Non-homogeneous infinite sites model under demographic change: mathematical description and asymptotic behavior of pairwise distributions

We developed a mathematical model, which makes possible to predict joint distributions of numbers of mismatches in two or more linked regions of the genome, based on the Infinite Sites Models, under mutation-drift equilibrium as well as under various patterns of population growth. With mutation rates varying in the region, one of the predictions is different correlation between numbers of mismatches in the two regions, depending on the pattern of the past population growth (constant, slowly growing, or rapidly growing). Also, for slower growth patterns of population sizes, the coalescence tree is not necessarily `starlikeʹ. Thus, the joint distribution of mismatches, predicted by the model, provides additional insights into the demographic history of the populations. We also developed expectations and variances of sample statistics under different growth scenarios. As an application we used a sample of mitochondrial sequences from hypervariable regions 1 and 2 (HV1 and HV2), representing major world populations (Europeans, Asians and Africans). The patterns of joint distributions of numbers of mismatches differ markedly from one population to another. In addition, there is a considerable variability in the proportion of numbers of mismatches between HV1 and HV2 sequences. The patterns of bivariate distributions from the HV1 and HV2 data in these data are consistent with those generated by the model involving a stepwise change in population size.