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

Ramon Grima, Philipp Thomas, Arthur V. Straube

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish
Article number084103
Number of pages16
JournalThe Journal of Chemical Physics
Volume135
Issue number8
Early online date22 Aug 2011
DOIs
Publication statusPublished - 28 Aug 2011

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