A coalescent dual process for a Wright–Fisher diffusion with recombination and its application to haplotype partitioning

Abstract

Duality plays an important role in population genetics. It can relate results from forwards-in-time models of allele frequency evolution with those of backwards-in-time genealogical models; a well known example is the duality between the Wright-Fisher diffusion for genetic drift and its genealogical counterpart, the coalescent. There have been a number of articles extending this relationship to include other evolutionary processes such as mutation and selection, but little has been explored for models also incorporating crossover recombination. Here, we derive from first principles a new genealogical process which is dual to a Wright-Fisher diffusion model of drift, mutation, and recombination. The process is reminiscent of the ancestral recombination graph, a widely-used multilocus genealogical model, but here ancestral lineages are typed and transition rates are regarded as being conditioned on an observed configuration at the leaves of the genealogy. Our approach is based on expressing a putative duality relationship between two models via their infinitesimal generators, and then seeking an appropriate test function to ensure the validity of the duality equation. This approach is quite general, and we use it to find dualities for several important variants, including both a discrete L-locus model of a gene and a continuous model in which mutation and recombination events are scattered along the gene according to * Corresponding author. Address: Department of Statistics, University of Warwick, Coventry CV4 7AL, United Kingdom. E-mail: p.jenkins@warwick.ac.uk continuous distributions. As an application of our results, we derive a series expansion for the transition function of the diffusion. Finally, we study in further detail the case in which mutation is absent. Then the dual process describes the dispersal of ancestral genetic material across the ancestors of a sample. The stationary distribution of this process is of particular interest; we show how duality relates this distribution to haplotype fixation probabilities. We develop an efficient method for computing such probabilities in multilocus models.

A dual process for the coupled Wright–Fisher diffusion

“…To derive a process that is dual to the coupled Wright-Fisher diffusion, a generator approach will be used as in Griffiths et al (2016), where the authors find a dual process for a multi-locus Wright-Fisher diffusion with recombination. In this section the method will be explained, in general terms.…”

“…Unlike the Kingman coalescent, which has a tree structure, the ASG is branching and coalescing: the ancestral tree is replaced by an ancestral graph containing true and virtual lineages and embedding the genealogy of the sample of individuals. For a complete survey on duality for Markov processes, see Jensen and Kurt (2014), and for a brief overview of duality in population genetics see Griffiths et al (2016) and the references therein.…”

“…where F is a duality function, to be determined. The derivation uses a generator approach as in Griffiths et al (2016) and Etheridge and Griffiths (2009). It is based on the duality relationship of the infinitesimal generators…”

“…From the duality relation (3), it is also possible to derive an expansion of the transition distribution of the diffusion X, see Barbour et al (2000), Etheridge and Griffiths (2009), Griffiths et al (2016), in terms of the limit of the transition densities of the dual process N. In the absence of mutation, the duality relation (3) can also be used to determine fixation probabilities. That is, the probability that the frequency of a given allele at a given loci is equal to 1.…”

“…Such probabilities may be studied by taking the limit, as t → ∞, in (3) and considering the recurrence/transience properties of the dual process N, see e.g. Foucart (2013), Griffiths et al (2016), González Casanova and Spanó (2018), Mano (2009) for studies of the Wright-Fisher process with selec-tion and frequency dependent selection, the multi-locus Wright-Fisher process with recombination, the Λ-Wright-Fisher process with selection and the Ξ -Wright-Fisher process with frequency dependent selection, respectively.…”