Coarse-grained Models of Biological Systems and Asymptotic Analyses on Population Dynamics

Abstract

In my PhD thesis, I use simplified mathematical models to study emergent phenomena of various biological systems. In Chapter 1, I give an overview of how the tools from theoretical physics and applied math may help understand biological phenomena. In particular, I will talk about coarse-grained modeling and asymptotic analysis. In Chapter 2, I study the effect of asymmetric segregation of key proteins on fitness of a microbial organism. Unicellular organisms often segregate their key proteins asymmetrically to their daughter cells in a stressed environment. Using simulations and an analytical method from condensed matter physics (Landau method), I study how population growth rate depends on asymmetry parameter a. I find that the phase transition of optimal asymmetry (i.e. value of a that maximizes population growth rate) is sharp if the key protein is deleterious and smooth if the protein is beneficial. In Chapter 3, I generalize the theoretical study in Chapter 2 by adapting a transport equation to set up a self-consistent equation of population growth rate and distribution of key protein concentration. This equation lets us explore the phase transition of optimal segregation ratio in greater detail and is generalized to make the model more biologically relevant – for instance, letting the segregation be stochastic or be adaptive to stress level. In Chapter 4, I study wave dynamics generated from cell signaling relays. The initiation of the wave and wave speed can vastly change depending on the dimensionality of the system that the cells reside in. In contrast, the wave dynamics are not sensitive to the cellular level details. In Chapter 5, I study how spatial structure alters sweep signature in a site-frequency-spectrum (SFS). Even when a spatial structure is negligible in terms of neutral dynamics, it can significantly slow down the spread of a sweeping beneficial allele. Through analytical and numerical methods, I show how spatial sweep leads to distinct scaling laws of SFS.