Coalescent Theory

Abstract

The Kingman coalescent describes the ancestral tree of a sample of homologous genes from a population. Mutations occurring to ancestors in the tree determine the gene types in the sample. The coalescent originated as a mathematical concept and has become important in biology because it provides a way of looking back in time that allows ancestral inference, for example about parameters such as the mutation rate, the time to the most recent ancestor of a sample (TMRCA), ages of mutations, the way population size has changed back in time, and geographical information. The coalescent is a description of the ancestral tree of a sample of individuals from a neutral Wright-Fisher population with large haploid population gene size N, constant over time. Random mating in the population is assumed. With the large population size, the ancestral tree is approximately binary, that is, only two ancestral lineages can join, or coalesce, at a time instant. Time is measured in a long scale of N generations, and technically N tends to infinity. With a large sample size, coalescence is fast, so much so that the coalescent with the sample size tending to infinity represents ancestry back in time of the whole population as a well-defined coalescent tree starting with an infinite number of genes. The Kingman coalescent is said to come down from infinity because of this property. There is a connection between diffusion process models of allele frequencies and the coalescent. In the infinitely-many-alleles model, mutations are always to a new type not existing previously. The allele frequencies in a sample arise from mutations on coalescent lineages that are duplicated forward in time. The famous Ewens’ sampling formula is the probability distribution of the allele frequencies in a sample under the infinitely-many-alleles model. Extensions of the original coalescent model include variable population size, recombination, selection, and spatial models. The ancestral recombination graph (ARG) describes the ancestry of a sample of genes when recombination is present. The ancestral selection graph (ASG) describes the ancestral tree when there are different selective advantages between allele types. The Lambda coalescent is a related model of ancestry where multiple individuals can coalesce at the same time. Coalescent theory has contributed to biology and mathematics, a nice interplay of disciplines.