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

Erik Aurell, Magnus Ekeberg+1 more|arXiv (Cornell University)|Jun 3, 2019

Mathematical and Theoretical Epidemiology and Ecology Models33 references6 citations

TL;DR

This paper introduces a multilocus Wright-Fisher diffusion model with mutation and pairwise interlocus selection, formulated via a stochastic differential equation (SDE) system with a Shahshahani gradient-like drift structure. It establishes weak convergence from Markov chains to the limiting diffusion and derives the explicit stationary density under Kingman’s house-of-cards mutation, enabling construction of Wright-Fisher models for genetic networks.

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.

Motivation & Objective

• To develop a multilocus diffusion model of asexual, haploid populations with interlocus pairwise selection and parent-independent mutation.
• To establish weak convergence of a sequence of Markov chains to a limiting SDE with a Shahshahani gradient-like drift structure.
• To solve the stationary Fokker-Planck equation explicitly under Kingman’s house-of-cards mutation model.
• To provide a framework for constructing Wright-Fisher models corresponding to genetic interaction networks.

Proposed method

• Derives a system of SDEs for multilocus allele frequencies with drift function structured as a Shahshahani gradient of a fitness potential.
• Applies weak convergence techniques from [40] and [38] to show convergence of Markov chains to the diffusion limit.
• Uses Girsanov's theorem and a change of drift technique to prove uniqueness of the solution to the martingale problem.
• Solves the stationary Fokker-Planck equation explicitly under Kingman’s mutation, yielding a closed-form invariant density.
• Employs conditional independence in locuswise multinomial sampling to analyze drift and diffusion components.
• Validates convergence via moment bounds and vanishing fourth-moment increments in the limit as population size $N \to \infty$.

Experimental results

Research questions

• RQ1 How can a multilocus Wright-Fisher model with pairwise interlocus interactions be formulated as a diffusion process with a structured drift?
• RQ2 What conditions ensure weak convergence of a sequence of Markov chains to the proposed SDE system?
• RQ3 Can the stationary distribution of such a model be derived explicitly under parent-independent mutation?
• RQ4 How does the Shahshahani gradient-like structure in the drift relate to fitness and selection in multilocus systems?
• RQ5 What is the role of the invariant density in constructing models for genetic networks with interlocus interactions?

Key findings

• The stationary Fokker-Planck equation is solved explicitly under Kingman’s house-of-cards mutation, yielding a closed-form invariant density.
• The limiting diffusion process is uniquely characterized via Girsanov’s theorem and a change of drift technique.
• The drift function exhibits a Shahshahani gradient-like structure derived from a fitness potential, enabling explicit modeling of interlocus selection.
• Weak convergence of the Markov chain sequence to the SDE is established under conditional independence and moment conditions on the drift and diffusion coefficients.
• The fourth-moment increments of the process vanish as $N \to \infty$, confirming the absence of jumps in the limit.
• The invariant density allows for the systematic construction of Wright-Fisher models corresponding to networks of interacting loci.