标题： 非线性化学福克-普朗克方程和化学朗之万方程的准确性如何？ 显示英文标题

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

摘要： 化学福克-普朗克方程和相应的化学朗之万方程是化学主方程的常用近似。 这些方程是从化学主方程的克兰默斯-莫亚尔展开式的不受控制的二阶截断推导出来的，因此它们的准确性仍有待澄清。 我们使用系统规模展开来证明，对于不满足详细平衡的反应系统，化学福克-普朗克方程对平均浓度以及平均浓度附近的浓度涨落方差的估计在$\Omega^{-3/2}$阶是准确的，而对于满足详细平衡的系统，至少在$\Omega^{-2}$阶是准确的，其中$\Omega$是系统的特征尺寸。 因此，化学福克-普朗克方程比化学主方程的线性噪声近似（线性福克-普朗克方程）更准确，后者导致的平均浓度估计在$\Omega^{-1/2}$阶是准确的，方差估计在$\Omega^{-3/2}$阶是准确的。 这种更高的准确性在小体积中实现的化学系统（如细胞内的生化反应）中尤为明显。 还得到了一个公式，用于近似计算化学福克-普朗克方程在浓度和方差预测中的相对误差大小，其中相对误差定义为化学福克-普朗克方程的预测与主方程的预测之间的差值除以主方程的预测。 对于二聚反应和酶催化反应，即使稳态仅由几十个分子表征，误差通常也小于百分之几。 显示英文摘要

摘要： 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.