Population genetics genealogies under selection - Jiang, H (2013)
In the presence of selection and mutation, the genealogy of a given sample configuration can be described by two classes of ancestral processes, namely the coalescent-in-a-random-background model of Kaplan et al. (1988) and the dual process with typed lines of Etheridge and Griffiths (2009). These two processes are based on the same forwards population genetics model. However, in the former model, selection is reflected in the ancestral frequencies in the population, while in the latter model, there are branching events that generate virtual ancestral lines. We simulate the dual processes with typed lines and derive the limits of the two ancestral processes under strong selection and under selection-mutation balance to address the question of to what extent the genealogy is distorted. The two ancestral processes generate the same limiting genealogy. In a two-allele population under strong selection, the disfavoured individuals in the sample are instantaneously converted to a random number of favoured individuals, and the limiting genealogy is governed by the usual Kingman’s coalescent. Under selection-mutation balance, all disfavoured individuals in the sample are instantaneously converted to the favoured type, and the limiting genealogy is determined by a time-changed Kingman’s coalescent. The proofs of these limiting processes are based on the convergence result of Mohle (1998, Lemma 1). The studies of selection-mutation balance are then extended to an additive selection model, where each individual is composed of L diallelic loci. In the corresponding dual process with typed lines, the evolution of the virtual lines on a faster timescale can be approximated by a deterministic process, while the evolution of the real lines is independent of the virtual lines. The structure in the limiting genealogy collapses to Kingman’s coalescent. We also let L tend to infinity, and obtain a full description of the limiting genealogy in the background selection model.