On the Time for Gene Silencing at Duplicate Loci

A simple approximate formula is found for the mean time for mutant genes to first fix at one or another of two duplicate loci. The fixation is a result of one-way mutation from the wild-type to the mutant alleles, but is retarded by strong selection against the doubly homozygous mutants. The two-dimensional diffusion that models the evolution of the mutant allele frequencies at the two loci is studied by means of a transformation to two univariate diffusions having different time scales. Most of the results, but not all, agree well with the numerical studies by previous authors. Some new simulation results are presented, which agree well with the analytical results and, therefore, cast doubt on some previous simulations.