coalescent

Coalescent theory is a retrospective stochastic framework in population genetics that describes how the genealogical relationships of a sample of gene copies merge as one traces their ancestry back in time to a common ancestor. The standard model, known as the Kingman coalescent, describes neutral evolution in a randomly mating population of constant effective size Ne. If a sample contains k lineages, the rate at which any two of them coalesce in the previous generation is 1/(2Ne), so the total rate of a coalescent event among the k lineages is lambda_k = C(k,2)/(2Ne). The waiting time until the next coalescent event is exponentially distributed with mean 1/lambda_k, and time is usually measured in generations or in 2Ne generations. Repetition of these events reduces the number of lineages until a single ancestral lineage remains, the most recent common ancestor (MRCA).

Mutations can be superimposed along the resulting tree under a Poisson process with rate theta per coalescent unit, where theta = 4Ne mu for diploids and mu is the per-generation mutation rate. This combination yields predictions for patterns of genetic variation, such as the site frequency spectrum, under neutrality.

Assumptions of the basic model include constant population size, random mating, and no selection or migration. Extensions incorporate changing population size, population structure, migration, recombination (giving the ancestral recombination graph), and selection. Coalescent theory underpins methods for inferring demographic history, estimating population parameters, detecting departures from neutrality, and simulating genealogies across genomes. Computational tools such as ms and msprime implement coalescent simulations and analyses.