SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 1, Page 151-180, January 2020.
A common method for analyzing the effects of molecular noise in chemical reaction networks is to approximate the underlying chemical master equation by a Fokker--Planck equation and to study the statistics of the associated chemical Langevin equation. This so-called system-size expansion involves performing a perturbation expansion with respect to a small dimensionless parameter $\epsilon =\Omega^{-1}$, where $\Omega$ characterizes the system size. For example, $\Omega$ could be the mean number of proteins produced by a gene regulatory network. In the deterministic limit $\Omega\rightarrow \infty$, the chemical reaction network evolves according to a system of ordinary differential equations based on classical mass action kinetics. In this paper we develop a phase reduction method for chemical reaction networks that support a stable limit cycle in the deterministic limit. We present a variational principle for the phase reduction, yielding an exact analytic expression for the resulting phase dynamics. We demonstrate that this decomposition is accurate over timescales that are exponential in the system size $\Omega$. This contrasts with the phase equation obtained under the system-size expansion, which is only accurate up to times $O(\Omega)$. In particular, we show that for a constant $C$, the probability that the system leaves an $O(\zeta)$ neighborhood of the limit cycle before time $T$ scales as $T\exp(-C\Omega b\zeta^2)$, where $b$ is the rate of attraction to the limit cycle. We illustrate our analysis using the example of a chemical Brusselator.

中文翻译：

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随机生化振荡器的相减少

SIAM应用动力系统杂志，第19卷，第1期，第151-180页，2020年1月。
分析化学反应网络中分子噪声影响的一种常用方法是通过Fokker-Planck方程近似基础化学主方程并研究相关化学Langevin方程的统计量。这种所谓的系统大小扩展涉及对较小的无量纲参数$ \ epsilon = \ Omega ^ {-1} $执行扰动扩展，其中$ \ Omega $表示系统大小。例如，\ Omega $可以是基因调节网络产生的平均蛋白质数量。在确定性极限\\ Omega \ rightarrow \ infty $中，化学反应网络根据基于经典质量作用动力学的常微分方程组演化。在本文中，我们开发了一种用于化学反应网络的相还原方法，该方法在确定性极限内支持稳定的极限循环。我们提出了相变的变分原理，为所得的相动力学产生了精确的解析表达式。我们证明了这种分解在系统规模$ \ Omega $呈指数级的时间范围内是准确的。这与在系统大小扩展下获得的相位方程相反，该相位方程仅在高达$ O（\ Omega）$时才是精确的。特别是，我们表明，对于恒定的$ C $，系统在时间$ T $缩放为$ T \ exp（-C \ Omega b \之前，离开极限循环的$ O（\ zeta）$邻域的概率zeta ^ 2）$，其中$ b $是对极限周期的吸引率。我们以一个化学Brusselator为例说明我们的分析。