Duality, Ancestral and Diffusion Processes in Models with Selection

TL;DR

This paper explores the ancestral selection graph in population genetics, deriving explicit distributions and analyzing fixation times through diffusion processes, providing new insights into genetic ancestry under selection.

Contribution

It introduces an explicit distribution for the ancestral process and links fixation times to the convergence of this process in models with selection.

Findings

Explicit distribution of particle number in ancestral process

Connection between fixation and stationary measure convergence

Analysis of fixation time conditional on fixation

Abstract

The ancestral selection graph in population genetics was introduced by KroneNeuhauser (1997) as an analogue of the coalescent genealogy of a sample of genes from a neutrally evolving population. The number of particles in this graph, followed backwards in time, is a birth and death process with quadratic death and linear birth rates. In this paper an explicit form of the probability distribution of the number of particles is obtained by using the density of the allele frequency in the corresponding diffusion model obtained by Kimura (1955). It is shown that the process of fixation of the allele in the diffusion model corresponds to convergence of the ancestral process to its stationary measure. The time to fixation of the allele conditional on fixation is studied in terms of the ancestral process.