A general extreme value theory model for the adaptation of DNA sequences under strong selection and weak mutation

Abstract

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.

Figure 1.—

The three domains of attraction under the generalized Pareto distribution (GPD). (A) The Gumbel domain corresponds to the GPD with κ = 0, and the Fréchet domain corresponds to the GPD with κ > 0. (B) The Weibull domain corresponds to the GPD with κ < 0.

Figure 2.—

Mean transition probabilities from Equation 16 as a function of the shape parameter κ of the GPD. Each curve in the top center plot represents the probabilities for one allele. The top curve provides the probabilities of fixing the allele of rank 1, the next is for the allele of rank 2, etc. The histograms show the transition probabilities for specific values of κ.

Figure 3.—

The effect of rank of the wild-type i and the shape parameter κ on the accuracy of the law of large numbers approximation used to calculate the mean transition probabilities. The dashed curves represent the probability of fixing the best allele (j = 1) as a function of the GPD shape parameter κ according to Equation 16. The solid lines are simulation results assuming a GPD(κ, 1) with sample size of 10,000. Simulations were performed using R (R Development Core Team 2006).

Figure 4.—

The expected fitness rank after a single step in adaptation. The dashed lines give the theoretical expectation according to Equation 22, and the solid lines give the averages of 10,000 simulated data sets. Simulations were performed using R (R Development Core Team 2006).