Evolution in random fitness landscapes: the infinite sites model

TL;DR

This paper analyzes the evolution of an asexually reproducing population on uncorrelated, unbounded random fitness landscapes in the infinite genome limit, revealing a novel diluted record process and effects of mutation rate on fitness distribution.

Contribution

It introduces an analytical solution for the infinite population limit and a systematic approximation for the diluted record process in evolving populations.

Findings

Population fitness grows indefinitely with unbounded fitness distributions.

A new diluted record process models fixation events at low mutation rates.

Mutation rate U reduces mean fitness by a factor of 1-U.

Abstract

We consider the evolution of an asexually reproducing population in an uncorrelated random fitness landscape in the limit of infinite genome size, which implies that each mutation generates a new fitness value drawn from a probability distribution $g(w)$. This is the finite population version of Kingman's house of cards model [J.F.C. Kingman, \textit{J. Appl. Probab.} \textbf{15}, 1 (1978)]. In contrast to Kingman's work, the focus here is on unbounded distributions $g(w)$ which lead to an indefinite growth of the population fitness. The model is solved analytically in the limit of infinite population size $N \to \infty$ and simulated numerically for finite $N$. When the genome-wide mutation probability $U$ is small, the long time behavior of the model reduces to a point process of fixation events, which is referred to as a \textit{diluted record process} (DRP). The DRP is similar to the standard record process except that a new record candidate (a number that exceeds all previous entries in the sequence) is accepted only with a certain probability that depends on the values of the current record and the candidate. We develop a systematic analytic approximation scheme for the DRP. At finite $U$ the fitness frequency distribution of the population decomposes into a stationary part due to mutations and a traveling wave component due to selection, which is shown to imply a reduction of the mean fitness by a factor of $1-U$ compared to the $U \to 0$ limit.