The Genetic Basis of Phenotypic Adaptation I: Fixation of Beneficial Mutations in the Moving Optimum Model

We study the genetic basis of adaptation in a moving optimum model, in which the optimal value for a quantitative trait increases over time at a constant rate. We first analyze a one-locus two-allele model with recurrent mutation, for which we derive accurate analytical approximations for (i) the time at which a previously deleterious allele becomes beneficial, (ii) the waiting time for a successful new mutation, and (iii) the time the mutant allele needs to reach fixation. On the basis of these results, we show that the shortest total time to fixation is for alleles with intermediate phenotypic effect. We derive an approximation for this "optimal" effect, and we show that it depends in a simple way on a composite parameter, which integrates the ecological parameters and the genetic architecture of the trait. In a second step, we use stochastic computer simulations of a multilocus model to study the order in which mutant alleles with different effects go to fixation. In agreement with the one-locus results, alleles with intermediate effect tend to become fixed earlier than those with either small or large effects. However, the effect size of the fastest mutations differs from the one predicted in the one-locus model. We show how these differences can be explained by two specific effects of multilocus genetics. Finally, we discuss our results in the light of three relevant timescales acting in the system-the environmental, mutation, and fixation timescales-which define three parameter regimes leading to qualitative differences in the adaptive substitution pattern.