The Accumulation of Beneficial Mutations and Convergence to a Poisson Process

TL;DR

This paper models a population with beneficial mutations and shows that, under certain conditions, the times at which beneficial mutations fixate follow a Poisson process, despite potential competition among mutations.

Contribution

It demonstrates that fixation times of beneficial mutations in a population converge to a Poisson process under specific mutation rates and selection strengths.

Findings

Fixation times converge to a Poisson process after appropriate scaling.

Multiple beneficial mutations can occur simultaneously and compete.

Results hold under conditions of low mutation rate and moderate selection strength.

Abstract

We consider a model of a population with fixed size $N$, which is subjected to an unlimited supply of beneficial mutations at a constant rate $\mu_N$. Individuals with $k$ beneficial mutations have the fitness $(1+s_N)^k$. Each individual dies at rate 1 and is replaced by a random individual chosen with probability proportional to its fitness. We show that when $\mu_N \ll 1/(N \log N)$ and $N^{-\eta} \ll s_N \ll 1$ for some $\eta < 1$, the fixation times of beneficial mutations, after a time scaling, converge to the times of a Poisson process, even though for some choices of $s_N$ and $\mu_N$ satisfying these conditions, there will sometimes be multiple beneficial mutations with distinct origins in the population, competing against each other.