Splitting trees with neutral Poissonian mutations I: Small families

TL;DR

This paper analyzes the allele frequency spectrum in neutral splitting trees with Poissonian mutations, providing explicit formulas and asymptotic results for small family sizes using coalescent and branching process tools.

Contribution

It introduces explicit formulas for allele counts in splitting trees with mutations and derives their asymptotic behavior, advancing understanding of genetic diversity in such models.

Findings

Explicit expectation formulas for allele counts A(k,t)

Almost sure limits of A(k,t)/N_t and A(t)/N_t

Expected homozygosity computed using tree dynamics

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, and the population counting process (N_t;t\ge 0) is a homogeneous, binary Crump--Mode--Jagers process. We assume that individuals independently experience mutations at constant rate \theta during their lifetimes, under the infinite-alleles assumption: each mutation instantaneously confers a brand new type, called 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,...,N_t. We mainly use two classes of tools: coalescent point processes and branching processes counted by random characteristics. We provide explicit formulae for the expectation of A(k,t) in a coalescent point process conditional on population size, which apply to the special case of splitting trees. We separately derive the a.s. limits of A(k,t)/N_t and of A(t)/N_t thanks to random characteristics. Last, we separately compute the expected homozygosity by applying a method 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.