## Synonyms

## Definition

Chemical master equation is the stochastic counterpart of the chemical kinetic equation based on the law of mass action. It describes the kinetics of chemical reactions in a rapidly stirred tank with small volume in terms of stochastic reaction times giving rise to fluctuating copy numbers of reaction species.

Consider a system of fixed volume V at constant temperature T. Let there be well-stirred mixture of N ≥ 1 molecular species {S_{1}, …,S_{N}} and M ≥ 1 reactions {R_{1}, …, R_{M}}. One specifies the dynamical state of this system by X(t) = (X_{1}(t), …, X_{N}(t)), where X_{i}(t) is the copy number of molecular species S_{i} in the system at time t.

One describes the time evolution of X(t) from some given initial state X(t_{0}) = x_{0}. Both single-molecule experimental measurements and theoretical investigations have shown that X(t) is a stochastic process because the time at which a particular reaction occurs is random.

_{j}, there is a corresponding rate function r

_{j}and a stoichiometry vector v

_{j}= (v

_{j1}, …, v

_{jN}), which are defined as

The stoichiometry vector {v_{ji}} can be obtained from the difference between the numbers of a molecular species that are consumed and produced in the reaction.

_{j}has the mathematical form:

Here k_{j} is the specific probability rate constant for reaction R_{j}, which is defined such that k_{j}dt is the probability that a randomly chosen combination of the reactants of R_{j} will react accordingly in the next infinitesimal time interval dt. k_{j} is intimately related to the rate constants in the traditional law of mass action kinetics.

The function h_{j}(x) in Eq. 3 measures the number of distinct combinations of R_{j} reactant molecules available in the state x. It is a combinatorial factor that can be easily obtained from the reaction R_{j}, i.e., \( {{\mathbf {h_j}(\rm x)}} = {\prod\nolimits_{{k = 1}}^N {\frac{{{x_k}!}}{{{m_{{jk}}}!({x_k} - {m_{{jk}}})!{^.}}}}} \). In the transitional mass-action kinetics, this term is related to the product of the concentrations of all reactants.

_{k}>> m

_{jk}, then approximately

*x*

_{ k }

*/V*) is the concentration of species

*k*, and the term in the parenthesis, \( k_j^{\prime} = \frac{{{k_j}{V^{{n - 1}}}}}{{\prod\limits_{{k = 1}}^N {{m_{{jk}}}!} }} \), is the reaction rate constant in the traditional chemical kinetics. Here, k

_{j}is regarded as “stochastic reaction constant,” while \( k_j^{\prime} \) is the corresponding reaction constant for Law of Mass Action.

The reaction rate constant is deduced only from experiments, or calculated based on Kramers-Marcus theory. It is usually a function of temperature but is independent of the volume of a reaction system. However, the stochastic reaction constant *k*
_{
j
} depends on the system volume and the temperature.

From the rate function given from above, the state vector X(t) is a Markov jump process on the nonnegative N-dimensional integer lattice.

Any Markov process can be described by two completely different, complementary mathematical methods. One follows its stochastic trajectories and one consider its probability distribution function changing with time. The Gillespie algorithm gives the former, and the chemical master equation is for the latter.

In the chemical master equation perspective, one no longer asks what are the copy number (or concentration) of species i at time t, but rather what is the probability of the system having x_{i} copies of species X_{i} at time t.

_{j}reacts according to R

_{j}in (t, t + dt), the probability of which is r

_{j}(x − v

_{j})dt. Thus,

_{j}in (t, t + dt), the probability P(x, t) will decrease. Thus,

## Characteristics

### Consistent with Law of Mass Action

^{V}(t) denote this stochastic process, where V is the volume of the system, and the initial condition (in the thermodynamic limit) is

_{0}). Kurtz has shown that the relationship between these two solutions is

It is important to note the mathematical subtlety of \( s \leq T \). The above result is only valid for finite time interval, even though it can be as long as one likes: The *ε* is a function of *T*; greater the *T*, larger the *ε*.

Compare the solution to the chemical master equation with initial distribution concentrated at *x*
_{
0
}, *P*(*x,t*|*x*
_{0}) and the deterministic kinetics c(*t*,*x*
_{0}); if the latter approaches to a steady-state value then the former cumulates probability at the steady state. In general, there is an agreement between the steady states of the deterministic kinetics and the peaks of the stationary distribution of the chemical master equation.

However, in the limit of infinitely large volume, the situation is quite different: The peaks of the stationary distribution of chemical master equation may not be consistent with the stable fixed points of the mass-action kinetics. This is exactly due to the \( s \leq T \) in Eq. 15. This can be explained by a theory of multiple timescales.

### Numerical Simulation: Gillespie Algorithm

We now introduce stochastic simulation methods that generate the random trajectories of X(t). Once we have enough sample trajectories of a stochastic process, we will be able to calculate the probability distribution function P(x, t) and all other statistical behaviors including the mean trajectory, variances, and correlations.

_{1}and a

_{2}from the uniform distribution in the unit interval, and then take

- 1.
Initialization: Let the initial state as X = X

_{0}, and t = t_{0}. - 2.
Simulation: Generate a pair of random numbers (\( \tau \), j) according to the probability density function (17).

- 3.
Update: Increase the time by \( \tau \), and replace the molecule numbers by X + v

_{j}. - 4.
Iterate: Go back to Step 2 unless the end of the simulation procedure.

The exact stochastic simulation algorithm consists with the chemical master equation and gives an exact sample trajectory of the real system.