Stochastic Nonlinear Dynamics of Cellular Biochemical

Systems:

Hong Qian

Department of Applied Mathematics

University of Washington

abstractI present the stochastic, chemical master equation as a unifying approach to the dynamics of biochemical reaction systems in a mesoscopic volume under

a living environment. A living environment provides a continuous chemical energy input that sustains the reaction system in a nonequilibrium steady state

with concentration fluctuations. We discuss nonlinear biochemical reaction systems such as phosphorylation-dephosphorylation cycle (PdPC) with

bistability. Emphasis is paid to the comparison between thestochastic dynamics and the prediction based on the traditional approach

based on the Law of Mass Action. We introduce the dirence between nonlinear bistability and stochastic bistability, the latter has no deterministic counterpart.

For systems with nonlinear bistability, there are three dirent time scales: (a) individual biochemical reactions, (b) nonlinear network dynamics approaching to attractors, and (c) cellular evolution. For mesoscopic systems with size of a

living cell, dynamics in (a) and (c) are stochastic while that with (b) is dominantly deterministic. Both (b) and (c) are emergent properties of a

dynamic biochemical network; We suggest that the (c) is most relevant to major cellular biochemical processes such as epigenetic regulation, apoptosis,

and cancer immunoediting. The cellular evolution proceeds with transitions among the attractors of (b) in a "punctuated equilibrium" manner.

An analytical theory for Darwin’s variations in mesoscopic scale?

Intrinsic variations = Stochasticity

Natural environmental selections = Bias

Here are some recent headlines:

Physics

Molecular Cellular Systems

EvolutionaryBiology

Chemistry

Stochastic Physical Chemistry (1940)

The Kramers’ theory and the CME clearly marked the domains of two areas of

chemical research: (1) The computation of the rate constant of a chemical reaction

based on the molecular structures, energy landscapes, and the solvent environment;

and (2) the prediction of the dynamic behavior of a chemical reaction system,

assuming that the rate constants are known for each and every reaction in the

system.

Basic Facts on Single Molecule Stochastic Transition

Time is in the waiting, the transition is instaneous!

A Bk1

k2

A

B GΔek ‡

The Biochemical System Inside CellsE

GF

Signal T

ransduction Pathw

ay

The kinetic isomorphism between PdPC and GTPase

PdPC with a Positive Feedback

Gene regulatory and

Biochemical signaling Networks

(B)

gene state 0 gene state 1f

ho[TF]

synthesis

degradationTF

k

g0 g1

(A)E E*

K + E* K†

k3[P]

k[K ] †

k2

k-2

k[K]

Another Kinetic Isophorphism

According to macroscopic chemical kinetics following the

Law of Mass Action

Simple Kinetic Model based on the Law of Mass Action

NTP NDP

Pi

E

P

R R*

.

].][[

],)[]][[(

,][

*

*

*

θβ

α

RPβJ

RREαJ

JJdt

Rd

2

χ1

21

activating signal:

acti

vati

on

leve

l: f

1 4

1

Bifurcations in PdPC with Linear and Nonlinear Feedback

= 0

= 1

= 2

hyperbolic delayed onset

bistability

According to mesoscopic chemical kinetics following the

Chemical Master Equation

A Markovian Chemical Birth-Death Process

nZ

k1nxnyk1(nx+1)(ny+1)

k-1nZ k-1(nZ +1)

k1

X+Y Zk-1

Chemical Master Equation Formalism for Chemical

Reaction SystemsM. Delbrück (1940) J. Chem. Phys. 8, 120.D.A. McQuarrie (1963) J. Chem. Phys. 38, 433.D.A. McQuarrie, Jachimowski, C.J. & M.E. Russell (1964) Biochem. 3, 1732.T.L. Hill & I.W. Plesner (1965) J. Chem. Phys. 43, 267; (1971) 54, 34.I.G. Darvey & P.J. Staff (1966) J. Chem. Phys. 44, 990; 45, 2145; 46, 2209. D.A. McQuarrie (1967) J. Appl. Prob. 4, 413. G. Nicolis & A. Babloyantz (1969) J. Chem. Phys. 51, 2632.R. Hawkins & S.A. Rice (1971) J. Theoret. Biol. 30, 579.T.G. Kurtz (1971) J. App. Prob. 8, 344; (1972) J. Chem. Phys. 57, 2976.J. Keizer (1972) J. Stat. Phys. 6, 67.D. Gillespie (1976) J. Comp. Phys. 22, 403; (1977) J. Phys. Chem. 81, 2340.

and more recently …

1-dimensional, 1-stable, 1-unstable fixed pts1-dimensional, 2-stable, 1-unstable fixed pts2-dimensional, 1-stable limit cycle via Hopf bifurcation

R R*

K

P

2R*0R* 1R* 3R* … (N-1)R* NR*

Markov Chain Representation

v1

w1

v2

w2

v0

w0

Bistability and Emergent Sates

Pk

number of R* molecules: k

defining cellular attra

ctors

Landscape and Lyapunov Property

kP )(xf )(ln1

)( xfV

x

)(x

V

kx

*

(x,)

Extrema value

0

0.3

0.6

0.9

1.2

1.5

3 4 5 6 7 8

e

1(e) 2(e)

the cusp

the critical point

*(e)(B)

4

6

8

10

0.01 0.1 1

xss

(A)

A fundamental difference of two types of landscapes

• For a detailed balance system, such as protein folding dynamics, the energy landscape is given a priori. It directs the dynamics of the system.

• For a system without detailed balance, can be considered as an “landscape for dynamics”. However, it is a consequence of the dynamics. That is why we call it emergent. It dynamics is non-local.

• Q: which one is the “fitness landscape”?

Biological Implications:

for systems not too big, not too small, like a cell …

Emergent Mesoscopic Complexity• It is generally believed that when systems become

large, stochasticity disappears and a deterministic dynamics rules.

• However, this simple example clearly shows that beyond the “infinite-time” in the deterministic dynamics, there is another, emerging stochastic, multi-state dynamics!

• This stochastic dynamics is completely non-obvious from the level of pair-wise, static, molecule interactions. It can only be understood from a mesoscopic, open driven chemical dynamic system perspective.

In a cartoon: Three time scales

ny

nx

appropriate reaction coordinate

ABpr

obab

ility

A B

chemical master equation

discrete stochastic model among attractors

emergent slow stochastic dynamics and landscape

cy

cx

A

B

fast nonlinear differential equationsmolecular s

ignaing t.s.

biochemica

l netw

ork t.s

.

cellular evolution t.s.

Ch

oi, P

.J.; Ca

i, L.; F

rieda

, K. an

d X

ie, X

.S.

Scie

nce

, 322

, 44

2- 4

46 (2

008

). Bistability in E. coli lac operon

switching

Bistability during the apoptosis of human brain tumor cell (medulloblatoma) induced by topoisomerase II inhibitor (etoposide)

Buckmaster, R., Asphahani, F., Thein, M., Xu, J. and Zhang, M.-Q.Analyst, 134, 1440-1446 (2009)

Chemical basis of epi-genetics:

Exactly same environment setting and gene, different internal

biochemical states (i.e., concentrations and fluxes). Could

this be a chemical definition for epi-genetics inheritance?

The inheritability is straight forward: Note that (x) is independent of volume of the cell, and x is the

concentration!steady state chemical concentration distribution

concentration of regulatory molecules

c1* c2*2

c1*

2

c2*

0

25

50

75

100

0 50 100 150 200

time

nu

mb

er o

f E

*

0

40

80

120

160

0 1 2 3 4 5

0.001

0.01

0.1

1

10

100

0 0.2 0.4 0.6 0.8 1

0

50

100

150

200

0 2500 5000 7500 10000

concentration of E *

V=100

V=200

(A)

(B) (C)

(D)

Ntot=100

Ntot=200

1000 2000500 15000

total number of molecule E

swit

chin

g t

ime

in m

sec 1030

1015

3x1022

1.0

3x107

10 hrs

1011 yrs

Another insight

If one perturbs such a multi-attractor stochastic system:

• Rapid relaxation back to local minimum following deterministic dynamics (level ii);

• Stays at the “equilibrium” for a quite long tme;

• With sufficiently long waiting, exit to a next cellular state.

The emergent cellular, stochastic “evolutionary” dynamics follows not

gradual changes, but rather punctuated transitions between

cellular attractors.

alternative attractor

localattractor

Relaxation process

abrupt transition

Relaxation, Wating, Barrier Crossing: R-W-BC of Stochastic

Dynamics

• Elimination

• Equilibrium

• Escape

An emerging “thermo”-dynamic structure in stochastic dynamics

The Thermodynamic Structure of Stochastic Systems

Two Origins of Irreversibility

Summary for Systems Biol.(1) As a physical chemistry approach to

cellular biochemical dynamics, mesoscopic reaction systems can be

modeled according to the CME: A new mathematical theory.

(2) A possible chemical bases of epi-genetic inheritance is proposed;

(3) Emerging landscape is introduced;(4) Beyond deterministic physics, there is stochastic diversity in evolutionary time!

Summary for Theoret. Physics (5) Nonlinear multi-attractors become

stochastic attractors. Infinite large systems exhibit nonequilibrium phase

transition with Maxwell construction and Lee-Yang theory;

(6) A nonequilibrium statistical “thermo- dynamics” emerges from stochastic

nonlinear dynamics; (7) Epigenetic switching is a form of

nonequilibrium phase transition?

Thank You!