Theoretical and Numerical Analyses of Deviations Between Kingman's Coalescent and the Wright-Fisher Model

[摘要] The Kingman Coalescent is a commonly used model in genetics, which is often justified with reference to the Wright-Fisher (WF) model. In this thesis we seek to attain a deeper understanding of the relationship between these two models, particularly by quantifying under what conditions the models are similar, and by understanding the ramifications of deviations between the models outside those conditions.In Chapter 2, we investigate one source of deviation between the two models, that they have different partition distributions. We find an asymptotic bound on sample size relative to effective population size under which the partition distributions are identical. We additionally find similar asymptotic bounds under which no triple mergers will occur in the Wright-Fisher model.Furthermore, we use numerical methods to show that these bounds are generally applicable at finite sample and population sizes. In Chapter 3, we investigate the deviation between the site frequency spectrum (SFS) under the WF model and the coalescent model. There are two sources of this deviation. One is that there is a mismatch in rates of merger between the two models. The other is the aforementioned difference in partition distributions. The mismatch in rates raises the probability of singletons under WF, but the difference in partition distributions lowers it. These two effects are opposing everywhere except at the tail of the frequency spectrum. The WF frequency spectrum only begins to significantly depart from that of the coalescent at sample sizes close to the population size. We examine the case where the sample size is assumed to be equal to the population size N and find the total variation distance between WF and coalescent to be only 1% for populations of size 20000. Therefore we conclude that the coalescent is a good approximation for WF for the site frequency spectrum of large samples.In Chapter 4, we introduce an algorithm which allows us to generate the SFS under the coalescent with a time-varying population size and mutation rate. Using this algorithm we explore the effects of a variable mutation rate on the SFS. We find that the SFS changes substantially as a result of varying mutation rates even for small samples.

[发布机构] University of Michigan

[效力级别] Wright-Fisher

[关键词] Coalescent;Wright-Fisher;Variable mutation rate;Genetics;Mathematics;Science;Applied and Interdisciplinary Mathematics