Probabilistic Models of Evolving Populations

生物学的一个中心问题是详细了解自然选择如何影响种群的进化。当种群中的一个个体获得有益的突变时，这种突变最终可能会传播到种群的很大一部分，甚至整个种群。当人群中同时存在许多不同的有益突变时，用数学方法对这一现象进行建模就变得特别具有挑战性。这个研究项目研究不断获得有益突变的进化种群的数学模型。因为，在这些模型中，假定人口以随机方式进化，概率论在分析中起着核心作用。这项研究旨在为重要的生物学问题提供数学见解。感兴趣的问题包括确定由于有益突变而导致的种群适应度增加的速率，了解特定时间种群中个体适应度水平的分布，以及了解如何描述种群样本的谱系。为了模拟正在进行选择的种群，研究者将考虑一个称为分支布朗运动的随机过程。每个粒子以给定的速率死亡，每个粒子根据一维布朗运动独立运动，粒子偶尔会分裂成两个。这里粒子代表群体中的个体，粒子沿实线的位置对应于个体的适应度。假设分支率取决于粒子的位置，因此适应度高的个体有更多的后代。据推测，从长期来看，粒子位置的经验分布近似于高斯分布。对这一结果的证明将为生物学文献中确立的观点提供一个数学上严谨的公式，即对于某些经历选择的种群，种群中个体的适应水平分布就像高斯行波一样进化。因为分支布朗运动可以用来模拟种群经历有益或有害突变，这项工作也可以揭示被称为穆勒棘轮的现象，它指的是由于有害突变的积累导致种群适应性的下降。研究者还将考虑一些包含重组效应的种群模型，以及描述从多个物种中采样的个体的家谱的嵌套聚结模型。

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.