Approximate probability distributions of the master equation

TL;DR

This paper introduces an analytical approximation method for the probability distributions described by master equations in mesoscopic systems, using orthogonal polynomials, with a focus on discrete support solutions that converge rapidly.

Contribution

It presents a novel approximation technique for master equation distributions employing orthogonal polynomials, offering two formulations and demonstrating superior convergence of discrete solutions.

Findings

Discrete approximations rapidly converge to true distributions.

Continuous approximations tend to oscillate and become negative with higher truncation.

The method yields simple analytical expressions for biological systems.

Abstract

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 orthogonal 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.