Exact solutions for the selection-mutation equilibrium in the Crow-Kimura evolutionary model

TL;DR

This paper derives exact analytical solutions for the steady-state distribution in the Crow-Kimura model of evolution, providing insights into the effects of different fitness landscapes on population genetics.

Contribution

It reformulates the eigenvalue problem into a generating function equation, enabling analytical solutions for specific fitness landscapes in the infinite sequence limit.

Findings

Analytical steady state distributions obtained for particular fitness landscapes

The approach is validated through numerical comparisons

Provides a new method for solving the Crow-Kimura model analytically

Abstract

We reformulate the eigenvalue problem for the selection--mutation equilibrium distribution in the case of a haploid asexually reproduced population in the form of an equation for an unknown probability generating function of this distribution. The special form of this equation in the infinite sequence limit allows us to obtain analytically the steady state distributions for a number of particular cases of the fitness landscape. The general approach is illustrated by examples; theoretical findings are compared with numerical calculations.