An Ancestral Recombination Graph for Diploid Populations with Skewed Offspring Distribution

Abstract

A large offspring-number diploid biparental multilocus population model of Moran type is our object of study. At each time step, a pair of diploid individuals drawn uniformly at random contributes offspring to the population. The number of offspring can be large relative to the total population size. Similar "heavily skewed" reproduction mechanisms have been recently considered by various authors (cf. e.g., Eldon and Wakeley 2006, 2008) and reviewed by Hedgecock and Pudovkin (2011). Each diploid parental individual contributes exactly one chromosome to each diploid offspring, and hence ancestral lineages can coalesce only when in distinct individuals. A separation-of-timescales phenomenon is thus observed. A result of Möhle (1998) is extended to obtain convergence of the ancestral process to an ancestral recombination graph necessarily admitting simultaneous multiple mergers of ancestral lineages. The usual ancestral recombination graph is obtained as a special case of our model when the parents contribute only one offspring to the population each time. Due to diploidy and large offspring numbers, novel effects appear. For example, the marginal genealogy at each locus admits simultaneous multiple mergers in up to four groups, and different loci remain substantially correlated even as the recombination rate grows large. Thus, genealogies for loci far apart on the same chromosome remain correlated. Correlation in coalescence times for two loci is derived and shown to be a function of the coalescence parameters of our model. Extending the observations by Eldon and Wakeley (2008), predictions of linkage disequilibrium are shown to be functions of the reproduction parameters of our model, in addition to the recombination rate. Correlations in ratios of coalescence times between loci can be high, even when the recombination rate is high and sample size is large, in large offspring-number populations, as suggested by simulations, hinting at how to distinguish between different population models. D IPLOIDY, in which each offspring receives two sets of chromosomes, one from each of two distinct diploid parents, is fairly common among natural populations. Mathematical models in population genetics tend to assume, however, that all individuals in a population are haploid, simplifying the mathematics. Mendel's laws describe the mechanism of inheritance as composed of two main steps, equal segregation (first law) and independent assortment (second law). The first law proclaims gametes are haploid, i.e., carry only one of each pair of homologous chromosomes. Most models in population genetics are thus models of chromosomes or gene copies. Mendel's second law proclaims independent assortment of alleles at different genes, or loci, into gametes. Linkage of alleles on chromosomes, resulting in nonrandom association of alleles at different loci into gametes, is of course an important exception to the second law.Coalescent processes (Kingman 1982a,b;Hudson 1983b;Tajima 1983) describe the ancestral relations of chromosome...

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