Birth and Death Processes with Neutral Mutations

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 the birth of particles or at a constant rate during their lives. In both models, we study the allelic partition of the population at time . We give closed-form formulae for the expected frequency spectrum at and prove a pathwise convergence to an explicit limit, as , of the relative numbers of types younger than some given age and carried by a given number of particles (small families). We also provide the 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. 1. Introduction We consider a general branching model, where particles have i.i.d. (not necessarily exponential) life lengths and give birth at a constant rate during their lives to independent copies of themselves. The genealogical tree thus produced is called splitting tree [1–3]. The process that counts the number of the alive particles through time is a Crump-Mode-Jagers process (or general branching process) [4] which is binary (births occur singly) and homogeneous (constant birth rate). We enrich this genealogical model with mutations. In Model I, each child is a clone of her mother with probability and a mutant with probability . In Model II, independently of other particles, each particle undergoes mutations during her life at constant rate (and births are always clonal). For both models, we are working under the infinitely many alleles model; that is, a mutation yields a type, also called allele, which was never encountered before. Moreover, mutations are supposed to be neutral; that is, they do not modify the way particles die and reproduce. For any type and any time , we call family the set of all particles that share this type at time . Branching processes (and especially birth and death processes) with mutations have many applications in biology. In carcinogenesis [5–10], they can model the evolution of cancerous cells. In [11], Kendall modeled carcinogenesis by a birth and death process where mutations occur during life according to an inhomogeneous Poisson process. In [8, 10], cancerous cells are modeled by a multitype branching process where a cell is of type if it has undergone mutations and where the more a cell has undergone mutations, the faster it