CLOSED-FORM ASYMPTOTIC SAMPLING DISTRIBUTIONS UNDER THE COALESCENT WITH RECOMBINATION FOR AN ARBITRARY NUMBER OF LOCI

Abstract

Obtaining a closed-form sampling distribution for the coalescent with recombination is a challenging problem. In the case of two loci, a new framework based on asymptotic series has recently been developed to derive closed-form results when the recombination rate is moderate to large. In this paper, an arbitrary number of loci is considered and combinatorial approaches are employed to find closed-form expressions for the first couple of terms in an asymptotic expansion of the multi-locus sampling distribution. These expressions are universal in the sense that their functional form in terms of the marginal one-locus distributions applies to all finite- and infinite-alleles models of mutation.

Figure 1

Illustration of L loci arranged linearly. The population-scaled recombination rate between loci l and l + 1 is ρ_l/2.

Figure 2

Illustration of sub-cases considered in the proof of Theorem 1. Here X ⊆ S(g), where g ⪰ h. Squares denote the loci in S(g), while shaded squares denote the loci in X (and hence also in S(g)). Circles denote the loci not in S(g) (and hence not in X). A squiggle denotes the recombination break interval l considered in each case. The squares to the left and to the right of the squiggle respectively denote the loci in $S(R_l^{-}(g))\text{ and }S(R_l^{+}(g))$. (a) Case with l < min(X). (b) Case with min(X) ≤ l < max(X).