Distribution approximations for the chemical master equation: comparison of the method of moments and the system size expansion

TL;DR

This paper compares two methods for approximating probability distributions in chemical master equations, highlighting their accuracy and limitations in modeling stochastic gene expression networks.

Contribution

It introduces a comparative analysis of the moment-based maximum entropy method and the system size expansion for CME distribution approximation.

Findings

Both methods accurately approximate CME distributions.

Each method has distinct advantages and limitations.

Performance varies with unimodal and multimodal distributions.

Abstract

The stochastic nature of chemical reactions involving randomly fluctuating population sizes has lead to a growing research interest in discrete-state stochastic models and their analysis. A widely-used approach is the description of the temporal evolution of the system in terms of a chemical master equation (CME). In this paper we study two approaches for approximating the underlying probability distributions of the CME. The first approach is based on an integration of the statistical moments and the reconstruction of the distribution based on the maximum entropy principle. The second approach relies on an analytical approximation of the probability distribution of the CME using the system size expansion, considering higher-order terms than the linear noise approximation. We consider gene expression networks with unimodal and multimodal protein distributions to compare the accuracy of the two approaches. We find that both methods provide accurate approximations to the distributions of the CME while having different benefits and limitations in applications.