The Effect of Recombination on the Speed of Evolution

It has been a puzzling question why some organisms reproduce sexually. Fisher and Muller hypothesized that reproducing by sex can speed up the evolution. They explained that in the sexual reproduction, recombination can combine beneficial alleles that lie on different chromosomes, which speeds up the time that those beneficial alleles spread to the entire population. We consider a population model of fixed size $N$, in which we will focus on two loci on a chromosome. Each allele at each locus can mutate into a beneficial allele at rate $\mu_N$. The individuals with 0, 1, and 2 beneficial alleles die at rates $1, 1-s_N$ and $1-2s_N$ respectively. When an individual dies, with probability $1-r_N$, the new individual inherits both alleles from one parent, chosen at random from the population, while with probability $r_N$, recombination occurs, and the new individual receives its two alleles from different parents. Under certain assumptions on the parameters $N, \mu_N, s_N$ and $r_N$, we obtain an asymptotic approximation for the time that both beneficial alleles spread to the entire population. When the recombination probability is small, we show that recombination does not speed up the time that the two beneficial alleles spread to the entire population, while when the recombination probability is large, we show that recombination decreases the time, which agrees with Fisher-Muller hypothesis, and confirms the advantage of reproducing by sex.