method. Slightly different results were obtained by Sancho eta/. ( 1982). They studied the qualitative properties of the stationary distribution as a function of the intensity and correlation time of the noise. Two results of their experiments on an electric circuit with a digital noise generator were not in accordance with the white noise limit theory: (i) for higher noise intensity the stationary distribution might be bimodal even for rather small correlation time; (ii) the location of the maximum of the stationary distribution depends on the noise characteristics.

To extend the validity range of noise-induced phenomena for a wider range of correlation times the dichotomous Markov noise has been used. The dichotomus Markov noise, also known as the random telegraph signal has a quite simple structure, therefore the stationary probability density can be calculated for an arbitrary value of the correlation time, and for any value of the noise intensity. The state of the Markovian dichotomous noise/, consists of two levels {A+, A_} only. The noise is characterised by the transition probability:

Pii(t) = 9(1, = il / 0 = j); i, je {A+, A_}.

A system subject to a dichotomous noise is described by the SDE

X, = h(X,) + l,g(X,) = f(X,, /,)

(5.168)

(5.169)

According to the evidence of calculations the transition behaviour is richer than under influence of white noise.


5.8.5 On the effects of external noise on oscillations

The effect of internal fluctuations have been studied in particular cases, as it was mentioned in Section 5.6.

The influence of external fluctuations on two-dimensional oscillatory systems has been studied by Ebeling & Engel-Herbert ( 1980). The stochastic counterpart of the appearence of a limit cycle is strongly connected with the formation of a 'probability crater' on the stationary probability surface. They studied particular cases when the system has the form

x = aH(x,y) + f(H- u)aH(x, y) ay ' ax

. = _ aH(x, y) + f(H- u) aH(x, y) y ax ' ay

(5.170)

H plays the role of a Hamiltonian (in a certain sense) andfis some nonlinear function of H and the external parameter u.

Assuming the existence of closed trajectories of (5.170): H(x, y) = H, H satisfies the equation

f(H; u)IH = 0.

The effect of Gaussian white noise was studied by associating

ul-+ ul(t) =ill+ O"lSt•

which assumes that fluctuations occur on a much smaller time scale than the macroscopic evolution. Equation (5.164) has to be replaced by the SDE

x = ~~ + f(H; il 1 )~~ + cr 1 ~~g(H)~(t) Y =- aaH + f(H; ill)aaH + crl H g(H)~(t)

X y y

Under certain assumptions the probability density P(x, y, t; ud reduces to P(H, t; u1 ). Deterministic and stochastic bifurcation sets coincide for particular f and g only. Depending on the special forms of the functions f and g, the bifurcation values presented by the deterministic approach shifts, or even new transitions are induced which are absent in the deterministic picture.

Schimansky-Geier et a/. ( 1985) investigated transition phenomena in a two-dimensional system due to external noise in one equation. The second equation of the deterministic system

dx1 (t)/dt = / 1 (x1 (t), x 2 (t)) dx2(t)/dt = / 2 (X 1 (t), X 2(t)) (5.172)

is supplemented with a term JK ~(t), adopting again the Gaussian white noise idealisation (where K is the parameter of noise strength). The stationary


Fokker-Planck equation in the two-dimensional case is obtained by using an antisymmetric tensor to convert the problem into the potential case.

Specifically, for the case of van der Pol oscillator:

x=y, y = -x + y(cx- y 2 ) + fi~(t)

they found that a crater-like distribution corresponding to a limit cycle appears only at a value of the bifurcation parameter that was definitely positive, instead of zero as in the deterministic model). More detailed culculations demonstrated the possible occurrence of asymmetrical proba'!>ility crater (Ebeling eta/., 1986).

The effect of additive as well as multiplicative noise by the centre-manifold approach was studied by Knobloch & Wiesenfeld (1983). The main point of their procedure is that in the vicinity of bifurcation points a reduced Fokker-Planck equation is sufficient to describe the dynamics (after a short relaxation time).

The centre-manifold approach is a powerful technique for reducing dimensions at a bifurcation point (Kelley, 1967).

The (deterministic) system

can be transformed to

x(t) = F(x(t), J.l); x(t) E !Rn

il = A(Jl0 )u + f(u, v, J.lo) v = B(J.10 )v + g(u, v, J.lo)

where dim u + dim v = n, Jlo is the bifurcation point, and

Re AA!lo) = 0; Re A8 (J.10 ) # 0.

The centre manifold is defined as

dfdt(v(t)- v0 (J.1)) = 0,

and the essential dynamics is described by

il = A (Jl0 )u + f(u, v0 (u), J.lo).

The (x, y) variables of the van der Pol oscillator

x=y j• = - x + y + Ax3 + Bx2 y

(5.173)

(5.174)

(5.175)

(5.176)

has been transformed to (u, v) in such a manner that the linearisation problem is in diagonal form at the bifurcation point. The reduced dynamical equation is

u = - (cx/~)u - (A/~)u3 + O(u5 )

The bifurcation diagram, (Fig. 5.16) is

(5.177)


u(t-+ oo)

Fig. 5.16 The pitchfork bifurcation. The stationary points of (5.177) are plotted as a function of ex. , stable; _____ , unstable.

Adding a term due to Gaussian white noise to the second equation of (5.176) and adopting the centre-manifold approach it can be calculated that the stationary solution of the reduced Fokker-Pianck equation is

P'1(u) = Nexp{ -g(cxu2 + A/2u4)} (5.178)

where N is a normalisation constant, and K is the intensity of the white noise. The noise-induced transition is illustrated in Fig. 5.17.

If the deterministic system is subject to multiplicative noise the stationary solution of the reduced Fokker-Pianck equation is

P'1(u) = Nlul(4 -:x~/Klexp{ -~~u2 }. (5.179)

and the noise-induced transition occur, as they are illustrated on the Fig. 5.18.

The response of chemical oscillators to external noise has been studied neither experimentally nor theoretically. However, noisy precursors of

p~t(u) p~t(u)

u u ex < 0 ex > 0

Fig. 5.17 The stationary probability density (5.178) in the case of additive noise. (a) ex < 0 (b) ex > 0.


p•'(n) I pst(n) I pst(n)

_Lu_Aufhu Fig. 5.18 The stationary probability density (5.179) in the case of multiplicative noise. The transition points between regimes depend explicitly on the noise intensity K.

different instabilities of periodic orbits are characteristic (Wiesenfeld, 1985), and the examination of the effect of external noise could be an efficient procedure to discriminate among possible models of chemical oscillations.

What is more interesting, at least from the theoretical point of view is that oscillatory behaviour might emerge as a result of the interaction between the system and the external noise applied. This phenomenon was earlier described in a radio-engineering context (Kuznetsov eta/., 1965). Studying the role of multiplicative coloured noise for the catalytic oxidation of CO on a platinum surface, de Ia Rubia eta/. (1982) demonstrated that a limit cycle is induced by external noise. Similarly, Treutlein & Schulten (1985) found noise induced limit cycles in the Bonhoffer-van der Pol model of neural pulses (see further Lefever & Turner ( 1984) ).

5.8.6 Internal and external fluctuations: a unified approach

It is a quite natural endeavour to search for a unified mathematical framework of describing internal and external fluctuations. Internal fluctuations used to be described by the Markovian master equation. Sancho & San Miguel (1984) offered two equivalent techniques for a unified theory, at least for single-variable systems, when internal fluctuations were modelled specifically by a one-step Markovian master equation, and external noise was considered by dichotomous noise.

The methods start from an evolution equation describing the internal fluctuation. In the next step a fixed value of an external parameter, namely the infinitesimal transition probability, is substituted by a stochastic process. This procedure can be done in the master equation or in the equation for the generating function. The first case leads to an integrodifferential equation, while the second model leads to a stochastic partial differential equation.

Another technique was adopted by Horsthemke & Lefever (1984a), to give a joint description of the two classes of fluctuations in order to predict the behaviour of systems of finite size. They started with a stochastic differential equation to describe internal fluctuations:


dXI = h(XI) + Ag(XI) + V - 112C1, (5.180)

Vis the volume of the system. To include finite correlation time the noise was described by the following Ornstein-Uhlenbeck process:

dC = -y(dt + crdW1•

The influence of the external noise is taken into account as another Ornstein-Uhlenbeck process given by

dC1 = -C~ + adW1•

It was shown that the impact of internal fluctuations increases when the speed of the external noise increases.

5.8. 7 Estimation of reaction rate constants using stochastic differential equations

Although there are several techniques for estimating the reaction rate constants based upon the deterministic model, these methods are usually rather complicated, and the results cannot be statistically characterised. That is why from time to time estimates based upon one or another stochastic model are suggested. Such a suggestion has earlier been described under the name 'fluctuation-dissipation theorem', and similar methods have been presented by Mulloolly (1971, 1972, 1973), Hilden (1974), and Matis and Hartley (1971).

Now we outline another method based upon a stochastic differential equation model of a certain internal noise of unspecified origin in a reaction not coming from the reaction itself. A possible way of doing this is to consider the stochastic differential equation

dX(t) = A(X(t))k dt + B(X(t)) dw(t), (5.181)

where X is the vector of concentrations, k is the vector of reaction rate constants,

A(x) = (p - OE) diag(x'")

and w is a Wiener process and B is a positive definite matrix valued function expressing local variance. Equation (5.181) is to be understood as being equivalent to the integral equation

I I

X(t) - X(O) = f A(X(s))k ds + f B(X(s)) dw(s), (5.182)

0 0

The integral with respect to the Wiener process is taken using the Ito approach that has an important advantage over the Stratonovich approach, namely the integral in this case will be a martingale. Definitions and


statements on these notions can be found in, for example, Arnold ( 1973) written for nonmathematicians, or in many books by Gikhman and Skorohod, e.g. in Gikhman & Skorohod (1974).

What is the reason for writing a model in the form (5.181 )? It turned out (Hangos & T6th, 1988) that this description makes it easy to estimate the reaction rate constants, because the linear dependence of the right-hand side on the reaction rate constants can be utilised. The estimate can be explicitly given as

T T

k = (f AT(X(t))A(X(t))dtr1 I A(X(t))dX(t) (5.183)

0 0

if the concentration versus time functions are exactly measured during the time interval [0, T] and if B(x) = idRM = I M can be assumed. This estimate is a maximum likelihood estimate, i.e. it gives that value k under which the actual measurements are the most probable. Maximum likelihood estimates in general (and in this case) have advantageous statistical properties (Rao, 1973; Zacks, 1971).

The estimated value k is normally distributed with mean k and with variance r 1 (where J is the Jacobian of the likelihood function with respect to k) in the limit T--+ oo.

It is a remarkable fact that formula (5.183) gives an estimate that is -from the numerical point of view - somewhere between the estimate obtained from the direct differential method and from the direct integral method. A comparison of different methods has been given by Hosten ( 1979).

However, a series of problems arises in connection with this estimation procedure. How can the circumstances be characterised in physicochemical terms among which the model (5.181) is realistic? Is it possible to obtain continuous records of all the concentrations with absolute accuracy? How is the estimate (5.183) to be modified if one has time discrete measurements loaded with error? Which of the rate constants can be determined if only several concentrations or several linear combinations are measured?

To have a stochastic differential equation model for both the internal and external fluctuations in complex chemical reactions would have advantages other than providing the framework for estimation. As it turned out from the analysis of Chapter I it would be necessary to have a common model of macroscopic phenomena that does include stochasticity, spatial processes (such as diffusion) and sources and sinks (such as reactions). These topics will be further analysed in Section 6.3.

5.8.8 Exercises

I. Check that the maximum likelihood estimate of the rate of a simple inflow described formally by the elementary reaction


0-+X

is (X(T) - X(O))/T. Hint. Start from writing down the likelihood function again. Cf. Arat6

(1982). 2. Calculate the maximum likelihood estimate of the reaction rate constants

of the reaction

X-+ Y-+ Z -+ U.

5.8.9 Problem

I. How is the maximum likelihood estimate modified if in the stochastic differential equation model if one has B(x) = B instead of B(x) = IM?

5.8.10 Open problems

I. Give sufficient and/or necessary conditions for the nonsingularity of the matrix

T

f AT(X(t))A(X(t))dt

0

in terms of the mechanism of the reaction.

5.9 Connections between the models

5.9.1 Similarities and differences: some remarks

Relations between the usual deterministic and stochastic model have been studied since the start of the subject. Early investigators gauged the quality of a stochastic model by the proximity of its behaviour to that of the corresponding deterministic one. If one considers that the CDS model takes into consideration the discrete character of the state-space and it does not neglect fluctuations then the appropriate question nowadays seems to be: in what sense and to what extent can the deterministic model be considered a good approximation of the stochastic one?

5.9.2 Blowing up

There exists an interesting (or, perhaps annoying) phenomenon in the theory of stochastic processes called blowing up. This means that a process defined by its infinitesimal quantities grows in such a quick way that within a finite interval of time the random variable characterising the process in a fixed time point takes on infinite values with greater than zero probabilities. This phenomenon can be found in the case of the usual (polynomial) CDS model of the reaction


2X-+ 3X.

It is not astonishing at all if one considers the deterministic model of the same reaction and verifies that all the solutions of the induced kinetic differential equation are such that they cannot be extended to the whole positive half line, or to use the stochastic terminology, the deterministic model blows up too (cf. T6th, 1986).

A very natural (but so far only historical) fact is that sufficient conditions not to blow up are known for the same mechanisms in the deterministic and in the stochastic case. (It may be useful to note that this kind of blowing up has nothing to do with shock waves, but it sometimes does depend on concentration ratios.)

5.9.3 Kurtz's results: consistency in the thermodynamic limit

The deepest and most far-reaching results on the relation of the usual deterministic and stochastic models are due to Kurtz. These results (and their generalisations by L. Arnold) can only be outlined in an informal way here.

As it has turned out that consistency in the mean does not hold in general, several people have presented a proof of the fact that the stochastic model of a certain simple special reaction tends to the corresponding deterministic model in the thermodynamic limit. This expression means that the number of particles and the volume of the vessel tend to infinity at the same time and in such a way that the concentration of the individual components (i.e. the ratio of the number and volume) tends to a constant and the two models will be close to· each other. In addition to this the fluctuation around the deterministic value is normally distributed as has been shown in a special case by Delbriick (later head of the famous phage group) almost fifty years ago (Delbriick, 1940). To put it into present-day mathematical terms: the law of the large numbers, the central limit theorem, and the invariance principle all hold. These statements have been proved for a large class of reactions: for those with conservative, reversible mechanisms. Kurtz used the combinatorial model, and the same model was used by L. Arnold (Arnold, 1980) when he generalised the results for the cell model of reactions with diffusion.

Kurtz's results are summarised in a very concise, but still appropriate for the chemist, form in Kurtz (1972). His book (Kurtz, 1981) contains more mathematical details, more recent results, and all the relevant references which form the necessary background.

5.9.4 Exercise

I. . Write down the differential equation the solution of which is close to the number of occurrences in the reversible Lotka-Volterra reaction.

Hint. Apply the differential equation just before (3.2) of Kurtz (1972) to this special case.


5.9.5 Problems

I. Prove the analogous statements to those in Problems 2-4 of Subsection 4.1.4.

Hint. Determine the points of the state-space that can be reached from a certain starting point with probability 1.

2. Check the importance of the reversibility assumption in Kurtz's proofs.

6 Chemical reaction accompanied by diffusion

6.1 What kinds of models are relevant?

The main body of this book deals with spatially homogeneous systems. It is well-known that macroscopic theories of chemical reactions neglect the spatial aspects of chemical reactions, which are generally considered as rearrangements of'points having internal structure'. Pure reaction kinetics is interested only in temporal processes. Though the details of a general theory of chemically reacting mixtures are beyond our scope, we cannot avoid making some comments on model of systems when the occurrence of diffusion is also taken into account.

Spatia-temporal change of the quantities of components of chemically reactive mixtures are described by reaction-diffusion equations. These equations may be induced from reaction kinetics, when the diffusion of chemical substances is also involved, or deduced from a general theory of mixtures, when only mass-conservation is taken into account. (For 'theories of reactive mixtures' in the deterministic, continuum context offered by the school of 'rational thermodynamics see, for example, Bowen ( 1969) or Samohyl (1982)).

Both deterministic and stochastic models exist to describe reactiondiffusion systems macroscopically. Deterministic, continuum models are used frequently. (A mathematically thorough book, written for mathematicians interested in chemical and biological applications, is Fife (1979).) The main assumptions of applicability of this model are:

(I) To each component i a point function P;(x, t) can be associated such that the number of particles of that component in the interval a < x < b at

Chemical reaction and d(tfusion 163

h

time t is given by f p;(x, t)dx. The density functions P; are continu-

ously twice differentiable in x, and once in t. (2) The temporal change of the functions P; is a deterministic process.

The assumptions just mentioned are most plausible in the thermodynamic limit, for example, just in the situation when CCD models of chemical kinetics are adequate. The basic equation to describe the temporal change of the vector p of spatial density functions p;:

iJpfiJt = oap + f(p)

where Dis the diffusion matrix, and f is the component formation vector. This equation, and its modifications, have very many applications in the field of chemistry, of chemical engineering science, and of theoretical developmental biology.

The stochastic approach to reaction-diffusion systems is not mathematically well-established. Though spatio-temporal stochastic phenomena ought to be associated with random fields, and not with stochastic processes, the usual investigation of such kinds of physicochemical problems starts from the master equation, and then it is extended by some heuristic procedure. From the physical point of view the role of spatial fluctuations is obviously important. It is well known that the density fluctuations are spatially correlated, and according to the modern theory of critical phenomena (e.g. Fisher, 1974; Wilson & Kogut, 1974) small fluctuations are amplified owing to spatial interactions causing drastic macroscopic effects.

Some possibilities for defining stochastic models of reaction-diffusion systems will be shortly studied. The connection between deterministic and certain stochastic models that can be established by the extension of Kurtz theorems will be mentioned (Arnold, 1980; Arnold & Theodosopulu, 1980).

Spatial structures in chemical systems, such as spatial periodicity, waves and fronts, began to be studied after the discovery of periodicity in space in the Belousov-Zhabotinskii reaction (Zaikin & Zhabotinskii, 1970). The onset of spatial patterns are generally explained in a deterministic framework, though Walgraef et a/. (1983) has offered a treatment based on a stochastic reaction-diffusion model.

It is widely accepted that mechanisms responsible for patterns and order in a reaction-diffusion system may also play a fundamental role in understanding certain aspects of biological order and pattern formation.

6.2 Continuous time, continuous state-space deterministic models

A quite general equation of reaction-diffusion problems is:


iJp;/iJt = L;p; + F;(x, t, p); i = I, ... , n, (6.1)

where the L; are uniformly elliptic differential operators on the vector variables x:

L . = -"' iJH;k. '- f iJxk,

This equation is capable of describing anisotropic state-dependent diffusion including drift and internal interactions due to chemical reactions (see Fife 1979).

The basic mathematical problems are to find the solution of the equation satisfying subsidiary conditions, or to determine its qualitative properties without solving the equation.

To formulate the usual basic problems the introduction of some notation is necessary. Q is the domain in m-dimensional space, its boundary, n. is supposed to have a unit normal which is a smooth function of the position on Q. Furthermore Qr = Q x (0, T) for T > 0.

Some basic problems in connection with (6.1 ):

- the initial boundary value problem: to find a continuous function p: Qr--+ ~·. bounded in x satisfying the initial condition p(x, 0) = p(x), and the boundary conditions on Q x (0, T). Various types of boundary conditions can be given: (a) Individuals are imported or exported at a given rate: ~ vk(x)H;k

= a;(x, t); (the Neumann condition). (b) the efflux is a linear function of p: ~ vk(x)H;k = a;(x, t) + b;(x, t)p:

(the Robin condition). Here v(x) is the unit outward normal at x, and h;> 0.

(c) P;(x, t) = c;(x, t) (the Dirichlet condition). - initial value problems: similar to the previous problems, with Q = ~m.

and without boundary conditions. - boundary value problems: to find a bounded function

p: fiX (- 00, 00)--+ ~·, satisfying one of the three boundary COnditions. - stationary problems: to find a time-independent solution of the boundary

value problem. - periodic problem: to find a solution of a boundary value problem which is

periodic in time under certain conditions. The asymptotic state is defined as a collection of solutions, all approaching

the same solution as t --+ oo. In the case of the occurrence of symmetries in reaction-diffusion systems, a family of solutions can be generated from any given solution by using an appropriate group of transformations. The application of the Lie theory of transformation groups of reaction-diffusion equations (e.g. Steeb & Strampp, 1981) can be a very efficient method to study the symmetry properties of the solutions.

Chemical reaction and diffusion 165

Let us consider the particular case L; = D;llp; + F;(p), where D; are nonnegative constants, l1 is the Laplacian operator, and F is independent of (x, t).

(6.2)

Let the group of transformations G be generated by rigid motions and reflections in x and translations in t. Since the solutions of (6.2) are invariant under these transformations, the solutions p of the original equation are equivalent with the solutions Tp, for TeG. More precisely, two solutions p1 ,

p2 are asymptotically equivalent for some Te G, if

lim supjp1 (x, t)- Tp 2 (x, t)l = 0. r-:n x

From a practical point of view it is the stable asymptotic states that are important. One of the main questions in the theory of reaction-diffusion systems is to determine which are the stable asymptotic states for a given reaction-diffusion system.

A solution p(x, t) is called permanent if it is defined, and is a solution, for all real t, negative as well as positive.

Familiar examples of solutions of the permanent type (the classes are not disjoint) are

- time periodic solutions # x-independent oscillatory solutions # wave trains # target patterns: p(x, t) = U(lxl, t), U is periodic in t # rotating spiral patterns: n = 2, x = r(cos 9, sin 9), p(x, t) = U(r,

9 - ct), U is periodic in the second argument. - travelling wave solutions: the form of the solutions: P(x, t) = p(x - ct),

c is a velocity vector # stationary solutions (c = 0, p is time-independent) # plane waves: p(x, t) = U(x - ct)v. vis a unit vector in the direction of c +wave trains: U is periodic +wave fronts: U is monotone and bounded +pulses: U(- oo) = U( oo ), U is not constant.

Remarks

1. For the one-component case the reaction-diffusion system can be considered as a gradient system, therefore the results of elementary catastrophe theory can be applied. Ebeling & Malchow (1979) analysed bifurcations in (pseudo)-one-component systems by this technique. They showed that stable homogeneous stationary states are also stable in reaction-diffusion systems.

2. The two-component model plays an important role in the analysis of


wave propagation in chemical systems. Many special systems have been studied by linear stability analysis. Diffusion has been taken into account, (i) as transport between two homogeneous compartments; (ii) as diffusion in a continuous system. The occurrence of nonuniform steady states, as well of chemical waves in the Brusselator model, was demonstrated by linear stability analyisis (see Nicolis & Prigogine, 1977).

3. A class of two-component models with one diffusion coefficient much smaller than the other was studied by Fife ( 1976). The concentrations of u and v of the components U and V satisfy the equation

oufot - E 2V 2u - f(u, v) = 0 ovfot- V 2 v- g(u, v) = 0.

The boundary condition is expressed as

E20ujon = Uo - u; OVjOn = Vo - V.

(6.3a) (6.3b)

The determination of the time-independent functions u(x) and v(x) is easier for the special case when/= 0 is expressible as an S-shaped curve of v versus u.

Fife demonstrated that in the first stage of the evolution sharply differentiated subregions, bounded by layers within which one of the concentrations has a large spatial gradient, are formed. At this time the initial distributions determine the shape of the subregions. The final distribution that is achieved with a much smaller time scale depends on the stoichiometry of the reaction, the larger diffusion coefficient, and the reference concentrations on the boundary, but is relatively independent of the initial distribution.

4. Bifurcations. In many situations the uniform solution of the reaction-diffusion equation exists, but is stable only for certain regimes of the parameters. Near to the transition zone between stability and instability of this uniform solution, other nonuniform, small-amplitude, solutions exist as well. If these solutions are stable, their appearance can be considered as bifurcation phenomena. The equation

oufot = Do2 ujox2 + (A + A.B)u + g(u)

is an example for illustrating diffusion-induced instability. The stationary solution of the kinetic equation (i.e. when D = 0) is destabilised by a nonuniform perturbation, but preserves its stability in the case of an xindependent perturbation (Segel & Jackson, 1972).

5. Solitary waves were found in a model of a biochemical reaction system catalysed by an allosteric protein, as a result of a threshold phenomenon and diffusion (Anan & Go, 1979). They studied the simplified equation:

osjot = -s- cp + V2s opfot = s + hp- p 3 + a + dV2p

In the spatially homogeneous system for certain parameter values of a, h and


c one stable stationary point, one unstable stationary point and one saddle point can be found. A separatrix (i.e. a trajectory flowing into the saddle point) goes very close to the stable stationary point. In response to a perturbation it could occur that the trajectory leaves the stable point and goes around the unstable steady state. Inclusion of diffusion can imply the occurrence of solitary waves.

6.3 Stochastic models: difficulties and possibilities

6.3.1 Introductory remarks

The fact, that it is necessary to take into account spatial fluctuations arose in consequence of a strange debate. Kuramoto (1973) criticised the statement of Nicolis & Prigogine ( 1971) that the stochastic kinetic model is not in accordance with the microscopic description based on the Boltzmann equation. In their comprehensive answer to Kuramoto's short paper Nicolis et a/. ( 1974) stated that the assumptions of Kuramoto were not natural, and the stochastic kinetic model is not appropriate to describe certain microscopic fluctuations. In fact, stochastic kinetics represents an intermediate level between 'microscopic reaction kinetics' operating with elementary collisions and 'phenomenological reaction kinetics' working with macroscopic quantities. In the course of this debate the view that chemical fluctuations lead 'locally' to a Poissonian distribution, but deviate from it 'globally' has emerged (Malek-Mansour & Nicolis, 1975). The debate seems to be incoherent, since three distinct models were compared. Not only a microscopic and a global stochastic model were involved, but also a-not well-defined-local stochastic model. Global and local kinetics can be compared only partially. In the case of deterministic models an ordinary differential equation would not be blamed for not being able to describe spatial changes if the elements of the domain of the unknown function were interpreted as points of time. The usual stochastic approach tends to associate a stochastic process with the partial differential equations of the deterministic model, and not a random field. However, three directions in the theory of random fields seem to be able to cope with such complexity: the theory of random measures (Prekopa, 1956, 1957a, b; Kendall, 1975), the theory of stochastic partial differential equations relating to trajectories (see Skorohod, 1978) and the theory of Hilbert space valued stochastic processes (Ichikawa, Arnold 1982; eta/., 1980 Dawson 1972, 1975). At present, this last direction seems to be the most fruitful: it seems to be broad enough to provide the framework for the generalisation of

(l) nonequilibrium thermodynamics to include randomness, (2) stochastic reaction kinetics to include spatial effects (such as diffusion)

correctly, and


(3) stochastic theories of diffusion to include sources and sinks (such as chemical reactions).

Numerous efforts have been made in the last decade to evaluate spatial correlations of fluctuations. The examination of specific models was given preference over the formulation of general models. Different approaches to stochastic modelling of reaction-diffusion systems will be shortly reviewed in this chapter.

6.3.2 Two-cell stochastic models

The simplest extension of stochastic models of homogeneous kinetic systems for describing spatial effects is the introduction of the two-cell model. According to this approach reactions take place within the homogeneous cells, while transport of matter between the cells is taken into account as a random exchange between cells.

The two-cell stochastic model of the Schlogl reaction describing 'firstorder phase transition' has been carefully studied (Borgis & Moreau, 1984b; Moreau & Borgis, 1984).

Let P(n 1 , n2 ; t) denote the absolute distribution function. Accordingly, at a fixed time t, the number of molecules in cell 1 is n1 , and in cell 2 is n2 •

Denoting with R; the operator of the reaction operating on n;, which describes the effect of chemical reactions the master equation can be described:

ojotP(n~> n2 ; t) = (R1 + R 2 )(P(n 1 , n2 ; t)) + D(n 1 + l)P(n1 + I, n2 - 1; t)- D(n 1 + n2 )P(n 1 , n2 ; t) + D(n2 + l)P(n 1 - 1, n2 + 1; t) (6.4)

Approximations or simulation experiments are necessary to find even the stationary solution. The behaviour of the system is particularly interesting for small values of D, since large D implies a homogeneous solution.

The simulation technique used earlier (Frankowicz & Gudowska-Nowak, 1982) was adopted by Borgis & Moreau ( 1984b) to calculate the stationary distribution. Its shape is shown by Fig. 6.1.

Let (X and y denote the stable steady states in single cells. The stationary distribution presents two peaks on the homogeneous steady states ((X, (X), (y, y) and two peaks on the inhomogeneous states ((X', y') and (y', (X'). (For D = 0, (X' = (X andy' = y, otherwise there is a shift toy and (X repectively.) In the plane (n 1, n2) there are only four regions around these peaks, where the stationary distribution differs significantly from zero.

An approximate master equation has been derived (Borgis & Moreau, 1984) based on the quasistationary assumption. Accordingly, exchanges between the regions ((X) and (y) are much slower than the relaxation inside these regions. This assumption led to a simple approximate equation for the


Fig. 6.1 Qualitative shape of the stationary distribution for small D.

probability P*(x, y; t) to find cell I in region (x) and cell2 in region (y). The definition of P*(x, y; t) is:

P*(x, y; t) = L L P(n 1 , n2 ; t). n, E(x) n,E(y)

The approximate solution of the evolution equation for P*(x, y; t) based on the hypothesis of low coupling between cells shows that the transition between two homogeneous stationary states occurs via an inhomogeneous transient state. Transient inhomogeneous structures may be also responsible for the stabilisation of an unstable state. Such kinds of phenomena are reminiscent of nucleation processes (Blanche, 1981; Frankowicz, 1984) and might be considered as a nonequilibrium analogue of an equilibrium phenomenon.

6.3.3 Cellular model

According to the basic assumption of this model, not only is the component space discrete, but the real space is also subdivided into mesoscopic cells. The meaning of the term 'mesoscopic' here is that the size of cells is larger than the size of the constituent molecules, but much smaller than the characteristic scale of the total system. While from a heuristic point of view the discrete state-space description of chemical reactions seems to be natural, the discretisation of the space can be qualified as a more or less forced technical procedure.

Subdividing the total system into JVd cells of equal size, the state of the


whole system can always be characterised by a finite-dimensional vector (dis the dimension). Let X;, denote the number of particles of species i in cell r (r is the central point of the cell), and P({X;,}. t) is the joint probability distribution of all X;,. Since the chemical reaction is modelled as in the homogeneous case, and diffusion is considered as a random walk between adjacent spatial cells, a Markovian process can be constructed in the space combined from the component space and the real space. The probability distribution P({X;,}, t) is modified due to chemical reactions and diffusion:

iJjiJt P({X;,}, t)Jt' =''reactions within cell r' + 'diffusion among cells'

While the first term is described according to the rules of homogeneous stochastic kinetics, the second term is as follows (Nicolis & Malek-Mansour, 1980):

LDJ2d[(X;, + J)P(X;,r+l- J, X;,+ J, {X;,}, t)- X;,P({X;,}, f)] s, i

where s denotes the first neighbours of cell r.

Remarks

1. Many difficulties occur in connection with the discretisation of the space. In practice a large set of rate constants and diffusion constants are necessary to evaluate the model. In general there is little chance of obtaining a solution of the model. Approximations, as by the mean-field approach (which neglects the spatial fluctuations), cumulant expansions and perturbative methods (see, for example, Nicolis, 1984) can be applied.

2. Kurtz (1981) introduced a different, better founded model. It was mentioned (Subsection 5.1.4) that reaction-diffusion systems can be derived by the semigroup operator approach giving the generator. For sake of simplicity Kurtz considered the reaction A + B -+C. The state of the system is described by the vector (k, /, X~o x 2 , ... , xko y 1 , y 2 , ... , y1), where k is the number of molecules of A and X~o x 2, ... , xk are their locations, I is the number of molecules B with coordinates y1 , y 2 , ••• , y1• According to the physical assumptions, the molecules undergo Brownian motion with normal reflection at the boundary, the probability of a reaction between a molecule A at x, and B at y in a time interval!l.t is C(x - y)M + o(M). Associating to the vector x e Rk the vector tt;E Rk- 1 by dropping X; the generator A of the process has the form

k I

Af(k, /, X, y) = L d./l.J(k, I, X, y) + L d,AyJ(k, I, X, y) i= I k I j= I

+ L L c(x;- y;)(f(k- I,/- 1, tt;. qj)- f(k, /, x, y)), i= lj= I

where fl. denotes the discrete Laplacian.


3. The connection between the stochastic cell model and the usual deterministic model of reaction-diffusion systems were given by Kurtz for the homogeneous case in the same spirit. To give the relationship among a Markovian jump process and the solution p(r, t) of the deterministic model a law of large numbers and a centra/limit theorem hold (Arnold & Theodosopulu, 1980; Arnold 1980).

Roughly speaking the law of large numbers is (at least for finite time in the thermodynamic limit):

limsupP(IIx(t)- p*(t)IIL > t) = 0, t > 0, 0 ~ t ~ T.

Problem

Rewrite the central limit theorem given by Kurtz for the homogeneous case Subsection 5.9.3 for inhomogeneous systems.

6.3.4 Other models

To avoid the drawbacks of the discretisation of space a particle density function u(r) can be introduced by the relation

X- (t) u(r) = lim -" -.

~·-o Av The joint distribution function P({X;,}, t) is converted formally into a probability in function space, P([u(r)], t), which is a functional of u(r); (see van Kampen, 1981, pp. 346-7). From a mathematical point of view this procedure is badly founded, since the probability density in function space is not defined.

The explicit use of probability in function space can be avoided by the method of compounding moments. For the reaction

A -+X, 2X-+ B

evolution equations can be derived for the first and second moment. For the first moment:

o,(u*(r, t)) = 1 - (u*(r, t)2) + DV2(u*(r, t)). (6.5)

Equations can also be derived for the factorial cumulants defined by the relation

o,[xlx2] = <xlx2>- (xl) (x2)- 012(nl); (6.6)

o,[u*(rl, t)u*(r2, t)] = -2{(u*(r1 , t)) + (u*(r2, t))}[u*(r1 , t)u*(r2, t)]

- {r1 - r2)(u*(r1 , t)) 2 + D(Vi + VD[u*(r1 , t)u*(r2, t)]. (6.7)

These kinds of equations are appropriate to describe the spatio-temporal


fluctuations in reaction-diffusion systems. As in the case of homogeneous systems, there are two kinds of stochastic

descriptions for reaction-diffusion systems as well: the 'master equation' approach and the stochastic differential equation method. Until now we have dealt with the first approach; however, stochastic partial differential equations are also used extensively. Most often partial differential equations are supplemented with a term describing fluctuations. In particular, timedependent Ginzburg-Landau equations describe the behaviour of the system in the vicinity of critical points (Haken, 1977; Nitzan, 1978; Suzuki, 1984). A usual formulation of the equation is:

ap(r, t)jat = ap(r, t) + DV2 p(r, t) - bp3 (r, t) + <p(r, t), (6.8)

where <p(r, t) is a Gaussian noise term satisfying

(<p(r, t)<p(r', t')) = 2&o(t- t')S.(r- r') (6.9)

with some function s .. The mathematical interpretation of such equations requires the clarification of the meaning of stochastic integrals, as in the case of stochastic ordinary differential equations. From a practical point of view certain kinds of scaling laws (which express a certain scaling invariance of some physical quantities) are useful for getting results for spatial and temporal correlations. Critical point scaling, due to the infinite correlation length, and initial instability scaling, due to the asymptotitic evaluation of initially exponential growing process, has been adopted for the spatiotemporal situation (Suzuki, 1984).

Remarks

Critical dynamic analysis using renormalisation techniques were presented by Walgraef eta/., 1982. A method based on the systematic study of the Taylor expansion of the stochastic potential was applied for reactiondiffusion systems exhibiting Hopfbifurcation (Fraikin & Lemarchand, 1985). The Poissonian representation technique of Gardiner & Chaturvedi (1977) is also an efficient procedure for evaluating the parameters of fluctuations. A more rigorous derivation of reaction-diffusion equations with fluctuations were given by DeMasi et al. (1985).

Our feeling is that the setting up of mathematically well-founded and practically acceptable models of stochastic reaction-diffusion systems still remains to be done.

6.4 Spatial structures

A variety of spatial structures such as chemical fronts or waves, periodic precipitates and regular spatial patterns at liquid interfaces have begun to be investigated since the first experiments on the formation of chemical spatial


structures were reported (Busse, 1969; Zaikin & Zhabotinskii, 1970). Standard experimental techniques based on automatic image-processing

devices capable of analysing spatial structures seem to be under development (see, for example, Bodet et a/., 1984; Kagan et al., 1984; Muller et a/., 1985, 1986).

We are far from understanding the detailed mechanisms leading to different types of structures. Even the possible classifications of the phenomena are still controversial (see, for example, Smoes, 1980).

Winfree (1972, 1973, 1974, 1980; and a number of other papers) has described several wave patterns: spirals, elongated spirals, elongated rings and scroll rings, all of which he attributes to three-dimensional scroll waves. (Winfree (1985) gives extensive references on the subject.) According to Winfree's classification, chemical waves are 'trigger waves' that are characterised by the involvement of mass transfer, and 'pseudo-waves' that occur without mass transfer. The evolution of three-dimensional toroidal scrolls has been demonstrated by Welsh et al. (1983). Detailed studies of trigger wave initiation was presented by Showalter et al. (1979). They demonstrated that an initially spatially homogeneous oscillatory chemical system is excitable, and coupling with diffusion can generate a trigger wave that propagates through the medium undamped at almost constant velocity. Chemical wave propagation exhibits trivial formal analogies to neural excitability. One-dimensional stripes in a variant of the BelousovZhabotinskii system are generated by a diffusion-independent mechanism (P6ta eta/., 1982), i.e. they are pseudo-waves.

A particular type of structure, the leading centres, can be generated either by some small auto-oscillatory regions of the medium or by a special type of sources (echo-sources) arising in a medium with nonoscillatory kinetics. Based on the experimental results of Agladze & Krinsky (1984) the first mechanism seems to be more probable for the Belousov-Zhabotinskiisystem.

Chemical reactions at liquid interfaces exhibit remarkable patterns. Chemical reactions are necessary for structure formation, and hydrodynamic and diffusion effects in the absence of reaction could not generate these patterns. However, different types of reactions led qualitatively to the same result (Avnir et al., 1984). Additionally, surface-driven convection might have a crucial role for the onset of convection patterns in chemically active medium (Muller et al., 1985).

Periodic precipitation phenomena, commonly called Liesegang rings, have been studied for a long time. According to recent experimental results (Kai et a/., 1982; Muller et al., 1982) patterned precipitation is a postnucleation phenomenon which occurs after a continuous distribution of colloid has been established. The phenomenon can be described at least formally by an instability mechanism based on the autocatalytic growth of colloid particles with diffusion (Ross, 1985).


Attempts to describe the formation of spatial structures in chemical systems have been made overwhelmingly by deterministic reaction-diffusion models. However, Walgraef eta/. (1983) emphasize the role of fluctuations as the origin of these patterns.

As temporal oscillation was the subject of intensive research in the nineteen-seventies, it is reasonable to predict that the understanding of the mechanisms for the formation of chemical spatial and spatia-temporal structures will be one of the most challenging areas of chemical kinetics.

6.5 Pattern formation and morphogenesis

The term 'pattern' is a rather vague notion to denote the arrangement of certain abstract elements. According to our subjective feeling there is no clear distinction between the terms 'structure' and 'pattern'. Temporal, spatial and spatia-temporal structures in thermochemical systems might be the result of the interaction of chemical components by diffusion and chemical reactions.

It is widely accepted that mechanisms responsible for patterns and order in a reaction-diffusion system may also play a fundamental role in understanding certain aspects of biological pattern formation. Though it is evident that chemical reactions and diffusion processes are everywhere and always present in biological systems, one question seems to be poorly discussed in the literature: are the pattern-forming reaction-diffusion systems the real basis of biological morphogenesis or do they offer some analogy for us to obtain insights into the mechanism of pattern formation?

Since chemical reactions and diffusion processes are the 'consequences' of the interactions among chemical constitutents, chemical patterns may be interpreted at the molecular level. A certain, but closely limited, class of biological patterns may be described in biochemical terms. Adopting the spirit of the hierarchical approach, patterns at 'higher than molecule' levels may be formulated as the products of interactions among 'higher than molecule' (e.g. cell) constituents.

Two points have to be emphasised here. First, the interactions in 'higher than molecule' populations can be formally described by models similar to those used in thermodynamics (i.e reaction-diffusion equations). Second, these interactions are just thermodynamic effects, often due to chemical reactions and diffusion. For example, the spatia-temporal patterns of interacting cells due to cellular communication are based partly upon the chemical nature of the communication (and partly on the topological arrangement of the cell population).

From the biological point of view, numerous specific mechanisms have been suggested to explain pattern formation, such as 'induction', 'lineage', 'prepattern', 'positional information' and 'polar co-ordinate' models. All these mechanisms were required to explain the question: how can genetic information be translated in a reliable manner to give specific and different


spatial patterns of cellular differentiation? This kind of pattern formation differs from molecular differentiation, since the latter is mainly concerned with the control of synthesis of specific macromolecules within cells rather than the spatial arrangement of cells.

Development-controlling prepattern mechanisms have been modelled in reaction-diffusion context. In the celebrated paper of Turing (1952) a model was presented in terms of reaction-diffusion equations to show how spatially inhomogeneous arrangements of material might be generated and maintained in a system in which the initial state is a homogeneous distribution. Two components were involved in the model, and the reactions were described by linear differential equations. The model in one spatial dimensions is:

X,= kiX + k2Y + DXYSS Y, = k3X + k4Y + DyYss

(6.10)

(X and Y denote deviations from the equilibrium values). The equations can be solved analytically, since they are linear. Specific relations have to be assumed among the six constants to obtain 'morphogenetic behaviour'.

We suppose k1 and k3 to be positive, k 2 to be negative, and k4 to be zero; also D,. > Dx; i.e. X catalyzes its own growth and that ofY, Y inhibits X, and the inhibitor di-ffuses faster than the activator (X). This implies that, from a localised centre of activity, Y will, at least in early stages of development, spread out farther than X. (Harrison, 1981)

An obvious defect of the Turing model is that, because of its linearity, it cannot describe saturation effects. To overcome this difficulty a two-variables model was introduced by Gierer & Meinhardt (1972). Although its terms can be identified with reactions and diffusion, the equations contain rational functions at the right-hand side. Denoting by a and h the complete concenirations of the activator and inhibitor (and not the deviations from the equilibrium values) the model is

a,= Ca2 /h- Ba + Daa,, +A h, = Ca2 - Eh + Dhh,.,

(6.11)

where A, B, C, D and E are independent of time. The model is able to describe not only the generation of an inhomogeneous structure, but also the regeneration occurring after partial lesion. The model 'has probably attracted more attention among experimental biologists than any other model of the reaction-diffusion type (this whole field of modelling being still regarded with much scepticism among biologists) .. .' (Harrison, 1981). The Gierer-Meinhardt model and its versions (Gierer, 1977; Meinhardt, 1978; Meinhardt & Gierer, 1980; Meinhardt, 1982; Meinhardt, 1985) describes regeneration and induction, dependence of the pattern formed on the tissue size and the generation of polarity, size regulation, periodic structures, and


offers a hierarchical mechanism for the development of higher organisms as well. The biochemical identification of the activator and inhibitor molecules is not easy-but this is another story.

The mechanical basis of morphogenesis has been emphasised recently (Oster eta/., 1980; Oster & Alberch, 1982; Odell eta/., 1984; Murray eta/., 1983). Embryonic cells differentiate into epithelial and mesenchymal cells during early ontogeny. An attempt was made to describe morphogenetic movements based on the mechanical properties of the cells.

Our feeling is that validity ranges of reaction-diffusion and mechanicsbased models of morphogenesis are complementary and so the models supplement each other.

7 Applications

7 .I Introductory remarks

We certainly do not want to give an extensive review of the possible applications of chemical kinetic models. The intention of this chapter is to demonstrate that the conceptual framework of chemical kinetics is appropriate to describe 'co-operation' and 'competition' among elements of 'otherthan-molecule' populations, as well as interactions among molecules. As was mentioned earlier (Section 1.5) chemical kinetics may serve as a metalanguage for modelling different kinds of population phenomena.

The examples to be presented illustrate the diversity of fields of applications, but they are mentioned in outline form only. Many biological phenomena used to be modelled by real or formal kinetic models. A biochemical control theory that is partially based on non-mass-action-type enzyme kinetics seems to be under elaboration, and certain aspects will be illustrated. A few specific models of fluctuation and oscillation phenomena in neurochemical systems will be presented. The formal structure of population dynamics is quite similar to that of chemical kinetics, and models referring to different hierarchical levels from elementary genetics to ecology are wellknown examples. Polymerisation, cluster formation and recombination kinetics from the physical literature will be mentioned briefly. Another question to be discussed is how electric-circuit-like elements can be constructed in terms of chemical kinetics. Finally, kinetic theories of selection will be mentioned.

7.2 Biochemical control theory

I. Enzyme kinetics traditionally deviates from mass-action kinetics. This deviation results from the early enzyme kinetic measurements, which were


apparently not in accordance with the mass-action kinetic law. Now it is clear, how to derive enzyme kinetic equations from mass-action kinetics by using quasi-steady-state approximation. Even deviations from the usual (Michaelis-Menten) enzyme kinetics can be clarified taking into consideration detailed structural effects.

The mass-action kinetic model of the interaction among four molecule populations, namely enzyme (E), substrate (S), complex (C) and product (P) is described by

defdt = -k1es + (k_, + k 2 )c ds/dt = -k1es + k_ 1c dcfdt = k 1es- (k_ 1 + k 2 )c dpfdt = k 2 c.

(7.1)

The usual initial concentrations are e(O) = e0 , s(O) = s0 , c(O) = c0 ,

p(O) = p0 . Since the enzyme molecules are always either in free (E) or complexed (C) form the enzyme concentration can be eliminated. Since the substrate molecules are always in original (S), complexed (C) or product (P) form they can be eliminated too:

e(t) = e0 + e0 - c(t), s(t) = s0 +Po + c0 - p(t) - c(t) (7.2)

The system of differential equations obtained by the elimination cannot be solved analytically. Assuming that the substrate concentration changes 'slowly', the complex concentration can be expressed by the quasi-steady equation

( ) _ (eo + c0 )s(t) c t - Km + s(t) '

where the Km Michaelis constant is defined as

K = k_, + k2 m- kl .

(7.3)

Renaming the common value of lds/dtl and dp/dt as the velocity of the reaction the approximate expression

(7.4)

can be obtained (it would be more precise to call V the initial velocity). The maximal value of V:

S 0 - oc

Since experimentalists have preferred always to interpret their results in terms of straight .lines, the connection among V, Vm • ., s0 and Km can be expressed as

Applications 179

1/V = (1/Vmax)(l + Km) So

(7.5)

The 1/V versus the 1/s0 function is called the Lineweaver-Burk plot.

Remark

The validity of the quasi-steady-state approximation, has already been mentioned in Subsection 4.8. 7. A detailed analysis of enzyme kinetics is given in Heineken et a/. (1967), Walter (1977) and Segel (1984). The strict mathematical basis of the assumption is based on a theorem by Tikhonov (1952). He investigated the assumptions leading to separation of the 'fast' and 'slow' components of the solutions of the system

dx(t)/dt = f(x(t), y(t)) &dy(t)fdt = g(x(t), y(t)).

(7.6)

The solution of the system (7.6) converges under certain conditions forE-+ 0 to the solutions of the system

dx(t)/dt = f{x(t), Y(x(t))},

where

0 = g(x, Y(x)).

Many generalisations of the theorem have been given. 2. Deviations from the Lineweaver-Burk plot may be accounted for by

considering the structural details of the enzyme molecule ('allosteric' theory: Monod et a/. (1965), 'induced fit' theory: Koshland et a/. (1966). The allosteric theory is based particularly on the co-operation among the subunits of the enzyme molecule. According to the simplest assumptions, only two subunits exist (i.e. the enzyme is a dimer), and both of them can be in two conformational states (inactive, T, and reactive, R). Furthermore symmetric conformation pairs only (i.e. TT and RR) may occur, and the transitions between the conformations are concerted. The detailed kinetic description of enzyme-substrate interaction containing the binding of substrates as well as the conformation changes (between completely substrate-free subunits) can be given in terms of mass-action kinetics:

dTofdt = - fTo + bRo dR0 /dt = fT0 - bR0 + k - R 1 - 2k + R0 S dR 1 /dt = -k- R 1 + 2k + R 0 S + 2k- R 2 - k + R 1S (7.7) dR2 /dt = -2k- R 2 + k + R 1S dSJdt = -2k + R0 S + k- R 2 -k + R 1S.

The subscripts to R and T refer to the number of bound substrate molecules, f and b denotes the forward and backward rate constants of the confor-


mation transition. The operation of the system is well characterised by the fraction of subunits to which ligand or ligands are bound in the steady state.

3. The simple enzyme kinetic system does not exhibit oscillatory behaviour. The existence, uniqueness and global asymptotic stability of a periodic solution of a Michaelis-Menten mechanism was proved (Dai, 1979) for the case when the reaction occurred in a volume bounded by a membrane. The permeability of the membrane to a given species was specified as the function of another species. The form of the model is

c dsfdt = - s + Asc + Be + 1 + p" (l - s)

Ddc/dt = s - Asc - Ec (7.8) dpfdt = F(c- p),

where s, c and p denotes the transformed substrate, complex and product concentrations, while A, B, C, D, E and Fare constants. To prove the uniqueness and global asymptotic stability of periodic solutions in the threedimensional system is difficult. The proof given by Dai utilises the procedure given by Hastings (1977) based on the Brouwer fixed point theorem.

4. Biochemical systems are able to show controlled behaviour. The existence of steady states or of oscillatory operation are characteristic examples of controlled behaviour.

A cyclic enzyme system, where two enzymes share a substrate in a cyclic manner, was assumed to be the control element of a feedback system to maintain the stationary concentration of the end product at a desired level against external perturbation (Okamoto & Hayashi, 1984). Switching behaviour is shown by a mass action kinetic model involving interactions among the active (£.) and inactive (EJ form of the cyclic enzyme, the inhibitor (Y1), the activator (Y2 ) and the end product (Y3 ). The model is:

dE.fdt = k3E;Y2 - k2E. Y1 dEJdt = k2E.Y1 - k3£1 Y2 d Yddt = k1y - k2E. Y1 dY2/dt = k4 x- k3E;Y2 dY3/dt = k3E1 Y2 - k 5 Y3.

(7.9)

The excitatory factor x and the inhibitory factory are assumed to be external components. The variable (xfy) is considered as the control parameter. Numerical computations showed that steady-state concentrations of E. and E; changed stepwise at (xfy) = 1. (Fig. 7.1 ). This switching scheme has been incorporated into a feedback system. The variables of the system are the input(/), the reactants and intermediates (X 1 , Y2 and Y3 ), the inhibitor Y1 ,

the precursor of Y2 molecule (A) and the active(£.) and inactive(£;) form of the enzyme. The kinetic model is:

Applications

0 0 2

o.____---....~~..... _ __, 0 1

y X

2

181

Fig. 7.1 Steady-state concentrations of Ea and E; as a function yfx. Rate constant: k 1 = k4 = k 5 = 1.0, k 2 = k3 = 5 x 104 . Initial concentration: Ea = E; = 0.5, Y 1 = Y2 = Y3 = 10.0. They-value was fixed at 20.0.

dXI/dt = I- kiXI dX2/dt = k1X1 - k2X2 dX3jdt = k2X2 - k3EaX3 dX4jdt = k3EaX3 - k4X4

dE.fdt = k1E;Y2- k6EaYI dEtfdt = k6EaYt - k1E;Y2 dYtfdt = ksX4- k6EaYI d Y2/dt = k 9 A - k 7 E; Y2

d Y3/dt = k 7 E; Y2 - k 8 Y3

(7.10)

Numerical calculations gave the evidence that the stationary value of X4 is highly insensitive even to a ten-fold increase of the input concentration /, i.e. the system simulates homeostatic control behaviour.

Remarks

Many model studies dealt with models involving feedback control of enzymatic activities (e.g. Morales & McKay, 1967; Hunding, 1974; Hastings et a/., 1977; Tyson & Othmer, 1978; Berding & Harbich, 1984). Furthermore, the interpretation of periodic and aperiodic phenomena in biochemical systems is still an open topic.

4. Rhythmic behaviour is characteristic of biological systems. The stability of most biological rhythmic phenomena can be understood by the concept of a limit cycle. The independence of the amplitudes and frequencies from the initial conditions might be associated with regular temporal patterns. The


mutual interaction among different biological oscillators might imply as well other organised temporal patterns. Oscillatory and chaotic behaviour in biochemical systems will be mentioned shortly. The two best-known examples of periodic operation in cellular metabolism are glycolitic oscillations and the periodic synthesis of adenosine 3', 5'-cyclic monophosphate (cAMP) oscillations in the slime mould Dictyostelium discoideum.

Glycolytic oscillations (Hess & Boiteux, 1971; Goldbeter, & Caplan, 1976) occur due to the allosterically controlled phosphofructokinase (PFK, EC 2.7.1.11) reaction. The simplest (perhaps oversimplified) model of glycolysis contains two variables only, namely the substrate (ATP) and the product (ADP) of the enzyme (Goldbeter & Nicolis, 1976). The hierarchical regulatory mechanism of glycolysis can be described by a more detailed model (Boiteux & Hess, 1984).

A model of periodic synthesis of cAMP (Goldbeter & Segel, 1977) proposed in connection with the aggregation of cellular slime moulds is based on the fact that extracellular cAMP binds to a receptor at the cell surface and activates adenylate cyclase by an autocatalytic ('positive feedback') step. Consequently, synthesis of intracellular cAMP is increased.

The possibility of producing more complex temporal patterns in biochemical systems has been underlined in a model that is formally similar to the two just-mentioned systems (Decroly & Goldbeter, 1982; Goldbeter & Decroly, 1983; Decroly & Goldbeter, 1984a; b). The model contains two instability generating steps, i.e. two autocatalytic reactions coupled in series, as shown in Fig. 7.2.

+ +

~ I v ·---1 ~-.~ ---s E, E,

Fig. 7.2 Model for the coupling of two autocatalytic enzyme reactions. The system admits three simultaneously stable periodic regimes for appropriate values of the parameters.

Assuming that the enzymes E 1 and E2 obey the concerted transition model, the kinetic equation for the transformed concentrations of S, P, and P2 is:

dsfdt = A - Bf(s, p 1)

dpddt = Cf(s, p,)- Dg(p1 , P2) dp2 /dt = Eg(p,, P2)- Fp 2 ,

(7.11)

where f and g are nonlinear functions, A is the constant input of the

Applications 183

s

s

s

Fig. 7.3 Trirhythmicity. The system evolves towards three distinct periodic regimes for three different sets of initial conditions.

substrate, B, C, D and E are functions of the parameters of the enzymatic system, F is a rate constant.

This model containing three variables is capable of describing complex temporal patterns. The possibility of a coexistence of two and three periodic regimes ('birhythmicity', 'trirhythmicity; Fig. 7.3) was demonstrated numerically. Multirhythmicity means that, depending on the initial conditions, distinct stable oscillations are observed. It might serve as a switching element among oscillatory regimes, and might have great biological relevance. Chaotic behaviour was illustrated numerically.

A beautiful zoo of different biochemical time patterns was found both experimentally and theoretically in studies of the glycolytic system under periodic perturbation (Hess & Markus, 1985; Markus et a/., 1985).

The system of differential equations

dx(t)/dt = f(x(t)), x(t) E IR", x(O) = x 0


might be supplemented with a term a*(t) = (0, 0, ... , a cos(cot + <p )) leading to

dx(t)fdt = f(x(t)) + a*(t).

It means that one component is perturbed by amplitude a, a cosinusoidal forcing term with frequency co and phase q>. Recently particular attention has been paid to the response of nonlinear chemical oscillators that are periodically perturbed. The celebrated Brusselator model

~[x] = [x2y- Bx +A-x]+ [acos(cot + q>)J dt y Bx- x 2y 0

has been studied (Tomita, 1982). The undriven system shows limit cycle behaviour in a certain region of the parameter space. It was clearly demonstrated that four different types of region might be recognised on the (a-co) plane (Fig. 7.4), namely entrainment (phase locking), quasiperiodic oscillation, periodic doubling cascade, and chaos have been found. Although elaborate mathematical techniques exist to analyse bifurcations from and to the regions just mentioned (e.g. Ioos & Joseph, 1980), numerical methods often offer a technically simpler, 'down-to-earth' treatment of dynamic problems. The analysis of systems, when the unperturbed system is higher than three-dimensional is particularly difficult (but not hopeless-see, for example, Rossler & Hudson, 1985). Periodically perturbed chemical systems were reviewed recently by Rehmus & Ross (1984).

IX

0.16

0.12

0.08

0.04

0 0 0.2: '0.6 :o.a 1.0 1.2 (J)

1/3 ~ 2/3: 3/2 2 3 1/2 3/4 4n

wlw0

Fig. 7.4 Phase diagram. The numbers indicate the harmonic periods appearing in the respective regions in the unit of forced period. A limit cycle of nonintegral period appears in the shaded region Q, and a chaotic response is found in the region indicated by J(.

Applications 185

Studying the time patterns in the perturbed gloycolytic system Markus et a/. (1985) examined their model using three variables, namely the concentration of ADP, a, and phosphoenolpyruvate, b, and the input flux phase <p = ro.t, where ro. is the frequency of external perturbation. The kinetic model is

da(t)fdt = VPFK - VPK

db(t)jdt = V;n - VPK

d<p(t)fdt = ro •. (7.12)

The enzyme rate of VPFK and VPK (PK stands for pyruvate kinase) are complicated functions of a, b and concentrations of other components that are not involved explicitly in the model (ATP, fructose-6-phosphate) V;n

consists of the mean input flux superimposed by the input flux oscillation. While other models of forced glycolysis (e.g. Richter (1984 based on the

Selkov model)) demonstrates the occurrence of entrainment (the frequency locking of an autonomous oscillator by externally imposed frequencies) Hess and Markus emphasised the failure to see entrainment leading to quasiperiodic or even chaotic dynamics, and the coexistence of complicated periodic attractors. Currently the biological function of complex temporal patterns are not well-understood. Some neurobiological examples will illustrate this problem.

7. 3 Fluctuation and oscillation phenomena in neurochemistry

0. The neurochemistry of synaptic transmission has been thoroughly investigated over the decades. Though the approach and techniques of neurochemistry is static rather than dynamic, kinetic models may prove useful for explaining both the chemical details and the regulatory aspects of synaptic transmission. First, a short review of synaptic transmission is given. Second, postsynaptic membrane noise as a source of information will be mentioned; then oscillatory and other temporal patterns connected to integrated synaptic activity will be analysed.

1. Information is transferred from one neuron to another at specialised points of contact, at synapses. Synapses are most often made between one axon of cell and the dendrite of another (but there are other kinds of synaptic junction too).

On the arrival of an impulse at the nerve terminal, a chemical transmitter substance is released from the presynaptic cell, crosses the synaptic cleft separating the two cells, and interacts with specific receptor sites on the postsynaptic membrane. The receptors are large protein molecules embedded in the semifluid matrix of the cell membrane. The binding of the transmitter to the receptors implies the opening and closing of 'membrane channels' controlling the flow of specific ions through the membrane.

Most of what is known about the mechanism of synaptic transmission


Vesicular· Jo<t~l=.:~~IV\ grid

Postsynaptic density

Synaptic vesicle Synaptopore

Fig. 7.5 Schematic illustration of the chemical synapse.

comes from experiments on a particular synapse: the neuromuscular junction of the frog, where acetylcholine (ACh) serves as the transmitter. The schematic illustration of a synapse is given in Fig. 7.5.

The metabolism of the ACh can be considered as a recycling process constituted by the following 'elementary' steps:

(I) ACh synthesis. The precursors of the ACh are the choline (Ch) and the acetyl coenzyme A (Ac-CoA). The reversible formation of ACh is catalysed by the choline acetyl transferase (ChAT; EC 2.3.1.6). Since the enzyme has both cytoplasmic and membrane-bound forms, consequently the synthesis can take place by two different mechanisms. The synthesis can be influenced by the formation reactions of the precursors and by their transport to the place of the synthesis. Both the sodium-dependent high affinity choline uptake (Barker & Mittag, 1975) and the transport of the mitochondrial Ac-CoA to the cytoplasm (Jope, 1979) may be the rate-limiting step of the synthesis.

(2) ACh release. According to the classical, vesicular hypothesis (Del Castillo & Katz, 1954) packets of transmitter molecules ('quantums') contained in synaptic vesicles are released by exocytosis. The vesicular hypothesis was in accordance with morphological studies demonstrating the existence of synaptic vesicles (De Roberties & Bennett, 1954) and explained the discrete nature of the postsynaptic response, namely the occurrence of miniature-endplate potentials (Fatt & Katz; 1952). Alternative hypotheses (e.g. Israel & Dunant, 1979; Tauc, 1979) are based on experimental observations of the release of cytoplasmic ACh.

(3) Cleft processes. ACh, when released, diffuses the synaptic cleft separating

Applications 187

the presynaptic and postsynaptic membranes. However, the synaptic cleft cannot be considered as an 'empty tube', since it contains different types of chemical molecules. Chemical reactions taking place in the cleft can strongly modify the mechanism of the information transfer.

(4) Transmitter-receptor interaction. While one class of channels in neural membranes responds to voltage differences across the cell membrane (i.e. they are voltage-gated), a second type of channel is chemically gated. Chemically gated channels are activated by the transmitter-receptor interaction taking place at the postsynaptic membrane surface. Conformational changes in membrane proteins form the basis for gating since they serve to open and close the channel by slight movements of critically placed portions of the molecule that unblock and block (at least partially) the pore. When channels are open, the resulting electric current can be measured. The opening and closing of chanels have a random character, therefore fluctuation in the electric current measured reflects certain biochemical details of the transmitter-receptor interaction.

(5) Hydrolysis of the postsynaptically bound ACh. The most rapid step of the ACh metabolism is the enzymatic decomposition of ACh. Acetyl choline esterase (EC 3.1.1. 7) is a very active enzyme. Rapid hydrolysis is the precondition of rapid recovery of the resting state after transmitting an impulse.

(6) Diffusion and (re)uptake of choline. Choline diffuses from the postynaptic cell towards the presynaptic cell. Choline resulting from different sources can be taken up by different mechanisms, but thin is beyond our present scope.

2. The possibility of the measurement of postsynaptic membrane noise (Katz & Miledi, 1972; De Felice, 1981) gives a deeper insight into the details of the chemical mechanism of transmitter-receptor interaction.

According to a simple assumption the transmitter-receptor complex has one 'closed' and one 'open' form. This assumption can be questioned and an algorithm can be given to determine the number of 'non-completely closed' conformations of the complex, and the kinetic rate constants of the conformation changes.

The measurable fluctuations are in the electrical potential or current of postsynaptic cells, and are induced by fluctuations of quantities of chemical components, particularly transmitter, receptor and intermediate molecules. Consequently, it is possible to give a relation between the quantities describing the electrical fluctuations in a postsynaptic cell and the stoichiometric and kinetic parameters of transmitter-receptor interaction.

Assumptions of the model:

(I) Transmitter-receptor interaction can be described by the following simple model scheme:


(7.13)

where x is the number of transmitters binding per one receptor molecule, T is the transmitter, R is the receptor, T.R is the conformation of the complex having zero ion-conductance, T.R; are conformations to be associated with non-zero ion-conductance, T.R 1 is the conformation with maximal conductance.

(2) The quantity of transmitter is maintained at a constant level. (3) The conductance is a linear function of the T.R; conformation quantity

vector (denoted by N):

g=LN, (7.14)

where g is the h-dimensional conductance vector (h is the number of ionic species considered), the dimension of the vector N is I + 2, the dimension of the matrix L is h x (I + 2).

(4) The conductances of the individual ions are summarized by g = grl. Here g is the total conductance, 1 is an h-dimensional vector with all coordinates equal to unity.

(5) A particular form of the function g(N) can be specified assuming one of the channel-models (Chen, 1976):

I

g(N) = L g,exp[- (I - j)E)kTJN;, (7.15) j=O

where g1 is the conductance vector of one channel being in the completely open state, Ni is the quantity of the complex T.Ri, T is the temperature and Ei is the activation energy of the conformation change T.Ri-+ T.Ri+ 1• This channel model is strongly motivated by the transition state theory of reaction kinetics.

This model for describing transmitter-receptor interaction' can be identified with the stochastic model of closed compartment systems. Adopting the fluctuation-dissipation theorem the conductance spectrum of the postsynaptic cell is determined by three qualitatively different factors:

(I) the length of the compartment system (implicitly); (2) the precise form of the function connecting the conductance and the

quantities of the different conformations (e.g. eqn. 7.15). (3) the rate constants.

The analysis of this model in Erdi & Ropolyi (1979) is an illustration of how noise analysis may be used as a tool for obtaining kinetic information.

3. At least three different neurochemical and neurophysiological oscillatory phenomena appear at different hierarchical levels of the cholinergic synaptic

Applications 189

transm1ss1on system. (Erdi & T6th, 1981; Erdi, 1983). 'Rapid' and 'slow' oscillation of free, presumably cytoplasmic, acetylcholine (ACh) have been described (Dunant et a/, 1977; Israel et a/., 1977). Another oscillatory phenomenon is the series of miniature-endplate potentials (Fatt & Katz, 1952) that correspond to the postsynaptic effects of pockets of transmitter, at least in the spirit of the classical vesicular hypothesis.

A complete theory of the hierarchical mechanism of dynamic synaptic activity would require the study of co-operation and competition among the three oscillators. Here only one model, established for explaining the 'slow oscillation' will be presented. The 'isolated' model shows periodic behaviour.

The skeleton model of the 'slow' oscillation in free acetylcholine can be formulated in terms of mass-action kinetics (Csaszar et a/., 1983). In principle, instead of setting up a lumped, skeleton model, a more complex model could be defined to take into account the details of subprocesses (synthesis, storage and release of ACh, cleft processes, transmitter-receptor interaction, diffusion, re-uptake). Experimental information, however, certainly would not be sufficient to parametrise such kinds of models.

According to the assumptions of the four-component model the state of the system is characterised by:

- cytoplasmic ACh concentration: X, - ACh concentration at the postsynaptic membrane surface, - choline (Ch) concentration near postsynaptic cell, - Ch concentration near presynaptic cell.

The subprocesses of the model:

(I) transmitter release, cleft processes, transmitter-receptor interaction; (2) ACh hydrolysis (the most rapid subprocess); (3) metabolic products (mostly Ch) diffuse to the vicinity of the presynaptic

cell; (4) (re)uptake of Ch; (5) autocatalytic synthesis of ACh.

A four-component formal chemical model

k 1 k 1 kJ k 4 ks X-+ Z, Z-+ W, W-+ Y, Y-+ X, 2X + Y-+ 3X

can be associated with the synaptic activity containing the five subprocesses. This mass-conserving model has a single nonlinearity and can exhibit sustained oscillations in numerical computations. Fixing the 'total mass', but changing the initial concentration the limit cycle behaviour can be visualised (Fig. 7.6(a)). Fixing the rate constants also, but changing only the 'total mass', a family of limit cycles (limit shells) can be derived (T6th, 1985) (Fig. 7.6(b)).

190

y 40 38

36

34

32

30 28

26

24

22

20 18

161

14

12

10

6

4

2

Mathematical models of chemical reactions

M = 300

o~.,~.~.,~ •• ~,.-.~,.--.~ .. -.~ •• -~.~.-•• ~.~.~ •• ~.~ •• -•• ~.~ •• -.~~w-270 280 290 300

(a)

0 40

(b)

Fig. 7.6(a) The Y versus W selection curve seems to tend from outside and from inside to a stable limit cycle. (b) The relatively stable limit cycles are changed as a function of the total mass of the system; the system has limit cycles within a certain (ange of M, the shape and location of these closed curves vary as the constraint is varied.

Adopting the neurochemical assumption according to which the choline uptake (not exclusively, however) controls the process, an independent choline oscillation is assumed. Therefore the kinetic equation is:

Applications

dx(t)fdt = -k1x(t) + k 4 y(t) + k 5x 2 (t)y(t) dz(t)/dt = k 1x(t)- k 2z(t) dw(t)fdt = k 2z(t) - k 3 w(t) dy(t)fdt = k 3 w(t)- k 4y(t)- k 5x 2 (t)y(t) + acos(wt + <p).

191

(7.16)

Numerical studies (Erdi & Barna, 1986a, b) gave the evidence of occurrence of 'abnormal' dynamic behaviour. Phase space plots and stroboscopic plots are obtained. The former are identified with two-dimensional projection of trajectories, the latter can be constructed by points taken at regular intervals of the period (figs 7.7 and 7.8).

Fig. 7.7 Phase plane diagram in different time regimes. (a) Approaching to the 'phantom attractor'. (b), (c) Intermediate regime, (d) Near the true attractor.

In general terms, many neural disorders are characterised by changing the 'normal' temporal patterns to 'abnormal' behaviour. This abnormal dynamics might be interpreted as a 'dynamical disease' occurring in a physiological control system operating within a range of control parameters (Guevara et a/., 1983).


/~::. " ,_ ... l

I 1.~.

'•·

(b)

/ .. . .

'::'

.

i. .. 1.". ·,.

·/···. '"-":.<,.~ .. ······-·

.. .. ·········· ...

~·

(c) I (d)

, •... ,_ · ..

··.:· .

. :.i) \.····· /

Fig. 7.8 Stroboscopic representation of perturbed variable. (a) Quasistable 'butterfly' shape. (b) Decay of the quasistable shape and start to tend to a new shape. (c) Tending to a new attractor. (d) Arriving at a fixed point of the Poincare plot.

7.4 Population genetics

I. Population genetics describes interactions among alleles. The system (in the simplest case it is a one gene locus) is composed of alleles A 1 , ••• , A •. The state of the system is characterised by the frequencies x 1, .•• , x. of the alleles. The frequency of the genotype A;Aj is 2x;x;; (i=Fj). Adopting the traditional assumptions the survival (or fitness) of A;Aj is W;; (w;; ~ 0; w;; = wj;). A scalar quantity

h

w*(t) = L w,.x,(t)x.(t) (7.17) r.j~ I

can be assigned to the whole population (mean fitness), and the normalisation LX; = I is valid. Selection equations in a deterministic framework can be· given by the differential equation:

h

dx;(t)fdt = X;(t)( L w;;xj(/) - w*(t)). j~l

(7.18)

Applications 193

The reconciliation of the Darwinian theory of natural selection and of Mendelian population genetics has been based on this model (Fisher, 1930; Wright, 1931).

According to Fisher's theorem, the mean fitness w* increases steadily along the trajectory of (7 .18). Technically it means that w* is Lyapunov function.

Kimura (1958) stated that the trajectories of (7.8) point in the direction of maximal increase of w*. Technically it would mean that (7 .18) is a gradient system. However, the system is not a gradient system, at least not in the traditional sense. Maximal increase implies that the direction of the trajectories is orthogonal to the contour lines of w*, and this is not the case in general. The 'spirit' of Kimura's maximum principle have been saved by a redefinition of orthogonality by Shahshahani ( 1979) who introduced a new Riemannian metric at every point of the state-space.

Denoting the state space by

• s.={x=(xl> ... ,x.); LX;= l,x) ~0}, (7.19)

i= I

the Shahshahani metric is given at pEs. by

(x, y)p = L (x;y;)/P; (7.20) i= I

(x andy are elements of the tangent space). Using this metric or (equivalently) making an appropriate transformation of co-ordinates, Akin (1979) presented a new geometrical framework of population genetics. One important prize from the work is the validity of Kimura's principle.

It can be shown more generally (Sigmund, 1984) that a differential equation

dx;(t)/dt = x;(/;(x(t)) - LX;(t)/;) (7.21) ;

is a Shahshahani gradient system, if and only if dx(t)/dt = f(x(t)) is a usual gradient system.

The basic selection model has been generalised in many different ways (Akin, 1979; but see Hofbauer, 1985). Mutations from Ai to A; with probability bii have been taken into account leading to the model:

(7.22)

For the case of equal mutation rate (i.e if bii is independent ofJ) the system is Shahshahani gradient system. For any other bii wii can be found such that the Shahshahani gradient property fails and periodic behaviour occurs via Hopf bifurcation. In terms of genetics this means that the frequency distribution of the alleles does not converge to a stable equilibrium point, but may exhibit oscillations. Even the occurrence of chaos seems probable in slightly


modified versions of the model. While the existence of oscillatory and even chaotic behaviour has become

more or less acceptable in the chemical and biomathematical communities, it is a serious challenge for population geneticists, since it contradicts their 'adaptive landscape' concept. However, if chemical kinetics could adopt the geometric approach of population dynamics, the interplay between the two disciplines could be productive.

2. Many stochastic models are used in population genetics to describe the effects such as random drift selective force, and mutation pressure (e.g. Karlin & McGregor, 1964; Crow & Kimura, 1970; Maruyama, 1977). Diffusion models are mostly used, and some of them can be interpreted in terms of the CCS model of chemical reactions. The models are in terms of the probability distribution of gene frequencies.

The tractable models are more or less restricted to the one locus situation. Let x, be the frequency of A genes at the time t. Pure random drift is superimposed by the reversible mutation

u A ¢a

The infinitesimal generator A is:

- _ x(l - x) d 2 d A = 4N dx2 + { v - (u + v)x} dx'

i.e. it is the infinitesimal generator of a simple, often-investigated diffusion of a single particle.

Many other models have been defined, and the problems of obtaining solutions are similar to those in the case of chemical reactions.

7.5 Ecodynamics

7.5.1 The theory of interacting populations

7.5.l.l Boulding ecodynamics

Observing the theory of dynamics of populations from the point of view of chemical kinetics, it is well-known that the celebrated Lotka-Volterra model has both chemical and ecological interpretations. It is quite reasonable that the formal theories of chemical kinetics and of 'mathematical' ecology are highly overlapping.

Ecodynamics (Boulding, 1978, 1983) motivated by the general system theory tries to become a general theory of ecosystems.

The ecosystem is the basic concept of Ecodynamics. It can be defined as a set of interacting populations of different species, the population of a species being defined as a set of objects, or elements of some kind, which are similar enough to be

Applications 195

interesting as an aggregate. This lands us immediately in the problem of taxonomy, which is the big skeleton in the closet science. If we get our taxonomies wrong, as the alchemists did-earth, air, fire and water are far too heterogeneous to be elementswe will never get very far. Physics and chemistry succeeded because they somehow got the right, or at least very useful, taxonomies, though even chemical elements exploded somewhat into isotopes and the Newtonian taxonomy of matter and energy got into trouble with Einstein. (Boulding, 1983)

Having determined the state variables by adopting an appropriate taxonomy, dynamic laws have to be defined:

To define an ecosystem, then, we have to take a bold and dangerous step and just assume that we can define significant species. We have the fact that the total number or quantity of a species, whether of hydrogen atoms, water molecules, horses, automobiles, or persons who believe in the second law of thermodynamics, follows the principle of what I have called the 'bathtub theorem'. It states that the total quantity or population of anything increases by the number of additions less the number of subtractions. Thus, the number of water molecules in a given area increases when hydrogen burns in oxygen (water molecules are 'born') and diminishes when the water molecule is dissociated (as, for instance, by an electric current).

The same principle goes for biological organisms of any kind, for human artefacts, and even for human ideas. Additions consist of births and in-migrants; subtractions of deaths and out-migrants. The rate of additions and subtractions to a population, however, is going to depend upon its own size and the size of the other populations of the ecosystem. (Boulding, 1983)

The 'ideology' of ecodynamics have partially been induced from the experiences of many ecological model experiments (and of course it was deduced from the principles of the general system theory). Some illustrations will be given.

7 .5.1.2 Compartmental ecokinetics

Compartmental analysis is a well-known technique of ecokinetics (an example: Blau eta/., 1975).

Let the ecosystem be a fish in an aquarium. The uptake and clearance of compounds by fish can be modelled in the simplest way by a twocompartment model treating the fish as a single homogeneous compartment. Assuming that:

(I) Uptake of chemical (through gills, skin etc.) can be taken into consideration by a single rate constant k 12 •

(2) The clearance of the chemical by different mechanisms can be characterised by a single rate constant k 21 •

(3) Constant rate of chemical addition and removal into and from the water compartment are involved.

(4) Kinetics is zero-order or first order.


Constant rate of chemical addition (a)

Water compartment

Constant rate chemical removal (r)

Fish compartment

Fig. 7.9 Two-compartment model of bioconcentration test ecosystem.

The two-compartment model is shown in Fig. 7.9. The kinetic differential equation system with the initial conditions is:

dc1{t)/dt =a- k 12c1(t) + k 21 c2 (t)- r dcz(t)Jdt = ktzCt(t)- kztCz(t) c1 (0) = Cw and c2 (0) = 0.

(7.23)

In practical cases dc1{t)/dt = 0 used to be assumed. The solution of the model can be easily given, but it is not very important for us at this moment.

The fish itself can be represented by two compartments where one of these compartments corresponds to a particular tissue in which the chemical compound may be preferentially concentrated. Two possible threecompartment models are shown in Fig. 7.10.

The two models differ only in that the second model a rate constant for clearance from the third compartment to the water compartment also was involved. Model discrimination by solving the inverse problem is a difficult task. 'Note that this discrimination takes place only among the set of postulated models. That is, the model selected may be the best of the originally postulated models but totally inadequate for characterizing the system' (Blau et a/., 1975). (Model discrimination and structural identifiability of compartment systems have been mentioned in Subsection 4.8.4).

7.5.1.3 Generalised Lotka-Volterra models

The examination of competitive interactions among different species has been one of the main topics of mathematical biology. The most often used mathematical model is still a generalisation of the Lotka- Volterra model; systems of polynomial ordinary differential equations expressible in terms of formal chemical reactions have also been investigated. The main problem is to find criteria for the coexistence of species. All species in the communities

Applications 197

Fish compartment

Metabolite on particular tissue

Fish compartments

Metabolite

Fig. 7.10 Three-compartment model of bioconcentration test ecosystem.

can coexist in three distinct ways; species may coexist at equilibrium, in persistent oscillation or in chaotic motion (see, for example, May & Leonard, 1975; Takeuchi & Adachi, 1983a, b; Freedman & Waltman, 1985). The generalised Lotka-Volterra system is:

dx;(t)fdt = X;(t)[b;- Laiix;(t)], i = I, 2, .. . n, j

and some of the most often used models are the:

Two-species-competing system:

dx 1(t)/dt = x 1(t)[l- x 1(t)- ax2 (t)] dx2 (t)fdt = x 2(t)[l- bx1(t)- x 2 (t)];

Two-prey, one-predator system:

dx1 (t)/dt = x 1(t)[l- x 1(t)- ax2(t)- ey(t)] dx2 (t)fdt = x 2 (t)[l- bx1(t)- x 2(t)- uy(t)] dy(t)/dt = y(t)[ -1 + dex 1 (t) + dux2 (t)];

Two-prey, two-predators system:

(7.24)

(7.25)

(7.26)


dx1 (t)fdt = x 1 (t)[I - x 1 (t)- ax2(t) - e 1y 1 (t) - e2y 2(t)] dx2(t)Jdt = x 2(t)[I- bx 1(t)- x 2(t)- u1YJ(t)- u2J2(t)] dYI (t)Jdt = y 1 (t)[ -I + d 1e1x 1 (t) + d1u1x 2 (t)) dy2 (t)/dt = Y2(t)[- I + d2e2x 1 (t) + d2u2x 2(t)].

(7.27)

Here X;(t), y;(t) are the population sizes for prey and predators, respectively, a> 0 and b > 0 represent competitive effects between two prey, e; > 0 and u; > 0 are coefficients of decrease of prey due to predation, and the d; are the transformation rates of i'h predator.

The possible equilibria for (7.26), for example, are denoted by (E+ + +), (E+ +0), (E+o+), (Eo++), (E+ 00), (E0 +0), Here (E+o+) denotes the equilibrium in which prey x 1 and predatory remain positive, and prey x 2 is extinct.

The two-species-competing system (7.25) cannot show oscillatory behaviour (in consequence of the Hanusse-Tyson-Light-P6ta theorem: it is different from the Lotka-Volterra model). The conditions of stability of equilibria were studied by Takeuchi & Adachi (l983b). Accordingly, the (E++) is globally stable if and only if a< I, b <I. (E+o) and (Eo+) are globally stable if and only if a~ I, b ~I and a~ I, b ~I respectively. For the case of a> I, b > I (E+o) and (E00) are locally stable.

Equation (7.26) is derived from (7.25) including predator. How does this inclusion affect the possibility and mechanism of coexistence? (Takeuchi & Adachi I983a, b) proved that the addition of a predator to a system with a stable equilibrium may give a bifurcation to a stable limit cycle at increasing rates of predation. They showed a + b < 2 implies (under certain technical conditions) the global stability of nonnegative (E + + +>· Supposing ab ~ I, in certain parameter regions Hopf bifurcation might occur. The assumption leading to a limit cycle can be further classified. According to numerical studies in the case of a > I, b > I the limit cycle can collapse into one of two single-prey equilibria when there is a further increase in the control parameter. In the case when a~ I, b ~ I (or a~ I, b ~ I) there is the possibility of turning into (spiral) chaos (Gilpin, I979).

The combined analytical and numerical analysis of the two-prey, twopredators, system (Takeuchi & Adachi, I983a, b) showed similar the following qualitative results: a stable equilibrium also can bifurcate to a stable limit cycle and the limit cycle turns into a nonperiodic oscillation of bounded amplitude with increasing cycle time.

Remarks

'Predator-mediated coexistence' of competing species has been found both in spatially homogeneous (see Takeuchi, I985) and in inhomogeneous systems (Hutson & Vickers, I983; Hutson & Moron, I985). The term means that the introduction of a predator may allow two prey to coexist which would not otherwise do so.

Applications 199

7 .5.1.4 The advantages of stochastic models: illustrations

Differences between deterministic and stochastic ecological models have been emphasised many times, explicity or implicitly. The main qualitative deviation in the behaviour of ecodynamic models can be interpreted in terms of survival and extinction.

u

a>1 b>1

Fig. 7.11 In the regions (A), (8), and (C) E+ ++exists. In (A) (or (B)) there exists at least one Hopf bifurcation parameter e* (or u*) for any straight line u = constant (or e = constant) if

where

is satisfied.

dfde(A 0 A 1 - ADI,~.• ¥- 0; or dfdu(A 0 A 1 - ADiu~u• ¥-0

A 0 = xf + x~. A, =(I - ab)xfx~ + du2x~y* + de*y*xf, A2 = d(e2 + u2 - (a+ b)eu)xfx~y*

Macroscopic ecological level equations have emerged from microscopic genetic equations, and there is a feedback from the macroscopic structure to modify the microscopic dynamics. The appearance and eventual stabilisation of a new mutant might have a crucial role in the evolutionary process.

Allen & Ebeling (1983) have argued for the necessity of a stochastic description of simple ecosystems connected to evolution. They considered interactions among a 'prey' species, X;, and a 'predator' species, Yk, where i and k denote phenotypes. 'Elementary' steps at the microscopic level modifying the number of interacting species:

(a) normal replication of species X;:

X;--+ X;+ I; with probability a;x;(l - x)N),


(b) capture and destruction of prey X; by predator Yk:

X;~ X;- I; with probability (s;k/V)X;Jk,

(c) normal reproduction of predator Yk:

Yk ~ Yk + 1; with probability (S;k/ V)x;yk,

(d) death of a predator Yk:

Yk ~ Yk - I; with probability DkYk,

(e) 'diffusion' of a prey or predator from the system:

X;~ X;+ I, Yk ~ Yk + I; with probability D;x;0 and Dky2,

(f) 'diffusion' of a prey or predator of the system:

X;~ X;- I, Yk - Yk ~ 1; with probabilities D;x; and Dkyk,

(g) 'abnormal' reproduction of X;:

X;Xj ~ X;Xj + I with probability Miix;(I - xjJN)

(h) 'abnormal' reproduction of the predator Yk:

YkYI ~ YkY1 + 1; with probability Mk;X;YJ! V.

To described the temporal evolution of the probability distribution of a new mutant population starting from a pre-existing situation a linearised birth and death equation can be derived:

dP(x2, t)Jdt = A(x?, yy .. . )((x2 - I)P(x2 - I, t) - x 2P(x2, t)) + D(x?, y? ... ) x ((x2 + I)P(x2 + I, t)- x 2P(x2, t)),

where x 2 denotes the numbers of the mutants, the function A contains all the processes tending to increase x 2 , linearised in x 2a, and the function D contains all the terms tending to decrease x 2 • Let us denote with A' and D' the value of these functions, in general. If A' < D' then the probability of extinction is I, while for A'> D', this probability·is D'JA'. The increase or decrease in efficiency of the mutant type is d = A' I D' - I. The probability of survival P,.,(t) is

d P,.,(t) = I + d- exp(- (A' - D')t) .

Evaluating this expression for a time t = n/A', where n is the number of reproduced generations during t:

d P,.,(n) = I + d- exp- (ndj(l +d))·

The general form of the function P,.,(n) is given by Fig. 7.12. Is shows that,

Applications 201

1--------------------------L Step function resulting from f deterministic analysis

Probability of survival

10% 20% 30% 40% SO% 60% 70% ao•to 90% 100% 110•t. d

Fig. 7.12 Probability of survival of a mutant of fitness d, for a number of generations, n.

while deterministic analysis results in very sharp distinction between favourable and unfavourable mutations, the stochastic analysis leads to a much 'smoother' picture.

Models of population growth in random environment have been constructed by inserting 'white noise' fluctuations in the deterministic growth equations (May, 1972; Capocielli & Ricciardi, 1974; Ricciardi, 1977; Nobile & Ricciardi, 1984a, b). The structure of these models is similar to those described in Section 5.8 ('external fluctuations') and we shall not discuss them here.

As was mentioned in Subsection 5.6.2, stochastic versions of the Lotka-Volterra model lead to qualitatively different results from the deterministic model. The occurrence of a similar type of results is not too surprising. A simple model for random predator-prey interactions in a varying environment has been studied, staring from generalised LotkaVolterra equations (De, 1984). The transition probability of 'extinction' is to be determined. The standard procedure is to convert the problem to a Fokker-Pianck equation (adopting continuous approximation) and to find an approximation procedure for evaluating the transition probabilities of extinction and of survival.

Stochastic models are appropriate when we are interested in the behaviour of mutant populations present in very small quantities only. In more general terms, the role of 'genetic drift' due to randomness during evolution has been emphasised (Crow & Kimura 1970) in the framework of 'neutral theory of evolution'.

The relative importance of natural selection and genetic drift, and more generally of the deterministi!= and stochastic factors in evolution. remains the most important unsolved problem in our understanding of the mechanisms that bring about biological evolution. (Dobzhansky eta/., 1977, p. 164)


7.5.2 An ecological case study

A relatively detailed example will be given to illustrate the methods offormal reaction kinetics in ecological modelling. (It is considered to be a case study, and the choice of this model is quite arbitrary.) The model is of the eutrophication process taking place in the largest lake in Central Europe, Lake Balaton, Hungary. There is a wide recreational area all around the lake, which is very popular with tourists from abroad and from within the country as well. The phosphorus input to the lake has greatly increased in recent years, and as a consequence of this the growth of the phytoplankton has also increased.

In order to understand is phenomenon, and to help to reverse the process of eutrophication, a modeling study was started in the late nineteenseventies.

In what follows we briefly describe the main characteristics of a deterministic and a stochastic model of the eutrophication process in Lake Balaton, using the concepts of formal reaction kinetics.

The phenomenon to be described here does not fit precisely into the framework of mass-action type reaction kinetics. However, the differences are of the same nature that are met within the area of chemical applications too.

First, one has time-dependent input (here: forcing) functions. Second, the rate of some of the elenentary reactions are not of the mass-action type. (Sometimes Michaells-Menten-type rate functions have been chosen, sometimes even more complicated ones consistent with present knowledge in ecology.

After a general introductory discussion of the stochastic model the two models will both be defined. Relevant, and ecologically meaningful, differences between the two models will be discussed at the end.

7.5.2.1 Arguments for a stochastic model

The exact definition of the model will be done very easily using the metalanguage of formal reaction kinetics. The stochastic model of the reaction to be defined below will be the stochastic model of the phenomena of eutrophication. Therefore the only questions worth treating here are:

(I) What are the special ecological reasons that make it plausible to use a stochastic model?

(2) Why are the arguments for the usual continuous time discrete state stochastic model so good for reaction kinetics here?

(3) What are the consequences of the stochastic model defined here, both from the scientific and from the management point of view?

Applications 203

Deterministic models (most commonly systems of nonlinear differential equations) have been used to model the dynamics of lake ecosystems (Park eta/., 1974; Chen & Orlob, 1975; Joergensen, 1976; Di Toro eta/., 1979). These models have been satisfactory to describe the main trends of ecological phenomena; however, deterministic models are not capable of reflecting the inherently random behaviour of ecosystems. This source of uncertainty is especially important in large multiparameter models.

The cases when there are qualitative differences between a deterministic model and the corresponding stochastic one have been enumerated previously. A single viewpoint should be emphasised here.

Models in general are aimed at describing the investigated phenomena, and making predictions of the behaviour of the system under different conditions. In terms of Schofield & Krutchkoff's (1974, p. I) statement: 'it is possible for the instantaneous pollution concentrations to be at ecologically dangerous levels, while the means or deterministic concentrations remain at acceptable levels.' We are interested in how far the evolution of a variable is reproducible under essentially the same conditions.

In general the method of confidence intervals (Zacks, 1971, Ch. I 0) may be suggested for approaching a problem of this kind. As it follows from the nature of the stochastic model, an increase in input load of phosphorus does not necessarily lead to an acceleration of the eutrophication process. The same input load, however, may lead to a major increase in algae biomass (as was probably the case in 1982).

It should be mentioned here that spatial inhomogeneities and fluctuations have been disregarded in the present model, although it is easy to incorporate some of them in the following-usual-way: let us divide the lake into four different basins and let us suppose that they are homogeneous. If linear diffusion is supposed to take place between them that can be modelled by first-order elementary reactions then we have a homogeneous kinetic model of a phenomenon taking place in an inhomogeneous medium. Here only the one-basin model will be treated.

Finally, let us mention an argument for the stochastic model coming from the endeavour to revitalise the science of ecological modelling. Large multiparameter systems aimed at a detailed description of lake eutrophication seem to have become less popular in recent years. This may either be the consequence of either the mathematical and computational problems involved or be due to the lack of measurements, or both. It is hoped that stochastic models constitute a possibility in this regard.

Very few stochastic lake models seem to exist in the literature. Ecological\ problems related to those investigated here are described stochastically by ! the following. tools: time series, random walk, diffusion processes, differential/ equations with random parameters (taken in the wide sense), and Markovia¢ pure jump processes.

As for the different types of stochastic models: time series, random walk


and diffusion models seem to be too simple to cope with the ecological details of a process. It is a question of judgement whether the fluctuation of the state variables are considered to be caused by the parameters (including initial conditions, external forcing functions and parameters in the narrow sense) or whether assumptions are made on the fluctuations of the state variables themselves. Here the second alternative has been chosen because of the importance of inner fluctuations. So one had to choose between stochastic differential equations (apparently never used in ecological modelling) and Markovian jump processes. The latter type was chosen because

(I) a certain amount of experience has accumulated in connection with these models in physics, in some areas of biology, and-last but not least-in reaction kinetics;

(2) randomness enters the picture without introducing further ecological parameters that require estimation (see, however, page 000);

(3) a model of this type corresponding to a deterministic model of the usual type (to a system of-non-mass-action-kinetic differential equations); can quickly be constucted;

(4) these models are easy to simulate (see Subsection 5.3.8 and Jernigan & Tsokos, 1980);

(5) estimation of the parameters from the measurements will be done more easily if such a model is adequate (cf. T6th & Erdi, 1977b; Erdi & Ropolyi, 1979).

7.5.2.2 A common description of the deterministic and stochastic models

Formal components and formal elementary reactions

The formal chemical components of the model are as shown in Table 7.1.

Table 7.1 Formal components of the lake eutrophication model.

Ecological meaning Notation

Autumn-spring phytoplankton Summer phytoplankton Autumn phytoplankton Blue-green algae Bacterioplankton Dead organic matter Dissolved inorganic phosphorus Dissolved inorganic nitrogen Organic matter in the sediment Exchangeable phosphorus in the sediment

A( I) A(2) A(3) A sa B OM p N s PS

Applications 205

Fig. 7.13 Vol'pert graph of the eutrophication process.

The structure of the model is usually visualised using the Vol'pert graph of the reaction shown in Fig. 7.13. The formal elementary reactions considered could have been given as a huge set of 40 elementary reactions-mainly with kinetics of the mass-action type. Instead of this formal and awkward definition only some of the formal elementary reactions will be discussed in detail here. As to the others the reader is referred to Herodek et a/. ( 1982) where all the necessary biological background is available, including the reasons for the selection of the forms of the rates of elementary reactions, the external forcing functions and the parameter values. We hope that the biological background there and the background in formal reaction kinetics accumulated from the present book is sufficient to exactly define the model.

First, a short description of the deterministic model is given.

The deterministic model

The deterministic model is the (nonautonomous, nonpolynomial) induced kinetic differential equation of the reactions in Fig. 7.13. This model was described in detail by Herodek et a/. (1982). Now we give a formal description of a small part of the model. As an example let us consider the time evolution of summer phytoplankton. Our assumption is that it takes part in the elementary reactions No. 2, 6, 13, 26, 34, therefore the equation for a2 is:

Here

206 Mathematical models o.fchemical reactions

R 2 (a 2 (t), p(t), n(t), t) = PMAX2 •TEMP2 (9(t))•U2 (p(t), n(t), /0 (t))•a 2 (t),

corresponding to the elementary reaction

A(2) + N + P-+ 2A(2) +(I - nr)•N +(I - pr)•P.

PM AX corresponds to the reaction rate constant, called maximal production rate in the present context, TEMP expresses a kind of dependence on time through temperature (a phenomenon common in nonisothermal reactions), it is called here temperature limitation, /0 is an external forcing function: it is the global radiation on the water surface, U2 is a joint limitation factor describing light- and nutrient-dependence of primary production, or to use the language of reaction kinetics: it expresses the deviation from mass-action type kinetics.

All the remaining rates can be (and have been) defined in the same way; see Herodek et a/. (1982).

The right-hand side of the induced kinetic differential equation consists of analogous terms. More precisely, each subprocess contributes, as usual, additively to the corresponding derivatives. Let us emphasise again that 9 and /0 are not state variables, they are forcing functions known from measurements.

Results obtained from the numerical solution of the differential equations are evaluated by Herodek et a/. (1982). It may be worth emphasising that recent results by Kutas & Herodek ( 1987) show the usefulness of the model in aiding decision. Comparisons between the two models will be given below.

The stochastic model

To define the stochastic counterpart of a deterministic model of a reaction, the elementary reactions and the reaction rates of the elementary reactions have to be known. As this has been given above, the stochastic model may be considered to have been defined. Several remarks still may be appropriate.

Defining the state-space as usual is quite realistic for the needs of reaction kinetics, because a change of state has a physical meaning: an elementary reaction has taken place and the change itself means changing by a single molecule. In the ecological context, how.ever, the unity of change has no natural biological meaning, it is an artificial unit to measure the changes. (This is the case with the deterministic model as well!) The meaning of this measure is arbitrary, and may be varied from one simulation to another. This is an extra parameter (or ten extra parameters if different components are measured in different units).

The model obtained is complicated enough not to be amenable to analytical investigations; therefore it had to simulated by the standard procedures described in Section 5.3. As the process is inhomogeneous in time

Applications 207

and measured forcing functions are to be taken into consideration, and some of the rates are not of the mass-action type, a separate simulation program has been written for this special problem in the language SIMULA '67 which is especially suitable for simulation purposes.

Qualitatively different consequences

Similarities and differences between the stochastic and deterministic models are as usual (approximate consistency in mean, decreasing relative variance with the increase of the number of the introduced 'functional particles', etc.). The most astonishing and ecologically relevant result found was that there is a marked difference between the stochastic mean and the deterministic result, and there is an even greater difference between realisations and the deterministic value of the biomass of blue-green algae. At the locations of maxima the maximum curve is ten times higher than the deterministic one. This reveals a very high instability of the system, not only in the model, but also in reality, as in two consecutive years with similar meteorological conditions and in a slightly different input situation the summer biomasses showed very high differences. This cannot easily be described by any deterministic model; however, if the measurements could be simulated with the deterministic model the same could be done with the stochastic one too.

It may be useful to know that simulation of a one-year-long period of this really complex phenomenon takes 2-3 minutes on an IBM 3031 computer, only three times as much as to simulate the deterministic model.

7.5.2.3 Exercise

I. Verify that the deterministic model by Herodek et a/. (1982) fulfils the conditions by Vol'pert (1972) which assure nonnegativity of the solutions starting from nonnegative initial values.

7.6 Aggregation, polymerisation, cluster formation

1. Many studies in different disciplines deal with kinetic phenomena where 'elementary units' (e.g. monomers) of a population interact to form greater structures. Cell aggregation, polymerisation showing gelation and an abstract growth model formulated in terms of chemical kinetics, will be presented here as characteristic examples.

2. The phenomenon of sorting out of embryonic cells has been modelled (Rogers, 1977) by adopting the celebrated Smoluchowski model. The original model described the coagulation of colloidal particles based on the principles of Brownian motion. Three main assumptions are postulated: the cells are discrete, randomly mobile and differentially cohesive and adhesive. Starting from a population containing two cell types, say a and b, assuming


that forces of adhesion of bb are essentially larger than for the other two, a model can be given for the temporal evolution of the concentration of different b-cell aggregates. Let c1 (t), c2 (t), ... denote the concentrations of single b-cells, double b-cell aggregates and so on. Assuming mass-action type interactions among the cells the system of differential equations can be derived:

'X

dck(t)/dt = 1/2 L auc;(t)cit) - ck(t) L akjcj(t). (7.28) i+j~k j~ I

It means that an increase in ck is due to the formation of groups of k b-cells by merging a group of i b-cells with a group of j b-cells and a decrease is given due to the formation of aggregates of size k + j (where j = 1, 2, ... ).

Equation (7.28) has no known closed form solution. In the particular case of aij =A =constant, introducing the transformed time t = At/2, and writing nk(t) instead of ck(t) we get

00

dnk(t)/dt = L n;(t)nj(t)- 2nk(t) L nj(t). i+j~k j~ I

The series of solutions of these equations is given by

nk(t) = n0 [(n0t/- 1/(1 + n0 t)k+ 1].

The successive maxima of nk occur at n0 = (k- 1)/2. (Fig. 7.14)

(7.29)

2. Reversible polymerisation is the combined process of formation and breakage of chemical bonds in a system of reaction polymers (see, for example van Dongen & Ernst (1984)).

314 N No

1/2

1/4

0

0 2 3 4

Fig. 7.14 The variation with time, N 0 , = CN0 t/2, of the number of k-aggregates of ceiJs of the internalising type, Nk. The successive maxima of Nk occur at No,= (k- 1)/2.

Applications 209

Reversible reactions

where an arbitrary ak denotes a cluster containing k monomeric units of a;. Two sets of rate constants are assigned, Kij describes the bimolecular coagulation process, while Fij is assigned to the unimolecular fragmentation process. The temporal evolution of the concentration ck of k-mers is given by the infinite set of differential equations:

dck(t)fdt = 1/2 L (Kijc;(t)cj(t)- Fijci+j(t)) i+ j~k X

- L (Kijck(t)cj(t) - Fkjck + j(t)) j~l

with the initial distribution

ck(O) = M()kl = ()kl·

(7.30)

(Here M is the initial concentration of monomers; for the sake of convenience M = I.)

Particular theories of polymerisation make different structural assumptions that result in different Kij and Fij. According to the detailed balance condition

Fijc; + j( oo) = kijcij( oo )cu( oo ). (7 .31)

The normalisation condition requires that the total fragmentation rate of a k-mer be proportional to the number of bonds:

1/2 L Fij = g(k- 1), i+ j~k

here g is the proportional constant. Since Fij and kij are interconnected, the definition of Kij implies the form of Fij as well as the stationary size distribution. The latter is expressed in terms of c1 ( oo) as:

(7.32)

where Nk is a combinatorial factor. The kind of mechanisms that lead to gelation characterised by infinite

clusters are not clear. The infinite cluster contains of course a finite fraction G(t) of the total mass (M(t) + G(t) = 1). Pre-gel and post-gel states separated by a gelation transition can be analysed in terms of a kinetic equation. Sol-gel transitions are similar to phase transition phenomena. It is not surprising that scale in variance principles elaborated in the theory of phase transition can be adopted for polymer systems. Modern percolation theory (see, for example Stauffer (1979)) offer a conceptual framework to treat cluster formation.

3. Certain growth models can also be formulated in terms of formal reactions. Let us consider a process in which an introduced seed grows on a lattice by creating particles at nearby sites, which then in turn grow.


Furthermore, multiply occupied sites are 'highly unfavourable'. The model scheme contains these two steps

e

2X;-+ X;.

Denoting with P({x;(t)}, t) = P(x 1 , x 2 , •• x., t) the probability of finding {x;} particles at time t the CDS model is

N

ofotP({x;(t)}, t) = k L xiP(x]' ... ' X; - I' ... XN, t) i.j= I

N

- P({x;}, t)) + e L X;(X; + I)P(x1 , ... , x., t) i= I

- X;(X;- I)P({x;}, t). (7.33)

The superiority of the stochastic over the deterministic approach was emphasised by Elderfield (1985), who calculated the parameters of fluctuation.

In the model of diffusion limited aggregation, growth occurs by capture of a diffusing particle. Two sets of components, X; (as previously) and diffusers (D;) are postulated. The reaction scheme is

D;+ ~-+X;+~ 2X;-+X; D;-+Di D;+X;-+X;.

The stochastic version of the model was studied by Elderfield (1985), who exactly calculated the parameters of the fluctuation.

Growth phenomena (e.g. cluster formation) are adequately described by cellular automata (e.g. Wolfram, 1986). Cellular automata are examples of systems constituting many identical components. While the components are simple, their collection is capable of complex dynamic behaviour. Cellular automata provide an alternative approach to spatia-temporal structure formation, as opposed to models based on differential equations.

A one-dimensional cellular automation consists of a line of sites, with each site carrying a value (0 and I in the simplest case). The value of each position is updated in discrete time steps by an identical deterministic rule depending on the neighbourhood of sites around it.

Reaction-diffusion systems, growth phenomena with inhibition effects and a wide variety of biological systems are well modelled by cellular automation models.

7. 7 Chemical circuits

I. General remarks. The theory of dynamic systems has three large fields of

Applications 211

applications: mechanics, chemical kinetics (and analogously population dynamics) and electronics. As we mentioned earlier (Chapter 2), chemical kinetics can not be 'mapped' into mechanics, though some analogous phenomena can be found.

Electronics applies 'linear' and 'nonlinear' elements for finding networks capable of showing complex behaviour. The question what can electronics offer to chemical kinetics (from the theoretical point of view) has two sides:

( 1) How can chemical phenomena be modelled by using electric and electronic devices?

(2) How can phenomena well-known in circuit theory be modelled in terms of chemical kinetics?

The first question was answered by 'Network Thermodynamics' (e.g. Oster et a/., 1973;. Schnakenberg, 1977; Peusner, 1985) adopting the 'bond graph' technique. This approach benefits from the formal correspondence between certain interpretations of nonequilibrium thermodynamics and of electrical network theory. Adopting the notions of 'chemical impedance', 'chemical capacity' and 'chemical inductance', chemical reactions as well transport processes can be represented by networks obeying Kirchhoff's current and voltage laws.

The second problem seems to be more interesting. Though ionic reactions and transport processes in solutions are much slower than electron transport in solids, the investigation of the realisability of'electric circuits' by chemical tools might be important from the point of view of theoretical biology. Furthermore, the possibility of constructing molecular computing devices can increase our interest in a theory of 'dynamic chemotronics'. A few concept introduced more or less in connection with circuits will be studied in terms of chemistry.

2. The term chemical amplification can be interpreted in at least two different senses. First, a trace concentration can be amplified by chemical methods to yield a higher order of magnitude of product concentrations. In contrast to the traditional analytical procedure, chemical amplification may be performed before (and not after) the measurement step. Second, the concept of amplification can be introduced as an abstract (and perhaps a bit artificial) analogy, where the x(t) concentrations are considered as the input signals, and f(x(t)) = dx(t)/dt are the output signals. The relationship of;/ox; > 0, expressing the occurrence of an autocatalytic reaction can be identified with 'positive feedback' phenomena.

Some methods cf chemical amplification can be used in analysis (Blaedel & Boguslaski, 1978).

Both inorganic and enzyme catalysis can be applied for carrying out sensitive and selective analysis by amplification. Another possibility is the use of 'cyclic reactions':


An interesting approach for achieving amplification is through a reaction mechanism that casts the analyte itself into a cycling role, so that a stoichiometric amount of product is produced and accumulated each time that the analyte is cycled or turned over. If the cycling is efficient without loss of the analyte to side reactions, and if the measured product is stable enough to permit accumulation, the amplification achieved by this approach may be large. To date, mainly cofactors for enzymecatalysed reactions have been amplified in this way, but in principle cycling is also generally applicable to other substances. (Blaedel & Boguslaski, 1978)

Searching for analogies between elements of electronic circuits and those of chemical reactions, Rossler ( 1974b) associated chemical concentrations to voltages across capacitors. Based on this approach he presented a (Michaelis-Menten-like) analogy of non-inverting as well inverting amplifiers.

3. A formal chemical analogy of linear RC-oscillators leading to harmonic oscillation was introduced (Seelig, 1971; Seelig & Gobber, 1971 ). A threecomponent system

dx(t)/dt = Kx(t) + j; x(t), j e IR 3

was introduced. As it is well-known, linear oscillators are structurally unstable and cannot be realised by physical devices.

4. The most famous electric oscillator, namely the van der Pol-Lienard oscillator given by

i(t) = u[y- x 3/3- x)] y(t) = - (1/u)x,

is not a mass-action type kinetic equation. The Brusselator model was qualified as a simple 'functional analogue' of the van der Pol equation, since it can also bifurcate to limit cycle oscillation. Furthermore, the periodically perturbed version of both oscillators might exhibit bifurcation from limit cycle to quasiperiodicity and even chaos (Cartwright & Littlewood, 1945; Tomita & Kai, 1979).

5. Multivibrators have three different classes. A bistable multivibrator has two stable states switching by perturbation of an external signal. A monostable multivibrator has one stable state; it is attained by every perturbation. An astable multivibrator has two locally stable states. They occur periodically without getting an external signal. Rossler (1972; but see 1974a, 1974b) suggested an astable multivibrator in terms of MichaelisMenten-like kinetics. Applying the 'inverse' of the usual approximative techniques of enzyme kinetics it can be decoded as (Erdi & T6th, 1976)

81-+ AI 2AI ~AI + 82 AI-+ A2 A2-+ 83 A 1 -:-A2 ~A3 A3 -+A1.

A realisation of its CDS model obtained by simulation experiments can be seen (Fig. 7 .15).

Applications 213

a,(t)

Fig. 7.15 Chemical multivibrator.

6. Logical circuits are one of the most important groups of digital devices. The fundamental logical elements are the AND, OR and NOT circuits. Combination of these elements can lead to complex logical networks.

The realisability of logical functions by chemical bistable systems was mentioned by Rossler (l974b):

The functions AND and OR are trivially realized by algebraic addition, in the presence of a threshold. Now any flip-flop is a threshold device. Hence any pair of 'convergent' reactions (leading to the same product) realizes these functions in a straightforward manner. If both influxes are required for the setting (or resetting, respectively) of the flip-flop, AND is realized, and if either is sufficient, OR.

7. The possible fabrication of a molecular electronic device (MED) resulted in serious interest in elaborating the principles of molecular computation (Conrad, 1985). Switching is possible even at molecular level by means of conformational changes. Three promising mechanisms of switching, namely electron tunnelling in short periodic arrays, soliton switching and soliton valving was reviewed (Carter, 1984).

7.8 Kinetic theories of selection

7.8.1 Prebiological evolution

7.8.1.1 Introductory remarks

Several, not completely disjoint phases of biological evolution can be distinguished:

(I) during chemical evolution the building blocks of biopolymers are formed spontaneously from small molecules;

(2) the building blocks just mentioned were self-organised into structures having the ability to exhibit self-replication;


(3) the evolution of species can be identified with the conventional subject of evolutionary theories.

The second phase, namely the self-organising mechanism of the molecular self-replication has been investigated by Eigen (1971) and his coworkers (Eigen & Schuster, 1979) unifying the detailed results of biochemical experiments with mathematical models of chemical kinetics. Theoretical studies have initially been motivated by the 'test-tube experiments' on RNA evolution (for an early review see Spiegelman (1971)). The molecular mechanism of RNA replication is still always being studied (Biebricher eta/., 1983).

Eigen's approach is based on the concepts of 'quasispecies' and 'hypercycles'. Quasispecies is considered as a set of molecules acting as a group. Not single molecules, but quasispecies are the target of evolution. Hypercycles are mentioned in the next Subsection.

7 .8.1.2 The hypercycle: the basic model

According to the founders of the concept (Eigen & Schuster, 1979, page v):

A. Hypercycles are a principle of natural self-organization allowing an integration and coherent evolution of a set of functionally coupled selfreplicative entitites.

B. Hypercyles are a novel class of nonlinear reaction networks with unique properties, amenable to a unified mathematical treatment.

C. Hypercycles are able to originate in the mutant distribution of a single Darwinian quasi-species through stabilization of its diverging mutant genes. Once nucleated hypercycles evolve to a higher complexity by a process analogous to gene duplication and specialization.

The original mathematical model is based on three assumed properties: 'metabolism', 'self-reproduction' and 'mutability'. The simplest deterministic equation derived is:

X;= (A;Q;- D;) + L W;kXk + F;. k~i

(7.34)

By X; the quantity of the population variable is described. Metabolism, according to its present interpretation spontaneous formation and decomposition, is expressed by the terms (A;Q;x;) and (D;x;) respectively. Selfreproduction is reflected by the fact that the rate of formation depends on x. Mutability is introduced by the quality factor Q;. It expresses the measure of the exactness of copying. The term L W;kxk describes the contribution to the

i~k

increase of X; by mutations. A conservation relation

IA;(1 - Q;)x; = L L w;kxk (7.35) ; ; k '#:;

Applications 215

holds for the 'error copies' obtained by mutation. F; describes the changes of the individuals by non-chemical interactions.

Specifically,

F; = F,(x;/ L xk) k

(7.36)

expresses that each species conributes to the total flow F, in proportion to its presence. Using the assumption of 'constant overall organisation'

and consequently

L xk = constant = en k

where E; =A;- D; is the 'excess productivity'. Introducing the notation

W;; = A;Q; - D;

for the (intrinsic) selective value, and

E*(t) = '[_Ekxk/'[_xk k k

(7.37)

(7.38)

(7.39)

for the average excess productivity the differential equation (7.34) can be transformed for the constraint of 'constant overall organisation' as follows:

Remarks

X;= (W;;- E*(t))x;(t) + L W;kxk(t). k.;. i

(7.40)

I. Generalisations of the model are possible within the framework of the hypercycle concept:

- kinetics of the 'metabolism' might have more complex nonlinearity; - self-reproduction can be realised via cyclic catalytic process; - more detailed description of the mutation can be given; - different kinds of constraint can be given instead of the 'constant overall

organisation': - discrete state-space stochastic models can be defined to describe pheno

mena, e.g. in finite populations.

Stochastic effects due both to finite population size and to other sources have been reviewed recently (Schuster (1985); for the details see McCaskill (1984) and Sigmund (1984). 'Random drift' or 'random selection' and 'random replication' are the most important stochastic phenomena which play a role in the evolution of polynucleotides. The former is important in small populations and in particular in the limit of kinetic degeneracy. The


term degeneracy means that there is no difference in the rate constants of the same type of reactions of the competing 'subunits'. Replication was considered as a multitype branching process, which is a particular class of stochastic process. This approach makes it possible to analyse the mutationevoked error propagation. (For further details see the reference given above.)

2. From the methodological point of view the hypercycle model is the most impressive example of the possibility of unifying detailed biochemical examinations and mathematical models of chemical reactions. Both the prebiotic relevance of the model and its mathematical structure were studied, sometimes critically (e.g. King, 1981; Miiller-Herold, 1983; Szathmiuy, 1984). It was suggested that the hypercyclic evolution is not as effective as conventional autocatalysis, and the model should be defined by multiple time singular perturbation theory.

Rossler (1981, l983b) presented the outline of the possibility of a 'deductive prebiology' based on the particular properties of chemical dynamic systems. As he emphasises, the creation of a chemical system with huge (say 1010 000) state variables is very easy, in contrast to electronic systems.

Just purchase some ten substances at a pharmacist's: a liter of distilled water, a gram of ammonium chloride, and some grams of sodium carbonate plus some other salts. This mixture, when shaken continually at 40"C, is bound to implement an equilibrium between some I 010 000 substances. With infinite observation time granted, the expectation value of all these state variables become non-zero. Mathematically this whole machine exists the very moment the ten substances have been poured together. (Rossler, 1983b)

Rossler argued that the very long-term qualitative dynamic behaviour of an evolving chemical system cannot be analysed. The attractor can be different from the present known attractors: dynamic systems describing evolution internally select a sequence of subsystems in which each generates a viable successor even if the environment changes during the process.

7.8.2 The origin of asymmetry of biomolecules

I. Substances that can exist in pairs of optical antipodes are found in nature almost exclusively only in one form: amino acids in proteins belong to the L

type, most of the sugars are o-type. The question of the formation of this asymmetry can be treated by two approaches: (i) biological asymmetry is the clear implication of certain well-defined physical asymmetries; (ii) asymmetry spontaneously emerges by some self-organising mechanism.

2. There are some physical factors which can be considered as possible candidates to cause biological asymmetry. The most often mentioned physical phenomenon is the weak parity violating interaction. It seems to be very appealing to explain biological asymmetry by the the general asymmetry of the physical world.

Applications 217

In a series of papers Kondepundi & Nelson (1983, 1984, 1985) have investigated the consequences of the energy difference A£ between Land D enantiomers causing differences also in the rate constants expressed by the factor g = AE/k8 T = (kLfk0 ) - 1. (The subscripts L and D of the rate constant k refer to the enantiomers; k8 is the Boltzmann constant.) The crucial point of this approach is the relative effect of the factor g and the fluctuations. The order of magnitude of g is about 10- 17• It is an open question whether small or perhaps very large volumes are necessary. Very different estimations are known for the time being, but g seems to be too small to imply biological asymmetry.

3. Theories suggesting the spontaneous formation of biological asymmetry are based on two-component chemical systems, which are symmetric for the two components (mirror-image reaction systems). Fluctuations in the initial conditions are amplified by the kinetic mechanism leading to macroscopic asymmetry. (For a review of critical evaluation of models for the amplification of chirality see Fajszi & Czege (1981).) The measure of the asymmetry can be expressed by

11 = (L - D)/(L + D).

One of the simplest models capable of chemical amplification is

dL(t)fdt = kLL(t)- r(L(t)- D(t)) dD(t)/dt = k 0 D(t) + r(L(t) - D(t)).

(7.41)

In this model there is no interaction among the enantiomers, r is the velocity of racemisation.

In case of kL = k0 = k the 11 versus g function is

11- g = (rfk) + j(rfk)2 + g 2 •

(7.42)

The simplest model taking into consideration heterochiral interactions was given by Franck ( 1953):

dL(t)/dt = (k - k 2D(t))L(t) dD(t)Jdt = (k - k 2 L(t))D(t).

(7.43)

The symmetric stationary state L* = D* = kjk2 is unstable; small differences in the initial conditions determine which component survives and which dies out. (For a similar but more complex reaction scheme see Seelig (1971).)

Since biological asymmetry due to physical asymmetry has not been verified, more realistic selection kinetic mechanisms are searched for to get a more detailed picture of the origin of optical chirality.

It may be useful to make comparisons (see Table 7.2) between the models of this chapter in order to see what we have done and obtained when applying the mathematical models of chemical reactions in some areas of science.

Table 7.2 Comparison of models.

Field of application Model type

Biochemical control Michaelis-theory Menten-like

equations

Neurochemical CDS fluctuations

Periodicity and ceo aperiodicity in neurochemical systems

Population genetics Deterministic selection equations

Ecokinetics ceo

Generalised ceo Lotka-Volterra

Methods

Qualitative, numerical

Numerical integration, phase space analysis

Stroboscopic method

Compartmental analysis

Bifurcation analysis

Results

Conditions for oscillation, periodic perturbation of oscillations may lead to chaos

Algorithm for calculating the transport coefficients

'Abnormal' temporal patterns occur due to the impairment of the control system

Examination of 'generalised' gradient systems

Model discrimination

Equilibrium --+ limit cycle --+ nonperiodic oscillation

Stochastic ecology CDS, stochastic differential equation

Eutrophication in CDS Lake Balaton

Aggregation of cells Infinite dimensional deterministic

Harmonic oscillation

Multi vibrators

ceo

CDS

Simulation

Analytic

Analytic

Simulation

Prebiological evolution

Deterministic, Qualitative

The origin of asymmetry of biomolecules

stochastic

Combination of deterministic and stochastic models

Shift in the extinction-survival property due to presence of small mutant populations

Explanation for the irregular behaviour of blue algae

Determination of duration necessary to reach the maximal degree of aggregation

Linear oscillators are structurally unstable

Selection of specific biomolecules

Kinetic mechanisms might lead to selection due to small fluctuations

References

AczeJ, J. (1966). Lectures on functional equations and their applications. Academic Press, New York.

Agladze, K. I. & Krinsky, V.I. (1984). On the mechanism of target pattern formation in the distributed Belousov-Zhabotinsky system. In: Selforganization, ed. V. I. Krinsky, pp. 147-9. (Springer Series in Synergetics, Vol. 28), Springer Verlag, Berlin.

Akin, E. ( 1979). The geometry of population genetics (Lecture Notes in Biomathematics, Vol. 31). Springer Verlag, Berlin.

Alberding, N., Frauenfelder, H. & Haenggi, P. (1978). Stochastic theory of ligand migration in biomolecules. Proc. Nat/. Acad. Sci. USA, 75, 26-9.

AHara, D. L. & Edelson, D. (1975). A computational modelling study of the low temperature pyrolysis of n-alkanes; mechanisms of propane, n-butane and npentane pyrolyses. Int. J. Chern. Kin., 7, 479-507.

Allen, P.M. & Ebeling, W. (1983). Evolution and the stochastic description of simple ecosystems. BioSystems, 16, 113-26.

Almasy, G. A., Veress, G. E., Vadnai, Sz. A., Ser, V. (1974). True property functions and multicomponent fractionation balances. Hung. J. Ind. Chern., 2, 117-48.

Aly, A. F., Rope, B. W. & Wise, J. J. (1965). Kinetics of xylene isomerisation, 77th AIChe National Meeting, Paper No. 566.

Amundson, N. R. (1966). Mathematical methods in chemical engineering. Matrices and their applications. Prentice-Hall, Englewood Cliffs, NJ.

Anan, Y. & Go, N. (1979). Solitary wave and spatially locked solitary pattern in a chemical reaction system. J. Theor. Bioi., 80, 171-83.

Andronov A. A., Lemitovich, J. A., Gordon, I. I. & Mayer, A. G. (1967). Bifurcation theory of dynamic systems on the plane. Nauka, Moscow (in Russian).

Anikonov, Yu. E. (1978). Several methods to study multidimensional inverse problems for differential equations, Nauka, Novosibirsk (in Russian).

Aranyi, P. (1976). Stochastic analysis of kinetics and equilibrium of systems consisting of several reacting molecules (Doctoral thesis), Budapest (in Hungarian).

Aninyi, P. & Toth, J. (1977). A full stochastic description of the Michaelis-Menten reaction for small systems. Acta Biochim. Biophys. Acad. Sci. Hung., 12, 375-88.

References 221

Arat6, M. (1982). Linear stochastic systems with constant coefficients. A statistical approach, Springer Verlag, Berlin.

Aris, R. (1963). Chern. Eng. Sci., 18, 553-Aris, R. (1964). Algebra of systems of second-order reaction. Ind. Eng. Chern. Fund.,

3, 28-64. Aris, R. ( 1965). Prolegomena to the rational analysis of systems of chemical reactions.

Arch. Rat/. Mech. Anal., 19, 81-99. Aris, R. ( 1968). Prolegomena to the rational analysis of systems of chemical reactions,

II. Some addenda. Arch. Rat/. Mech. Anal., 27, 356-64. Aris, R. (1969). Mathematical aspects of chemical reactions. Ind. Eng. Chern. Fund.,

61, 17-29. Aris, R. (1972). Mobility, permeability and the pseudo steady-state hypothesis.

Mathematical Biosciences, 13, (1-8). Aris, R. ( 1972). Some problems in the analysis of transient behaviour and stability of

chemical reactors. Adv. in Chemistry, 109, 578-604. Aris, R. ( 1979). Chemical reactors and some bifurcation phenomena. Ann. N. Y.

Acad. Sci., 316, 314-31. Aris, R. & Gavalas, G. R. (1966). On the theory of reactions in continuous mixtures.

Phil. Trans. Roy. Soc., London, A260, 351-93. Aris, R. & Mah, R. H. S. (1963). Independence of chemical reactions.lnd. Eng. Chern.

Fundamentals, 2, 90-4. Arneodo, A., Coullet, P. & Tresser, C. (1980). Occurrence of strange attractors in

three-dimensional Volterra equations. Phys. Letters, 79A, 259-63. Arneodo, A., Coullet, P., Peyraud, J. & Tresser, C. (1982). Strange attractors in

Volterra equations for species in competition. J. Math. Bioi., 14, 153-7. Arnold, L. (1973). Stochastic differential equations. Oldenbourg Verlag, Munich. Arnold, L. (1980). On the consistency of the mathematical models of chemical

reactions. In: Dynamics of synergetic systems, H. Haken (Springer Series in Synergetics, Vol. 6) pp. 107-18. Springer Verlag, Berlin.

Arnold, L. ( 1981 ). Qualitative theory of stochastic non-linear systems. In: Stochastic nonlinear systems, eds L. Arnold, & R. Lefever, pp. 100-13. Springer Verlag, Berlin.

Arnold, L. & Kliemann, W. (1981). Qualitative theory of stochastic systems, In: Probabilistic analysis and related topics, ed. A. T. Bharucha-Reid, Vol. 3. Academic Press, New York.

Arnold, L. & Kotelenez, P. (1981). Linear stochastic evolution equation models for chemical reactions. In: Stochastic nonlinear systems, eds L. Arnold & R. Lefever, pp. 174-84. Springer Verlag, Berlin.

Arnold, L. & Theodosopulu, M. (1980). Deterministic limit of the stochastic model of chemical reactions with diffusion. Adv. Appl. Prob., 12, 367-79.

Arnold, L., Horsthemke, W. & Lefever, R. (1978). White and coloured external noise and transition phenomena in non-linear systems. Z. Physik, 829, 367-73.

Arnold, L., Curtain, R. F. & Kotelenez, P. (1980). Nonlinear stochastic evolution equations in Hilbert space, Report No. 17 (June 1980), Forschungsschwerpunkt Dynamische Systeme, Bremen University.

Arnold, V. I. (1975). Ordinary differential equations, Nauka, Moscow (in Russian). Atlan, H. & Weisbuch, G. (1973). Resistance and inductance-like effects in chemical

reactions: influence of time delays. lsr. J. Chern., 1 I, 479-88. Avnir, D., Kagan, M. L., Korloff, R. & Peleg, S. (1984). Spatial structures formed by

chemical reactions at liquid interfaces: phenomenology, model simulations and pattern analysis. In: Non-equilibrium dynamics in chemical systems, C. Vidal & A. Pacault, (Springer Series in Synergetics, Vol. 27), pp. 118-21. Springer Verlag, Berlin.

222 References

Badii, R. & Politi, A. ( 1985). Statistical description of chaotic attractors: The dimension function. J. Stat. Phys., 40, 725-49.

Bailey, J. E. (1972). Lumping analysis of reactions in continuous mixtures, Chern. Eng. J., 3, 52-61.

Bailey, N. T. J. (1964). The elements of stochastic processes with applications to the natural sciences, John Wiley, New York.

Balakotaoiah, V. & Luss, D. (1982). Exact steady-state multiplicity criteria for 2 consecutive or parallel reactions in lumped-parameter system. Chern. Eng. Sci., 37, 433-45.

Baras, F. & Frankowicz, M. (1984). Transient bimodality: a new mechanism of stochasticity in explosive chemical reactions, In: Non-equilibrium dynamics in chemical systems, eds C. Vidal & A. Pacault (Springer Series in Synergetics, Vol. 27), p. 250. Springer Verlag, Berlin.

Barker, L. A. & Mittag, T. W. (1975). Comparative studies of substrates and inhibitors of choline transport and choline acetyltransferase. J. Pharmac. Exp. Theor., 192, 86-94.

Bartholomay, A. F. (1962a). Enzymatic reaction-rate theory: a stochastic approach. Ann. N. Y. Acad. Sci., 96 (4), 897-912.

Bartholomay, A. F. (1962b). A stochastic approach to chemical kinetics with application to enzyme kinetics. Biochemistry, 1 (2), 223-30.

Bastiansen, 0. (ed.) (1964). The law of mass action. A centenary volume 1864-1964, Universitetsforlaget, Oslo.

Bataille, J., Edelen, D. G. B. & Kestin, J. (1978). Nonequilibrium thermodynamics of the nonlinear equations of chemical kinetics, J. Non-Equilib. Thermodyn., 3, 153-68.

Bazsa, Gy. & Beck, M. (1971). Autocatalysis, autoinhibition, self-catalysis-selfinhibition.-Specific kinetic effects of reaction products and reactants. Kern. Kozlem., 36, 167-(in Hungarian).

Bellman, R. (1960). Introduction to matrix analysis. McGraw-Hill, New York. Benson, S. W. (1960). Foundations of chemical kinetics. McGraw-Hill, New York. Berding, C. & Harbich, T. ( 1984). On the dynamics of a simple biochemical control

circuit. Bioi. Cyb., 49, 209-19. Berge, P., Pomeau, Y. & Vidal, Ch. (1983). L'ordre dans le chaos. Vers une approche

deterministe de Ia turbulence. Herman, Paris. Berne, B. J. & Pecora, R. (1976). Dynamic light scattering. Wiley-lnterscience. Berne, B. J., Deutch, J. M. & Hynes, J. S. (1968). Light scattering from chemically

reacting mixtures. J. Chern. Phys., 49, 2864-5. Beumee, J. G. B., Eno, L. & Rabitz, H. (1985). Pointwise feature sensitivity analysis.

J. Comp. Phys., 57, 318-25. Biebricher, C. K., Eigen, M. & Gardiner, W. C. jr. (1983). Kinetics of RNA

replication. Biochemistry, 22, 2544-59. Billingsley, P. (1974). Statistical inference for Markov processes, Midway reprint.

University of Chicago Press, Chicago. Binder, K. (ed.) (1979). Monte Carlo method in statistical physics. Springer Verlag,

Berlin. Blaedel, N. J. & Boguslaski, R. C. (1978). Chemical amplification in analysis: a

review, Anal. Chern., 50, 1026-32. Blau, G. E., Neely, W. B. & Branson, D. R. (1975). Ecokinetic: A study of the fate

and distribution of chemicals in laboratory ecosystems. A/Che J., 21, 854-61. Blomberg, C. (1981). Some properties of stochastic equations for coupled chemical

reactions far from equilibrium. J. Stat. Phys., 25, 73-109. Blum, L. & Salsburg, Z. W. (1968). Light scattering from a chemically reactive fluid,

I. J. Chern. Phys., 48, 2292-300.

References 223

Blum, L. & Salsburg, Z. W. (1969). Light scattering from a chemically reactive fluid, II. J. Chern. Phys., 50, 1651- .

Bluman, G. W. & Cole, J. P. (1974). Similarity methods for differential equations. Springer Verlag, New York.

Bodenstein, M. (1913). Eine Theorie der photochemischen Reaktionsgeschwindigkeiten. Z. Phys. Chern. 85, 329-97.

Bodet, J. M., Vidal, C., Pacault, A. & Argoul, F. Experimental study of target patterns exhibited by the BZ reaction. In: Non-equilibrium dynamics in chemical systems, ed. C. Vidal & A. Pacault (Springer Series in Synergetics, Vol. 27), pp. 102-6. Springer Verlag, Berlin.

Bogdanov, Yu. S. & Syroid, Yu. B. (1983). Differential equations. Vysheishaya Shkola, Minsk (in Russian).

Boissonade, J., Roux, J. C., Saadaoui, H. & De Kepper, P. (1984). Inhomogeneities induced by imperfect stirring in a CSTR effect on bistability. In: Non-equilibrium dynamics in chemical systems, ed. C. Vidal, & A. Pacault (Springer Series in Synergetics, Vol. 27), pp. 172-7. Springer Verlag, Berlin.

Boiteux, A. & Hess, B. ( 1984). Experimental and theoretical studies of the regulatory hierarchy in glycolysis. In: Selforganization, ed. V. I. Krinsky (Springer Series in Synergetics, Vol. 28), pp. 240-54. Springer Verlag, Berlin.

Borgis, D. & Moreau, M. (1984a). On the stochastic theory of a bistable chemical reaction. Physica, 123A, 109-30.

Borgis, D. & Moreau, M. (1984b). Two cell stochastic model ofSchlogl reaction with small diffusional coupling, J. Stat. Phys., 34, 631-51.

Boudart, M. (1968). Kinetics of chemical processes. Prentice-Hall, Englewood Cliffs, NJ.

Boulding, K. E. ( 1978). Ecodynamics: A new theory of societal evolution. Sage Publications, Beverly Hills, CA.

Boulding, K. E. (1983). Ecodynamics, Interdisciplinary Sci. Rev., 8, 108-13. Bowen, J. R., Acrivos, A. & Oppenheim, A. K. (1963). Chern. Eng. Sci., 18, 177. Bowen, R. ( 1968). On the stoichiometry of chemically reacting materials. Arch. Rat/.

Mec-h. Anal., 29, 114-24. Bowen, R. ( 1969). The thermochemistry of a reacting mixture of elastic materials with

diffusion. Arch. Rat/. Mech. Anal., 34, 97-127. Bradley, J. N. (1975). Fast reactions. Clarendon Press, Oxford. Bray, W. C. (1921). A periodic reaction in homogeneous solution and its relation to

catalysis. J. Am. Chern. Soc., 43, 1262-Briggs, G. E. & Haldane, J. B. S. (1925). A note on the kinetics of enzyme action,

Biochem. J., 19, 338-9. Brinkley, S. R. jr. (1946). Note on the conditions of equilibrium for systems of many

constituents. J. Chern. Phys., 14, 563-4. Brockett, R. W. (1970). Finite Dimensional Linear Systems, Addison-Wesley, Reading

(MA). Brush, S. (1976). The kind of motion we call heat, Vols. I, 2. North Holland,

Amsterdam. Brush, S. ( 1982). Changes in the concept of time during the second scientific

revolution. In: Ludwig Boltzmann Gesamtausgable, Vol. 8, pp. 325-328. Akademische Druck und Verlagsanstalt, Graz.

Budnikov, P. B. & Gisling, A. M. (1965) Solid phase reactions. Izd. Ministerstva po Stroitelstvu, Moscow (in Russian).

Bulsan, A. R., Schieve, W. C. & Gran, R. F. (1978). Phase transitions induced by white noise in bistable optical system. Phys. Letters, 68A, 294-8.

Busse, H. G. (1969). A -spatial periodic homogeneous chemical reaction. J. Chern. Phys., 73, 750-

224 References

Callen, H. B. & Greene, R. F. (1952). On a theorem of irreversible thermodynamics. Phys. Rev., 86, 702-17.

Callen, H. B. & Welton, T. A. (1951). Irreversibility and generalized noise. Phys. Rev., 83, 34-40.

Capocielli, R. M. & Ricciardi, L. M. (1974). Growth with regulation in random environment, Kybernetik, 15, 145-57.

Caram, H. & Scriven, L. E.: (1976). Nonunique reaction equilibria in non-ideal system. Chern. Eng. Sci., 31, 163- .

Carter, F. L. (1984). The molecular device computer: point of departure for large scale cellular automata. Physica, IOD, 175-94.

Cartwright, M. L. & Littlewood, J. E. (1945). On nonlinear differential equations of the second order. J. London Math. Soc., 20, 180-9.

Chen, C. W. & Orlob, G. T. (1975). Ecologic simulation for aquatio ecosystems. In: Systems analysis and simulation in ecology, vol. III, ed B. R. Patten, pp. 476-588. Academic Press, New York.

Chen, Y. der (1976). Differentiation of channel models by noise analysis. Biophys. J., 16, 965-71.

Chen, Y. der (1978). Noise analysis of kinetic systems and its applications to membrane channels. Adv. Chern. Phys., 37, 67-97.

Christiansen, J. A. (1953). Advances in Catalysis, 5, 314-315. Clarke, B. (1974a). Graph theoretic approach to the stability analysis of steady state

chemical reaction networks. J. Chern. Phys., 60 (4), 1481-92. Clarke, B. ( 1974b ). Stability analysis of a model reaction network using graph theory.

J. Chern. Phys., 60 (4), 1493-501. Clarke, B. (1980). Stability of complex reaction networks. Adv. Chern. Phys. (ed. I.

Prigogine and S. A. Rice), 43, 1-215. Cobb, L. (1978). Stochastic catastrophe models and multimodal distributions. Behav.

Sci., 23, 360-74. Cobelli, C. & Romanine-Jacur, G. (1976). Controllability, observability and struc

tural identifiability of multi input and multi output biological compartmental systems. IEEE Trans. on Biomed. Systems, BME23 (2), 93-100.

Coddington, E. A. & Levinson, N. (1955). Theory of ordinary differential equations, McGraw-Hill, New York.

Coleman, B. D. (1964). Thermodynamics of materials with memory. Arch. Rat/. Mech. Anal., 17, 1-46.

Coleman, B. D. & Noll, W. (1960). An approximation theorem for functionals, with applications in continuum mechanics, Arch. Rat/. Mech. Anal., 6, 355-70.

Collet, P. & Eckmann, J. P. (1980). Iterated maps of the interval as dynamical systems, Birkhaeuser, Boston.

Conrad, M. (1985). On design principles for a molecular computer. Comm. ACM, 28, 464-80.

Costanza, V. & Scinfeld, J. H. (1981). Stochastic sensitivity analysis in chemical kinetics. J. Chern. Phys., 74, 3852-8.

Creel, C. L. & Ross, J. (1976). Multiple stationary states and hysteresis in a chemical reaction. J. Chern. Phys., 65, 3779-89.

Crow, J. F. & Kimura, M. (1970). An introduction to population genetics theory. Harper & Row, New York.

Csaszar, A., Jicsinszky, L. & Turanyi, T. (1981). Generation of model reactions leading to limit cycle behaviour. React Kinet. Catal. Lett., 18 65-71.

Csaszar, A., Erdi, P., Jicsinszky, L., Toth, J. & Turanyi, T. (1983). Several exact results on deterministic exotic kinetics. Z. phys. Chern. (Leipzig), 264, 449-63.

Czerlinski, (1966). Chemical relaxation. An introduction to the theory and application of stepwise perturbation. Edward Arnold, London; Dekker, New York.

References 225

Dai, L. S. (1979). On the existence, uniquene~s and global asymptotic stability of the periodic solution of the modified Michaelis-Menten mechanism. J. Di.ff. Eqs., 31, 392-417.

Dambrine, S. & Moreau, M. (198Ia). On the stationary distribution in a stochastic chemical process without detailed balance. J. Stat. Phys., 26, 137-48.

Dambrine, S. & Moreau, M. (198lb). Note on the stochastic theory of a self-catalytic chemical reaction, I. Physica, 106A, 559-73.

Dambrine, S. & Moreau, M. (1982). Note on the stochastic theory of a self-catalytic chemical reaction, II. Physica, 106A, 574-88.

Darvey, I. G. & Staff, P.-J. (1966). Stochastic approach to first order chemical reaction kinetics. J. Chern. Phys., 44, 990-7.

Darwin, C. ( 18 ). His life told in an autobiographical chapter and selected series of his published letters, ed. F. Darwin. London.

Dawson, D. A. (1972). Stochastic evolution equations. Mathematical Biosciences, 15, 287-316.

Dawson, D. A. (1975). Stochastic evolution equations and related measure processes, J. Multivari. Anal., 5, 1-52.

De, S. S. (1984). Random predator-prey interactions in a varying environment: extinction or survival, Bull. Math. Bioi., 46, 175-84.

Decroly, 0. & Goldbeter, A. (1982). Birhythmicity, chaos, and other patterns of temporal selforganization in a multiply regulated biochemical system. Proc. Nat/. Acad. Sci. USA, 79, 6917-21.

Decroly, 0. & Goldbeter, A. (1984a). Multiple periodic regime and final state sensitivity in a biochemical system. Phys. Lett. 105A, 259-62.

Decroly, 0. & Goldbeter, A. (1984b). Coexistence entre trois regimes periodiques stables dans un systeme biochimique a regulation multiple, C. R. A cad. Sci. (Paris), 298 (18), Serie II, 779-782.

Defay, R. (1931). Bull. Acad. Roy. Be/g., 17 (5), 940-De Felice, L. J. (1981). Introduction to membrane noise. Plenum Press, New York. De Kepper, P. & Horsthemke, W. (1978). Etude d'un reaction chimique periodique.

Influence de Ia lumiere et transitions induites par un bruit externe, C. R. A cad. Sci. (Paris), C287, 251-4.

Delbriick, M. (1940). Statistical fluctuations in autocatalytic reactions, J. Chern. Phys., 8, 120-4.

Del Castillo, J. & Katz, B. (1954). Quanta) components of the endplate potential. J. Physiol. (London), 124, 560-73.

DeMasi, A., Ferrari, P. A. & Lebowitz, J. L. (1985). Rigorous derivation of reaction diffusion equations with fluctuations, Phys. Rev. Lett., 55, 1947-9.

Demidowitsch, W. B.: (1966). Eine Verallgemeinerung des Kriteriums von Bendixson. Z. Angew. Math. u. Mech. 46, 145-6.

Denbigh, K. G. (1981) Three concepts of time, Springer Verlag, Berlin. DeRobertis, E. D.P. & Bennett, M.S. (1957). Some features of the submicroscopic

morphology of synapses in frog and earthworm. J. Biophys. Biochem. Cytol., 1, 47-58.

Deuflhard, P., Bader, G. & Nowak, U. (1981). Larkin-A software package for the numerical simulation of large systems arising in chemical reaction kinetics, In: Modelling of chemical reaction systems, eds K. H. Ebert, P. Deuflhard & W. Jaeger (Springer Series in Chemical Physics, Vol. 18), pp. 38-51. Springer Verlag, Berlin.

Dirac, P. A. M. (1929). Quantum mechanics of many-electron systems. Proc. Roy. Soc. London, A123, 243-65.

Di Toro, D. M. & van Straten, G. (1979). Uncertainty in the parameters and predictions of phytoplantkon models. IIASA WP 79-27.

Dobzhansky, Th., Ayala, F. J., Stebbing, G. L. Valentine, J. W. (1977). Evolution, W.

226 References

H. Freeman, San Francisco. Doob, J. L. (1953). Stochastic processes. John Wiley, New York. Dulac, H. (1937). Recherche des cycles Limites, C. R. Acad. Sci. (Paris), 1703-6. Dunant, Y., Israel, M., Lesbats, B. & Manaranche, R. (1977). Oscillation of

acetylcholine during nerve activity in the torpedo electric organ, Brain Res., 125, 123-40.

Dynkin, E. B. (1965). Markov processes, vols. 1-2, Springer, Berlin. Ebeling, W. (1978). Nonequilibrium transitions and stationary probability distri

butions of stochastic processes. 68A, 430-Ebeling, W. & Engel-Herbert, H. (1980). The influence of external fluctuations on

self-sustained temporal oscillations. Physica, 104A, 378-86. Ebeling, W., Herzel, H., Richert, W. & Schimansky-Geier, L. (1986). Influence of

noise on Dulling-Van der Pol oscillator. Z. Angew. Math. u. Mech. 66, 141-6. Ebeling, W. & Malchow, H. (1979). Bifurcations in a bistable reaction-diffusion

system. Ann. Physik, 7, Folge, 36, 121-34. Ebeling, W. & Schimansky-Geier, L. (1979). Stochastic dynamics of a bistable

reaction system. Physica, 98A, 587-600. Ebert, J. (1981). A sensitive transitive closure algorithm. lnf Proc. Letters, 12, 255-8. Ebert, K. H., Deuflhard, P. & Jaeger, W. (eds) (1981). Modelling of chemical reaction

systems (Springer Series in Physical Chemistry, Vol. 18). Springer-Verlag, Berlin. Edelen, J. (1973). Asymptotic stability, Onsager fluxes and reaction kinetics. Int. J.

Engng. Sci., 11, 819-39. Edelson, D. (1976). A simulation language and compiler to aid computer solution of

chemical kinetic problems. Comput. Chem., 1, 29-33. Edelson, D., Field, R. J. & Noyes, R. M. (1975). Mechanistic details of the

Belousov-Zhabotinskii oscillations. Int. J. Chem. Kin., 7, 417-32. Edelstein, B. (1970). A biochemical model with multiple steady states and hysteresis.

J. Theor. Bioi., 29, 57-62. Eigen, M. (1971 ). Selforganization of matter and the evolution of biological

macromolecules,' Naturwissenschaften, 58, 465-523. Eigen, M. & De Mayer, L. (1963). In: Techniques of organic chemistry, 2nd edn, eds S.

L. Friess & B. S. Lewis, vol. VIII, Part II. Interscience, New York. Eigen, M. & Schuster, P. (1979). The hypercycle. A principle of natural self-

organization. Springer Verlag, Berlin. Einstein, A. (1905). Ann. Physik, 17, 549-Einstein, A. (1910). Ann. Physik, 3, 1275-Elderfield, D. (1985). Field theories for kinetic growth models. J. Phys. A, Math.

Gen., 18, L773-80. Emanuel, N. M. & Knorre, D. G. (1969). A course on chemical kinetics, Vysshaya

Shkola, Moscow (in Russian). Engel, W. G. (1979). Lagrange-Hamilton's formalism and the chemical reaction, I.

Lagrange's equation of motion for a single reaction. Hamiltonian treatment. The concept of chemical inertia, An. Acad. brasil. Cienc., 51, 195-201.

Engel, W. G. (1980). Lagrange-Hamilton's formalism and the chemical reaction, II. The differential equations of chemical kinetics. Chemical Lagrangian functions and inertia. Chemical inertia and chemical affinity. Chemical resistance. Catalysis. An. Acad. brasil. Cienc., 52, 437-43.

Engel, W. G. (1982). Lagrange-Hamilton's formalism and the chemical reaction, III. Simultaneous reactions, conservative and nonconservative simultaneous reaction systems. The chemical field and the chemical phase space. An. A cad. brasil. Cienc., 54, 27-33.

Engel-Herbert, H., Ebeling, W. & Herzel, H. (1985). The influence of fluctuations on sustained oscillations, In: Temporal order (Proc. Symp. Oscill. Heter. Chern. Bioi.

References 227

Syst., Bremen, Germany, 1984), (Springer Series in Synergetics, Vol. 29), pp. 144-52. Springer Verlag, Berlin.

Epstein, I. R. (1979). Coexistence, competition and hypercyclic interaction in some systems of biological interest, Biophys. Chern., 9, 245-50.

Epstein, I. R. (1983). Oscillations and chaos in chemical systems. Physica, 7D, 47-56. Erdi, P. (1983). Hierarchical thermodynamic approach to the brain. Int. J. Neurosci.,

20, 193-216. Erdi, P. & Barna, G. (1984). Self-organizing mechanism for the formation of ordered

neural mappings. Bioi. Cyb., 51, 93-101. Erdi, P. & Barna, G. (1985). Self-organization of neural networks: noise induced

transition. Phys. Letters, 107A, 287-90. Erdi, P. & Barna, G. (1986a). Pattern formation in neural systems, I. Autorhythmic

ity, entrainment, quasiperiodicity and chaos in neurochemical systems. In: Cybernetics and systems '86, ed. R. Trappl, pp. 335-42.

Erdi, P. & Barna, G. (1987). Self-organization in nervous systems. Some illustrations. In: Mathmatical topics in population biology, morphogenesis and neurosciences. Lecture Notes in Biomathematics, Vol. 71. Springer Verlag, Berlin.

Erdi, P. & Ropolyi, L. (1979). Investigation of transmitter-receptor interactions by analyzing postsynaptic membrane noise using stochastic kinetics. Bioi. Cyb., 32, 41-5.

Erdi, P. & T6th, J. (1976a). Stochastic reaction kinetics-'Non-equilibrium thermodynamics of the state spaces?' React. Kinet. Catal. Lett., 4, 81-5.

Erdi, P. & T6th, J. (1976b). On the stochastic formulation of the thermodynamics of the chemical reaction. A kemia ujabb eredmenyei, 31, 177-298 (in Hungarian).

Erdi, P. & T6th, J. (1977). On the theory of reacting mixtures. In: Proc. 3rd Conf. Appl. Chern., Veszpn!m, pp. 1-8.

Erdi, P. & T6th, J. (1979). Some comments on Prigogine's theories, React. Kinet. Catal. Lett., 11, 371-5.

Erdi, P. & T6th, J. (1981). Oscillatory phenomena at the synapse, Adv. Physiol. Sci. (Math. Comp. Methods in Physiol.), 34, 113-21.

Erdi, P., Sipos, T. & T6th, J. (1973). Stochastic simulation of complex chemical reactions by computer. Magy. Kem. Foly., 79, 97-108 (in Hungarian).

Erdi, P., T6th, J. & Hars, V. (1981). Some kinds of exotic phenomena in chemical systems. Colt. Math. Soc. Janos Bolyai 30, Vols. I, II. North Holland, Amsterdam, pp. 205-29.

Erickson, R. V. (1970). Functions of Markov chains. Ann. Math. Statist., 41,843-50. Erugin, N. P. (1972). Reader in differential equations, Nauka i Tekhnika, Minsk (in

Russian). Escher, C. (1981). Bifurcation and coexistence of several limit cycles in models of

open two-variable quadratic mass-action systems. Chern. Phys., 63, 337-48. Fajszi, Cs. & Czege, J. (1981). Critical evaluation of mathematical models for the

amplification of chirality. Origin of Life, 11, 143-62. Farkas, H., Kertesz, V. & Noszticzius, Z. (submitted). Explodator and bistability.

React. Kinet. Catal. Lett. Farkas, H. & Noszticzius, Z. (1985a). Generalized Lotka-Volterra schemes and the

construction of two-dimensional Expodator cores and their Liapunov-function via critical Hopf bifurcations. J. Chern. Soc. Faraday 2, 81, 1487-505.

Farkas, H. & Noszticzius, Z. (1985b) Use of Liapunov-function in dissipative and explosive models. Ber. Bunsenges. Phys. Chern., 89, 604-5.

Farkas, M. (ed.) (1981). Qualitative theory of differential equations. Coli. Math. Soc. Janos Bolyai 30, Vols. I, II. North-Holland, Amsterdam.

Farkas, M. & Hatvani, L. (eds) (1975). Differential equations. Coli. Math. Soc. Janos Bolyai 15. North Holland, Amsterdam.

228 References

Fatt, P. & Katz, B. ( 1952). Spontaneous subthreshold activity at motor nerve endings. J. Physiol. (London), 117, 109-28.

Feher, G. & Weismann, M. (1973). Fluctuation spectroscopy: determination of chemical reaction kinetics from the frequency spectrum of the fluctuations, Proc. Nat/. Acad. Sci. USA, 70, 870-5.

Feinberg, M. (1972a). 1-41. On chemical kinetics of a certain class. Arch. Rat/. Mech. Anal., 46, 1-41.

Feinberg, M. (1972b). Complex balancing in general kinetic systems. Arch. Rat/. Mech. Anal., 49, 187-94.

Feinberg, M. ( 1977). Mathematical aspects of mass action kinetics. In: Chemical reactor theory: A review, eds N. Amundson & L. Lapidus, pp. 1-78. Prentice-Hall, Englewood Cliffs, NJ.

Feinberg, M. (1980). Chemical oscillations, multiple equilibria, and reaction network structure. In: Dynamics and modelling of reactive systems, eds W. Stewart, W. H. Ray & C. Conley, pp. 59-129. Academic Press, New York.

Feinberg, M. (1981). Reaction network structure, multiple steady states, and sustained composition oscillations: A review of some results. In: Modelling of chemical reaction systems, eds K. H. Ebert, P. Deuflhard & W. Jaeger, (Springer Series in Chemical Physics, Vol. 18), pp. 56-68. Springer Verlag, Berlin.

Feinberg, M. Lectures on chemical reaction networks, to be issued as a technical report of the Mathematics Research Center, University of Wisconsin-Madison.

Feinberg, M. & Horn, F. J. M. (1974). Dynamics of open chemical systems and the algebraic structure of the underlying reaction network, Chem. Eng. Sci., 29, 775-87.

Feinberg, M. & Horn, F. J. M. (1977). Chemical mechanism structure and the coincidence of the stoichiometric and kinetic subspaces. Arch. Rat/. Mech. Anal., 66, 83-97.

Field, R. J. & Burger, M. (eds) (1984). Oscillations and travelling waves in chemical systems, John Wiley, New York.

Field, R. J. & Noyes, R. M. (1974). Oscillations in chemical systems, IV. Limit cycle behaviour in a model of a real chemical reaction, J. Chem. Phys., 60, 1877-84.

Field, R. J., Koros, E. & Noyes, R. M. (1972). Oscillations in chemical systems, II. Thorough analysis of temporal oscillations in the Ce-BrO-malonic acid system. J. Am. Chem. Soc., 94, 8649-64.

Fife, P. C. (1976). Pattern formation in reacting and diffusing systems, J. Chem. Phys., 64, 554-64.

Fife, P. C. (1979). Mathematical aspects of reacting and diffusing systems, Lecture Notes in Biomathematics, Vol. 28, Springer Verlag, Berlin.

Filippov, A. F. (1979). Problems on ordinary differential equations, 5th edn. Nauka, Moscow (in Russian).

Fisher, R. A. (1958). The genetical theory of natural selection, 2nd edn. Dover, New York. (1st edn.: 1930.)

Fisher M. E. (1974). Renormalization group in the theory of critical behaviour. Rev. Mod. Phys. 46, 597.

Fraikin, A. (1984). Stochastic analysis of a complex bifurcation, In: Non-equilibrium dynamics in chemical systems, eds C. Vidal & A. Pacault. Springer, Berlin.

Fraikin, A. & Lemarchand, M. (1985). Stochastic analysis of a Hopf bifurcation: Master equation approach. J. Stat. Phys., 41, 531-51.

Franck, U. (1978). Chemical oscillations. Ang. Chemie, 17, 1-15. Frank, F. C. (1953). Biochim. Biophys. Acta, 11, 453- . Frank-Kamenetskii, D. A. (1940). Conditions of applicability of Bodenstein's

method in chemical kinetics. Zh. Fiz. Khimii, 14 (5/6), 695-700. Frank-Kamenetskii, D. A. (1947). Diffusion and heat transfer in chemical kinetics. Izd.

References 229

Akad. Nauk. SSSR. Moscow (in Russian). Frankowicz, M. (1984). Relaxation processes in spatially distributed bistable chem

ical systems: deterministic and stochastic analysis. In: Non-equilibrium dynamics in chemical systems, eds C. Vidal & A. Pacault (Springer Series in Synergetics, Vol. 27}, p. 232. Springer Verlag, Berlin.

Frankowicz, M. & Gudowska-Nowak, E. (1982). Stochastic simulation of a bistable chemical system: The two-box model. Physica, 116A, 331-44.

Frankowicz, M., Malek-Mansour, M. & Nicolis, G. (1984). Stochastic analysis of explosive behaviour: a qualitative approach. Physica, 123A, 237-45.

Frankowicz, M. & Nicolis, G. (1983). Transient evolution towards a unique stable steady state: stochastic analysis of explosive behaviour in a chemical system. J. Stat. Phys., 33, 595-609.

Fraser, J. T. (1972). The study of time. Springer Verlag, Berlin. Fraser, J. T. (1978). The study of time, III. Springer Verlag, Berlin. Fraser, J. T. & Lawrence, W. (1975). The study of time, II. Springer Verlag, Berlin. Freedman, H.l. & So, J. W.-H. (1985). Global stability and persistence of simple food

chains. Mathematical Biosciences, 76, 69-86. Freedman, H. I. & Waltman, P. (1985). Persistence in a model of three competitive

populations. Mat. Biosci., 73, 89-101. Frehland, E. (1982). Stochastic transport processes in discrete biological systems.

Springer Verlag, Berlin. Ganapathisubramanian, N. & Noyes, R. M. (1982). A discrepancy between experi

mental and computational evidence for chemical chaos, J. Chern. Phys., 76, 1770-4.

Ganapathishubramanian, N. & Showalter, K. (1983). Critical slowing down in the bistable iodate-arsenic (III) reaction. J. Phys. Chern., 87, 1098-9. (Additions and corrections: ibid., p. 4014.

Ganapathisubramanian, N. & Showalter, K. (1984a). Bistability, mushrooms, and isolas. J. Chern. Phys., 80, 4177-84.

Ganapathisubramanian, N. & Showalter, K. (1984b). Washout effects in pumped tank reactors. J. Am. Chern. Soc., 106, 816-17.

Gans, P. J. (1960). Open first-order processes. J. Chern. Phys., 33, 691-4. Gaponov-Grekhov, A. V. & Rabinovich, M. I. (1983). Nonstationary structures

Chaos and order, Synergetics of the brain, Proc. Int. Symp., Schloss Elmau, Bavaria, 1983 (Springer Series in Synergetics, Vol. 23), pp. 227-37. Springer Verlag., Berlin.

Garcia-Colin, R. S., Novarra, 0. & Bernardo, M. (1973). The many-body approach to chemical kinetics. Rev. Mexicone de Fisica, 22, 351-73.

Gardiner, C. W. (1983) Handbook of stochastic methods for physics, chemistry, and the natural sciences (Springer Series in Synergetics, Vol. 13). Springer Verlag, Berlin.

Gardiner, C. W. & Chaturvedi, S. (1977). The Poisson representation: a new technique for chemical master equations. J. Stat. Phys., 17, 429-68.

Gaspard, P. & Nicolis, G. (1985). Chaotic dynamics, Physica/ia Magazine, 7, 151-95. Gavalas, G. R. ( 1968). Nonlinear differential equations of chemically reacting systems.

Springer Verlag, Berlin. Gear, C. W. (1971). Numerical initial value problems in ordinary differential equations.

Prentice-Hall, Englewood Cliffs, NJ. Geiseler, W. & Bar-Eli, K. (1981). Multistability in flow systems of the

Belousov-Zhabotinskii and related systems. In: Modelling of chemical reaction systems, eds K. H. Ebert, P. Deufthard & W. Jaeger (Springer Series in Chemical Physics, Vol. 18), pp. 268-75. Springer Verlag, Berlin.

Georgakis, C. & Aris, R. (1975). Diffusion, reaction and the pseudosteady-state hypothesis. Math. Biosci., 25, 237-58.

230 References

Gibbs, J. W. (1876). On the equilibrium of heterogeneous substances, Trans. Connect. Acad., 3, 108-248.

Gibbs, J. W. (1948). The collected works of Gibbs J. W., (Vol. 12). Yale University Press, New Haven, CT.

Gierer, A. (1977). Biological features and physical concepts of pattern formation exemplified by hydra. Curr. Top. Dev. Bioi., 11, 17-59.

Gierer, A. & Meinhardt, H. (1972). A theory of biological pattern formation. Kybernetik, 12, 30-9.

Gikhman, I. 1., & Skorohod, A. V. (1974). The theory of stochastic processes, Vols I-III, Springer Verlag, Berlin.

Gillespie, D. T. (1976). A general method for numerically simulating the stochastic time evolutions of coupled chemical reactions. J. Chern. Phys., 22, 403-34.

Gillespie, D. T. (1977). Exact stochastic simulation of coupled chemical reactions. J. Phys. Chern., 81, 2340-65.

Gillespie, D. T. ( 1978). Monte Carlo simulation of random walks with residence time dependent transition probability rates. J. Comput. Physics, 28, 395-407.

Gillespie, D. T. ( 1979). A pedestrian approach to transitions and fluctuations in simple nonequilibrium chemical systems. Physica, 95A, 69-103.

Gillespie, D. T. (1980). Transition time statistic in simple bi-stable chemical systems, Physica, lOlA, 535-51.

Gilipin, M. (1979). Spiral chaos in a predator-prey model, Amer. Naturalist, 113, 306-8.

Glansdorff, P. & Prigogine, I. (1971). Thermodynamic theory of structure, stability and fluctuations. Wiley-lnterscience New York.

Glass, L. (1977). Global analysis of nonlinear chemical kinetics. In: Statistical mechanics, Part B: Time-dependent processes, (Modern theoretical chemistry, Vol. 6), ed B. J. Berne, pp. 311-49. Plenum Press, New York.

Glasstone, S. K., Laidler, J. & Eyring, H. (1941). The theory of rate processes, McGraw-Hill, New York.

Goh, B. S. (1975). Stability, vulnerability and persistence of complex ecosystems, Eco/. Mod., 1 105-16.

Goldbeter, A. & Caplan, S. R. (1976). Oscillatory enzymes, Ann. Rev. Biophys. and Bioenerg., 5, 449-76.

Goldbeter, A. & Decroly, 0. (1983). Temporal self-organization in biochemical systems: periodic behaviour vs. chaos, Am. J. Physio/., 245, R478-83.

Goldbeter, A. & Nicolis, G. (1976). An allosteric enzyme model with positive feedback applied to glycolytic oscillations. In: Progress in Theoretical Biology, ed R. Rosen, Vol. 4, pp. 65-160. Academic Press, New York.

Goldbeter, A. & Segel, L.A. (1977). Unified mechanism for relay and oscillation of cyclic AMP in Dictyostelium discoideum, Proc. Nat/. Acad. Sci. USA, 74, 1302-15.

Golden, S. (1969). Quantum statistical foundation of chemical kinetics. Clarendon, Oxford.

Gordon, P. G. (1969). Error bounds and spectral densities. In: Stochastic processes in chemical physics, ed K. Shuler, pp. 79-99. John Wiley, New York.

Goselle, U. M. (1984). Reaction kinetics and diffusion in condensed media. Progr. React. Kinet., 13 (2).

Grabert, H., Haenggi & P., Oppenheim, I. (1983). Fluctuations in reversible chemical reactions, Physica, 117A, 300-16.

Gray, P. & Scott, S. K. (1983a). Autocatalytic reactions in the isothermal, continuous stirred tank reactor. Isolas and other forms of multistability. Chern. Eng. Sci., 38, 29-43.

Gray, P. & Scott, S. K. (1983b). Absence of restrictions on permissible path for changing the efficiencies of operation of flow reactors. J. Phys. Chern., 87, 1835-8.

References 231

Gray, P. & Scott, S. K. (1985). Sustained oscillations and other exotic patterns of behaviour in isothermal reactions. J. Phys. Chern., 89, 22-32.

Greene, R. F. & Callen, H. B. (1951). On the formalism of thermodynamic fluctuation theory. Phys. Rev., 83, 1231-5.

Guevara, M. R., Glass, L., Mackey, C. & Shries, A. Chaos in neurobiology, IEEE Trans. Systems. Man, and Cybernetics, SMC-13, 790-7.

Guldberg, C. M. & Waage, P. (1867). Etudes sur /es affinites chimiques, Bregger & Christie, Christiania.

Haag, G. (1978). Transition factor method for discrete master equations and applications to chemical reactions. Z. Physik B, 29, 153-9.

Haag, G. & Haenggi, P. (1979). Exact solutions of discrete master equations in terms of continued fractions, Z. Physik B, 34, 411-17.

Haag, G. & Haenggi, P. (1980). Continued fraction solutions of discrete master equations not obeying detailed balance, II., Z. Physik B, 39, 269-79.

Haenggi, P., Grabert, H., Talkner, P. & Thomas, H. (1984). Bistable systems: master equation versus Fokker-Pianck modeling, Phys. Rev., 29, 371-8.

Hahn, H. S., Nitzan, A., Ortoleva, P. & Ross, J. (1974). Threshold excitations, relaxation oscillations, and effect of noise in an enzyme reaction. Proc. Nat/. A cad. Sci. USA, 71, 4067-71.

Haken, H. ( 1977). Synergetics. An introduction. Springer Series in Synergetics. Springer Verlag, Berlin.

Haken, H. (1983). Advanced synergetics: instability, hierarchies of self-organizing systems and devices (Springer Series in Synergetics). Springer Verlag, Berlin.

Hammersley, J. M. & Handscomb, D. C. (1964). Monte Carlo methods. Methuen, London.

Hangos, K. M. & T6th, J. (1988). Maximum likelihood estimation of reaction rate constants. Comp. Chern. Eng. 12.

Hanusse, P. (1972). De l'existence d'un cycle limite dans l'evolution des systemes chimiques ouverts. C. R. Acad. Sci. (Paris), 274, 1245-7.

Hanusse, P. ( 1973). Simulation of chemical systems using Monte-Carlo method. C. R. Acad. Sci. (Paris), 277, 93- (in French).

Hanusse, P. (1976). Contribution a )'etude des systemes dissipatifs chimiques. Simulation par une methode de Monte-Carlo. Etude d'une reaction oscillante (Thesis), Bordeaux.

Hanusse, P. (1979). Stochastic simulation of chemical systems out of equilibrium. In: Synergetics. Far from equilibrium, eds A. Pacault & C. Vidal (Springer Series in Synergetics), pp. 70-4. Springer Verlag, Berlin.

Harrison, L. G. (1981). Physical chemistry of biological morphogenesis, Chern. Soc. Rev., 10, 491-528.

Hars, V. ( 1976). On some questions of stochastic reaction kinetics (Thesis}, Budapest. Hars, V. & T6th, J. (1981). On the inverse problem of reaction kinetics. Col/. Math.

Soc. Jimos Bolyai 30. North Holland, Amsterdam, pp. 363-79. Hastings, S. P. (1977). On the uniqueness and global asymptotic stability of periodic

solutions for a third order system. Rocky Mountain J., 7, 513- . Hastings, S., Tyson, J. & Webster, P. (1977). Existence of periodic solutions for

negative feedback control systems, J. Di.ff. Eqs., 25, 39-64. Hayman, H. J. G. (1971). Orthonormal chemical reactions and chemical relaxation.

lsr. J. Chern., 9, 419-27. Hearon, J. Z. (1953). The kinetics of linear systems with special reference to periodic

reactions, Bull. Math. Biophys., 15, 121-41. Hearon, J. Z. (1963). Theorems on linear systems, Ann. N. Y. Acad. Sci., 108, 36-68. Heineken, Tschuhiya & Aris, R. (1967). On the mathematical status of pseudo-steady

state hypothesis. Mathematical Biosciences, 1, 95-113.

232 References

Heller, A. (1965). On stochastic processes derived from Markov chains, Ann. Math. Statist., 1286-91.

Herman, I. (1978). On reaction kinetics with continuous components (Thesis), Budapest.

Herodek, S., Kutas, T. & Csaki, P. ( 1982). Simulation of phytoplankton dynamics in Lalte Balaton, /SEM Journal, 4, 97-126.

Hess, B. & Boiteux, A. (1971). Oscillatory phenomena in biochemistry. Ann. Rev. Biochem., 40, 237-58.

Hess, B. & Markus, M. (1985). The diversity of biochemical time patterns. Ber. Bunsenges Phys. Chem., 89, 642-51. Higgins, J. (1967). The theory of oscillating reaction, Ind. Eng. Chem., 59, 18- .

Higgins, J. (1968). Some remarks on Shear's Liapunov function for systems of chemical reactions. J. Theor. Bioi., 21, 293-304.

Hilden, H. (1974). Die Schaetzung des Parameters im linearen Todesprozess, Lecture presented at the 8th International Biometric Conference, Constan~, Roumania.

Hill, T. L. (1969). Thermodynamics of small systems, Benjamine, New York. Hill, T. L., Chen, Y. der (1972). Fluctuation and noise in kinetic systems. Biophys. J.,

12, 948- . Hirsch, M. & Smale, S. (1974). Differential equations, dynamical systems, and linear

algebra, Academic Press, New York. Hirschfelder, J. 0. ( 1957). Pseudostationary state approximation in chemical kinetics,

J. Chem. Phys., 26 (2), 271-3. Hodgkin, A. L. & Huxley, A. F. (1952). A quantitative description of membrane

current and its application to conduction and excitation in nerve. J. Physiol. (London), 1176, 500-44.

Hofbauer, J. (1981). On the recurrence oflimit cycles in the Volterra-Lotka equation, Non/in. Analysis, Theory, Methods and Applications, 5, 1003-7.

Hofbauer, J. (1985). The selection mutation equation, J. Math. Bioi., 23, 41-53. Hohenberg, P. C. & Halperin, B. I. ( 1977). Theory of dynamic critical phenomena.

Rev. Mod. Phys., 49, 435-79. Holden, A. V. (1981). Membrane current fluctuations and neural information

processing. Adv. Physiol. Sci., 30, 23-41. Holden, A. V. (1985). Why is the nervous system not as chaotic as it should be? In:

Dynamic phenomena in neurochemistry and neurophysics: theoretical aspects, ed. P. Erdi, pp. 6-11. Central Research Instirate of the Hungarian Academy Science, Budapest.

Holden, A. V. (ed.) (1986). Chaos (Nonlinear Science: Theory and Applications), Manchester University Press, Manchester.

Holden, A. V. & Yoda, M. (1981 ). The effects of ionic channel density on neuronal function. J. Theoret. Neurobio/., 1, 60-81.

HQ)derith, J. & Smirnov, N. I. (1968). On some problems in the simulation of chemical reactors, Annales Univ. Scient. Bp. de R. Eotvos, Sec. Chimia, 10, 107-17.

Hongler, M. 0. (1979a). Exact solutions of a class of nonlinear Fokker-Pianck equations, Phys. Letters, 75A (3-4).

Hongler, M. 0. (1979b). Exact time dependent probability density for a nonlinear non-Markovian stochastic process, Helv. Phys. Acta, 52, 280-

Hongler, M. 0. (1982). Diffusion in a class of double-well potentials-exact results, Phys. Letters, 91A, 396-8.

Hongler, M. 0. & Zheng, W. M. (1982). Exact solution for the diffusion in bistable potentials. J. Stat. Phys., 29, 317-27.

Hongler, M. 0. & Zheng, W. M. (1983). Exact results for the diffusion in a class of assymetric bistable potentials. J. Math. Phys., 24, 336-40.

Hooyman, G. J. (1961). Proc. Nat/. Acad. Sci. USA, 47,1169-1173.

References 233

Hormander, L. (1964). Linear partial differential operators. Springer Verlag, Berlin. Horn, F. (1971). General first order kinetics, Berichte der Bunsen-Gesellschaft fur

physika/ische Chemie, 75 (II), 1191-201. Horn, F. (1973a). Necessary and sufficient conditions for complex balancing in

chemical kinetics. Arch. Rat/. Mech. Anal., 49, 172-86. Horn, F. (1973b). On a connexion between stability and graphs in chemical kinetics,

I. Stability and the reaction diagram. Proc. Roy. Soc., London, Ser. A, 334, 299-312.

Horn, F. (1973c). On a connexion between stability and graphs in chemical kinetics, II. Stability and the complex graph. Proc. Roy. Soc., London, Ser. A, 334, 313-30.

Horn, F. (1973d). Stability and complex balancing in mass-action systems with three short complexes. Proc. Roy. Soc., London, Ser. A, 334, 331-42.

Horn, F. (1973e). Formal kinetics, (unpublished manuscript), pp. 1-213. Horn, F. (1974). The dynamincs of open reaction systems. SIAM-AMS Proceedings,

8, 125-37. Horn, F. & Jackson, R. (1972). General mass action kinetics. Arch. Rat/. Mech. Anal.,

47,81-116. Horsthemke, W. & Brenig, I. (1971). Non-linear Fokker-Pianck equation as an

asymptotic representation of the master equation. Z. Physik, B27, 341-8. Horsthemke, W. & Hanson, L. (1984). Nonequilibrium chemical instabilities in

continuous flow stirred tank reactors: the effect of stirring. J. Chern. Phys., 81, 4363-8.

Horsthemke, W. & Lefever, R. (1984a). Finite size effects and external noise in nonequilibrium systems, Phys. Letters, 106A, 10-12.

Horsthemke, W. & Lefever, R. (1984b). Noise induced transitions (Theory and applications in physics, chemistry, and biology), (Springer Series in Synergetics, Vol. 15). Springer Verlag, Berlin.

Hosten, L. K. (1979). A comparative study of short cut procedures for parameter estimation in differential equations. Comp. Chern. Eng., 3, 117-

Howard, L. N. & Kopell, N. (1974). Wave-trains, shock fronts, and transition layers in reaction-diffusion equations, SIAM-AMS-Proc., 8, 1-12.

Hsii, I.-D. (1976). Existence of periodic solutions for the BelousovZaikin-Zhabotinskii reaction by a theorem of Hopf. J. Diff. Eqs, 20, 399-403.

Hudson, J. L., Hart, M. & Marinko, D. (1979). An experimental study of multiple peak periodic and nonperiodic oscillations in the Belousov-Zhabotinskii Reaction. J. Chern. Phys., 71, 1601-

Hudson, J. L. & Rossler, 0. F. (1984a), Chaos in simple three- and four-variable systems. In: Modelling of patterns in space and time, eds W. Jaeger & D. Murray, (Lecture Notes in Biomathematics, Vol. 55), pp. 135-45. Springer Verlag, Berlin.

Hudson, J. L. & Rossler, 0. E. (1984b). A piecewise-linear invertible noodle map. Physica, liD, 239-42.

Hunding, A. (1974). Limit cycles in enzyme systems with nonlinear feedback. Biophys. Struct. Mech., 1, 47-54.

Hutchinson, P. & Luss, D. (1970). Chern. Eng. J., 1, 129- . Hutson, V. & Moron, W. (1985). Persistence in systems with diffusion. In: Dynamics

of macrosystems, eds J.-P. Aubin, D. Saari & K. Sigmund, (Lecture Notes in Economy and Mathematical Systems), pp. 43-8. Springer Verlag, Berlin.

Hutson, V. & Vickers, G. T. (1983). A criterion for permanent coexistence of species with an application to a two-prey one-predator system. Mathematical Biosciences, 63, 253-69.

Ichikawa, A. (1982). Stability and optimal control of stochastic evolution equations, In: Distributed parameter control systems. Theory and application, ed. S. G. Tsafestas, pp. 147-77. Pergamon Press, Oxford.

234 References

Ioos, G. & Joseph, D. D. (1980). Elementary stability and bifurcation theory, Springer Verlag, Berlin.

Israel, M. & Dunant, Y. (1979). On the mechanism of acetylcholine release. Prog. Brain Res., 49, 125-39.

Israel, M., Lesbats, B., Manaranche, R., Marsoe, J. & Mastour-Frachon, P. (1977). Related changes in amounts of ACh and A TP in resting and active nerve electroplaque synapse. J. Neurochem., 28, 1259-67.

Ito, K. (1951). On a formula concerning stochastic differentials. Nagoya Math. J., 3, 55-

Ito, K. & Mckean, H. P. (1965). Diffusion processes and their sample paths. Springer Verlag, Berlin.

Jachimowski, C. J. McQuarrie, D. A. & Russell, M. E. (1964). A stochastic approach to enzyme-substrate reactions. Biochemistry, 3 (II), 1732-6.

Jacquez, J. A. (1972). Compartmental analysis in biology and medicine. Elsevier, Amsterdam.

Janssen, H. K. (1974). Stochastisches Reaktionsmodell fiir einen NichtgleichgewichtPhaseniibergang. Z. Physik, 270, 67-73.

Jernigan, R. W. & Tsokos, C. P. (1980). Simulation of a nonlinear stochastic ecology model. Appl. Math. and Comp., 7, 9-25.

Jernigan, R. W., Turner, J. C. & Tsokos, C. P. (1980). Parameter and moment bounds for a nonlinear stochastic ecology model. App/. Math. and Comp., 7, 27-53.

Jope, R. S. (1979). High affinity choline transport and acetyl coenzyme A production and their roles in the regulation of acetylcholine synthesis. Brain Res. Rev., 1, 313-44.

Jergensen, S. E. (1976). an eutrofication model for a lake. Ecol. Model., 2, 147-65. Jouguet, E. (1921). Notes de mecanique chimique. J. Ec. Polyt. (Paris), 2de serie, 21

erne cahier. Kac, M. (1959). Probability and related topics in physical sciences. John Wiley, New

York. Kagan, M. L., Peleg, S., Tchiprout, A. & Avnir, D. (1984). Image analysis in the study

of dissipative spatial patterns. In: Non-equilibrium dynamics in chemical Systems, eds C. Vidal & A. Pacault (Springer Series in Synergetics, Vol. 27), pp. 118-121. Springer Verlag, Berlin.

Kai, S., Miiller, S. & Ross, J. (1982). Measurements of temporal and spatial sequences of events in periodic precipitation processes. J. Chern. Phys., 16, 1392-

Kamke, E. (1959). Di.fferentia/g/eichungen. Losungsmethoden und Losungen, I. Gewohnliche Di.fferentia/ig/eichungen, 6th revised edn. Leipzig.

Karlin, S. & McGregor, J. (1957). The classification of birth and death processes. Trans. Amer. Math. Soc., 86, 366-400.

Karlin, S. & McGregor, J. (1964). On some stochastic models in genetics. In: Stochastic models in medicine and biology, ed J. J. Gurland, pp. 245-79. University of Wisconsin Press, Wisconsin, Madison.

Karnapp, D. & Rosenberg, R. (1968). Analysis and simulation of multiport systems. MIT Press, Cambridge, MA.

Katz, B. & Miledi, R. (1972). The statistical nature ofacethylcholine potential and its molecular components. J. Physiol. (London), 224, 665-69.

Kaufman, A. N. (1984). Dissipative Hamiltonian systems: a unifying principle. Phys. Letters, IOOA, 419-22.

Kawczyilski, A. L. (1981). Construction of a model of chaos. Bull. Acad. Polon. Sci., 29 (7-8), 319-27.

Kawczyilski, A. L. (1984). Chemical model of chaos. Pol. J. Chern. Keizer, J. (1976). An observation on the effect of fluctuations far from equilibrium, J.

References 235

Stat. Phys., 15, 477-83. Keller, J. B. (1976). Inverse problems. Amer. Math. Monthly, 83, 107-18. Kelley, A. (1967). The stable, center-stable, center, center-unstable and unstable

manifolds. J. Dijf. Eqs., 3, 546-70. Kendall, D. G. (1975). Foundations of a theory of random sets. In: Stochastic

geometry, eds D. G. Kendall, Harding, pp. 322-75. John Wiley, London. Kerker, M. (1974). Brownian movements and molecular reality prior to 1900. J.

Chern. Ed., 51, 764-8. Kerner, E. H. (1964). Dynamical aspects of kinetics. Bull. Math. Biophys., 26, 333-49. Kertesz, V. (1984). Global mathematical analysis of the explodator. Nonline. Anal.

Theory Meth. App/., 8, 941-62. Khintchine, A. Ja. (1934). Korrelationstheorie der stationiiren stochastischen Pro

zesse. Math. Ann., 109, 604-10. Kimura, M. (1958). On the change of population fitness by natural selection.

Heredity, 12, 145-67. King, G. A. M. (1981). Growth of a hypercycle and comparison with conventional

autocatalysis. BioSystems, 13, 225-34. King, R. B. (1980). Chemical applications of topology and group theory. 8.

Topological aspects of oscillating chemical reactions. Theor. Chimica Acta (Berlin), 56, 269-96.

King, R. B. (l982a) Topological aspects of oscillating chemical reactions. In: Instabilities, bifurcations and fluctuations in chemical systems, eds L. E. Reichl & W. C. Schieve, pp. . University of Texas Press, Austin, Texas.

King, R. B. (1982b). The flow topology of chemical reaction networks. J. Theor. Bioi., 98, 347-68.

King, R. B. (1983). Chemical applications of topology and group theory. 14. Topological aspects of chaotic chemical reactions. Theor. Chimica Acta (Berlin), 63, 323-38.

King, R. B. ( 1986). Kinetic logic as a qualitative approach for the study of oscillating and chaotic systems. In: Perspectives in biological dynamics and theoretical medicine, Proc. N.Y. Acad. Sci. Conf. (9-11 April 1986). Abstract No. P-13.

Kingman, J. F. C. (1969). Markov population processes. J. Appl. Prob., 6, 1-18. Kitahara, K., Metiu, H. & Ross, J. (1975). A stochastic theory of cluster growth in

homogeneous nucleation. J. Chern. Phys., 63, 3156-60. Klonowski, W. (1983). Simplifying principles for chemical and enzyme reaction

kinetics. Biophys. Chern., 18, 73-87. Knobloch, E. & Wiesenfeld, K. A. (1983). Bifurcation in fluctuating systems: the

center manifold approach. J. Stat. Phys., 33, 611-37. Kodama, A. & Oka, M. (1983). On integral curves obtained from differential

equations in the Michaelis-Menten mechanism. Math. Japonica, 28, 737-50. Kokovin, G. A. (1982). Matematicheskie metody khimicheskoi termodinamiki

(Mathematical methods of chemical thermodynamics). Nauka SO, Novosibirsk, (in Russian).

Kolmogorov, A. N. (1936). Sulla teoria di Volterra della lutta per l'esistenza. Giorn. /Stituto /tal. Attuari, 7, 74-80.

Kondepundi, D. K. & Nelson, G. W. (1983). Chiral symmetry breaking in nonequilibrium systems. Phys. Rev. Lett., 50, 1023-6.

Kondepundi, D. K. & Nelson, G. W. (1984). Chiral symmetry breaking and their sensitivity in nonequilibrium chemical systems. Physica, 125A, 465-96.

Kondepundi, D. K. & Nelson, G. W. (1985). Chiral symmetry breaking in nonequilibrium chemical systems: time scales for chiral selection. Phys. Letters. (1985).

Korn, G. A. & Korn, T. M. (1968). Mathematical handbook for scientists and engineers. Definitions, theorems and formulas for reference, 2nd enlarged and revised

236 References

edn. McGraw-Hill, New York. Korzukhin, M. D. (l97la). II. Mathematical modelling of the kinetics of homo

geneous chemical systems. In: Oscillatory processes in biological and chemical systems, pp. 231-42. Chast 2-aya, Pushchino na Oke (in Russian).

Korzukhin, M. D. (l97lb). III. Mathematical modelling of the kinetics of homogeneous chemical systems. In: Oscillatory processes in biological and chemical systems, pp. 242-251. Chast 2-aya, Pushchino na Oke (in Russian).

Koshland, D. E., Nemethy, G. & Filmer, D. (1966). Comparison of experimental binding data and theoretical models in proteins containing subunits. Biochemistry, 5, 365- .

Krambeck, F. J. (1970). The mathematical structure of chemical kinetics in homogeneous single-phase systems. Arch. Rat/. Mech. Anal., 38, 317-47.

Krambeck, F. J. (1984). Accessible composition domains for monomolecular systems. Chem Eng. Sci., 39, 1181-4.

Kramers, H. A. ( 1940). Brownian motion in a field of force and the diffusion model of chemical reactions. Physica, 7, 284-304.

Krasnov, M. L., Kiselev, A. I. & Makarenko, G. I. (1978). Problems on ordinary differential equations. Vysshaya Shkola, Moscow (in Russian).

Krieger, I. M. & Gans, P. J. (1960). First-order processes. J. Chem. Phys., 32,240-50. Kubo, R. (1957). Statistical-mechanical theory of irreversible processes. J. Phys. Soc.

Japan, 12, 570-86. Kubo, R., Matsuo, K. & Kitahara, K. (1973). Fluctuation and relaxation of

macrovariables. J. Stat. Phys., 9, 51-96. Kuramoto, Y. (1973). Fluctuations around steady states in chemical kinetics. Progr.

Theor. Phys., 49, 1782-3. Kurtz, T. G. (1972). The relationship between stochastic and deterministic models for

chemical reactions, J. Chem. Phys., 57, 2976-8. Kurtz, T. G. (l98la). Approximation of discontinuous processes by continuous

processes. In: Stochastic nonlinear systems, eds L. Arnold & R. Lefever, pp. 22-35. Springer Verlag, Berlin.

Kurtz, T. G. (l98lb). Approximation of population processes. SIAM, Philadelphia, PA.

Kutas, T. & Toth, J. (1985). A stochastic model of phytoplankton dynamics in the Lake Balaton. J. Statist. Comp. and Simulation, 21, 241-64.

Kuznetsov, P.l., Stratonovich, P. L. & Tikhonov, V.I. (1965). The effect of electrical fluctuations on a valve oscillator. In: Nonlinear transformations of stochastic processes, eds P. I. Kuznetsov, P. L. Stratonovich & V. I. Tikhonov, pp. 223-Pergamon Press, Oxford.

Laidler, K. J. (1965). Chemical kinetics. McGraw-Hill, New York. Lamba, P. & Hudson, J. L. (1985). Experimental evidence of multiple oscillatory

states in a continuous reactor. Chem. Eng. Commun., 32, 369-75. Lampard, D. G. (1954). Generalization of the Wiener-Khinchine theorem to

nonstationary processes. J. Appl. Phys., 25, 802-3. Kutas, T. & Herodek, S. (1987). Effects of load reductions on the water quality of a

large shallow lake. Ecol. Mod., 39, 85-99. Lang, S. (1962). Introduction to differentiable manifolds. Wiley-Interscience, New

York. Langevin, P. (1908). Sur Ia theorie du movement brownien. C. R. Acad. Sci. (Paris),

146, 530-3. Langford, W. F. (1984). Numerical studies of torus bifurcations. Intern. Series of

Num. Math., 70, 285-95. Langford, W. F. Hopfbifurcation at a hysteresis point. Coli. Math. Soc. Janos Bolyai

(submitted).

References 237

Lapidus, L. & Amundson, N. R. (eds) (1977). Chemical reactor theory: a review. Prentice-Hall, Englewood Cliffs, NJ.

Lax, M. (1960). Fluctuations from the nonequilibrium steady states. Rev. Mod. Phys., 32, 26- .

Lax, M. (1966a). Classical noise, III: Nonlinear Markoff processes. Rev. Mod. Phys., 38, 359-79.

Lax, M. (1966b). Classical noise, IV: Langevin methods, Rev. Mod. Phys., 38, 541-60.

Lefever, R. & Turner, J. W. (1984) Hopf bifurcation and ripple induced by external multiplicative noise. In: Non-equilibrium dynamics in chemical systems, eds C. Vidal & A. Pacault (Springer Series in Synergetics, Vol. 27), pp. 166-71. Springer Verlag, Berlin.

Leichester, H. M. & Klickstein, H. S. (1952). A source book in chemistry 1400-1900. McGraw-Hill, New York.

Lemarchand, H. (1980). Asymptotic solution of the master equation near nonequilibrium transition: the stationary solutions. Physica, lOlA, 518-34.

Lemarchand, H. & Fraikin, A.: Stochastic description of various types of bifurcations in chemical systems. In: Non-equilibrium dynamics in chemical systems, eds C, Vidal, A. Pacault (Springer Series in Synergetics, Vol. 27), pp. 205-9. Springer Verlag, Berlin.

Lengyel, B., Prekopa, A. & Torok, F. (1956). Ober Kinetik und Gleichgewicht der Aquilibrierungs-Reaktion von linearen Methylpolysiloxanen, I. Z. phys. Chem., 206, 161-8.

Leontovich, M. (1935). The basic equations of the kinetic theory of gases from the viewpoint of stochastic processes. Zh. Eksp. Teor. Fiz., 5, 211-31.

Levy, P. (1948). Processus stochastiques et mouvement brownien. Gauthier-Villars, Paris.

Li, Genyuan (1984). A lumping analysis in mono- or/and bimolecular reaction systems. Chem. Eng. Sci., 29 (7 -8), 1261-70.

Lielmezs, J. (1965). On linear dependence of chemical reactions and normalized stoichiometry. Chem. Eng. Sci., 20, 363-

Liljenroth, F. (1918). Starting and stability phenomena of ammonia-oxidation and similar reactions. Chem. and Met. Eng., 19, 287-

Lindblad, P. & Degn, H. (1967). A compiler for digital computation in chemical kinetics and an application to oscillatory reaction schemes, Acta Chim. Scand., 21, 791-800.

Lobry, C. & Lozi, R. (1981). In: Nonlinear phenomena in chemical dynamics, eds C. Vidal & A. Pacault, p. 67.

Lorenz, P. (1963). Deterministic nonperiodic flow, J. Athm. Sci., 20, 130-41. Los, J. & Los, M. W. (eds) (1974). Mathematical models in economics. North Holland,

Amsterdam. Lotka, A. ( 1920). Undamped oscillations derived from the law of mass action. J. Am.

Chem. Soc., 42, 1595-Lovasz, L. (1979). Combinatorial problems and exercises. Akademiai Kiado,

Budapest; North Holland, Amsterdam. Luss, D. ( 1980). Steady state multiplicity and criteria for chemically reacting systems.

In: Dynamics and modelling of reactive systems, eds W. Stewart, W. H. Ray & C. Conley, pp. 131-59. Academic Press, New York.

Luss, D. (1981 ). Steady state multiplicity of chemically reacting systems. In: Modelling of chemical reaction systems, eds K. H. Ebert, P. Deuflhard & W. Jaeger (Springer Series in Chemical Physics, Vol. 18), pp. 305-11. Springer Verlag, Berlin.

Luss, D. & Hutchinson (1971). Chem. Eng. J., 2, 172-8. McCaskill, J. S. (1984). A stochastic theory of macromolecular evolution. Bioi. Cyb.,

238 References

50, 63-73. McQuarrie, D. A. (1967). Stochastic approach to chemical kinetics. J. Appl. Prob., 4,

413-78. McQuarrie, D. A. (1968). Stochastic approach to chemical kinetics. Corrigendum. J.

Appl. Prob., 5, 484. McQuarrie, D. (1969). In: Stochastic processes in chemical physics, ed. K. E. Shuler,

John Wiley, New York. Magde, D. (1977). Concentration correlation analysis and chemical kinetics. In:

Chemical relaxation in molecular biology, eds J. Pecht & R. Riegler. Springer Verlag, Berlin.

Malek-Mansour, M. & Nicolis, G. (1975). A master equation description of local fluctuations. J. Stat. Phys., 13, 197-217.

Malinvaud, E. (1969). Methodes statistiques de /'econometric. Dunod, Paris. Markus, M., Muller, S.C. & Hess, B. (1985). Observation of entrainment, quasipe

riodicity and chaos in glycolyzing yeast extracts under periodic glucose input. Ber. Bunsenges. Phys. Chem., 89, 651-4.

Maruyama, T. (1977). Stochastic problems in population genetics (Lecture Notes in Biomathematics, Vol. 17). Springer Verlag, Berlin.

Matheson, J., Walls, D. F. & Gardiner, C. W. (1975). Stochastic models of first-order nonequilibrium phase transitions in chemical reactions. J. Stat. Phys., 12, 21-34.

Matis, J. H. & Hartley, H. 0. (1971). Stochastic compartmental analysis: model and least squares estimates from time series data. Biometrics, 27, 77-102.

Matsuno, K., Lindeberg, K. & Shuler, K. I;. (1978). Stochastic theory of nonlinear rate processes with multiple stationary states, II, Relaxation time from a metastable state. J. Stat. Phys., 19, 65-75.

Matveev, N. M. (1983). Problems and exercises on ordinary differential equations. Vysheishaya Shkola, Minsk (in Russian)

May, R. M. (1972). Stability in random fluctuating versus deterministic environments. Am. Nat., 107, 621-50.

May, R. M. & Leonhard, W. J. (1975). Nonlinear aspects of competition between three species, SIAM J. Appl. Math., 29, 243-53.

Maynard Smith, J. (1982). Evolutionary game theory. Cambridge University Press, Cambridge.

Medgyessy, P. (1977). Decomposition of superpositions of density functions and discrete distributions. Akademial Kiad6, Budapest.

Meinhardt, H. (1978). Models for the ontogenetic development of higher organisms. Rev. Physiol. Biochem. Pharmacol., 80, 47-104.

Meinhardt, H. (1982). Models of biological pattern formation. Academic Press, London.

Meinhardt, H. (1982b) Models of biological pattern/ormation, London. Meinhardt, H. (1985). Mechanisms of pattern formation during development of

higher organisms: a hierarchical solution ofa complex program. Ber. Bunsenges. Phys. Chem., 89, 691-9.

Meinhardt, H. & Gierer, A. (1980). A. Generetion and regeneration of structures during morphogenesis. J. Theor. Bioi., 85, 429-50.

Michaelis, L. & Menten, M. L. (1913). Die Kinetik der Invertinwirkung. Biochem. Z., 49, 333-69.

Micheau, J. C., Horsthemke, W. & Lefever, R. (1984). Sensitivity of biphotonic systems to light intensity fluctuations: experimental evidence in the thermoluminescence offluoresceine in boric acid glasses. J. Chem. Phys., 81, 2450-7.

Moelwigh-Hughes, E. A. (1971). The chemical static and kinetics of solutions. Academic Press, London.

Monod, J., Wyman, J. & Changeux, J. P. (1965). On the nature of allosteric

References 239

transitions: a plausible model, J. Mol. Bioi., 12, 88-118. Morales, M. & McKay, D. (1967). Biochemical oscillations in 'controlled' systems.

Biophys. J., 1, 621-5. Moreau, M. (1975). Formal studies of a chemical reaction by Grad expansion in the

Boltzmann equation. I. II. Physica 79A, 18-38, and 39-51. Moreau, M. & Borgis, D. (1984). Stochastic theory of a bistable chemical reaction in

non-homogeneous liquids. In: Non-equilibrium dynamics in chemical systems, eds C. Vidal & A. Pacault (Springer Series in Synergetics, Vol. 27), pp. 200-4. Springer Verlag, Berlin.

Moss, F. & Weiland, G. V. (1982). Multiplicative noise in the Vinen equation for turbulence superfluid 4 He. Phys. Rev., 25, 3389-92.

Moyal, J. E. (1949). Stochastic processes and statistical physics. J. Roy. Stat. Soc., Bll, 150- .

Muller, S.C., Kai, S. & Ross, J. (1982). Curiosities in periodic precipitation patterns. Science, 216, 635-7.

Muller, S. L., Plesser, T. & Hess, B. (1985a). Surface tension driven convection in chemical and biochemical solution layers. Ber. Bunsenges. Phys. Chem., 89, 654-8.

Muller, S. L., Plesser, T. & Hess, B. (1985b). The structure of the core of the spiral wave in the Belousov-Zhabotinskii reaction. Science, 230, 661-6.

Muller, S. L., Plesser, T. & Hess, b. (1986). Two-dimensional spectrophotometry and pseudocolor representation of chemical reaction patterns. Naturwissenschaften, 73, 165-79.

Muller-Herold, U. (1983). What is a hypercycle? J. Theor. Bioi., 102, 569-84. Mulloolly, P. (1971). Maximum likelihood estimation for stochastic first order

reactions. Bull. Math. Bioi., 33, 83-96. Mulloolly, P. (1972). Maximum likelihood estimation for stochastic rth order

reactions. J. App/. Prob., 9, 32-42. Mulloolly, P. (1973). Maximum likelihood estimation for stochastic rth order

reactions, II. J. App/. Prob., 10, 441-6. Murray, J. D., Oster, G. F. & Harris, A. K. (1983). A mechanical model for

mesenchymal morphogenesis. J. Math. Bioi., 17, 125-9. Neher, E. & Stevens, C. F. (1977). Conductance fluctuations and ionic poses in

membranes. Ann. Rev. Biophys. Bioeng., 6, 345-81. Nemes, I., Vidoczy, T., Botar, L. & Gal, D. (1977a). A possible construction of a

complex chemical reaction network, I. Definitions and procedure for construction, Theoret. Chim. Acta (Berl.), 45, 215-23.

Nemes, I., Vidoczy, T., Botar, L. & Gal, D. (1977b). A possible construction of a complex chemical reaction network, II. Applications. Theoret. Chim. Acta (Bert.), 45, 225-33.

Nemes, I., Vidoczy, T. & Gal, (1977c). A possible construction of a complex chemical reaction network, III. The systematization of the elementary processes. Theoret. Chim. Acta (Berl.), 46, 243-50.

Nicolis, G. (1984). Bifurcations, fluctuations and dissipative structures. In: Nonlinear phenomena in physics and biology, eds R. H. Eu, B. L. Jones, R. M. Micura & M. M. Rangiekar. Plenum Press, New York.

Nicolis, G., Allen, P. & Van Nypelseer, A. (1974). Some remarks on the theory of fluctuations around nonequilibrium states. Prog. Theor. Phys., 52, 1481-97.

Nicolis, G. & Malek-Mansour, M. (1980). Systematic analysis of the multivariate master equation for a reaction-diffusion system. J. Stat. Phys., 22, 495-512.

Nicolis, G. & Prigogine, I. (1971). Fluctuations in nonequilibrium systems. Proc. Nat/. Acad. Sci. USA, 68,2102-7.

Nicolis, G. & Prigogine, I. (1977). Self-organization in nonequilibrium systems. Wiley-Interscience, London.

240. References

Nicolis, G. & Portnow, I. (1973). Chemical oscillations. Chern Rev., 73, 365-84. Nicolis, G. & Turner, J. W. (1977). Stochastic analysis of a nonequilibrium phase

l.ransition: some exact results. Physica, 89A, 326-38. Nicolis, G., Dewell, G. & Turner, J. W. (eds) (1984). Order and fluctuations in

equilibrium and non-equilibrium statistical mechanics. Wiley-Interscience, London. Nitzan, A. (1978). Chemical instabilities as critical phenomena. Phys. Rev., 17,

1513-28. Nitzan, A., Ortoleva, P., Deutch, J. & Ross, J. (1974a). Fluctuations and transitions

at chemical instabilities. The analogy to phase transitions, J. Chern. Phys., 61, 1056-74.

Nitzan, A., Ortoleva, P. & Ross, J. (1974b). Nucleation in systems with multiple stationary states, Faraday Symp. the Chern. Soc., No.9 241-53.

Nobile, A. G. & Ricciardi, I. (1984a). Growth with regulation in fluctuating environments, I. Alternative logistic-like diffusion models, Bioi. Cyb., 49, 179-88.

Nobile, A. G. & Ricciardi, L. (1984b). Growth with regulation in fluctuating environments, II. Intrinsic lower bound to population size. Bioi. Cyb., 50, 285-99.

Noszticzius, Z., Farkas, H., Schelly, Z. A. & Savage, C. Explodator: a new skeleton mechanism for the chalate driven oscillators. J. Chern. Phys., 80, 6062-70.

Nowak, U. & Deuflhard, P. (1985). Numerical identification of selected rate constants in large chemical reaction systems. Appl. Num. Math., 1, 59-75.

Noyes, R. M. (1974). Kinetics of complex reactions. In: Techniques of chemistry, Vol. 6, Pt. I, ed. E. S. Lewis, pp. 489-538. John Wiley,

Noyes, R. (1978). Generalized kinetics of chemical change: some conditions for validity of the steady state approximation. Supplement of the Progr. Theor. Phys., 64, 295-306.

Noyes, R. (1984). In: Non-equilibrium dynamics in chemical systems, eds C. Vidal & A. Pacault (Springer Series in Synergetics, Vol. 27), pp. 118-21. Springer Verlay, Berlin.

Nyquist, H. (1928). Thermal agitation of electrical activity in conductors. Phys. Rev., 32 110-3.

Odell, G. R., Oster, G., Burnside, B. & Alberch, D. (1984). The mechanical basis of morphogenesis. Develop. Bioi., 85, 446-

0kamoto, M. & Hayashi, K. (1984). Optimal control mode of a biochemical feedback system. BioSystems, 16, 315-21.

Olsen, L. F. (1983). An enzyme reaction with a strange attractor. Phys. Letters, 94A, 454-7.

Olsen, L. F. & Degn, H. (1977). Chaos in an enzyme reaction. Nature, 267, 177-8. Olsen, L. F. & Degn, H. (1978). Oscillatory kinetics of the perioxidase-oxidase

reaction in an open system. Experimental and theoretical studies. Biochim. Biophys. Acta, 523, 321-34.

Olsen, L. F. & Degn, H. (1985). Chaos in biological systems. Quart. Rev. Biophys., 18, 165-225.

Oppenheim, 1., Shuler, K. E. & Weiss, G. M. (1969). Stochastic and deterministic formulation of chemical rate equations. J. Chern. Phys., 50, 460-6.

Oppenheim, 1., Shuler, K. E. & Weiss, G. M. (1977). Stochastic theory of nonlinear rate processes with multiple stationary states. Physica, 88A, 191-214.

Orlov, V. N. & Rozonoer, L. I. (1981). Zh. Vychisl. i Mat. Fiz., 21, 1192-205. Orlov, V. N. & Rozonoer, L. I. (1984a). The macrodynamics of open systems and the

variational principle of the local potential-I, Journal of the Franklin Institute, 318, 283-314.

Orlov, V. N. & Rozonoer, L. I. ( 1984b ). The macrodynamics of open systems and the variational principle of the local potential-11. Journal of the Franklin Institute, 318, 315-47.

References 241

Oster, G. & Alberch, D. (1982). Evolution and bifurcation of developmental programs. Evolution, 36, 444-59.

Oster, G., Perelson, A. & Katchalsky, A. (1973). Network thermodynamics: dynamic modelling of biophysical systems. Quart. Rev. Biophys., 6, 1-134.

Oster, G. F. & Perelson, A. S. (1974). Chemical reaction dynamics, I. Arch. Rat/. Mech. Anal., 55, 230-74.

Oster, G., Odell, G. R. & Alberch, D. (1980). Mechanics, morphogenesis and evolution. Lect. Math. Life Sci., 13, 165-225. (Providence, RI.).

Othmer, H. G. (1976). Nonuniqueness of equilibria in closed reacting systems. Chern. Eng. Sci., 31, 993-1003.

Othmer, H. G. (1981). The interaction of structure and dynamics in chemical reaction networks. In: Modelling of chemical reaction systems, eds K. H. Ebert, P. Deuflhard & W. Jaeger (Springer Series in Chemical Physics, Vol. 18), pp. 2-19. Springer Verlag, Berlin.

Othmer, H. G. (1985). Synchronization, phase-locking and other phenomena in coupled cells. In: Temporal order, Proc. Symp. Oscill. Heter. Chern. Bioi. Syst., Bremen, Germany, 1984 (Springer Series in Synergetics, Vol. 29), pp. 130-43. Springer Verlag, Berlin.

Ovsyannikov, L. V. (1969). Group properties of differential equations. University of British Columbia, Vancouver (in Russian: 1962).

Ovsyannikov, L. V. (1982). Group analysis of differential equations. Academic Press, New York (in Russian: 1978).

Park R. eta/. (1974). A generalized model for simulating lake ecosystems. Simulation, 23, 33-50.

Passy, U. & Wilde, D. J. (1968). A GP algorithm for solving chemical equilibrium problems, SIAM Rev., 16, 363-

Pecht, J. & Riegler, R. (eds) (1977). Chemical relaxation in molecular biology. Springer Verlag, Berlin.

Peil, J. ( 1979). Inverse problems in growth dynamics. In: Inverse and Improperly Posed Problems in Differential Equations, ed G. Anger, pp. 197-213. Akademie Verlag, Berlin.

Peneloux, A. (1949). C. R. Acad. Sci. (Paris), 288, 1727-Peschel, M. & Breitenecker, F. (1984). Socio-economic consequences of the Volterra

approach to nonlinear systems. In: Cybernetics and Systems, '84, ed R. Trapp), pp. 423-8. Elsevier Science Publishers B. V. North-Holland. Amsterdam, New York, Oxford.

Peschel, M. & Mende, V. (1985). The predator-prey model: Do we live in a Volterra world? Akademie Verlag.

Perelson, A. S. & Wallwork, D. (1976). The arbitrary dynamic behaviour of open chemical reaction systems. J. Chern. Phys.

Petheo, A. & Schay, G. (1954). Mathematische Diskussion der Anwendung des Hess' schen Satzes. Acta Chim. Acad. Sci. Hung., 4, 21-35.

Petho, A. (1959). Pub/. Math. lnst. Hung. Acad. Sci., 3, 101-Petho, A. (1964). Zur Theorie der Stochiometrie chemischer Reaktionassysteme.

Wissenschaftliche Zeitschrift der Technischen Hochschule fur Chemie, LeunaMerseburg, 6, 13-15.

Petho, A. (1965). Die Theorie der 'einfachen' chemischen Reaktionen in der Stochiometrie. Z. phys. Chern. (Frankfurt), 45, 89- .

Petho, A. (1965b). On 'simple' chemical reactions in stoichiometry. Chern. Eng. Sci., 20, 791-

Petho, A. (1967a). In: Symp. on Chemical Reaction Dynamics, Padua, 1966, Consiglio Nazionale delle Ricerce, Roma, pp. 195-9. . . .

Petho, A. ( 1967b ). On a class of solutions of algebraic homogeneous hnear equations.

242 References

Acta Math. Acad. Sci. Hung., 18, 19-23. Petho A. (1968). Kemiai reakciok egy osztalyanak algebrai targyalasa. Magy. Kem.

Foly., 14, 488-91. Petho A. & Gyory K. (1965). On solutions of linear systems of equations containing

'many' zeros. Mat. Lapok, 16, 267-73 (in Hungarian). Peusner, L. (1985). Studies in network thermodynamics. Elsevier, Amsterdam. Pikovsky, A. S. (1981). A dynamic model for periodic and chaotic oscillations in the

Belousov-Zhabotinsy reaction. phys. Letters, 85A, 13-16. Pikovsky, A. S. & Rabinovich, M. I. (1981). Stochastic oscillations in dissipative

systems. Physica, 2D, 8-24. Pinter, L. & Hatvani, L. (1977-80). Solution of Prob. 10 of the 1979 Miklos

Schweitzer Competition, Mat. Lapok, 28 (4), 349-50. · Pismen, L. M. (1984). Dynamics of lumped chemically reacting systems near singular

bifurcation points. Chern. Eng. Sci., 39, 1063-77. Pismen, L. M. Dynamics of lumped chemically reacting systems near singular

bifurcation points-H. Almost Hamiltonian dynamics. Chern. Eng. Sci., 40, 905-16.

Polushchik, E. M. (1983). Sophus Lie 1842-1899. Nauka LO., Leningrad (in Russian).

Ponomarev, K. K. (1962). Formulating and solving differential equations of engineering problems. Gosudarstvennoe, Moscow (in Russian).

Pota, Gy. (1981). On a theorem of overshoot-undershoot kinetics. React. Kinet. Catal. Lett., 17, 35-9.

Pota, Gy. (1983). Two-component bimolecular systems cannot have limit cycles: a complete proof. J. Chern. Phys., 78, 1621-2.

Pota, Gy. (1985). Irregular behaviour of kinetic equations in closed chemical systems. J. Chern. Soc. Faraday Trans. 2, 81, 115-21.

Pota, Gy., Bazsa, Gy. & Beck, M. (1982). Study of pseudo-waves in periodic reactions. Acta Chim. Acad. Sci. Hung., llO, 277-81.

Prekopa, A. (1953). Investigation of the degradation process of long chain molecules using probabilistic methods, MTA Alk. Mat. Int. Kozl., 2, 103-23 (in Hungarian).

Prekopa, A. (1956). On stochastic set functions, I. Acta Math. A cad. Sci. Hung., 7 (2), 215-63.

Prekopa, A. (1957a). On stochastic set functions, II. Acta Math. Acad. Sci. Hung., 8 (3-4), 337-74.

Prekopa, A. (1957b). On stochastic set functions, III. Acta Math. Acad. Sci. Hung., 8 (3-4), 375-400.

Prigogine, I. (1978). Time, structure, and fluctuations, Science, 201, 775-85. Prigogine, I. (1980). From being to becoming, Freeman, San Francisco. Prigogine, I. & Defay, R. (1947). On the number of independent constituents and the

phase rule. J. Chern. Phys., 15, 614-15. Prigogine, I. & Lefever, R. (1968). Symmetry breaking instabilitites in dissipative

systems, II. J. Chern. Phys., 48, 1695-700. Prigogine, I. & Stengler, K. (1984). Order out of chaos. Heinemann, London. Primas, H. (1983). Chemistry, quantum mechanics, and reductionism. Springer Verlag,

Berlin. Rabai, Gy., Bazsa, G. & Beck, M. T. (1979). Design of reaction systems exhibiting

overshoot-undershoot kinetics. J. Am. Chern. Soc., 131, 6746-8. Rabinovitch, B. (1969). The Monte Carlo method. Plotting the curve of complex

reactions. J. Chern. Ed., 46, 262-8. Rabitz, H., Kramer, M. & Docal, D. (1983). Sensitivity analysis in chemical kinetics.

Ann. Rev. Phys. Chern., 34, 419-fil. Rao, C. R. (1973). Linear statistical inference and its applications. J. Wiley and Sons,

References 243

Inc., New York-London-Sydney-Toronto. Reddy, V. T. N. (1975). Existence of steady-state in stochastic Volterra-Lotka model.

J. Stat. Phys., 13, 61-4. Rehmus, & Ross, J. (1984). Periodically perturbed chemical systems. In: Oscillations

and travelling waves in chemical systems, eds R. J. Field & M. Burger, pp. 287-332. John Wiley, New York.

Reichl, L. E. & Schieve, W. C. (eds) (1982). Instabilities, bifurcations, and .fluctuations in chemical systems. University of Texas Press, Austin.

Renyi, A. ( 1953). Treating chemical reactions using the theory of stochastic processes. MT A Alk. Mat. Int. Kozl., 2, 83-101 (in Hungarian).

Reti, P. & Ropolyi, L. (1984). Analogies between point mechanics and chemical reaction kinetics, React. Kinet. Catal. Lett., 25, 109-13.

Reti, P. & Ropolyi, L. (1986). On the geometrical structure of equilibrium chemical systems. Utilization of analogies between point mechanics and reaction kinetics. Nuovo Cimento B94, 16-28.

Ricciardi, L. M. (1977). Diffusion processes and some related topics in biology. Lect. Notes in Biomathematics. Springer, Berlin.

Richter, P. (1984). Entrainment of chemical oscillators and resonance dissipation. Physica, IOD, 353-68.

Ringland, J. & Turner, J. S. One-dimensional behaviour in a model of the Belousov-Zhabotinskii reaction, Phys. Letters (in press).

Rodiguin, N. M. & Rodiguina, E. N. (1964). Consecutive chemical reactions. Van Nostrand, New York.

Roehner, B. & Valent, G. (1982). Solving the birth and death process with quadratic asymptotically symmetric transition rates. SIAM J. Appl. Math., 42, 1020-46.

Rogers, T. D. (1977). 23-41. Local models of cell aggregation kinetics. Bull. Math. Bioi., 39, 23-41.

Romine, W. 0. jr. (1976). Studies in fluctuation analysis: I. On relation between rate constants and relaxation times in chemical systems and the multivariate power spectrum of noise and fluctuations, J. Chern. Phys., 64, 2350-8.

Rosenblatt, M. (1971). Markov processes. Structure and asymptotic behaviour. Springer Verlag, Berlin. (1971).

Rosenbrock, H. & Storey, C. (1966). Computational techniques for chemical engineering. Pergamon Press, Oxford.

Ross, J. (1985). Non-linearities in chemical reactions. Temporal and spatial structures. Efficiency, Ber. Bunsenges. Phys. Chern., 89, 604-19.

Ross, J. & Mazur, P. Some deduction from a formal statistical mechanical theory of chemical kinetics. J. Chern. Phys., 35, 19-

Rossler, 0. E. (1972). A principle for chemical multivibration. J. Theor. Bioi., 36, 413-17.

Rossler, 0. E. ( 1974a). Chemical automata in homogeneous and in reaction-diffusion kinetics. In: Physics and mathematics of the nervous system, Lecture Notes in Biomathematics, Vol. 4, eds M. Conrad, W. Giittinger & M. Delcin, pp. 399-418. Springer Verlag, Berlin.

Rossler, 0. E. (1974b), A synthetic approach to exotic kinetics (with examples). In: Physics and mathematics of the nervous system, Lecture Notes in Biomathematics, Vol. 4, eds M. Conrad, W. Giittinger, & M. Delcin, pp. 546-82. Springer Verlag, Berlin.

Rossler, 0. E. (1976a). Chaotic behaviour in simple reaction systems. Z.Naturforsch., 31a, 259-64.

Rossler, 0. E. (1976b). Different types of chaos in two simple differential equations. Z. Naturforsch., 31b, 1664-70.

Rossler, o". E. (1978). Chaotic oscillations in a 3-variable quadratic mass action

244 References

system. In: Proc. Int'l Symp. Math. Topics in Biology (Sept. 11-12, 1978, Kyoto, Japan) pp. 131-5. Kyoto Research Institute for Mathematical Science, Kyoto.

Rossler, 0. E. (1979). Chaos and strange attractors in chemical kinetics. In: Synergetics. Far from equilibrium, eds A. Pacault C. Vidal (Springer Series in Synergetics, Vol. 3), pp. 107-113. Springer Verlag, Berlin.

Rossler, 0. E. (1981). Chaos and chemistry. In: Nonlinear phenomena in chemical dynamics, eds C. Vidal & A. Pacault (Springer Series in Synergetics, vol. 12), pp. 79-87. Springer Verlag, Berlin.

Rossler, 0. E. (1983a). The chaotic hierarchy. Z. Naturforsch., 38a 788-801. Rossler, 0. E. (1983b). Deductive prebiology. In: Molecular evolution and the

protobiologica/ paradigm eds K. Matsuno, K. Dose, K. Harada & B. L. Rohlfing. Plenum Press, New York.

Rossler, 0. E. & Hudson, D. L. (1985). A piecewise linear hierarchy. Symp. on Math. Bioi., Kyoto (Abstract) pp. 18-20.

Rossler, 0. E. & Wegman, K. (1978). Nature, 271, 89-90. Roux, J .-C. ( 1983). Experimental studies of bifurcations leading to chaos in the

Belousov-Zhabotinsky reaction, Physica, 7D, 57-68. Roux, J. C., Rossi, A., Bachelart, S. & Vidal, C. Experimental observations of

complex dynamical behaviour during a chemical reaction. Physica, 2D, 395-403. Riickenstein, E. & Vavanellos, (1975). Kinetics of solid state reactions. A/ChE J., 21,

756-763. Rudin, W. (1978). Principles of mathematical analysis, McGraw-Hill, Inc. New York. Ruelle, D. (1981). Chemical kinetics and differentiable dynamical systems. In:

Nonlinear phenomena in chemical dynamics, eds C. Vidal & A. Pacault (Springer Series in Synergetics, Vol. 12), pp. . Springer Verlag, Berlin.

Rumschitzky, D. S. (1984). On the theory of multiple steady states in isothermal CSTRs (Ph.D. thesis), University of Rochester, Rochester, New York.

Rumschitzky, D. & Feinberg, M. Reaction network structure and the possibility of multiple steady states in complex isothermal CSTRs, Chern. Eng. Sci. (in preparation).

Samohyl, I. (1982). Racionalni termodinamika chemicky reagijucich smes. Academia. Prague.

Sancho, J. M. & San Miguel, M. (1982). External non-white noise: theory and experiment. Physica, 116A, 560-2.

Sancho, J. M. & San Miguel, M. (1984). Theory of external two-state Markov noise in the presence of internal fluctuations, J. Stat. Phys., 37, 151-72.

Sayasov, Yu. S. & Vasil'eva, A. B. (1955). Foundations and conditions of the applicability of the method of quasistationary concentrations by Semenov and Bodenstein, Zh. Fiz. Khimii, 29 (5), 802-10.

Schay, G. & Petho, A. (1962). Ober die mathematischen Grundlagen der Stochiometrie, Acta Chim. Acad. Sci. Hung., 32, 59-67.

Schimansky-Geier, L., Tolstopjatenko, A. V. & Ebeling, W. (1985). Noise induced transitions due to external additive noise. Phys. Letters, 108A, 329-32.

Schlogl, F. ( 1972). Chemical reaction models for nonequilibrium phase transitions. Z. Physik, 253, 147-61.

Schmitz, R. A., Graziani, K. R. & Hudson, J. L. (1977). Experimental evidence of chaotic states in the Belousov-Zhabotinsky reaction. J. Chern. Phys., 67 (7) 3040-4.

Schnakenberg, J. (1977). Thermodynamics network analysis of biological systems. Springer Verlag, B<:rlin.

Schnakenberg, J. (1979). Simple chemical reaction systems with limit cycle behaviour. J. theor. Bioi., 81, 389-400.

Schneider, K. R. (1969). Ober die periodischen Losungen einer Klasse nichtlinearer

References 245

autonomer Differentialgleichungssysteme dritter Ordnung. Z. Angew. Math. u. Mech., 49, 441-3.

Schneider, K. R. (1980). Andronov-Hopf-Bifurkation in Faile Stetiger Parameterabhiingigkeit, ZAMM, 60, 19-24.

Schneider, K. R., Wegner, B. & T6th, J. (1987). Qualitative analysis of a model for synaptic slow waves. J. Math. Chern., 1, 219-34.

Schofield, W. R. & Krutchkoff, R. G. (1974). Stochastic model for an aquatic ecosystem. WRRC Bull., 60, Virginia Polytechnic Institute and State University, Bladesburg. Virginia.

Schubert, A. ( 1976). Kinetics of homogeneous reactions. Miiszaki Konyvkiad6, Budapest (in Hungarian.)

Schulmeister, Th. (1978). Chaos in einem Lotka-Schema mit Depot, Stud. Biophys. (Berlin), 12, 205-6 and microfiche 3/14-28.

Schuster, P. (1985). Effects of finite population size and other stochastic phenomena in molecular evolution. In: Complex systems-Operational Approaches. Springer Series in Synergetics. Vol. 31., Springer Verlag, Berlin.

Schuster, P. & Sigmund, K. (1983). Replicator dynamics. J. Theor. Bioi., 100, 333-8. Schuster, P. & Sigmund, K. (1984). Random selection-a simple model based on

linear birth and death processes. Bull. Math. Bioi., 46, 11-17. Schuster, P. & Sigmund, K. (1985). Dynamics of evolutionary optimization. Ber.

Bunsenges. Phys. Chern., 89, 668-82. Schwarz, F. & Steeb, W.-H. (1983). Symmetries and first integrals for dissipative

systems. J. Phys. A, Math. Gen., 17, L819-23. Seelig, F. F. (197la). System-theoretic model for the spontaneous formation of

optical antipodes in strongly asymmetric yield. J. Theor. Bioi., 31, 335-61. Seelig, F. F. (1971 b). Mono- or bistable behaviour in a weakly or strongly open

chemical reaction system. J. Theor. Bioi., 32, 93-106. Seelig, F. F. & Gobber, F. (1971). Stable linear reaction oscillator. J. Theor. Bioi., 30,

485-96. Segel, L.A. (1984). Modelling dynamic phenomena in molecular and cellular biology.

Cambridge University Press, Cambridge (UK). Seller, P. (1967a). Algebraic complexes which characterize chemical networks. SIAM

J. App/. Math., 15, (1), 13-68. Sellers, P. (1967b). Stoichiometric equations are mathematical equations. SIAM J.

App/. Math., 15 (3), 637-40. Sellers, P. (1971). An introduction to a mathematical theory of chemical reaction

networks, I. Arch. Rat/. Mech. Anal., 44, 23-40. Semenov, N. (1939). On the kinetics of complex reactions, J. Chern. Phys., 7, 683-99. Shahshahani, S. ( 1979). A new mathematical tool for the study of linkage and

selection. Mem. Am. Math. Soc, 211. Shapiro, A. Horn, F. J. M. (1979). On the possibility of sustained oscillations,

multiple steady states, and asymmetric steady states in multicell reaction systems. Mathematical Biosciences, 44, 19-39.

Shear, D. (1967). An analog of the Boltzmann H-theorem (a Liapunov function) for systems of coupled chemical reactions. J. Theor. Bioi., 16, 212-28.

Showalter, K., Noyes, R. M. & Bar-Eli, K. (1978). A modified Oregonator model exhibiting complicated limit cycle behaviour in a flow system. J. Chern. Phys., 69, 2514-24.

Showalter, K., Noyes, R. M. & Turner, H. Detailed studies of trigger wave initiation and detection. J. Am. Chern. Soc., 101, 7463-9.

Sibirskii, K. S. (1976). Algebraic invariants of di.ff'erential equations and matrices. Stiintsa, Kishinev (in Russian).

Sibirskii, K. S. (1982). Introduction to the algebraic theory of invariants of differential

246 References

equations. Stiintsa, Kishinev (in Russian). Siegert, A. J. F. (1949). On the approach to statistical equilibrium. Phys. Rev., 76,

1708-14. Sigmund, K. (1986). A survey of replicator equations. In: Complexity Language, and

Life: Mathematical Approaches, eds J. L. Casti & A. Karlquist, Springer Verlag, Biomathematics. Vol. 18.

Simon, E. M. (1971). Light scattering of fluctuations of a biological conformation change, I. Simplified stochastic model. J. Chern. Phys., 54, 4738-45.

Simonyi, M. & Mayer, I. (1975). A critical survey of the transition state theory. Acta Chim. Acad. Sci. Hung., 87, 15-32.

Sinanoglu, 0. ( 1975). Theory of chemical reaction networks. All possible mechanisms of synthetic pathways with given number of reaction steps or species. J. Am. Chern. Soc., 97, 2309-20.

Sinanoglu, 0. (1976). Acta Chim. Turcica, 3, 155- . Sinha, A. S. (1975). On Volterra-Lotka systems. Int. J. System Sci., 6, 49-55. Sipos, T., Toth, J. & Erdi, P. (l974a). Stochastic simulation of complex chemical

reactions by digital computer, I. The model. React. Kinet. Catal. Lett., 1, 113-17. Sipos, T., Toth, J. & Erdi, P. (l974b). Stochastic simulation of complex chemical

reactions by digital computer, II. Applications. React. Kinet. Catal. Lett., 1, 209-13.

Skorohod, A. V. (1978). Random fields. In: Textbook on probability theory and mathematical statistics, V. S. Korolyuk, pp. 272-292. Naukova Dumka, Kiev (in Russian).

Skumanich, M. & Rabitz, H. (1982). Feature sensitivity analysis in chemical kinetics. Comm. in J. of Mol. Sci., 2, 79-92.

Smale, S. (1976). On the differential equations of species in competition. J. Math. Bioi., 3, 5-7.

Smoes, M. L. (1980). Chemical waves in the oscillatory Zhabotinsky system. A transition from temporal to spatio-temporal organization. In: Dynamics of synergetic systems, ed. H. Haken (Springer Series in Synergetics, Vol. 6), pp 80-95. Springer Verlag, Berlin.

Smoluchowski, von M. (1916). Drei Vortriige iiber Diffusion, Brownsche Bewegung und Koagulation von Kolloidteilchen I. Physik. Zeitsch., 17, 557-84.

Sole, M. (1972). Alternative stochastic formulations of first-order reaction kinetics. Coli. Czech. Chern. Comm., 37, 1055-60.

Spiegelman, S. (1971). Quart. Rev. Biophys., 4, 215- . Srejder, Yu. (1965). Monte Carlo methods. Fizmatgiz, Moscow (in Russian). Stauffer, D. (1979). Scaling theory of percolation clusters. Phys. Rev., 54, l-74. Steeb, W. H. (1978). The Lie derivative, invariance conditions, and physical laws, Z.

Naturforsch., 33a, 742-8. Steeb, W. H. (1979). Generalized Liouville equation, entropy and dynamic systems

containing limit cycles. Physica, 95A, 181-90. Steeb, W. H. & Strampp, N. (1981). Symmetries of nonlinear reaction-diffusion

equations and their solutions. Physica, 3D, 637-43. Stewart, W. E., Ray, W. H. & Conley, Ch. C. (eds) (1980). Dynamics and modelling of

reactive systems. Academic Press, New York. Stratonovich, R. L. (1963). Stratonovich, R. L. (1967). Topics in the theory of random nose, Vol. l. Gordon &

Breach, New York. Stratonovich, R. L. ( 1985). Nonlinear nonequilibrium stochastic thermodynamics.

Nauka, Moscow (in Russian). Stucki, J. W. (1978). Stability analysis of biochemical systems-a practical guide.

Prog. Biophys. Molec. Bioi., 33, 99-187.

References 247

Suzuki, M. (1976a). Scaling theory of nonequilibrium systems near the instability point. I. General aspects of transient phenomena, Progr. Theor. Phys., 56, 77- .

Suzuki, M. (1976b). Scaling theory of nonequilibrium systems near the instability point. II. Anomalous fluctuation theorems in the extensive region. Progr. Theor. Phys., 56, 477-

Suzuki, M. (1980). Scaling theory of instability, nonlinear fluctuation and formation of macroscopic structure. In: Synergeticsfar from equilibrium, eds A. Pacault & C. Vidal, (Springer Series in Synergetics, Vol. 3), pp. 94-106. Springer Verlag, Berlin.

Suzuki, M. (1981). Chem. Phys., Adv. eds I. Prigogine, S. A. Rice, Suzuki, M. (1984). Fluctuation and formation of macroscopic order in non

equilibrium systems. Progr. Theor. Phys. Suppl., 79, 125-40. Sverdlove, R. (1977). Inverse problems for dynamical systems in the plane. In:

Dynamical systems, eds A. R. Bednarek & L. Cesari, pp. 499-502. Academic Press, New York.

Swinney, H. L. (ed.) (1983). Physica, 7D. Swinney, H. L. & Roux, J. C. (1984) Chemical chaos. In: Non-equilibrium dynamics in

chemical systems, eds C. Vidal & A. Pacault (Springer Series in Synergetics, Vol. 27), pp. 127-40. Springer Verlag, Berlin.

Szabo, Z. ( 1969). Kinetic characterization of complex reaction systems. In: Comprehensive chemical kinetics, eds C. H. Bamford & C. F. H. Tipper, Vol. 2, The theory of chemical kinetics. Elsevier, Amsterdam. pp 1-80.

Szathmary, E. (1984). Hyperbolic growth in molecuar and biological populations, I. Single species dynamics lacking direct linear growth rates. Abstracta Botanica, 8, 137-61.

Sz.-Nagy, B. & Hatvani, L. (1987). Differential equations: Qualitative theory. Coli. Math. Soc. Janos Bolyai 47, Vols. I, II. North-Holland, Amsterdam.

Takeuchi, Y. ( 1985). The effects of diffusion on stability in Volterra models. Abstracts oflntl. Symp. on Math. Bioi., Kyoto, Nov. 13-15, pp. 107-108.

Takeuchi, Y. & Adachi, N. (1983a). Existence and bifurcation of stable equilibrium in two-prey, one-predator communities, Bull. Math. Bioi., 45, 877-900.

Takeuchi, Y. & Adachi, N. (1983b). Oscillations in prey-predator models. In: Population biology, eds H. I. Friedman & H. F. Strobek, Lecture Notes in Biomathematics, vol. 52, pp. 320-6. Springer Verlag, Berlin.

Tauc, L. ( 1979). Are vesicles necessary for release of acetylcholine at cholinergic synapses? Biochem. Pharmacol., 27, 3493-8.

Thorn, R. (1975). Structural stability and morphogenesis, W. A. Benjamin, Reading, MA.

Thomas, R. ( 1979). Kinetic logic. A boolean approach to the analysis of complex regulatory systems. Lecture Notes in Biomathematics, Vol. 29, Springer Verlag, Berlin.

Thomas, R. ( 1981 ). On the relation between the logical structure of systems and their ability to generate multiple steady states or sustained oscillations. In: Numerical methods in the study of critical phenomena eds J. Della Dora, J. Demengot & B. Lasalle (Springer Series in Synergetics, Vol. 9), pp. 180-93. Springer Verlag, Berlin.

Tikhonov, A. N. (1952). Systems of differential equations containing a small parameter at the derivatives. Mat. sb., 31 (73), 575-85.

Tikhonov, A. N., Vasilyeva, A. B. & Sveshnikov, A. G. (1985). Differential equations, 2nd edn. Nauka, Moscow (in Russian).

Tisza, L. (1966). Generalized thermodynamics. The MIT Press, Cambridge. MA. Tisza, L. & Quay, P. M. (1963). Ann. Phys. (N.Y.) 25, 48-Tomita, K. (1982). Chaotic response of nonlinear oscillators. Phys. Rev., 86, I 13-67 · Tomita, K. & Kai, T. (1979). Chaotic response of a limit cycle. J. Stat. Phys., 21,

65-86.

248 References

Tomita, K. & Tsuda, I. (1979). Chaos in the Belousov-Zhabotinsky reaction in a flow system, Phys. Letters, 71A, 489-92.

Toth, J. (1978). What is essential to exotic kinetic behaviour? React. Kinet. Catal. Lett., 9, 377-81.

Toth, J. (1979) Gradient systems are cross-catalytic. React. Kinet. Catal. Lett., 12, 253-7.

Toth, J. (l98la). Poissonian stationary distribution in a class of detailed balanced reactions. React. Kinet. Catal. Lett., 18, 169-73.

Toth, J. (198lb). On the global deterministic and stochastic model of formal reaction kinetics and their applications. MTA SZTAKI Tanulmimyok, 129, 1-63 (in Hungarian).

Toth, J. (1985). A mass action kinetic model of neurochemical transmission. In: Dynamic phenomena in neurochemistry and neurophysics: Theoretical aspects, ed P. Erdi, pp. 52-5. MT A KFKI, Budapest.

Toth, J. (1986). Sufficient conditions against blowing up in the differential equations of chemical kinetics, Poster presented at the MID IT '86 Workshop on Structure, Coherence and Chaos, Lyngby, Denmark, August.

Toth, J. (1987). Bendixson type theorems with applications, Z. Angew. Math. u. Mech., 67, 31-5.

Toth, J. Structure of the state space in stochastic kinetics. Proc. 5th Pann. Symp. on Math. Stat. (submitted a).

Toth, J. Contribution to the general treatment of random processes used in chemical reaction kinetics. Proc. lOth Prague Conf on Jnf Theory, Stat. Decision Functions and Random Processes (submitted b).

Toth, J. & Erdi, P. (1977a). On the theory of pure reaction kinetics. In: Proc. 3rd Conf App/. Chem., Veszprem, pp. 71-6.

Toth, J. & Erdi, P. (1977b). Determination of reaction rate constants of complex chemical reactions from equilibrium fluctuations. Proc. 5th Symp. Comp. Chem. Engng., The High Tatras.

Toth, J. & Erdi, P. (1978). Models, problems and applications of formal reaction kinetics. A kemia ujabb eredmenyei, 41, 226-352 (in Hungarian).

Toth, J. & Hars, V. (1986a). Orthogonal transforms of the Lorenz- and Rossler-1 equation. Physica, 19D, 135-44. ~Toth, J. & Hars, V. (1986b). Specification of oscillating chemical reactions starting

from a given linearized form, Theor. Chimica Acta, 70, 143-50. Toth, J. & Torok, T. L. (1980). Poissonian stationary distribution: a degenerate case

of stochastic kinetics. React. Kinet. Catal. Lett., 13, 167-71. Toth, J. & Torok, T. L. (1981). Stationary distributions in stochastic kinetics. Adv.

Physiol. Sci. (Math. Comp. Methods in Physiol.), 34, 103-ll. Toth, J., Torok, T. L. & Erdi, P. (1983). Significance of the Poisson distribution in the

stochastic model of complex chemical reactions. Alk. Mat. Lapok, 9, 175-95. (in Hungarian).

Treutlein, M. & Schulten, K. 1985. Noise-induced limit cycle on the bonhoffer-Van der Pol model of neural pulses. Ber. Bunsenges. Phys. Chem., 89, 710-19.

Truesdell, C. A. (1969). Rational thermodynamics. McGraw-Hill, New York. Truesdell, C. A. & Noll, W. (1965). The nonlinear field theories of mechanics. In:

Encyclopedia of physics, ed. S. Fliigge, Vol. III/3. Springer Verlag, Berlin. Tuljapurkar, S. D. & Semura, J. S. (1979). Liapunov functions: geometry and

stability. J. Math. Bioi., 8, 25-32. Turing, A. M. (1952). The chemical basis of morphogenesis, Phil. Trans. Roy. Soc.,

8273, 37-72. Turner, J. S. (1974). Finite fluctuations and multiple steady states far from

equilibrium, Bull. Math. Bioi., 36 (2), 205-13.

References 249

Turner, J. S., Roux, J. C., McCormick, W. D. & Swinney, H. L. (1981). Alternating periodic and chaotic regimes in a chemical reaction-experiment and theory. Phys. Letters, 85A, 9-12.

Turner, J. W. ( 1979). Stationary and time dependent solutions of master equations in several variables. In: Dynamics of synergetic systems, ed. H. Haken (Springer Series in Synergetics, Vol. 6), pp. 255-9. Verlag, Berlin.

Tyson, J. J. (1976). The Belousov-Zhabotinsky reaction. Lecture Notes in Biomathematics, Vol. 10. Springer Verlag, Berlin.

Tyson, J. J. (1978). On the appearance of chaos in a model of the Belousov reaction. J. Math. Bioi., 5, 351-62.

Tyson, J. J. & Light, J. C. (1973). Properties of two-component bimolecular and trimolecular chemical reaction systems, J. Chern. Phys., 59, 4164-73.

Tyson, J. J. & Othmer, H. (1978). The dynamics of feedback control circuits in biochemical pathways. Progr. Theor. Bioi., 5, 1-62.

Uhlenbeck, G. E. & Ornstein, L. S. ( 1930). On the theory of Brownian motion. Phys. Rev., 36, 823-41.

Vajda, S. (1979). Comments on structural identifiability in linear time-invariant systems. IEEE Trans. on Automatic Control, AC-24, 495-

Vajda, S. (1981). Structural equivalence oflinear systems and compartmental models. Mathematical Biosciences, 55, 39-

Vajda, S. (1982). On parameter and structural identifiability-Nonunique observability, reconstructibility and equivalence of linear compartmental models. IEEE Trans. Autom. Control, 27, 1136.

Vajda, S. (1983). Structural identifiability of dynamical systems. Int. J. Systems Sci., 14 (II), 1229-47.

Vajda, S. & Tuninyi, T. (1986). Principal component analysis for reducing the Edelson-Field-Noyes model of the Belousov-Zhabotinsky reaction. J. Chern. Phys., 90, 1664-70.

Vajda, S., Valko, P. & Godfrey, K. R. (1987). Direct and indirect least squares methods in continuous-time parameter estimation. Automatica, 23, 707-18.

Vajda, S., Valko, P. & Tuninyi, T. (1985). Principal component analysis of kinetic models. Int. J. Chern. Kinet., 55-81. Van Dongen, P. G. J. & Ernst, M. H. (1981). Kinetics of reversible polymerization, J. Stat. Phys., 37, 301-24.

Van Kampen, N. G. (1961). A power series expansion of the master equation. Can. J. Phys., 39, 551-67.

Van Kampen, N. G. (1965). Fluctuation in nonlinear systems. In: Fluctuation phenomena in solids, ed. A. E. Burgers. Academic Press, New York.

Van Kampen, N. G. (1976). The equilibrium distribution of a chemical mixture. Phys. Letters, 59A, 333-4.

Van Kampen, N. G. (1981). Stochastic processes in physics and chemistry. NorthHolland, Amsterdam.

Van Vliet, K. M. & Fasset, J. R. (1965). Fluctuations due to electronic transition and transport in solids. In: Fluctuation phenomena in solids, ed, A. E. Burgers. Academic Press, New York.

Vasil'ev, V. M., Vol'pert, A. I. & Khudyaev, S. I. (1973). On the method of quasistationary concentrations for the equations of chemical kinetics. Zh. Vych. Mat. Fiz., 13, 683-97. (in Russian).

Venugopalan, M. (ed.) (1971). Reactions under plasma conditions, Vols. I-11. John Wiley, New York. . .

Vidal, C. & Pacault, A. (eds) (1981). Nonlinear phenomena in chemical dynamics (Springer Series in Synergetics, Vol. 12). Springer Verlag, Berlin. .

Vidal, C. & Pacault, A. (eds) (1984). Non-equilibrium dynamics in chem1cal systems

250 References

(Springer Series in Synergetics, Vol. 27). Springer Verlag, Berlin. Viswanathan, S. & Aris, R. (1975). On the concept of the steady state in chemical

reactor analysis. Chern. Eng. Commun., 2, 1-4. Vol'pert, A. I. (1972) Differential equations on graphs. Mat. sb., 88, 578-88 (in

Russian). Vol'pert, A. 1., Gel'man, E. A. & Ivanova, A. N. (1974). Questions of the qualitative

theory of the equations of chemical kinetics. In: Chimreactor-5, Ufa, Vol. III, pp. 105-9. (in Russian).

Vol'pert, A. T. & Khudyaev, S. I. (1975). Analysis in the class of discontinuous functions and the equations of mathematical physics. Nauka, Moscow.

Volterra, V. (1931). Lecons sur Ia lutte pour Ia vie. Gauthier-Villars, Paris. Vulpe, N. I. ( 1976). Polynomial basis of the comitants of differential systems and their

applications in the qualitative theory. Stiintsa, Kishinev (in Russian). Waddington, C. H. (1962). New Patterns in genetics and development. Columbia

University Press, New York. Walgraef, D., Dewel, G. & Borckmans, P. (1982). Nonequilibrium phase transitions

and chemical instabilities, Adv. Chern. Phys, 49, 311-57. Walter, Ch. (1977). Contribution to the enzyme models. In: Mathematical models of

biological discovery, Lecture Notes in Biomathematics, Vol. 13, pp. 39-94. Springer Verlag, Berlin.

Walwork, D. & Perelson, A. S. (1976). Restrictions on chemical kinetics: Models. J. Chern. Phys., 65, 284-92.

Wang, M. C. & Uhlenbeck, G. E. (1945). On the theory of the Brownian motion, II. Rev. Mod. Phys., 17, 323-42.

Wegmann, K. & Rossler, 0. E. (1979). Z. Naturforsch., 33a, 1179-Wegscheidger, R. (1900). Ober die allgemeinste Form der Gesetze der chemischen

Kinetik homogener Systeme. Z. phys. Chern., 35, 513-87. Wegscheider, R. (1901/2). Ober simultane Gleichgewichte und die Beziehungen

zwischen Thermodynamik und Reaktionskinetik homogener Systeme. Z. phys. Chern., 39, 257-303.

Wei, J. (1962). Axiomatic treatment of chemical reaction systems. J. Chern. Phys., 36 1578-84.

Wei, J. & Prater, C. D. (1962). The structure and analysis of complex reaction systems. Adv. in Cata/., 13, 203-392.

Weidlich, W. & Haag, G. ( 1983). Concepts and models of a quantitative sociology (The dynamics of interacting populations), (Springer Series in Synergetics, Vol. 14). Springer Verlag, Berlin.

Welsh, B. J., Gomatom, J. & Burgers, A. E. (1983). Three-dimensional chemical waves in the Belousov-Zhabotinsy reaction. Nature, 304, 611-14.

West, G. (1977). Vorlaesungen iiber Stabilitaetstheorie chemischer Reaktionssysteme (manuscript), Philipps-Universitaet, Marburg/Lahn.

West, G. (1981). Qualitative Untersuchungen zur allgemeinen Reaktionskinetik erster Ordnung. Z. Phys. Chern., Leipzig, 262, 625-34.

Whittle, P. (1968). Equilibrium distribution for an open migration process. J. App/. Prob., 5, 567-71.

Wiener, N. (1930). Generalized harmonic analysis. Acta Math., 55, 117-258. Wiesenfeld, K. A. (1985). Noisy precursors of nonlinear instability. J. Stat. Phys., 38,

1071-96. Wiesenfeld, K. A. & Knobloch, E. ( 1982). Effect of noise on the dynamics of a

nonlinear oscillator. Phys. Rev., 26, 2946-53. Wilhelmy, L. F. (1850). Ober das Gesetz nach welchem die Einwirkung der Siiuren

auf den Rohrzicker stattfindet. Poggendorf Annalen der Physik und Chemie, 81, 413-33.

References 251

Willamowski, K.-D. (1978a). Contributions to the theory of mass action kinetics, II. Representation of closed and open kinetics. Z. Naturforsch., 33a, 983-8.

Willamowski, K.-D. (1978b). Contributions to the theory of mass action kinetics, III. Some dynamic properties of second order mass action kinetics. Z. Naturforsch., 33a, 989-92.

Willamowski, K.-D. & Rossler, 0. E. (1978). Contributions to the theory of mass action kinetics, I. Enumeration of second order mass action kinetics. Z. Naturforsch., 33a, 827-33.

Willamowski, K.-D. & Rossler, 0. E. (1980). Irregular oscillations in a realistic abstract quadratic mass action system. Z. Naturforsch., 35a, 317-18.

Williams, M. M. R. (1974). Random processes in nuclear reactors. Pergamon Press. Wilson, K. G. & Kogut, J. (1974). The renormalization group and the £-expansion.

Phys. Rep., 12c, 75-199. Winfree, A. T. (1972). Spiral waves of chemical activity, Science, 175, 643-36. Winfree, A. T. (1973). Scroll-shaped waves of chemical activity in three dimensions.

Science, 181, 937-7. Winfree, A. (1974). Rotating chemical reactions. Sci. Am., 230, 82-95. Winfree, A. T. ( 1980). The geometry of biological time, Lecture Notes in Biomathema

tics, Vol. 8. Springer Verlag, Berlin. Winfree, A. T. (1984a). The prehistory of the Belousov-Zhabotinsky oscillator. J.

Chern. Educ., 61, 661-3. Winfree, A. T. (1984b). Organizing centers for chemical waves in two and three

dimensions. In: Oscillations and travelling waves in chemical systems, eds. R. J. Field & M. Burger, pp. 441-472. John Wiley, New York.

Winfree, A. T. & Strogatz, S. H. (1985). Organizing centres for three-dimensional chemical waves. Nature, 311, 611-15.

Wolfe, R. J. (1978). Temporal oscillation in a kinetic model of the BelousovZhabotinskii reaction. Arch. Rat/. Mech. Anal., 67, 225-50.

Wolfram, S. (1986). Theory and applications of cellular automata. World Scientific, Singapore.

Wright, S. (1931 ). Evolution in Mendelian populations. Genetics, 16, 97-159. Wulfman, C. E. (1979). Limit cycles as invariant functions of Lie groups. J. Phys. A.

Math. Gen., 12, L73-5. Yablonskii, G. S. & Bykov, V. I. (1977). On the nonstationary kinetics of catalytic

reactions. AN SSSR SO Institute of Catalysis, Novosibirsk. Yeh, Y. & Keeler, R. N. (1969). Experimental study of reaction kinetics by light

scattering, I. The polarized Rayleigh component. J. Chern. Phys., 51, 1120-7. Yeh, Y. & Keeler, R. N. (1970). Experimental study of reaction kinetics by light

scattering, II. Helixcoil transition of the coplymer deoxy-adenylatdeoxythymidate (dAT). J. Chern. Phys., 52, 6218-24.

Yourgrau, W. & Row, C. J. G. (1957). Supp/. Nuovo Cim., 5, 472-Zacks, S. (1971 ). The theory of statistical inference. John Wiley, New York. Zaikin, A. W. & Zhabotinskii, A.M. (1970). Concentration wave propagation in two

dimensional liquid-phase self-oscillating systems. Nature, 225, 535-7. Zeman, C. E. (1971). Time in science and philosophy. Zhabotinskii, A.M. & Korzukhin, M. D. (1971). I. Mathematical modelling of the

kinetics of homogeneous chemical systems. In: Oscillatory processes in biological and chemical systems., pp. 223-31. Chast 2-aya, Pushchino na Oke (in Russian).

Zurek, W. H. Schieve, W. C. (1980). Deviations from the Poisson behaviour of equilibrium fluctuations in a closed system. J. Stat. Phys., 22, 289-96.

Index

'abnormal' dynamic behaviour 191 absorption of drugs 12 acetylcholine 186, 187, 188, 189 activator xxii, 175-6 activity 4, 8, 9 activity coefficient 4 acyclicity 31, 45, 46, 67, 78 adenosine 5'-diphosphate xv, 31, 185 adenosine 5'-triphosphate xv, 31, 185 aggregation 71, 207, 218-19 algae 204 algebraic description of a reaction 29 algebraic structure 21 algebraic structure of reaction networks

I amplification of fluctuations 129 analogue computer 15 anomalous fluctuation 138 asymmetry of biomolecules 216-17 atom xv, 4, 22, 24, 29, 30 atom-free stoichiometry 26, 28 atomic balance 22, 24, 26 atomic component 24 atomic matrix xvi, 22, 24, 29 atomic reaction 24, 29 atomic stoichiometry 24 atomic structure 21, 22, 24, 26 autonomous ordinary differential

equation 18, 19 autonomous polynomial differential

equation 35

Balaton 202 Belousov-Zhabotinskii reaction 39, 58,

59, 60, 61, 81, 163, 173 bifurcate II bifurcation parameter xxii, 56, 123 bifurcation phenomenon 13, 166, 178 bifurcation point xxii, 54

Biltzmann equation 3, 5 biochemical control theory 177, 218-19 biochemistry 5 biological memory 8-9 biomathematical model 12 biomathematics 107, 196 biomodality 138 biomolecular reaction 2, 3, 55, 107 biomolecule 121-3, 216 birth and death process 112, 140, 141,

142 bistable reaction 135 blowing up 159 Bodenstein principle 88 Boulding ecodynamics 194 boundedness 32 Brownian motion xx, 98, 100, 101, 116,

130, 207 Brusselator 166

canonic reaction 65, 67, 74 ceo model 19, 159 CDS model 19, 159 cell 5, 21, 174 cellular automata 210 cellular model 169 central limit theorem 171 centre-manifold approach 154, 155 change of scale 87 chaos II, 19, 20, 38, 49, 59, 60, 61 chaotic behaviour 12, 60, 183 chaotic chemical reaction 62 Chapman-Enskog approximation 3 Chapman-Kolmegorov equation 71,

96,99 charge 27 charge transfer 12 chemical amplification 211 chemical circuit 210-13

Index

chemical thermodynamics 18 chemistry 3, 7, 93 chronon 15 closed compartmental system 30 closed generalized compartmental

system 30 cluster formation 6, 207 cluster growth 6 coexistence 44, 196-8 cold flame 54 cold plasma 5 collision 2, 3, 4, 5, 23 coloured noise 151, 152 combinatorial model I 04 compartmental analysis 12 compartmental ecokinetics 195 compartmental system 12, 30, 69, 73,

107 competition 12, 177 competitive phenomenon 13, 196 complete 29 completeness 28 complex xvii, 12, 30 complex vector xvii, 23 component xx, I, 22, 24, 26, 27, 29, 30,

34, 163 composition I, 7 concentration xiv, xxiv, 5, 7, 8-9, 10,

26, 33 concentration vector xiv, xvii, 10 condensed matter 6 condensed phase reaction 4 conformation 5, 79 conformation change 8-9, 127, 188 comformational change 5 connectedness of the state-space 29 conservative 27, 28, 29, 30, 32 conservative reaction 30, 71 conservativity 27, 28, 30, 32, 67, 78 consistency in the thermodynamic limit

160 constituent 33 constitutive functional xv, 17, 18, constitutive quantity xv, 7, 16, 17, 18 construction of material 28 continuous component xvii, 17, 78 continuous flow stirred tank reactor 50 continuous mechanics 16 continuous semigroup xx continuous state model 7, 33 continuous state-space 16, 97, 163 continuous time 10, 15,.33, 91, 163

cooperation 12, 177 cooperative phenomenon 13 core 73 correlation function xviii, 116 cosmology 16 covalent bond xiv, 5 creation of mass 27 critical point 6, 95, 135, 172 critical species 50 cross-catalytic mechanism 81 cross-effect 34 curve 19 cyclic phenomenon 16

DCD model 19 decomposition (of density functions

and discrete distributing) 140 deficiency xvii, 29, 43, 78 deformation tensor 8-9 deoxyribonucleic acid xviii, 127, 128 dependence 33, 35 dependent 23, 24 designing oscillatory reactions 56 destruction 27, 28 detailed balance 3, 45, 46 determination 14

253

deterministic 8-9, 18, 19, 21 deterministic dynamic system 48 deterministic model 6, 7, II, 33, 34, 46,

149, 162, 163, 206 deterministic process 18 dichotomous noise xix, !52 difference equation 19 differential-difference equation 10, 103 differential equation 7, 12, 19, 35, 54,

55 differential system 19 diffusion 4, 6, 8-9, 14, 16, 97, 162-75 diffusion matrix xxii, 97 diffusion process 97, 100, Ill digital computer 15 dilute gas 4, 8-9 direct problem 38, 63 directed edge 30 discrete 15, 16, 19, 21 discrete state I 0, 91 discrete state-space 7, 10, 16, 18, 99 discrete time 15, 18 discrete-time model 15 dissipative process 7, 116 dissipative structure 13 drift 201

254

drift vector 97 dynamic system xv, 14, 18, 85

ecodynamics 194-6 ecokinetics 218-19, 195, 197 ecological case study 202 ecological model 12 ecological system 12 econometrics 77 ecosystem 195-7, 199-201 electric field 5 electrolytes 4, 8-9 electron 24, 27 elementary reaction xiii, 2, 22, 23, 24,

25, 26, 27, 29, 30, 33, 34, 35 elementary reaction vector xvi, xvii, 22,

23,24 elliptic differential operator xxiii, 164 empty reaction 65 endpoint xvii, 172 energetic interaction 16 energy level 12 entropy, as 'chemical component' 126 entry point xvii, 72 enzyme xv, xxiii, 4, 5, 26, 27, 182, 187 enzyme kinetics 5, 119, 147, 177-9 enzyme reaction 4, 182 enzyme-substrate interaction xv, xxiv,

26, 179 enzyme system 8-9 epidemiology 13 equilibrium 7, 8-9, II, 16 equilibrium distribution 4, 118 equilibrium fluctuation 93, 117, 126 equilibrium measurement 10 equilibrium point xvi, 137 ergodic properties 12 estimation of reaction rate constant 71,

157 eutrophication 218, 219 eutrophication process 202 evolution 199, 201, 213 evolution law 19 exotic behaviour II exotic chemical system II exotic kinetics II exotic reaction 49 experimental observation 25 explodator 56, 58 external fluctuation 95, 146, 156, 201 external noise 10, 96, 147, 153

fading memory 6

Index

fast reactions 125-8 feature sensitivity analysis 84 Feinberg-Horn-Jackson graph 70 Field-Kt!ros-Noyes-model xvi, 39 field theory 16 first integral xvi, 86 first-order endpoint xvii, 72 first-order Markov process 18 first-order phase transition 134 first-order reaction 23, 30 fluctuation 6, 7, 8-9, II, 91, 185 fluctuation around the equilibrium

point 7, 84 fluctuation-dissipation theorem 7, 92,

94, 95, 115, 116, 125, 148, 157 fluctuation phenomenon 10, 93, 128 fluctuation theory II Fokker-Pianck equation 98, 110, Ill,

136, 149, 150, 154, 155 formal 14, 30 formal linear combination 24, 30 formal reaction kinetics 202 forward rate 3 Fourier transform xxi, 117 'free-energy-like' function Ill functional 17 functional differential equation 8-9

gain-loss equation II gas-phase reaction 3, 6 Gaussian process 139 general solution 25, 26 generalised compartmental system 30,

69-70, 72 generalised Lotka-Volterra model xvii,

196, 198, 218-19, 201 generation 15 genetic model 12 Gierer-Meinhardt model xxii, 175 glycolitic oscillation 182 glyoxylate cycle 31 gradient system xvii, 80, 82, 193 group of general linear transformations

xvi, 87 group of transformations xxii, xxiii, 165 growth phenomena 209-10

harmonic oscillation 218-19 heterogeneous reaction 4 hierarchial level 12 historyxv, 16, 17,18,21 homogeneity 14

Index

homogeneous reaction kinetics 49 homogeneous spatial distribution 7 hot plasma 5 hyperchaos II hypercycle 214, 215

identifiability 62 incidence relation 31 independence 22, 24 independence of components 28 independence of elementary reactions

28 independent component 22, 25 independent elementary reactions 22,

24, 28 induced kinetic differential equation

xvi, 35, 39, 55, 86, 88 industrial chemical system II infinite system 7 infinitesimal transition probability xx,

II, 99, 103 infinitesimal transition probability

matrix xiii, 102 inhibitor xxiii, 175-6 inner fluctuation 204 instability 13, 91, 129 instability point 7, 8-9, 128 interacting populations 194 internal component 29 internal fluctuation 10, 95, 156 internal state 5 internal structure 3, 6 inverse problem 38, 63, 72 invertibility, 18 ion 23, 24 ion reaction 4 irreversible 23 irreversible phenomenon 16 irreversible process 16 isola 53 isomerisation 79 isotope exchange 12 iterated map 19

jump process 10, 110, 112

kinetic condition 12 kinetic differential equation xiv, xvi, 7,

19, 35, 64, 65, 67 kinetic equation 7, 8-9, 33 kinetic experiment 50, 54, 60 kinetic gradient system 80

kinetic initial value problem 64 kinetic logic 47, 67 kinetic model 14, 19, 50, 61 kinetic theories of selection 213 kinetic theory of phase transition 6 kinetics 35

255

Kolmogorov equation 5, 97, 98, 99 Krameri-Moyal-Stratonovich equation

98, Ill, 115 Kurtz-type infinitesimal transition

probabilities xx

Lake Balaton 202, 218-19 lake models 203 Iangevin equation 148 Laplace transform xx, 105, 108 Laplacian operator xxiii law of atomic balance 22, 24, 26, 29, 30 law of definite proportions 28 law of large numbers 171 law of mass balance 27 law of multiple proportions 28 length xv, 23, 30 ligand migration 121, 122 limit cycle 12, 131, 154, 189, 198, 212 Lindeberg condition 97 linear 33 linear combination 23, 24, 25, 26, 29,

30 linear nonequilibrium thermodynamics

116, 127 linearly dependent 23 linearly independent 24, 25 linkage class xvii, 43 local sensitivity coefficient 83 logical circuit 213 Lohmann mechanism 31 Lohmann reaction 30 long-term behaviour 12 Lorenz equation xvi, xvii, xviii, 59, 62,

64-5 . Lotka-Volterra mechanism 30 Lotka-Volterra model 27, 54, 55, 129,

194 Lotka-Volterra reaction 30, 44, 62, 113 lumping xiv, 6, 34, 78-9

macrophysics 16 macroscopic 3, 5, 137, 194 macroscopic approach I macroscopic average value 7 macroscopic continuum 8-9 macroscopic effect 7

256

macroscopic structure 13, 95 macroscopic system 6, 10, 92 macroscopic theory 16 macroscopic variable II Markov chain 18 Markov process xx, 10, 18, 19, 96 Markovian jump process 10 Markovian property 18 mass 27 mass action I, 2, 3, 177, 178, 179 mass action kinetic II, 32, 34, 35 mass balance 27 mass-conserving 27 master equation 10, II, 104, 106, 107,

108, Ill, 120, 130, 136-7, 168 mathematical analysis 33 mathematical model 14 mathematical system theory 16 maximum of the stationary distribution

xxii, xxiii, 143 mechanics 16, 84-6 mechanism 32, 41 membrane channel 123, 124, 185 membrane noise 95, 123, 124, 125, 187 memory 6, 8-9, 17 memory-free 18 mesoscopic 8-9, 94, 163 metalanguage 12, 177, 202 Michaelis-Menten reaction xv, 26, 39,

119, 178, 180 microbiology 13 microphysics 16 microreversibility 3 microscopic I, 3, 10, 167 mobility 4, 8-9 molecular computation 213 molecular weight distribution 17 molecularity xiii, 23 monomolecular reaction 2 Monte Carlo methods 112 morphogenesis 174, 176 motion xv, 14, 116 multimodality II, 143, 144 multistability 49, 53 multistage reactor 12 multistationarity II, 49, 50, 133, 136,

143, 144 multivibrator 212,218-19 mushroom 53 mutarotation of simple sugars 2

natural selection 20 I nature of determination 14, 18, 19

nature of time 16 negative cross-effect 7, 64, 66 network I, 12 network thermodynamics 211 neurochemical fluctuation 218-19 neurochemical oscillation 188-90 neurochemistry 185

Index

noise-induced transition 96, 147, 149, 151

nonequilibrium statistical mechanics 3, 5, 8-9, 93

nonequilibrium thermodynamics 16, 115, 116, 167

nonlinear 2, 57 nonlinear equation II nonlinear phenomenon II nonlinear science I nonlinear system 13, 147, 148 nucleation process 169

occupation number 6 oil 16 order of an elementary reaction 23 order of a reaction (with respect to a

component) 2 ordinary differential equation 2, 10, 18 Oregonator model 26, 27 origin of asymmetry of biomolecules

218-19 Ornstein-Uhlenbeck process 152, 157 oscillation 185, 193 oscillatory reaction II, 38, 54, 56 oscillatory solution 165 overall reaction 22 overshoot-undershoot 57 oxyacid 6

parameter estimation 74 parameter sensitivity 83 particle xix, 16, 24 particle number xxiii, 7 pattern formation 163, 174 Pauli equation 5 periodic perturbation 183 periodic solution II, 19, 38, 165 periodicity 54, 56 phase transition 6, 8-9, 13, 131, 209 phenomenological description 6 phenomenological kinetics 5 phenomenological theory 3 phosphorus 204 physical chemistry 22, 34 physicochemical background 2

Index

physics 93 phytoplankton 204 plasma 4, 5, 8-9, 147 plasmachemistry 5 Poincare map, plot 19, 20, 192 Poisson distribution 138, 139, 142 Poisson process xxi, I 00, I 0 I polymer 4, 6, 8-9, 17 polymerisation 207, 208 polynomial differential equation 7, 64 population dynamics 16, 134 population genetics 192, 194, 218-19 population growth 20 I positive equilibrium point xvi, 54 potential xvii, xx, 80 prebiological evolution 213, 218-19 prebiotic chemical model 12 predator 197, 200 prediction 203 pressure xx, 2, 21 prey 197, 200 primary structure 4, 5 principle of quasistationarity 88 probability xx, 5, II, 18 process 16, 18 process-time 14 process without after-effect 10, 96 product xvi, xix, xxiv, 26 product complex vector xvi, xvii, 23 protein 4, 5 pseudo-wave 173

qualitative analysis 12 qualitative property 7, 12, 19, 164 qualitative theory 12, 37, 57 quantitative sociology 13 quantum mechanics 3, 15 quantum statistical mechanics 3 quantum theory 15 quasicomponent 6, 76 quasithermodynamic behaviour 41, 45

radical 4, 23, 27 random environment 201 random variable 14 rate 35 rate constant xvii, xix, xxi, 2, 7, 108,

117, 119, 128, 141 rate equation 2 rate law 3 rate of reaction 2 reactant complex vector xv, xvi, xvii, 23

reaction 21, 22, 26, 27, 31, 33 reaction-diffusion system 163, 166 reaction kinetics I, 7, 16, 21

257

reaction rate 2, 4, 7, 8-9, 16, 17, 157 reactor 12, 21 recurrence 12 reference category 5 regular behaviour II relaxation time xxii, 134, 136 release 186 reservoir 26 reverse rate 3 reversibility 3 reversible 23 Riemannian manifold xvi, 85 Rossler equation xvi, 63

salt 6 scaling theory 137 secondary structure 5 second-order endpoint xvii, 72 second order mechanism 23 second-order phase transition 133 sediment 204 selection 12, 213 self-replication 213 semigroup operator 99 sensitivity 83-4 separation 35 short complex 23, 32 simulation method 112-15, 168 site xv, 16, 17, 18 size 7, 8-9 small fluctuation 7 small system 91, 95, 110 sociobiological model 12 socioeconomic situation 13 solid 4, 6 solid phase 6 solid phase reaction xiii, 6 solid state 8-9 solitary wave 166 solution 35 source of information 7 spatial course 16 spatial homogeneity 14 spatial structure 163, 172-4 spatial transport 12 spatio-temporal phenomenon 16, 101,

163 species xiii, I, 5, 21 spectral density function xx, 117 spectrum xx, 117, 119, 124

258

spectrum line xix, 126 stability 96 stable stationary point II, 12 stablesteady state xxii, 47 state xix, xx, xxi, 6, 14, 16, 17, 18,77 state-space xv, 14, 16, 19, 29, 96-7, 159 state variable 6, 16, 17, 18 stationarity 12 stationary distribution xxii, xxiii, II,

109, 133, 142, 169 statistical mechanical treatment 5 statistical mechanics 6 steady state approximation 178 stochastic description I 0 stochastic differential equation 147, 157 stochastic dynamic system xxii, 147,

148 stochastic ecology 218-19 stochastic model xiii, 6, 7, 10, 19, 29,

91, 99, 105, 162, 167, 202, 206 stochastic process xx, xxi, xxii, 5, 6, 18,

91, 96 stochastic reaction kinetics, logical

status 91 stochastic thermodynamics 93 stochastic thermostatics 93 stoichiometric coefficient xiii, xvi, xvii,

xxi, 23, 33, 34 stoichiometric matrix xvi, xvii, 22, 23,

29 stoichiometric space xvi, xvii, 23, 32 stoichiometry I, II, 21, 26, 28, 29, 35 stress tensor 8-9 structural characterisation 62 structural identifiability 82 structurally stable 7 structure 14, 26, 32 structure of the state space 14, 16 subconservative 28, 29 subconservativity 28, 31, 32, 67 substrate xvi, xx, xxiv, 4, 27 superconservative 28, 29 superconservativity 32 supply 26 surface reaction 4 sustained oscillation II symmetry 6, 84-8, 164 synapse 185 synergetics 13, 95 system 17, 18 system theory 16, 73

temporal change 2

temporal evolution 5, 10 temporal process 14, 16 ternary interaction 12 test-tube 21 theoretical reaction kinetics 12 theory of dissipative structures 13 theory of irreversible processes 16 thermodynamic 21

Index

thermodynamic flow 16 thermodynamic fluctuation theory II,

139 thermodynamics II, 15, 16, 91, 93, 95 thermomechanics 6, 8-9, 17 time xiv, 15, 19 time delay 8-9 time evolution xiv, xv, 21, 26, 33 time homogeneous Markov process 19 topochemistry 8-9 traditional belief 40 trajectory 19, 40, 55 transformation group xvi, xxii, xxiii,

84, 85 transient bimodality 144, 145 transition probability xix, 19 transition state theory 4, 5 transmitter 125, 186, 188 transmitter-receptor interaction 187 transport process 4, 8-9, 14, 16, 21 transport theory 3 travelling wave solution 165 trigger wave 173 trimolecular reaction 2 Turing model 175 two-cell model xxii, xxiii, 168

unimodal distribution II unimodality 138, 140 unique stationary point II uniqueness 67 unstable stationary point II, 12, 167

vector of concentrations 34 velocity I, 2, 7 vesicular hypothesis 186 Vol'pert graph 30, 31, 205 Vol'pert's theorem 45

weak reversibility 29, 43, 44 well-mixed 21 white noise xxi, 148, 150 Wiener process xx, 148, 157 Wiener-Khintchine relation 116, 117 without after-effect I 0

Index

XYZ models 19

zero complex xvi, xviii, xxi, 30 zero deficiency 69

zero deficiency theorem 42, 62 zero divergence 86 zeroth approximation 7, 8-9

259

erdi+toth massaction

Documents