Length of adaptive walk on uncorrelated and correlated fitness landscapes

TL;DR

This paper analytically and numerically investigates the length of adaptive walks in rugged fitness landscapes, considering correlations and extreme value distributions, with implications for microbial evolution.

Contribution

It provides a comprehensive analytical framework for the length of adaptive walks on correlated and uncorrelated landscapes across all extreme value domains.

Findings

Walk length depends on initial fitness and mutation distribution.

Correlations among fitnesses alter walk length behavior in the Fréchet domain.

Numerical results show differences from analytical predictions in large fitness difference regimes.

Abstract

We consider the adaptation dynamics of an asexual population that walks uphill on a rugged fitness landscape which is endowed with large number of local fitness peaks. We work in a parameter regime where only those mutants that are single mutation away are accessible, as a result of which the population eventually gets trapped at a local fitness maximum and the adaptive walk terminates. We study how the number of adaptive steps taken by the population before reaching a local fitness peak depends on the initial fitness of the population, the extreme value distribution of the beneficial mutations and correlations amongst the fitnesses. Assuming that the relative fitness difference between successive steps is small, we analytically calculate the average walk length for both uncorrelated and correlated fitnesses in all extreme value domains for a given initial fitness. We present numerical results for the model where the fitness differences can be large, and find that the walk length behavior differs from that in the former model in the Fr\'echet domain of extreme value theory. We also discuss the relevance of our results to microbial experiments.