Evolutionary trajectories in rugged fitness landscapes

Abstract

We consider the evolutionary trajectories traced out by an infinite population undergoingmutation–selection dynamics in static, uncorrelated random fitness landscapes. Startingfrom the population that consists of a single genotype, the most populated genotype jumpsfrom one local fitness maximum to another and eventually reaches the global maximum.We use a strong selection limit, which reduces the dynamics beyond the first time stepto the competition between independent mutant subpopulations, to study thedynamics of this model and of a simpler one-dimensional model which ignores thegeometry of the sequence space. We find that the fit genotypes that appear along atrajectory are a subset of suitably defined fitness records, and exploit severalresults from the record theory for non-identically distributed random variables.The genotypes that contribute to the trajectory are those records that are notbypassed by superior records arising further away from the initial population. Severalconjectures concerning the statistics of bypassing are extracted from numericalsimulations. In particular, for the one-dimensional model, we propose a simple relationbetween the bypassing probability and the dynamic exponent which describesthe scaling of the typical evolution time with genome size. The latter can bedetermined exactly in terms of the extremal properties of the fitness distribution.