The chemical Langevin equation

TLDR

The chemical master equation accurately models the stochastic dynamics of a well‑stirred mixture of N species interacting through M reaction channels, and the chemical Langevin equation—first studied by Kurtz—provides a Langevin‑type approximation to this master equation. The study demonstrates that, whenever two explicit dynamical conditions hold, the chemical master equation yields a Langevin‑type approximation whose accuracy depends on time‑varying conditions rather than a static system size, thereby offering a new perspective on the origin and magnitude of noise and clarifying the link between the stochastic master equation and the deterministic reaction‑rate equation.

Abstract

The stochastic dynamical behavior of a well-stirred mixture of N molecular species that chemically interact through M reaction channels is accurately described by the chemical master equation. It is shown here that, whenever two explicit dynamical conditions are satisfied, the microphysical premise from which the chemical master equation is derived leads directly to an approximate time-evolution equation of the Langevin type. This chemical Langevin equation is the same as one studied earlier by Kurtz, in contradistinction to some other earlier proposed forms that assume a deterministic macroscopic evolution law. The novel aspect of the present analysis is that it shows that the accuracy of the equation depends on the satisfaction of certain specific conditions that can change from moment to moment, rather than on a static system size parameter. The derivation affords a new perspective on the origin and magnitude of noise in a chemically reacting system. It also clarifies the connection between the stochastically correct chemical master equation, and the deterministic but often satisfactory reaction rate equation.