Looking down in the ancestral selection graph: A probabilistic approach to the common ancestor type distribution

Abstract

In a (two-type) Wright-Fisher diffusion with directional selection and two-way mutation, let $x$ denote today's frequency of the beneficial type, and given $x$, let $h(x)$ be the probability that, among all individuals of today's population, the individual whose progeny will eventually take over in the population is of the beneficial type. Fearnhead [Fearnhead, P., 2002. The common ancestor at a nonneutral locus. J. Appl. Probab. 39, 38-54] and Taylor [Taylor, J. E., 2007. The common ancestor process for a Wright-Fisher diffusion. Electron. J. Probab. 12, 808-847] obtained a series representation for $h(x)$. We develop a construction that contains elements of both the ancestral selection graph and the lookdown construction and includes pruning of certain lines upon mutation. Besides being interesting in its own right, this construction allows a transparent derivation of the series coefficients of $h(x)$ and gives them a probabilistic meaning.

arXiv commentTheoretical Population Biology, in press

References in corpus (3)

• Coalescence in a random background
• Mutation, selection, and ancestry in branching models: a variational approach
• The common ancestor process revisited

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• Lines of descent in the deterministic mutation-selection model with pairwise interaction
• On the stationary distribution of the block counting process for population models with mutation and selection
• The historical Moran model
• The mutation process on the ancestral line under selection