Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-28T03:12:25.614Z Has data issue: false hasContentIssue false
  • English
  • Français

The ages of alleles and a coalescent

Published online by Cambridge University Press:  01 July 2016

Peter Donnelly  and
Simon Tavaré
Peter Donnelly*
Affiliation:
University College of Swansea
Simon Tavaré*
Affiliation:
University of Utah
*
∗Present address: Department of Statistical Science, University College London, London WC1E 6BT, UK.
∗∗Postal address: Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

A new coalescent is introduced to study the genealogy of a sample from the infinite-alleles model of population genetics. This coalescent also records the age ordering of alleles in the sample. The distribution of this process is found explicitly for the Moran model, and is shown to be robust for a wide class of reproductive schemes.

Properties of the ages themselves and the relationship between ages and class sizes then follow readily.

Keywords

GENEALOGYINFINITE-ALLELES MODELEWENS SAMPLING FORMULA
Type
Research Article
Information
Advances in Applied Probability , Volume 18 , Issue 1 , March 1986 , pp. 1 - 19
Copyright
Copyright © Applied Probability Trust 1986 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abramowitz, M. and Stegun, L. A. (1972) Handbook of Mathematical Functions. Dover, New York.Google Scholar
Cannings, C. (1974) The latent roots of certain Markov chains arising in genetics: A new approach. I. Haploid models. Adv. Appl. Prob. 6, 260–290.CrossRefGoogle Scholar
Crow, J. F. (1972) The dilemma of nearly neutral mutations; how important are they for evolution and human welfare. J. Hered. 63, 306–316.CrossRefGoogle ScholarPubMed
Donnelly, P. J. (1984) The transient behavior of the Moran model in population genetics. Math. Proc. Camb. Phil. Soc. 95, 349–358.CrossRefGoogle Scholar
Donnelly, P. J. (1985) Dual processes and an invariance result for exchangeable models in population genetics. J. Math. Biol. CrossRefGoogle Scholar
Engen, S. (1975) A note on the geometric series as a species frequency model. Biometrika 62, 694–699.CrossRefGoogle Scholar
Ewens, W. J. (1972) The sampling theory of selectively neutral alleles. Theoret. Popn Biol. 3, 87–112.CrossRefGoogle ScholarPubMed
Griffiths, R. C. (1980) Lines of descent in the diffusion approximation of neutral Wright-Fisher models. Theoret. Popn Biol. 17, 37–50.CrossRefGoogle ScholarPubMed
Kelly, F. P. (1977) Exact results for the Moran neutral allele model. Adv. Appl. Prob. 9, 197–201.CrossRefGoogle Scholar
Kelly, F. P. (1979) Reversibility and Stochastic Networks. Wiley, New York.Google Scholar
Kingman, J. F. C. (1975) Random discrete distributions. J. R. Statist. Soc. B37, 1–22.Google Scholar
Kingman, J. F. C. (1980) Mathematics of Genetic Diversity. CBMS-NSF Regional Conference in Applied Mathematics 34, Society for Industrial and Applied Mathematics, Philadelphia.CrossRefGoogle Scholar
Kingman, J. F. C. (1982a) On the genealogy of large populations. J. Appl. Prob. 19A, 27–43.CrossRefGoogle Scholar
Kingman, J. F. C. (1982b) The coalescent. Stoch. Proc. Appl. 13, 235–248.CrossRefGoogle Scholar
Kingman, J. F. C. (1982c) Exchangeability and the evolution of large populations. In Exchangeability in Probability and Statistics , ed. Koch, G. and Spizzichino, F., North-Holland, Amsterdam, 97–112.Google Scholar
Patil, G. P. and Taillie, C. (1977) Diversity as a concept and its implications for random communities. Bull. Internat. Stat. Inst. 47, 497–515.Google Scholar
Saunders, I. W., Tavare, S., and Watterson, G. A. (1984) On the genealogy of nested subsamples from a haploid population. Adv. Appl. Prob. 16, 471–491.CrossRefGoogle Scholar
Sawyer, S. (1977) On the past history of an allele now known to have frequency p. J. Appl. Prob. 14, 439–450.CrossRefGoogle Scholar
Sawyer, S. and Hartl, D. (1984) A sampling theory for local selection. J. Genet. Google Scholar
Serfling, R. J. (1980) Approximation Theorems of Mathematical Statistics. Wiley, New York.CrossRefGoogle Scholar
Tavaré, S. (1984) Line-of-descent and genealogical processes and their applications in population genetics models. Theoret. Popn Biol. 26, 119–164.CrossRefGoogle ScholarPubMed
Watterson, G. A. (1976a) Reversibility and the age of an allele. I. Moran's infinitely many neutral alleles model. Theoret. Popn Biol. 10, 239–253.CrossRefGoogle ScholarPubMed
Watterson, G. A. (1976b) The stationary distribution of the infinitely many neutral alleles diffusion model. J. Appl. Prob. 13, 639–651.CrossRefGoogle Scholar
Watterson, G. A. (1984) Lines of descent and the coalescent. Theoret. Popn Biol. 26, 77–92.CrossRefGoogle Scholar
Watterson, G. A. and Guess, H. A. (1977) Is the most frequent allele the oldest? Theoret. Popn Biol. 11, 141–160.CrossRefGoogle ScholarPubMed