Metastable behavior in Markov processes with internal states

TL;DR

This paper develops a perturbation framework to analyze metastable behavior in stochastic processes with internal states, deriving analytical approximations for stationary densities and switching times, and illustrating with a gene expression model.

Contribution

It introduces a new analytical approximation method for metastable switching times in Markov processes with internal states, improving understanding of long-term behavior.

Findings

Derived analytical approximation for stationary density and switching times.

Validated the approach with a gene expression model exhibiting bistability.

Showed limitations of diffusion approximations for long-time metastable behavior.

Abstract

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.