Phase Reduction of Stochastic Biochemical Oscillators

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 diffusion approximation 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 diffusion approximation, 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 $T\exp\big(-C\Omega b\zeta^2 \big)$, where $b$ is the rate of attraction to the limit cycle. We illustrate our analysis using the example of a chemical Brusselator.