Genetics of the biparental Moran model

TL;DR

This paper analyzes the genetic composition dynamics in a fixed-size population with biparental inheritance, proving convergence properties and limiting distributions of ancestral genetic contributions as population size grows.

Contribution

It introduces a biparental Moran model and rigorously characterizes the asymptotic distribution of ancestral genetic weights in large populations.

Findings

Proportions of genetic contributions converge almost surely to a random variable.

As population size increases, the scaled ancestral weights converge in law to a mixture of a point mass at zero and an exponential distribution.

Weights of different ancestors are shown to be independent in the limit.

Abstract

Our goal is to study the genetic composition of a population in which each individual has 2 parents, who contribute equally to the genome of their ospring. We use a biparental Moran model, which is characterized by its xed number N of individuals. We x an individual and consider the proportions of the genomes of all individuals living n time steps later, that come from this individual. We rst prove that when n goes to innity, these proportions all converge almost surely towards the same random variable. We then rigorously prove that when N then goes to innity, this random variable multiplied by N (i.e. the stationary weight of any ancestor in the whole population) converges in law towards the mixture of a Dirac measure in 0 and an exponential law with parameter 1/2, and that the weights of several given ancestors are independent.