The chemical Langevin equation: a path integral view of Gillespie's derivation

TL;DR

This paper presents a novel path integral formulation of the chemical master equation, demonstrating how Gillespie's conditions lead to the chemical Langevin equation and comparing different derivation approaches.

Contribution

It introduces an original path integral perspective of the CME and clarifies the relationship between Gillespie's conditions and the CLE derivation.

Findings

Path integral description of the CME constructed

Gillespie's conditions lead to a path integral equivalent of the CLE

Comparison shows qualitative differences between derivation methods

Abstract

In 2000, Gillespie rehabilitated the chemical Langevin equation (CLE) by describing two conditions that must be satisfied for it yield a valid approximation of the chemical master equation (CME). In this work, we construct an original path integral description of the CME, and show how applying Gillespie's two conditions to it directly leads to a path integral equivalent to the CLE. We compare this approach to the path integral equivalent of a large system size derivation, and show that they are qualitatively different. In particular, both approaches involve converting many sums into many integrals, and the difference between the two methods is essentially the difference between using the Euler-Maclaurin formula and using Riemann sums. Our results shed light on how path integrals can be used to conceptualize coarse-graining biochemical systems, and are readily generalizable.