Waiting with and without Recombination: The Time to Production of a Double Mutant

R. A. Fisher and H. J. Muller argued in the 1930s that a major evolutionary advantage of recombination is that it allows favorable mutations to be combined within an individual even when they first appear in different individuals. This effect is evaluated in a two-locus, two-allele model by calculating the average waiting time until a new genotypic combination first appears in a haploid population. Three approximations are developed and compared with Monte Carlo simulations of the Wright–Fisher process of random genetic drift in a finite population. First, a crude method, based on the deterministic accumulation of single mutants, produces a waiting time of 1/Nμ2with no recombination and 1/13RNμ2with recombination between the two loci, whereμis the mutation rate,Nis the haploid population size, andRis the recombination rate. Second, the waiting time is calculated as the expected value of a heterogeneous geometric distribution obtained from a branching process approximation. This gives accurate estimates forNμlarge. The estimates for small values ofNμare considerably lower than the simulated values. Finally, diffusion analysis of the Wright–Fisher process provides accurate estimates forNμsmall, and the time scales of the diffusion process show a difference betweenR=0 and forR⪢0 of the same order of magnitude as seen in the deterministic analysis. In the absence of recombination, accurate approximations to the waiting time are obtained by using the branching process for highNμand the diffusion approximation for lowNμ. For lowNμthe waiting time is well approximated by 1/8N2μ3. WithR⪢0, the following dependence onNμis observed: ForNμ>1 the waiting time is virtually independent of recombination and is well described by the branching process approximation. ForNμ≈ the waiting time is well described by a simplified diffusion approximation that assumes symmetry in the frequencies of single mutants. ForNμ⪡1 the waiting time is well described by the diffusion approximation allowing asymmetry in the frequencies of single mutants. Recombination lowers the waiting time until a new genotypic combination first appears, but the effect is small compared to that of the mutation rate and population size. For largeNμ, recombination has a negligible effect, and its effect is strongest for smallNμ, in which case the waiting time approaches a fixed fraction of the waiting time forR=0. Free recombination lowers the waiting time to about 45% of the waiting time for absolute linkage