The Rate of Compensatory Evolution

A two-locus model is presented to analyze the evolution of compensatory mutations occurring in stems of RNA secondary structures. Single mutations are assumed to be deleterious but harmless (neutral) in appropriate combinations. In proceeding under mutation pressure, natural selection and genetic drift from one fitness peak to another one, a population must therefore pass through a valley of intermediate deleterious states of individual fitness. The expected time for this transition is calculated using diffusion theory. The rate of compensatory evolution, kappa c, is then defined as the inverse of the expected transition time. When selection against deleterious single mutations is strong, kappa c becomes independent on the recombination fraction r between the two loci. Recombination generally reduces the rate of compensatory evolution because it breaks up favorable combinations of double mutants. For complete linkage, kappa c is given by the rate at which favorable combinations of double mutants are produced by compensatory mutation. For r > O, kappa c decreases exponentially with r. In contrast, kappa c becomes independent of r for weak selection. We discuss the dynamics of evolutionary substitutions of compensatory mutants in relation of WRIGHT's shifting balance theory of evolution and use our results to analyze the substitution process in helices of mRNA secondary structures.