Analysis of singularly perturbed stochastic chemical reaction networks motivated by applications to epigenetic cell memory

TL;DR

This paper develops a rigorous mathematical framework for analyzing stochastic models of chromatin modifications, elucidating how time-scale differences influence epigenetic cell memory through singularly perturbed Markov chains.

Contribution

It extends existing theory to characterize stationary distributions and mean first passage times in singularly perturbed Markov chains, with applications to epigenetic systems.

Findings

Characterizes limiting stationary distribution via reduced Markov chain

Provides an algorithm to determine poles of mean first passage times

Shows how erasure rates influence system behavior

Abstract

Epigenetic cell memory, the inheritance of gene expression patterns across subsequent cell divisions, is a critical property of multi-cellular organisms. In recent work [10], a subset of the authors observed in a simulation study how the stochastic dynamics and time-scale differences between establishment and erasure processes in chromatin modifications (such as histone modifications and DNA methylation) can have a critical effect on epigenetic cell memory. In this paper, we provide a mathematical framework to rigorously validate and extend beyond these computational findings. Viewing our stochastic model of a chromatin modification circuit as a singularly perturbed, finite state, continuous time Markov chain, we extend beyond existing theory in order to characterize the leading coefficients in the series expansions of stationary distributions and mean first passage times. In particular, we characterize the limiting stationary distribution in terms of a reduced Markov chain, provide an algorithm to determine the orders of the poles of mean first passage times, and determine how changing erasure rates affects system behavior. The theoretical tools developed in this paper not only allow us to set a rigorous mathematical basis for the computational findings of our prior work, highlighting the effect of chromatin modification dynamics on epigenetic cell memory, but they can also be applied to other singularly perturbed Markov chains beyond the applications in this paper, especially those associated with chemical reaction networks.