A General Extreme Value Theory Model for the Adaptation of DNA Sequences Under Strong Selection and Weak Mutation

Recent theoretical studies of the adaptation of DNA sequences assume that the distribution of fitness effects among new beneficial mutations is exponential. This has been justified by using extreme value theory and, in particular, by assuming that the distribution of fitnesses belongs to the Gumbel domain of attraction. However, extreme value theory shows that two other domains of attraction are also possible: the Fréchet and Weibull domains. Distributions in the Fréchet domain have right tails that are heavier than exponential, while distributions in the Weibull domain have right tails that are truncated. To explore the consequences of relaxing the Gumbel assumption, we generalize previous adaptation theory to allow all three domains. We find that many of the previously derived Gumbel-based predictions about the first step of adaptation are fairly robust for some moderate forms of right tails in the Weibull and Fréchet domains, but significant departures are possible, especially for predictions concerning multiple steps in adaptation.