A diffusion model for the fate of tandem gene duplicates in diploids

Suppose one chromosome in one member of a population somehow acquires a duplicate copy of the gene, fully linked to the original gene's locus. Preservation is the event that eventually every chromosome in the population is a descendant of the one which initially carried the duplicate. For a haploid population in which the absence of all copies of the gene is lethal, the probability of preservation has recently been estimated via a diffusion approximation. That approximation is shown to carry over to the case of diploids and arbitrary strong selection against the absence of the gene. The techniques used lead to some new results. In the large population limit, it is shown that the relative probability that descendants of a small number of individuals carrying multiple copies of the gene fix in the population is proportional to the number of copies carried. The probability of preservation is approximated when chromosomes carrying two copies of the gene are subject to additional, fully non-functionalizing mutations, thereby modelling either an additional cost of replicating a longer genome, or a partial duplication of the gene. In the latter case the preservation probability depends only on the mutation rate to null for the duplicated portion of the gene.