Quasi-potential analysis of multi-variate stochastic differential equations

Abstract

Genetic circuit motifs based on two transcription factors can model cell-fate decisions critical for embryonic development and adult homeostasis. One important such motif is the self-activating toggle switch allows for tri-stable configuration and is believed to be responsible for stem-cell differentiation in multi-cellular organisms. To aid observations and experiments, a theoretical framework for studying these motifs using potential theory from classical physics is sometimes utilized.

This thesis aims to be an expository and pedagogical introduction to this topic. Starting from first principles, I derive the deterministic equations describing these systems. Then, I derive the sources of noise and stochasticity based on basic probability theory. Stochastic differential equations are derived for these systems. Finally, I introduce and implement vector field decomposition methods used to arrive at quasi-potentials from the literature for polynomial systems with demonstrations on example systems. The application of these methods to genetic switches fails and is discussed in Chapter 4.