A Wright-Fisher graph model and the impact of directional selection on genetic variation

TL;DR

This paper introduces a Wright-Fisher graph model incorporating directional selection and mutation, deriving the stationary distribution and allele frequency spectrum, and demonstrating that directional selection reduces genetic variation.

Contribution

It presents a novel multi-allele Wright-Fisher model with a graph-based state space and provides rigorous bounds showing how directional selection impacts genetic variation.

Findings

Directional selection reduces genetic variation magnitude.

Derived explicit stationary distribution for mutation-selection-drift equilibrium.

Provided bounds for genetic variation measures under multiple loci.

Abstract

We introduce a multi-allele Wright-Fisher model with non-recurrent, reversible mutation and directional selection. In this setting, the allele frequencies at a single locus track the path of a hybrid jump-diffusion process with state space given by the vertex and edge set of a graph. Vertices represent monomorphic population states and edge-positions mark the biallelic proportions of ancestral and derived alleles during polymorphic segments. We derive the stationary distribution in mutation-selection-drift equilibrium and obtain the expected allele frequency spectrum under large population size scaling. For the extended model with multiple independent loci we derive rigorous upper bounds for a wide class of associated measures of genetic variation. Within this framework we present mathematically precise arguments to conclude that the presence of directional selection reduces the magnitude of genetic variation, as constrained by the bounds for neutral evolution.