Pad\'{e} approximants and exact two-locus sampling distributions

TL;DR

This paper introduces a computational method using Padé approximants to derive exact two-locus sampling distributions in population genetics, extending to models with natural selection and arbitrary recombination rates.

Contribution

A novel computational technique for asymptotic expansions of two-locus sampling distributions, enabling exact solutions for all recombination rates using Padé approximants.

Findings

Finite asymptotic series can recover exact distributions

Method applies to models with natural selection

Automated computation of higher-order terms

Abstract

For population genetics models with recombination, obtaining an exact, analytic sampling distribution has remained a challenging open problem for several decades. Recently, a new perspective based on asymptotic series has been introduced to make progress on this problem. Specifically, closed-form expressions have been derived for the first few terms in an asymptotic expansion of the two-locus sampling distribution when the recombination rate $\rho$ is moderate to large. In this paper, a new computational technique is developed for finding the asymptotic expansion to an arbitrary order. Computation in this new approach can be automated easily. Furthermore, it is proved here that only a finite number of terms in the asymptotic expansion is needed to recover (via the method of Pad\'{e} approximants) the exact two-locus sampling distribution as an analytic function of $\rho$; this function is exact for all values of $\rho\in[0,\infty)$. It is also shown that the new computational framework presented here is flexible enough to incorporate natural selection.