Stochastic biochemical systems and the chemical master equation

Stochastic biochemical systems and the chemical master equation 

not surprising that when we deal with macroscopic systems with numbers of molecules of the order of the Avogadro constant (6.022 xlO23), we do not need to concern ourselves with stochastic fluctuations. In a biological cell, however, many reacting species occur with only a few, or a few hundred, copies (see 

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It is widely appreciated that chemical and biochemical reactions in the condensed phase are stochastic. It has been more than 60 years since Delbriick studied a stochastic chemical reaction system in terms of the chemical master equation. Kramers theory, which connects the rate of a chemical reaction with the molecular structures and energies of the reactants, is established as a central component of theoretical chemistry [77], Yet study of the dynamics of chemical and biochemical reaction systems, in terms of either deterministic differential equations or the stochastic CME, is not the exclusive domain of chemists. Recent developments in the simulation of reaction systems are the work of many sorts of scientists, ranging from control engineers to microbiologists, all interested in the dynamic behavior of biochemical reaction systems [199, 210],... 

For biochemically driven reactions, embedding heat bath provides the source of stochastic dynamics. The stochastic model, in the form of the chemical master equation, is an infinite system of mathematically coupled ordinary differential equations (Vellela and Qian, 2009). Assuming that tia and b are the number of substrate molecules, which are fixed for a fixed volume, and p (f) is the probability of having nX molecule at time t. The stochastic model equations are... 

Over the last years it has become clear that the dynamics of most biological phenomena can be studied via the techniques of either nonlinear dynamics or stochastic processes. In either case, the biological system is usually visualized as a set of interdependent chemical reactions and the model equations are derived out of this picture. Deterministic, nonlinear dynamic models rely on chemical kinetics, while stochastic models are developed from the chemical master equation. Recent publications have demonstrated that deterministic models are nothing but an average description of the behavior of unicellular stochastic models. In that sense, the most detailed modeling approach is that of stochastic processes. However, both the deterministic and the stochastic approaches are complementary. The vast amount of available techniques to analytically explore the behavior of deterministic, nonlinear dynamical models is almost completely inexistent for their stochastic counterparts. On the other hand, the only way to investigate biochemical noise is via stochastic processes. 

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