Probabilistic Models of Evolving Populations

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A central question in biology is to understand in detail how natural selection affects the evolution of a population. When one individual in a population acquires a beneficial mutation, that mutation may eventually spread to a large fraction of the population, or even the entire population. Modeling this phenomenon mathematically becomes particularly challenging when there can be many different beneficial mutations in the population at a time. This research project studies mathematical models of evolving populations that repeatedly acquire beneficial mutations. Because, in these models, populations are assumed to evolve in a random way, the theory of probability plays a central role in the analysis. The research aims to provide mathematical insight into important biological problems. Questions of interest include determining the rate at which the fitness of the population increases as a result of beneficial mutations, understanding the distribution of the fitness levels of individuals in the population at a given time, and understanding how to describe the genealogy of a sample from the population. To model populations undergoing selection, the investigator will consider a stochastic process called branching Brownian motion. Each particle dies at a given rate, each particle moves independently according to one-dimensional Brownian motion, and particles occasionally split into two. Here particles represent individuals in a population, and the position of the particle along the real line corresponds to the individual's fitness. It will be assumed that the branching rate depends on the position of the particle, so that individuals with higher fitness have more offspring. It is conjectured that in the long-run, the empirical distribution of the positions of the particles is approximately Gaussian. A proof of this result would provide a mathematically rigorous formulation of the idea, well-established in the biology literature, that for certain populations undergoing selection, the distribution of the fitness levels of individuals in the population evolves like a Gaussian traveling wave. Because branching Brownian motion can be used to model populations experiencing either beneficial or deleterious mutations, this work could also shed light on a phenomenon known as Muller's ratchet, which refers to the decrease in the fitness of a population resulting from the accumulation of deleterious mutations. The investigator will also consider some population models that incorporate the effects of recombination, as well as a nested coalescent model that describes the genealogy of a collection of individuals sampled from multiple species.