Birth and death processes with neutral mutations

TL;DR

This paper reviews recent results on branching processes with neutral mutations under the infinitely many alleles model, providing explicit formulas and convergence results for allelic diversity and family sizes over time.

Contribution

It introduces new closed-form formulas and convergence results for allelic partition and family size distributions in branching processes with neutral mutations.

Findings

Explicit formulas for expected allele frequency spectrum at time t

Pathwise convergence of relative numbers of types by age and size

Distributional convergence of largest and oldest family sizes

Abstract

In this paper, we review recent results of ours concerning branching processes with general lifetimes and neutral mutations, under the infinitely many alleles model, where mutations can occur either at birth of individuals or at a constant rate during their lives. In both models, we study the allelic partition of the population at time t. We give closed formulae for the expected frequency spectrum at t and prove pathwise convergence to an explicit limit, as t goes to infinity, of the relative numbers of types younger than some given age and carried by a given number of individuals (small families). We also provide convergences in distribution of the sizes or ages of the largest families and of the oldest families. In the case of exponential lifetimes, population dynamics are given by linear birth and death processes, and we can most of the time provide general formulations of our results unifying both models.