Splitting trees with neutral mutations at birth

TL;DR

This paper models the genealogy of a population with neutral mutations at birth using splitting trees, deriving the expected allelic frequency spectrum and asymptotic properties of alleles with fixed mutation counts.

Contribution

It introduces a novel analysis of allelic partition and mutation counts in splitting trees with non-exponential lifetimes and neutral mutations.

Findings

Computed the expected allelic frequency spectrum.

Established almost sure convergence results for the spectrum.

Derived asymptotic properties of alleles with fixed mutation counts.

Abstract

We consider a population model where individuals behave independently from each other and whose genealogy is described by a chronological tree called splitting tree. The individuals have i.i.d. (non-exponential) lifetime durations and give birth at constant rate to clonal or mutant children in an infinitely many alleles model with neutral mutations. First, to study the allelic partition of the population, we are interested in its frequency spectrum, which, at a fixed time, describes the number of alleles carried by a given number of individuals and with a given age. We compute the expected value of this spectrum and obtain some almost sure convergence results thanks to classical properties of Crump-Mode-Jagers (CMJ) processes counted by random characteristics. Then, by using multitype CMJ-processes, we get asymptotic properties about the number of alleles that have undergone a fixed number of mutations with respect to the ancestral allele of the population.