  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.
