Thermodynamic behavior of living systems: a biophysical approach to stochastic single-cell gene expression dynamics

Abstract

Representing the state of a cell through its gene expression profile, we consider observable phenotypes as (metastable) attractor states of the underlying biochemical gene expression kinetics. The traditional deterministic dynamics, however, does not capture the possibility of spontaneous transitions between attractors (i.e., phenotype switching). In this work, we frame this picture in terms of stochastic dynamical systems and models: a quasi-potential ``landscape" for an arbitrary dynamical system emerges as a large deviations rate function for the density of a singularly perturbed stochastic differential equation (SDE). This quasi-potential, which exists even for nonequilibrium biochemical dynamics in living cells, admits the most probable path between any two attractors and the characteristic time scale on which these transitions occur. We discuss the implications of this framework for the population dynamics at the level of cell cultures or tissues, specifically its applications to cancer population dynamics. We also consider two different frameworks for analyzing data from single-cell experiments through the lens of phenotypic attractors and state transitions. The first draws on the mathematics of thermodynamics, namely convex analysis and the Legendre-Fenchel transform, to derive an energy-like representation of an ergodic system from statistical measurements of the system. The latter is a maximum likelihood framework for inferring cell proliferation and phenotype switching rates from single-cell data, which we extend from a previous work to novel single-cell experiments using DNA-barcode lineage tracing. In addition to this work, we outline possible directions suitable for future research projects.