  A perturbation framework is developed to analyze metastable behavior in
stochastic processes with random internal and external states. The process is
assumed to be under weak noise conditions, and the case where the deterministic
limit is bistable is considered. A general analytical approximation is derived
for the stationary probability density and the mean switching time between
metastable states, which includes the pre exponential factor. The results are
illustrated with a model of gene expression that displays bistable switching.
In this model, the external state represents the number of protein molecules
produced by a hypothetical gene. Once produced, a protein is eventually
degraded. The internal state represents the activated or unactivated state of
the gene; in the activated state the gene produces protein more rapidly than
the unactivated state. The gene is activated by a dimer of the protein it
produces so that the activation rate depends on the current protein level. This
is a well studied model, and several model reductions and diffusion
approximation methods are available to analyze its behavior. However, it is
unclear if these methods accurately approximate long-time metastable behavior
(i.e., mean switching time between metastable states of the bistable system).
Diffusion approximations are generally known to fail in this regard.
