A numerical framework for genetic hitchhiking in populations of variable size

Abstract

Natural selection on beneficial or deleterious alleles results in an increase or decrease, respectively, of their frequency within the population. Due to chromosomal linkage, the dynamics of the selected site affect the genetic variation at nearby neutral loci in a process commonly referred to as genetic hitchhiking. Changes in population size, however, can yield patterns in genomic data that mimic the effects of selection. Accurately modeling these dynamics is thus crucial to understanding how selection and past population size changes impact observed patterns of genetic variation. Here, we model the evolution of haplotype frequencies with the Wright-Fisher diffusion to study the impact of selection on linked neutral variation. Explicit solutions are not known for the dynamics of this diffusion when selection and recombination act simultaneously. Thus, we present a method for numerically evaluating the Wright-Fisher diffusion dynamics of two linked loci separated by a certain recombination distance when selection is acting. We can account for arbitrary population size histories explicitly using this approach. A key step in the method is to express the moments of the associated transition density, or sampling probabilities, as solutions to ordinary differential equations. Numerically solving these differential equations relies on a novel accurate and numerically efficient technique to estimate higher order moments from lower order moments. We demonstrate how this numerical framework can be used to quantify the reduction and recovery of genetic diversity around a selected locus over time and elucidate distortions in the site-frequency-spectra of neutral variation linked to loci under selection in various demographic settings. The method can be readily extended to more general modes of selection and applied in likelihood frameworks to detect loci under selection and infer the strength of the selective pressure.

Local fitness and epistatic effects lead to distinct patterns of linkage disequilibrium in protein-coding genes

“…Below, I extend this approach to model arbitrary diploid selection, which encompasses dominance, epistasis, and other forms of selective interactions between two loci. In a concurrent study to this paper, F riedlander and S teinrücken (2022) developed numerical solutions to the same moments system, which they used to describe selected haplotype trajectories and the distortion of neutral diversity at loci variably linked to beneficial alleles that sweep to high frequencies under non-equilibrium demography.…”

“…This requirement of extra lineages for nonzero recombination and selection means that the system in (2) is not closed, so that we need a moment-closure approximation to solve for ψ n . As in R agsdale and G ravel (2019), a jackknife approximation is used to estimate ψ n + l , for l = 1 or 2, from ψ n (following the single-locus closure introduced in J ouganous et al ., 2017), so that , although other accurate closure approximations are possible (F riedlander and S teinrucken , 2022). This emits a closed approximate system, …”

“…This makes performing forward simulations that span the range of all such selective interaction scenarios burdensome. Instead, closed, numerical approaches to compute expectations for two-locus statistics under a wide range of selection models, as presented here and in Friedlander and Steinrücken (2022), allow us to explore this highly parameterized space of models far more efficiently, and it opens the possibility for performing likelihood-based inference using signed LD or other two-locus summaries of the data. For example, inferring the joint distribution of dominance and selection is underpowered using the SFS alone, but because signed LD is sensitive to the levels of dominance (Figure 4), inferring the DFE with dominance may be feasible using the joint distribution of allele frequencies and LD.…”

On the prospect of achieving accurate joint estimation of selection with population history

“…Computational and statistical challenges remain for both inference approaches based on diffusion approximations and on forward simulations. Under a diffusion approximation, Friedlander and Steinrücken (2022) and Ragsdale (2021) demonstrated efficient numerical approaches for calculating the statistics of pairs of sites as noted above. Building the corresponding statistical inference framework, however, remains challenging.…”

“…Helpfully, Cvijovic et al (2018) obtained analytical expressions for the SFS at linked neutral sites experiencing BGS for a non-recombining locus under demographic equilibrium. Furthermore, while general analytical solutions for even single-locus, single-population scenarios have not yet been obtained, Friedlander and Steinrücken (2022) recently described a numerical framework in which a system of ordinary differential equations can be solved to obtain the expected SFS and linkage disequilibrium (LD) around a selected region, for a Wright–Fisher two-locus model with mutation, recombination, selection, and changing population size. The development of future likelihood methods leveraging such a numerical approach may be utilized to jointly infer parameters of complex demographic histories with selection.…”