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Background:  We investigate mathematically the accumulation of beneficial mutations in a population of N genomes comprising many linked sites. The factors included in the model are mutation (described by the rate per site, m), selection (selection coefficient, s), and recombination (rate per genome, r  s). The model is relevant for several aspects of HIV pathogenesis and anti-HIV therapy, including reversion of deleterious alleles acquired due to transmission bottleneck, adaptation of virus to an individual host, emergence of strains resistant to drug cocktails, and antigenic escape of virus from multi-epitope vaccines.

Methods:  To describe multi-locus evolution, we generalized the analytic method we developed recently for an asexual population to the case with recombination. Just as for the asexual case, we found that the distribution of genomes over the mutant allele number moves in time as a “solitary wave” that is quasi-deterministic in the middle but has a stochastic edge on the high-fitness side. We derived general expressions for the shape and the speed of the wave (reversion speed) in terms of the model parameters. We verified the accuracy of the results both analytically and by Monte-Carlo simulation.

Results:  The model predicts the existence of a critical point in the population size, N_c ~ 1/[r log(s/r)], where the speed and the dominant factors of evolution change sharply. Above the critical point, N > N_c, the reversion speed does not depend on mutation events, is dominated by highly-fit recombinants, and is on the order of the speed in the limit of large r. For example, r= 10^-4, s = 10^-1, m = 10^-4, N = 10⁶ yield a half-reversion time of 140 days, comparable to 70 days at r 1. Below the critical point in N (in our example, N_c ~ 10³), the reversion speed depends on the back mutation rate and is close to the much smaller value predicted for an asexual population. The entire population, in this regime, eventually becomes a quasi-clone, and the few differences between positions of deleterious alleles in individual genomes make recombination inefficient.

Conclusions:  Our results predict that the accelerating effect of infrequent recombination on virus evolution is surprisingly strong. The recombination effect can be eliminated by depletion of virus below a threshold. These theoretical findings are crucial for understanding and controlling evolution of virus resistance to replication inhibitors and vaccines.

Keywords: recombination; selection; mutation