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

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Title: How accurate are the nonlinear chemical Fokker-Planck and chemical Langevin equations?
Authors: Grima, R
Thomas, P
Straube, AV
Item Type: Journal Article
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 Ω(-3∕2) for reaction systems which do not obey detailed balance and at least accurate to order Ω(-2) for systems obeying detailed balance, where Ω 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 Ω(-1∕2) and variance estimates accurate to order Ω(-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.
Issue Date: 28-Aug-2011
Date of Acceptance: 27-Jul-2011
URI: http://hdl.handle.net/10044/1/42208
DOI: https://dx.doi.org/10.1063/1.3625958
ISSN: 1089-7690
Publisher: AIP Publishing
Journal / Book Title: Journal of Chemical Physics
Volume: 135
Issue: 8
Copyright Statement: © 2011 American Institute of Physics
Keywords: Models, Chemical
Multivariate Analysis
Reproducibility of Results
q-bio.QM
cond-mat.mes-hall
cond-mat.stat-mech
Chemical Physics
02 Physical Sciences
03 Chemical Sciences
09 Engineering
Publication Status: Published
Conference Place: United States
Article Number: ARTN 084103
Appears in Collections:Mathematics
Applied Mathematics and Mathematical Physics
Faculty of Natural Sciences



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