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

### 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 reactionsystems 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.

© 2011 American Institute of Physics

Received 15 June 2011
Accepted 27 July 2011
Published online 22 August 2011

Acknowledgments:
R.G. acknowledges support by SULSA (Scottish Universities Life Science Alliance).

Article outline:

I. INTRODUCTION
II. PERTURBATIVE EXPANSION OF THE CME
A. The multivariate system-size expansion of the CME
B. Time-evolution equations for the moments
III. PERTURBATIVE EXPANSION OF THE CFPE
IV. COMPARISON OF THE PREDICTIONS OF THE CFPE AND THE CME
A. Estimating the absolute and relative errors in CFPE predictions
B. The CFPE is more accurate than the linear-noise approximation
C. The CFPE is highly accurate for equal-step reactions involving one species
D. CFPE is highly accurate for multispecies reactions obeying detailed balance
V. APPLICATIONS
A. Dimerization
B. Enzyme catalysis: The Michaelis-Menten mechanism
VI. DISCUSSION AND CONCLUSION

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2011-08-22

2016-09-28

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