  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.
