Tractable diffusion and coalescent processes for weakly correlated loci

TL;DR

This paper introduces simplified multilocus genetic models that approximate complex standard models at high recombination rates, enabling closed-form sampling distributions and easier computations.

Contribution

The authors develop two new multilocus models, a diffusion and a coalescent, which are simpler yet accurately capture key properties of traditional models for large recombination rates.

Findings

Sampling distribution is a linear combination of Gaussian moments.

Models agree with standard models up to second order in inverse recombination rate.

New models enable closed-form solutions for sampling distributions.

Abstract

Widely used models in genetics include the Wright-Fisher diffusion and its moment dual, Kingman's coalescent. Each has a multilocus extension but under neither extension is the sampling distribution available in closed-form, and their computation is extremely difficult. In this paper we derive two new multilocus population genetic models, one a diffusion and the other a coalescent process, which are much simpler than the standard models, but which capture their key properties for large recombination rates. The diffusion model is based on a central limit theorem for density dependent population processes, and we show that the sampling distribution is a linear combination of moments of Gaussian distributions and hence available in closed-form. The coalescent process is based on a probabilistic coupling of the ancestral recombination graph to a simpler genealogical process which exposes the leading dynamics of the former. We further demonstrate that when we consider the sampling distribution as an asymptotic expansion in inverse powers of the recombination parameter, the sampling distributions of the new models agree with the standard ones up to the first two orders.