An asymptotic sampling formula for the coalescent with Recombination

TL;DR

This paper derives an asymptotic sampling formula for the coalescent with recombination in population genetics, providing approximate closed-form solutions for large recombination rates applicable to various sample configurations.

Contribution

It introduces the first asymptotic expansion of the two-locus sampling formula in inverse powers of the recombination rate, extending the classical Ewens sampling formula.

Findings

Derived closed-form expressions for the first terms in the asymptotic expansion

Applicable to arbitrary sample sizes and configurations

Provides practical approximations for large recombination rates

Abstract

Ewens sampling formula (ESF) is a one-parameter family of probability distributions with a number of intriguing combinatorial connections. This elegant closed-form formula first arose in biology as the stationary probability distribution of a sample configuration at one locus under the infinite-alleles model of mutation. Since its discovery in the early 1970s, the ESF has been used in various biological applications, and has sparked several interesting mathematical generalizations. In the population genetics community, extending the underlying random-mating model to include recombination has received much attention in the past, but no general closed-form sampling formula is currently known even for the simplest extension, that is, a model with two loci. In this paper, we show that it is possible to obtain useful closed-form results in the case the population-scaled recombination rate $\rho$ is large but not necessarily infinite. Specifically, we consider an asymptotic expansion of the two-locus sampling formula in inverse powers of $\rho$ and obtain closed-form expressions for the first few terms in the expansion. Our asymptotic sampling formula applies to arbitrary sample sizes and configurations.