Master equations are common descriptions of mesoscopic systems. Analytical solutions to these equations can rarely be obtained. We here derive an analytical approximation of the time-dependent probability distribution of the master equation using ort
hogonal polynomials. The solution is given in two alternative formulations: a series with continuous and a series with discrete support both of which can be systematically truncated. While both approximations satisfy the system size expansion of the master equation, the continuous distribution approximations become increasingly negative and tend to oscillations with increasing truncation order. In contrast, the discrete approximations rapidly converge to the underlying non-Gaussian distributions. The theory is shown to lead to particularly simple analytical expressions for the probability distributions of molecule numbers in metabolic reactions and gene expression systems.
Few analytical methods exist for quantitative studies of large fluctuations in stochastic systems. In this article, we develop a simple diagrammatic approach to the Chemical Master Equation that allows us to calculate multi-time correlation functions
which are accurate to a any desired order in van Kampens system size expansion. Specifically, we present a set of Feynman rules from which this diagrammatic perturbation expansion can be constructed algorithmically. We then apply the methodology to derive in closed form the leading order corrections to the linear noise approximation of the intrinsic noise power spectrum for general biochemical reaction networks. Finally, we illustrate our results by describing noise-induced oscillations in the Brusselator reaction scheme which are not captured by the common linear noise approximation.
The linear noise approximation is commonly used to obtain intrinsic noise statistics for biochemical networks. These estimates are accurate for networks with large numbers of molecules. However it is well known that many biochemical networks are char
acterized by at least one species with a small number of molecules. We here describe version 0.3 of the software intrinsic Noise Analyzer (iNA) which allows for accurate computation of noise statistics over wide ranges of molecule numbers. This is achieved by calculating the next order corrections to the linear noise approximations estimates of variance and covariance of concentration fluctuations. The efficiency of the methods is significantly improved by automated just-in-time compilation using the LLVM framework leading to a fluctuation analysis which typically outperforms that obtained by means of exact stochastic simulations. iNA is hence particularly well suited for the needs of the computational biology community.
