Evolution of Human Immunodeficiency Virus Under Selection and Weak Recombination

To predict emergence of drug resistance in patients undergoing antiretroviral therapy, we study accumulation of preexisting beneficial alleles in a haploid population of N genomes. The factors included in the model are selection with the coefficient s and recombination with the small rate per genome r (r ≪ s\documentclass[10pt]{article}\usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \pagestyle{empty} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}\sqrt{\overline{{\mathit{k}}}}\end{equation*}\end{document}, where \documentclass[10pt]{article} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \pagestyle{empty} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}\overline{{\mathit{k}}}\end{equation*}\end{document} is the average number of less-fit loci per genome). Mutation events are neglected. To describe evolution at a large number of linked loci, we generalize the analytic method we developed recently for an asexual population. We show that the distribution of genomes over the deleterious allele number moves in time as a “solitary wave” that is quasi-deterministic in the middle (on the average) but has stochastic edges. We arrive at a single-locus expression for the average accumulation rate, in which the effects of linkage, recombination, and random drift are all accounted for by the effective selection coefficient s lnNr/lnNs(2)\documentclass[10pt]{article} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \pagestyle{empty} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}\overline{{\mathit{k}}}\end{equation*}\end{document}/r. At large N, the effective selection coefficient approaches the single-locus value s. Below the critical size N(c) ∼ 1/r, a population eventually becomes a clone, recombination cannot produce new sequences, and virus evolution stops. Taking into account finite mutation rate predicts a small, finite rate of evolution at N < N(c). We verify the accuracy of the results analytically and by Monte Carlo simulation. On the basis of our findings, we predict that partial depletion of the HIV population by combined anti-retroviral therapy can suppress emergence of drug-resistant strains.