## 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 language | English |
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Article number | 084103 |

Number of pages | 16 |

Journal | The Journal of Chemical Physics |

Volume | 135 |

Issue number | 8 |

Early online date | 22 Aug 2011 |

DOIs | |

Publication status | Published - 28 Aug 2011 |