The Allelic Partition for Coalescent Point Processes

Abstract. Assume that individuals alive at time t in some population can be ranked in such a way that the coalescence times between consecutive individuals are i.i.d. The ranked sequence of these branches is called a coalescent point process. We have shown in a previous work [14] that splitting trees are important instances of such populations. Here, individuals are given DNA sequences, and for a sample of n DNA sequences belonging to distinct individuals, we consider the number Sn of poly-morphic sites (sites at which at least two sequences dier), and the number An of distinct haplotypes (sequences diering at one site at least). It is standard to assume that mutations arrive at constant rate (on germ lines), and never hit the same site on the DNA sequence. We study the muta-tion pattern associated with coalescent point processes under this assumption. Here, Sn and An grow linearly as n grows, with explicit rate. However, when the branch lengths have innite expectation, Sn grows more rapidly, e.g. as n ln(n) for critical birth{death processes. Then, we study the frequency spectrum of the sample, that is, the numbers of polymorphic sites/haplotypes carried by k individuals in the sample. These numbers are shown to grow also linearly with sample size, and we provide sim-ple explicit formulae for mutation frequencies and haplotype frequencies. For critical birth{death processes, mutation frequencies are given by the harmonic series and haplotype frequencies by Fisher's logarithmic series