How accurate are the non-linear chemical Fokker-Planck and chemical Langevin equations?

TL;DR

This paper evaluates the accuracy of the chemical Fokker-Planck and Langevin equations, showing they provide highly accurate estimates of mean concentrations and fluctuations, especially in small biochemical systems, through a systematic size expansion.

Contribution

The authors derive order-of-magnitude accuracy estimates for the chemical Fokker-Planck equation using the system-size expansion, clarifying its precision compared to the master equation.

Findings

Chemical Fokker-Planck estimates are accurate to order Ω^{-3/2} for non-detailed balance systems.

For systems obeying detailed balance, accuracy improves to order Ω^{-2}.

Errors in predictions are typically less than a few percent in small biochemical systems.

Abstract

The chemical Fokker-Planck equation and the corresponding chemical Langevin equation are commonly used approximations of the chemical master equation. These equations are derived from an uncontrolled, second-order truncation of the Kramers-Moyal expansion of the chemical master equation and hence their accuracy remains to be clarified. We use the system-size expansion to show that chemical Fokker-Planck estimates of the mean concentrations and of the variance of the concentration fluctuations about the mean are accurate to order $\Omega^{-3/2}$ for reaction systems which do not obey detailed balance and at least accurate to order $\Omega^{-2}$ for systems obeying detailed balance, where $\Omega$ is the characteristic size of the system. Hence the chemical Fokker-Planck equation turns out to be more accurate than the linear-noise approximation of the chemical master equation (the linear Fokker-Planck equation) which leads to mean concentration estimates accurate to order $\Omega^{-1/2}$ and variance estimates accurate to order $\Omega^{-3/2}$. This higher accuracy is particularly conspicuous for chemical systems realized in small volumes such as biochemical reactions inside cells. A formula is also obtained for the approximate size of the relative errors in the concentration and variance predictions of the chemical Fokker-Planck equation, where the relative error is defined as the difference between the predictions of the chemical Fokker-Planck equation and the master equation divided by the prediction of the master equation. For dimerization and enzyme-catalyzed reactions, the errors are typically less than few percent even when the steady-state is characterized by merely few tens of molecules.