Evolution in random fitness landscapes: the infinite sites model

We consider the evolution of an asexually reproducing population in an uncorrelated random fitness landscape in the limit of infinite genome size, which implies that each mutation generates a new fitness value drawn from a probability distribution g(w). This is the finite population version of Kingman’s house of cards model [J.F.C. Kingman, J. Appl. Probab. 15, 1 (1978)]. In contrast to Kingman’s work, the focus here is on unbounded distributions g(w) which lead to an indefinite growth of the population fitness. The model is solved analytically in the limit of infinite population size N → ∞ and simulated numerically for finite N . When the genome-wide mutation probability U is small, the long time behavior of the model reduces to a point process of fixation events, which can be described as a randomly diluted record process (DRP). We develop a systematic analytic approximation scheme for the DRP. At finite U the fitness frequency distribution of the population decomposes into a stationary part due to mutations and a traveling wave component due to selection, which is shown to imply a reduction of the mean fitness by a factor of 1− U compared to the U → 0 limit. Evolution in random fitness landscapes: the infinite sites model 2