genealogy

Wakeley J 2020 Developments in coalescent theory from single loci to chromosomes. Theor Popul Biol 133:56-64.

  • it took some time to understand the temporal structure behind Ewens' formula
  • working at first outside population genetics, Kingman (1975) introduced the Poisson-Dirichlet distribution, which also applies to allele frequencies in a population under infinite-alleles mutation when the frequencies are ordered largest to smallest
  • just prior to the introduction of coalescent theory, a closely related forward-time theory of lines of descent was developed by Griffiths (1980)
  • the comprehensive synthetic work of Tavaré (1984) placed the theories of coalescence, lines of descent and ages of alleles within a single framework
  • properties of the ancestral process of gene genealogies are obtained from allelic models in the limit as θ tends to zero
  • this highlighted the earlier work of Felsenstein (1971) which established a recursive equation for sampling probabilities of numbers of alleles at two different time points in the absence of mutation
  • Felsenstein (1971) considered the probability that i alleles present in the population now will all still be present at some future time
  • Kimura (1955) had shown previously using diffusion theory that this rate is equal to i(i − 1)/2 on the diffusion time scale
  • Felsenstein (1971) showed that a genealogical approach based on G gives the same answer
  • i(i − 1)/2 is also the rate of decay of the probability that i alleles are present in a sample of size i
  • following Kimura (1955), Felsenstein (1971) considered these results in relation to the rate of loss of i alleles at some distant future time
  • it is remarkable how close this came to the backward-time coalescent process without mutation, in which i(i − 1)/2 is the total rate of coalescence when there are i ancestral lineages
  • Felsenstein (1971) did not consider what would now be called the branching structure of the gene genealogy
  • Watterson (1975) was the first to present gene genealogies and their backward-time construction, through a series of n − 1 independent intervals and with the familiar random scattering of neutral mutations on the branches [of?] the gene genealogy, in a way that unambiguously captures our modern notion of coalescent theory
  • key aspects of the theory which are missing in Watterson (1975) compared to Kingman (1982a,b,c) are the description of the detailed relationships among n labeled samples, that is the state space of gene genealogies, and the proof of convergence to the coalescent process