On a Stochastic PDE Model for Epigenetic Dynamics

Abstract We propose and analyze a stochastic model to investigate epigenetic mutations, i.e., modifications of the genetic information that control gene expression patterns in a cell but do not alter the DNA sequence. Epigenetic mutations are related to environmental fluctuations, which leads us to consider (additive) noise as the driving element for such mutations. We focus on two applications: firstly, cancer immunotherapy involving macrophages' epigenetic modifications that we call tumor microenvironment noise-induced polarizations, and secondly, cell fate determination and mutation of the flower Arabidopsis thaliana. Due to the technicalities involving cancer biology for the first case, we present only a general review of this topic and show the details in a separate manuscript since our principal concerns here are the mathematical results that are important to validate our system as an appropriate epigenetic model; for such results, we rely on the theory of Stochastic PDE, theory of large deviations, and ergodic theory. Moreover, since epigenetic mutations are reversible, a fact currently exploited to develop so-called epi-drugs to treat diseases like cancer, we also investigate an optimal control problem for our system to study the reversal of epigenetic mutations. Mathematics Subject Classification: 92D10 , 35R60 , 60H15 , 37N25 , 49N90