Mutation-selection balance with recombination: convergence to equilibrium for polynomial selection costs

TL;DR

This paper analyzes a continuous-time model of genotype evolution with mutation, selection, and recombination, proving convergence to a unique mutation-selection balance equilibrium under polynomial costs.

Contribution

It introduces a specific polynomial cost model with interaction effects and proves convergence to equilibrium using a novel Lyapunov function.

Findings

Existence and uniqueness of equilibrium under polynomial costs.

Convergence of the dynamical system to the equilibrium from all initial states.

Application of complex reaction network ideas and Lyapunov functions to evolutionary dynamics.

Abstract

We study a continuous-time dynamical system that models the evolving distribution of genotypes in an infinite population where genomes may have infinitely many or even a continuum of loci, mutations accumulate along lineages without back-mutation, added mutations reduce fitness, and recombination occurs on a faster time scale than mutation and selection. Some features of the model, such as existence and uniqueness of solutions and convergence to the dynamical system of an approximating sequence of discrete time models, were presented in earlier work by Evans, Steinsaltz, and Wachter for quite general selective costs. Here we study a special case where the selective cost of a genotype with a given accumulation of ancestral mutations from a wild type ancestor is a sum of costs attributable to each individual mutation plus successive interaction contributions from each $k$-tuple of mutations for $k$ up to some finite ``degree''. Using ideas from complex chemical reaction networks and a novel Lyapunov function, we establish that the phenomenon of mutation-selection balance occurs for such selection costs under mild conditions. That is, we show that the dynamical system has a unique equilibrium and that it converges to this equilibrium from all initial conditions.