The Speed of Adaptation in Large Asexual Populations

In large asexual populations, beneficial mutations have to compete with each other for fixation. Here, I derive explicit analytic expressions for the rate of substitution and the mean beneficial effect of fixed mutations, under the assumptions that the population size $N$ is large, that the mean effect of new beneficial mutations is smaller than the mean effect of new deleterious mutations, and that new beneficial mutations are exponentially distributed. As $N$ increases, the rate of substitution approaches a constant, which is equal to the mean effect of new beneficial mutations. The mean effect of fixed mutations continues to grow logarithmically with $N$. As a consequence, the speed of adaptation, measured as the change of log fitness over time, also grows logarithmically with $N$. Moreover, I derive a simple formula that determines whether at given $N$ beneficial mutations are expected to compete with each other or go to fixation independently. Finally, I verify all results with extensive numerical simulations.