Mean time to resolution of gene duplication

The mean time to resolution of gene duplication (Tr) is studied in this paper under the double null recessive (DNR) and haplo-insufficient (HI) models within the same analytical and simulation framework. We show that when population size is not too small (more precisely Nμ > 0.1), Tr for unlinked duplication is usually larger than that for linked and Tr for unlinked duplication under the HI model might be greatly prolonged, which were consistent with previous observations. Furthermore, by analytical approach we here indicate the primary underlying mechanism is that the frequency of the original (or wild-type) chromosomal haplotype of the linked duplication decreases nearly exponential to zero with time while that of the unlinked decreases quickly to an quasi-equilibrium; and this phenomenon is particularly profound under the HI model, because the quasi-equilibrium frequency of the original chromosomal haplotype (x0) under the HI model is higher than that under the DNR model. These results suggest that recombination and HI model might jointly contribute to the marked prolongation of Tr even in a modest population. The prolonged Tr and higher quasi-equilibrium frequency of the original allele at both duplicated loci might have offered more opportunities for the emergence of novel genes.