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The Chemical Master Equation: From Reactions to

Complex Networks

Massimo Stella

April 21, 2015

Abstract

This project investigates the chemical master equation and its links to complex networks. The

report is composed of two parts: an introduction, deriving the chemical master equation from

some basic results of statistical mechanics and probability theory, and a second part, relating the

formalism of master equations to growing network models and random walks on graphs. At the

end of the first part, further analytical and numerical results about Markov processes are reported

and discussed.

1 The Physics behind the Chemical Master Equation

The mathematical modelling of chemically reacting gaseous systems, via the framework of

Markovian stochastic processes, relies on some delicate hypotheses from statistical mechanics

[3]. In this section, we review these basic results, with the aim of outlining a physically coherent

approach to the mathematics of the chemical master equation for chemical kinetics.

1.1 Some Physical Premises

Historically, the modelling of chemical reactions as stochastic processes was introduced in [2]

and became increasingly popular in the 1950s and 1960s. However, it was only in the nineties,

with the work of Gillespie [1], that a rigorous microphysical derivation of such approach was

provided, in order to demonstrate its a priori modelling validity. Before that date, in fact, it

was possible to perform such fidelity check only a posteriori, through comparisons with real or

molecular dynamics experiments [2, 13].

1

Following the physical approach of [1], we use a frequentist probability interpretation, i.e.

probability is the fraction of trials in which an event E occurs. Such approach is viable in the

context of chemical kinetics, were there are very high numbers of molecules engaging in the

very same reactions [3, 5]. In addition, it allows to derive results that should otherwise be

postulated (by using Kolmogorov and De Finetti axioms [4]), such as the following:

1. Addition Law: If events A and B are mutually exclusive (i.e. they never occur at the same

time), then the total probability of “either A or B” is given by P (A[B) = P (A)+P (B);

2. Multiplication Law: The joint probability of two events A and B happening at the same

time is P (A\B) = P (A,B) = P (A)·P (B|A), where P (B|A) is the conditional probability

of B happening, given the occurrence of A.

In our case, events are going to be chemical reactions at molecular level [1]. Therefore, let

us consider a gas comtaining molecules of N 2 N di↵erent species, S1, S2, ..., SN , interacting

through M chemical reaction channels R1, ..., RM and all contained in a recipient of constant

volume V . Let Xi(t) be a variable related to the number of molecules of type Si, in the system,

at time t � 0, with i 2 I := (1, 2, ..., N). We focus principally on the bimolecular elementary

reaction channels of the form Si + Sj ! Sk + ..., with i, j, k 2 I.

We restrict our analysis to close-to-ideal gases in thermodynamic equilibrium. In other

words, we consider the molecules as distinguishable, non-puntiform1 hard spheres, of given

mass and radius, interacting mainly by collisions, with other types of long range interactions

being neglectable in both frequency and intensity terms. Furthermore, the thermodynamic

equilibrium implies the existence of well defined temperature parameter T for the whole system.

Also, it means that Boltzmann’s molecular chaos hypothesis (i.e. Stosszahlansatz ) is valid: the

particle velocities are both uncorrelated and independent of position, mainly because of thermal

fluctuations [5]. These physical premises lead to two mathematical propositions [1, 3]:

• Spatial homogeneity : the probability of finding any randomly selected molecule inside any

subregion �V of the volume V equals �V/V ; in mathematical terms the molecule positions

are independent2 random variables, uniformly distributed over the domain V .

• Maxwell-Boltzmann velocity distribution: denoted as kB Boltzmann’s constant [5], then 1Ideal gases require for particles to be treated as puntiform mass points. Furthermore, the distinguishability

of particles refers to the possibility of identifying each particle in time, according to its Newtonian trajectory, given an initial “labelling”. This concept looses any validity in quantum mechanics, where there is no quantum counterpart of the idea of trajectory [5].

2Two random variables X and Y are independent (or pairwise independent) i↵ their joint probability distri- bution factorises, in formulas P (X \ Y ) = P (X,Y ) = P (X)P (Y ) [4].

2

the probability of finding a molecule of mass m with velocity between v and v + dv is3:

pMB(v)dv =

✓ m

2⇡kBT

◆3/2 exp

�m |v|

2

2kBT

! . (1)

In mathematical terms, the above equation means that each Cartesian velocity component

of a randomly selected molecule is a normally distributed random variable, with zero mean

and variance kBT/m. Additionally, all such components are independent variables.

These two points are often referred to as the system being “well-stirred”, so that molecules

are well mixed though the whole spatial domain and in thermal equilibrium. It has to be

underlined that the above findings emerge from a deterministic chaotic (mixing) behavior of

molecules at microscopic level, in a scenario close to ideality and in thermal equilibrium. It is

ultimately this physical concept of “molecular chaos” that provides the “unreasonable e�cacy”

of a mathematical stochastic tractation of such systems [5, 3, 14].

1.2 Towards the Chemical Master Equation

We want to determine the evolution law for the species population vector4 X(t) = (X1(t), ..., XN(t)),

compatibly with the two above definitions of molecule positions and velocities and focusing on

bimolecular reactions. In order to perform such task, we have to determine the probability

⇡µ(t, dt) that two molecules, randomly selected at time t, react in the next dt time interval,

accordingly to the bimolecular channel µ. However, according to the above physical discussion,

in order for a bimolecular reaction to occur, two (spherical) molecules i and j have to collide

with each other first. Additionally, their collision must be e�cient [1].

Denoted with uµ(t, dt) the probability of a collision (defined analogously to ⇡µ(t+ dt), but

for a collision event) and with Pµ the probability of a chemical reaction to be triggered, then:

⇡µ(t, dt) = uµ(t, dt) · Pµ. (2)

In other words, the probability ⇡µ(t, dt) that an e�cient collision (i.e. a reaction) happens in

the time interval [t, t+ dt) is equal to the product of the collision probability uµ(t, dt) with the

conditional probability Pµ = P (trigger a reaction|collision). 3In statistical mechanics, given a Cartesian vector v = (v

x

, v y

, v z

), the di↵erential element dv, sometimes denoted also as d3v, is equal to dv

x

dv y

dv z

. 4Because of the intrinsic stochasticity of our chemical system, we have to consider X(t) as an N -dimensional

random variable, having outcomes o defined on a subset of NN . Rather than considering the time evolution of X(t), we are more interested in determining the probability P (X(t) = o), evolving over time.

3

In order to compute uµ we can resort to the following:

Theorem 1. [1] Let {Ci}i2N be a set of mutually exclusive and collectively exhaustive events,

partitioning the sample space. Let the event A be mutually exclusive to {Ci}i2N. Then:

P (A) = X

i

P (Ci) · P (A|Ci) (3)

Proof. The Cis represent a partition of the whole sample space, so that actually A can be

decomposed onto the set {Ci} in terms of mutually exclusive subsets, i.e. A = [i(A\Ci). This

means that P (A) = P ([i(A\Ci)) = P

i P (A\Ci), from the addition law. Also, P

i P (A\Ci) = P

i P (A,Ci) = P

i P (Ci)·P (A|Ci), with the last passage being due to the multiplication law.

The above theorem is valid also in the continuous case (i.e. when i is a real index, defined

on the set K), with the sum substituted by an integral, with a proper measure.

We consider Cv0 , v 0 2 R3, being the event that two randomly selected molecules (in the

channel Rµ) at time t have a relative velocity v 0 =vj�vi. Given the simmetries of the Maxwell-

Boltzman velocity distribution (explicitly depending only on the modulus of velocity), a simple

change of reference frame and the random variable transformation theorem for statistically

independent random variables [3, 4, 5] lead to

P (Cv0) =

✓ m⇤

2⇡kBT

◆3/2 exp

� m⇤ �� v

0��2

2kBT

! , (4)

where m⇤ = mimj/(mi+mj) is the reduced mass of the two reactant molecules (in the channel

Rµ). In the reference frame of the j-th molecule, the i-th molecule moves on the straight

path connecting i and j at speed v 0 , covering a length

�� v

0�� dt in the time interval [t, t + dt).

Additionall

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