Deterministic evolution of an asexual population under the action of beneficial and deleterious mutations on additive fitness landscapes

TL;DR

This paper develops an exact mathematical model for the evolution of an asexual population with both beneficial and deleterious mutations on additive fitness landscapes, revealing a non-Poissonian stationary distribution.

Contribution

It introduces an exact eigenfunction expansion method to solve for the population's frequency distribution including beneficial mutations, extending previous models that ignored them.

Findings

Stationary distribution is non-Poissonian and related to Bessel functions.

Provided approximations for the stationary distribution and relaxation time.

Exact solutions useful for semi-deterministic evolutionary modeling.

Abstract

We study a continuous time model for the frequency distribution of an infinitely large asexual population in which both beneficial and deleterious mutations occur and the fitness is additive. When beneficial mutations are ignored, the exact solution for the frequency distribution is known to be a Poisson distribution. Here we include beneficial mutations and obtain exact expressions for the frequency distribution at all times using an eigenfunction expansion method. We find that the stationary distribution is non-Poissonian and related to the Bessel function of the first kind. We also provide suitable approximations for the stationary distribution and the time to relax to the steady state. Our exact results, especially at mutation-selection equilibrium, can be useful in developing semi-deterministic approaches to understand stochastic evolution.