UC San Diego

Epigenetic cell memory (ECM), the inheritance of gene expression patterns across subsequent cell divisions, is a critical property of multi-cellular organisms in cell differentiation and cell fate determination. The theoretical study presented in this dissertation is motivated by the desire to obtain mechanistic insights into the duration of this epigenetic cell memory and the dynamics of mediators (such as histone modifications and DNA methylation) that contribute to this memory.

Stochastic dynamical systems involved in ECM can be described using Stochastic Chemical Reaction Networks (SCRNs), a class of continuous time Markov chain models frequently used to study biochemical systems undergoing a series of reactions which change the numbers of molecules of a finite set of species in a probabilistic manner over time. This dissertation studies three aspects of the long-run and transient behavior of SCRNs. First, comparison theorems are developed which provide stochastic ordering results for SCRNs. These results provide sufficient conditions for establishing monotonic dependence on parameters of (mean) first passage times and stationary distributions of SCRNs. Second, some results on singular perturbations for continuous time Markov chains are developed. The leading terms in rigorous series expansions of stationary distributions and mean first passage times are characterized using a reduced Markov chain. Finally, we introduce the concept of coclique level structure and derive closed form formulas for both upper and lower bounds for the mean first passage times for SCRNs having such structures.

These newly developed theoretical tools not only allow us to set a rigorous mathematical basis for studying the effects of chromatin modification dynamics on epigenetic cell memory, but they can also be applied to other continuous time Markov chain models, especially those associated with (bio)chemical reaction networks.