Chemical master equation

Basic articles for chemical master equations are D.A. McQuarrie, J. Appl. Prob. 4, 413 (1967) and E.W. Montroll in Energetics in Metallurgical Phenomena III (W.M. Mueller ed., Gordon and Breach, New York 1967). 

A major difficulty is that such hierarchies of molecular models are not exactly known. Recent work by Gillespie (2000, 2002) has established such a hierarchy for stochastic models of chemical reactions in a well-mixed batch reactor. This hierarchy is depicted in Fig. 3b. In particular, it was shown that the chemical master equation is deduced to a chemical Langevin equation when the population sizes are relatively large. Finally, the deterministic behavior can be... 

Stochastic biochemical systems and 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. 

Figure 11.1 A schematic that illustrates the analogy between the theories for mechanical motions and for chemical dynamics. Newton s law of motion, governing a collection of particles with positions x (t), X2(t), , Xj/(t), arises from Schrodinger s equation for the wave function f in the limit h - 0. Similarly, the chemical master equation for p(n, n2, , ftat, t) yields the law of mass action in the limit V -> oo.

Recall that in Section 10.3 we worked out a detailed theory for the equilibrium distribution for the reaction A + 11 C. Here the task is to determine the governing differential equation (chemical master equation) for the dynamics of the state probabilities in Equation (11.5). 

Chemical master equation for Michaelis-Menten kinetics... 

Here to illustrate the procedure for non-linear systems, we work out the chemical master equation for the Michaelis-Menten reaction of Equation (11.19). We let p(m, n, t) be the probability that m, S, and n ES molecules occur in the enzyme reaction system at time t. The probability p(m, n, t) satisfies the CME... 

To use a computer to simulate a stochastic trajectory of the chemical master equation such as described in Figure 11.4, one must establish the rules of how to move the system from one grid point to its neighboring points. The essential idea is to draw random moves from the appropriate distribution and to assign random times (also drawn from the appropriate distribution) to each move. Thus each simulation step in the simulation involves two random numbers, one to determine the associated time step and one to determine the grid move. 

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],... 

Fig. 6.3. Distributions of x in between two steady states (A, C), and dynamic fluctuations between these two states (B, D). The steady-state distributions (A and C) were calculated using (6.4) in the text. The fluctuations in x were calculated using a Gillespie-type Monte Carlo algorithm to the chemical master equation (Beard and Qian, 2008). Parameters Panels (a) and (b), fci = 2.7, = 0.6, ks = 0.25, fc4 =...

D- T. Gillespie. A rigorous derivation of the chemical master equation, Physica A, 188 404-425,1992. 

Networks of coupled chemical reactions in a dilute solution should be described by a chemical master equation whenever fluctuations are relevant due to small numbers of at least one of the involved species. The master equation contains the rate constants of all possible reactions. The solution of the chemical master equation gives the dynamics of the probability of flnding a certain number of molecules of each species at a given time for a given initial condition. This leads to the stochastic trajectory of the network by recording the time at which each particular reaction took place with its concomitant change of the number of molecules. 

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... 

In the chemical master equation, the steady-state probability distribution of the equUihrium steady state is a Poisson distribution. For Schlogl s model steady-state probability distributions become... 

From chemical master equations under nonequilibrium steady state, the ratio between the probability of M forward turnovers p drit = M) and that of M backward turnovers p(drit= —M) is... 

The continued fraction representation of the transition factor has been applied for solving one-variable chemical master equations (Haag Hanggi, 1979, 1980). For simple birth and death processes with birth and death rate functions / and x the nearest neighbour transition gj is ... 

Gardiner, C. W. Chaturvedi, S. (1977). The Poisson representation a new technique for chemical master equations. J. Stat. Phys., 17, 429-68. 

Given the simplicity of the current system, it was possible to analytically solve the resulting chemical master equation. However, this is not always the case and one is limited to simulating individual realizations of the stochastic process in order to reconstruct the probability distributions out from several simulations. Below, we introduce the celebrated Gillespie algorithm (Gillespie 1977) to simulate the stochastic evolution of continuous-time discrete-state stochastic processes, like the one analyzed in the present chapter. 

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