On a Multilocus Wright-Fisher Model with Mutation and a Svirezhev-Shahshahani Gradient-like Selection Dynamics

TL;DR

This paper introduces a multilocus Wright-Fisher diffusion model incorporating mutation and pairwise interlocus selection, deriving explicit stationary densities and enabling network-based genetic modeling.

Contribution

It presents a novel multilocus diffusion model with a gradient-like selection structure and explicit stationary densities derived from Markov chain limits.

Findings

Explicit stationary density formula for the model.

Connection between Wright-Fisher models and genetic networks.

Derivation from Markov chain weak convergence.

Abstract

In this paper we introduce a multilocus diffusion model of a population of $N$ haploid, asexually reproducing individuals. The model includes parent-dependent mutation and interlocus selection, the latter limited to pairwise relationships but among a large number of simultaneous loci. The diffusion is expressed as a system of stochastic differential equations (SDEs) that are coupled in the drift functions through a Shahshahani gradient-like structure for interlocus selection. The system of SDEs is derived from a sequence of Markov chains by weak convergence. We find the explicit stationary (invariant) density by solving the corresponding stationary Fokker-Planck equation under parent-independent mutation, i.e., Kingman's house-of-cards mutation. The density formula enables us to readily construct families of Wright-Fisher models corresponding to networks of loci.