Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-24T20:29:30.032Z Has data issue: false hasContentIssue false
  • English
  • Français

Coalescent lineage distributions

Part of: Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Robert C. Griffiths
Robert C. Griffiths*
Affiliation:
University of Oxford
*
∗ Postal address: Department of Statistics, University of Oxford, 1 South Parks Rd, Oxford OX1 3TG, UK. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study identities for the distribution of the number of edges at time t back (i.e. measured backwards) in a coalescent tree whose subtrees have no mutations. This distribution is important in the infinitely-many-alleles model of mutation, where every mutation is unique. The model includes, as a special case, the number of edges in a coalescent tree at time t back when mutation is ignored. The identities take the form of expected values of functions of Zt=eiXt, where Xt is distributed as standard Brownian motion. Associated identities are also found for the distributions of the time to the most recent common ancestor, the time until loss of ancestral lines by coalescence or mutation, and the age of a mutation. Hypergeometric functions play an important role in the identities. The identities are of mathematical interest, as well as potentially being formulae to use for numerical integration or simulation to compute distributions that are usually expressed as alternating-sign series expansions, which are difficult to compute.

Keywords

Age of a mutationcoalescent lineage distributioncoalescent processhypergeometric functioninfinitely-many-alleles modelJacobi polynomialline of descent

MSC classification

Primary: 92D15: Problems related to evolution
Secondary: 60G99: None of the above, but in this section
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 38 , Issue 2 , June 2006 , pp. 405 - 429
Copyright
Copyright © Applied Probability Trust 2006 

References

Abramowitz, M. and Stegun, I. (1972). Handbook of Mathematical Functions. Dover, New York.Google Scholar
Donnelly, P. J. and Tavaré, S. (1987). The population genealogy of the infinitely many neutral alleles model. J. Math. Biol. 25, 381–391.Google Scholar
Ethier, S. N. and Griffiths, R. C. (1993). The transition function of a Fleming–Viot process. Ann. Prob. 21, 1571–1590.CrossRefGoogle Scholar
Ewens, W. J. (1972). The sampling theory of selectively neutral alleles. Theoret. Pop. Biol. 3, 87–112.CrossRefGoogle ScholarPubMed
Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. 2, 2nd edn. John Wiley, New York.Google Scholar
Griffiths, R. (1980). Lines of descent in the diffusion approximation of neutral Wright–Fisher models. Theoret. Pop. Biol. 17, 37–50.CrossRefGoogle ScholarPubMed
Griffiths, R. C. (1984). Asymptotic line-of-descent distributions. J. Math. Biol. 21, 67–75.CrossRefGoogle Scholar
Griffiths, R. C. (2003). The frequency spectrum of a mutation, and its age, in a general diffusion model. Theoret. Pop. Biol. 64, 241–251.Google Scholar
Griffiths, R. C. and Lessard, S. (2005). The Ewens sampling formula and related formulae: combinatorial proofs, extensions to variable population size and applications to ages of alleles. Theoret. Pop. Biol. 68, 167–177.CrossRefGoogle Scholar
Griffiths, R. C. and Li, W.-H. (1983). Simulating allele frequencies in a population and the genetic differentiation of populations under mutation pressure. Theoret. Pop. Biol. 32, 19–33.CrossRefGoogle Scholar
Griffiths, R. C. and Tavaré, S. (1998). The age of a mutation in a general coalescent tree. Stoch. Models 14, 273–295.Google Scholar
Hudson, R. R. (1991). Gene genealogies and the coalescent process. In Oxford Surveys in Evolutionary Biology, eds Futuyama, D. and Antonovics, J., Vol. 7, 2nd edn, Oxford University Press, pp. 1–44.Google Scholar
Kingman, J. F. C. (1982). The coalescent. Stoch. Process. Appl. 13, 235–248.Google Scholar
Kingman, J. F. C. (1993). Poisson Processes (Oxford Stud. Prob. 3). Clarendon Press, Oxford.Google Scholar
Nordborg, M. (2001). Coalescent theory. In Handbook of Statistical Genetics, eds Balding, D. J., Bishop, M. and Cannings, C., John Wiley, Chichester, pp. 179–208.Google Scholar
Saunders, I. W., Tavaré, S. and Watterson, G. A. (1984). On the genealogy of nested subsamples from a haploid population. Adv. Appl. Prob. 16, 471–491.CrossRefGoogle Scholar
Tavaré, S. (1984). Line-of-descent and genealogical processes, and their application in population genetics models. Theoret. Pop. Biol. 26, 119–164.Google Scholar
Watterson, G. A. (1984). Lines of descent and the coalescent. Theoret. Pop. Biol. 26, 239–253.CrossRefGoogle Scholar
Watterson, G. A. (1996). Motoo Kimura's use of diffusion theory in population genetics. Theoret. Pop. Biol. 49, 154–188.Google Scholar