The rate of multi-step evolution in Moran and Wright–Fisher populations

Several groups have recently modeled evolutionary transitions from an ancestral allele to a beneficial allele separated by one or more intervening mutants. The beneficial allele can become fixed if a succession of intermediate mutants are fixed or alternatively if successive mutants arise while the previous intermediate mutant is still segregating. This latter process has been termed stochastic tunneling. Previous work has focused on the Moran model of population genetics. I use elementary methods of analyzing stochastic processes to derive the probability of tunneling in the limit of large population size for both Moran and Wright–Fisher populations. I also show how to efficiently obtain numerical results for finite populations. These results show that the probability of stochastic tunneling is twice as large under the Wright–Fisher model as it is under the Moran model.