1. What is the nature of adaptation? We have only a rudimentary understanding of the genetic changes that underlie adaptation. Alleles affecting quantitative traits exhibit a broad spectrum of effects, but what does this spectrum look like? Because the gamma distribution has a flexible shape, let’s assume that the magnitude of mutational effects on a trait is gamma distributed: CV = 0.1CV = 1 (exponential) CV = 10 where CV is the “Coefficient of Variation”:

2. If there is a gamma distribution of effects among new mutations, what is the distribution among those alleles that spread to fixation? We can answer this question by weighting the gamma distribution, f(s), by the fixation probability, 2s. Carrying out the integration, the distribution of effects among fixed mutations is: But this is the same as a gamma distribution with r replaced by r+1.

3. The distribution of fixed alleles becomes gamma with: CV fixed is always less than one, implying that the distribution of alleles underlying adaptation should be exponential to bell-shaped. Original distribution Fixed distribution CV = 0.5 CV = 1 CV = 5 CV = 0.45 CV = 0.71 CV = 0.98 Even if there are many more ways for a mutation to have small benefits than large benefits (CV >> 1), the distribution observed among fixed alleles is predicted to be nearly exponential.