A brief introduction to the chemical master equation

The chemical master equation (CME) for a given system invokes the same rate constants as the associated deterministic kinetic model. Yet the CME is more fundamental than the deterministic kinetic view. Just as Schrodinger s equation is the fundamental equation for modeling motions of atomic and subatomic particle systems, the CME is the fundamental equation for reaction systems. Remember that Schrodinger s equation is not a model for a specific mechanical system. Rather, it is a theoretical framework upon which models for particular systems can be developed. In order to write down a model for an atomic system based on Schrodinger s equation, one needs to know how to write down the Hamiltonian a priori. Similarly, the CME is not a model for a specific biochemical reaction system it is a theoretical framework. To determine the CME model for a reaction system, one must know what are the possible elementary reactions and the associated rate constants. 

As far as we know, the CME first appeared in the work of Max Delbriick in 1940, who studied a small chemical reaction system in terms of a stochastic model 

Deterministic dynamics of biochemical reaction systems can be visualized as the trajectory of (ci(t), c2(t), , c v(0) in a space of concentrations, where d(t) is the concentration of ith species changing with time. This mental picture of path traced out in the N-dimensional concentration space by deterministic systems may prove a useful reference when we deal with stochastic chemical dynamics. In stochastic systems, one no longer thinks in terms of definite concentrations at time t rather, one deals with the probability of the concentrations being xu x2, , Wy at time t  

The CME is the equation for the probability function p, or equivalently if the system s volume is constant, for the probability function p(ni, n9, , jv, t)where is the number of molecules of species i. With given concentrations (ci, c2, , c v) at a time t, deterministic kinetic models give precisely what the concentrations will be at time t + St. According to the stochastic CME, however, the concentrations at t + St can take many different values, each with certain probability. 

Furthermore, the CME framework has been shown to be consistent with the general theory of non-equilibrium thermodynamics [109], and the recently developed 

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