# The allelic partition for coalescent point processes - Mathematics > Probability

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Abstract: Assume that individuals alive at time $t$ in some population can be ranked insuch a way that the coalescence times between consecutive individuals arei.i.d. The ranked sequence of these branches is called a coalescent pointprocess. We have shown in a previous work that splitting trees are importantinstances of such populations. Here, individuals are given DNA sequences, andfor a sample of $n$ DNA sequences belonging to distinct individuals, weconsider the number $S n$ of polymorphic sites sites at which at least twosequences differ, and the number $A n$ of distinct haplotypes sequencesdiffering at one site at least. It is standard to assume that mutations arriveat constant rate on germ lines, and never hit the same site on the DNAsequence. We study the mutation pattern associated to coalescent pointprocesses under this assumption. Here, $S n$ and $A n$ grow linearly as $n$grows, with explicit rate. However, when the branch lengths have infiniteexpectation, $S n$ grows more rapidly, e.g. as $n \lnn$ for criticalbirth-death processes. Then, we study the frequency spectrum of the sample,that is, the numbers of polymorphic sites-haplotypes carried by $k$ individualsin the sample. These numbers are shown to grow also linearly with sample size,and we provide simple explicit formulae for mutation frequencies and haplotypefrequencies. For critical birth-death processes, mutation frequencies aregiven by the harmonic series and haplotype frequencies by Fisher logarithmicseries.

Author: ** Amaury Lambert PMA**

Source: https://arxiv.org/