Exactly reduced chemical master equations

Abstract

In the small-number limit, we must abandon the description of chemical systems in terms of continuous concentration variables which evolve according to deterministic rate equations in favor of a discrete stochastic formulation. The probability distribution for the molecular populations however does obey a deterministic equation called the chemical master equation. Any desired population statistic (mean, standard deviation, etc.) can be obtained from the probability distribution. Unfortunately, the master equation consists of a huge set of differential equations, and it is thus in general impractical to use it directly. In this paper, we review the ideas underlying discrete population modeling and the chemical master equation. We then develop methods for reducing the chemical master equation to a much smaller set of differential equations by exploiting the same time-scale separation which leads to the emergence of a hierarchy of attracting manifolds in the mass-action case. Finally, we develop a method for generating an initial condition for the reduced model based on a generalization of the stationary reactant approximation. © 2006 Springer-Verlag Berlin Heidelberg.