Abstract
We consider a neutral dynamical model of biological diversity, where individuals live and reproduce independently. They have i.i.d. lifetime durations (which are not necessarily exponentially distributed) and give birth (singly) at constant rate b. Such a genealogical tree is usually called a splitting tree [9], and the population counting process (Nt;t≥0) is a homogeneous, binary Crump–Mode–Jagers process.We assume that individuals independently experience mutations at constant rate θ during their lifetimes, under the infinite-alleles assumption: each mutation instantaneously confers a brand new type, called an allele, to its carrier. We are interested in the allele frequency spectrum at time t, i.e., the number A(t) of distinct alleles represented in the population at time t, and more specifically, the numbers A(k,t) of alleles represented by k individuals at time t, k=1,2,…,Nt.We mainly use two classes of tools: coalescent point processes, as defined in [15], and branching processes counted by random characteristics, as defined in [11,13]. We provide explicit formulae for the expectation of A(k,t) conditional on population size in a coalescent point process, which apply to the special case of splitting trees. We separately derive the a.s. limits of A(k,t)/Nt and of A(t)/Nt thanks to random characteristics, in the same vein as in [19].Last, we separately compute the expected homozygosity by applying a method introduced in [14], characterizing the dynamics of the tree distribution as the origination time of the tree moves back in time, in the spirit of backward Kolmogorov equations.
Highlights
We consider a general branching population, where individuals reproduce independently of each other, have i.i.d. lifetime durations with arbitrary distribution, and give birth at constant rate duringindividuals are given a type, called allele or haplotype
Individuals are given a type, called allele or haplotype. They inherit their type at birth from their mother, and change type throughout their lifetime, at the points of independent Poisson point processes with rate θ, conditional on lifetimes
The type conferred by a mutation is each time an entirely new type, an assumption known as the infinitely-many alleles model
Summary
We consider a general branching population, where individuals reproduce independently of each other, have i.i.d. lifetime durations with arbitrary distribution, and give birth at constant rate during. Individuals are given a type, called allele or haplotype They inherit their type at birth from their mother, and (their germ line) change type throughout their lifetime, at the points of independent Poisson point processes with rate θ, conditional on lifetimes (neutral mutations). Bertoin [2, 3, 4] has set up a very general framework for Galton–Watson processes with mutations, where he has considered the allelic partition of the whole population from origination to extinction, and studied various scaling limits for large initial population sizes and low mutation probabilities. Branching processes have been used in the study of multistage carcinogenesis In this setting, the emphasis is put on the waiting time until a target mutation occurs, see [6, 18] and the references therein. In a companion paper [5], we will discuss the part of the frequency spectrum corresponding to the largest or/and oldest families (the age of a family being that of their original mutation)
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