Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-11T05:26:20.205Z Has data issue: false hasContentIssue false
  • English
  • Français

The infinitely-many-neutral-alleles diffusion model by ages

Published online by Cambridge University Press:  01 July 2016

S. N. Ethier
S. N. Ethier*
Affiliation:
University of Utah
*
∗Postal address: Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA. E-mail address: ma.ethier@science.utah.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We discuss two formulations of the infinitely-many-neutral-alleles diffusion model that can be used to study the ages of alleles. The first one, which was introduced elsewhere, assumes values in the set of probability distributions on the set of alleles, and the ages of the alleles can be inferred from its sample paths. We illustrate this approach by proving a result of Watterson and Guess regarding the probability that the most frequent allele is oldest. The second diffusion model, which is new, assumes values in the set of probability distributions on the set of pairs (x, a), where x is an allele and a is its age. We illustrate this second approach by proving an extension of the Ewens sampling formula to age-ordered samples due to Donnelly and Tavaré.

Keywords

MEASURE-VALUED DIFFUSIONERGODICITYSTATIONARITYREVERSIBILITYEWENS'SAMPLING FORMULAAGES OF ALLELES
Type
Research Article
Information
Advances in Applied Probability , Volume 22 , Issue 1 , March 1990 , pp. 1 - 24
Copyright
Copyright © Applied Probability Trust 1990 

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.)

Footnotes

Research supported in part by NSF grant DMS-8704369.

References

Donnelly, P. and Tavare, S. (1986) The ages of alleles and a coalescent. Adv. Appl. Prob. 18, 1–19. Correction 18, 1023.Google Scholar
Donnelly, P. 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. (1979) Limit theorems for absorption times of genetic models. Ann. Prob. 7, 622–638.Google Scholar
Ethier, S. N. (1981) A class of infinite-dimensional diffusions occurring in population genetics. Indiana Univ. Math. J. 30, 925–935.Google Scholar
Ethier, S. N. and Griffiths, R. C. (1987) The infinitely-many-sites model as a measure-valued diffusion. Ann. Prob. 15, 515–545.Google Scholar
Ethier, S. N. and Griffiths, R. C. (1988) The two-locus infinitely-many-neutral-alleles diffusion model. Unpublished.Google Scholar
Ethier, S. N. and Kurtz, T. G. (1981) The infinitely-many-neutral-alleles diffusion model. Adv. Appl. Prob. 13, 429–452.Google Scholar
Ethier, S. N. and Kurtz, T. G. (1986) Markov Processes: Characterization and Convergence. Wiley, New York.Google Scholar
Ethier, S. N. and Kurtz, T. G. (1987) The infinitely-many-alleles model with selection as a measure-valued diffusion. Lecture Notes in Biomathematics 70, 72–86. Springer-Verlag, Berlin.Google Scholar
Ewens, W. J. (1972) The sampling theory of selectively neutral alleles. Theoret. Popn. Biol. 3, 87–112.CrossRefGoogle ScholarPubMed
Feller, W. (1971) An Introduction to Probability Theory and Its Applications, Vol. II. Wiley, New York.Google Scholar
Fleming, W. H. and Viot, M. (1979) Some measure-valued Markov processes in population genetics theory. Indiana Univ. Math. J. 28, 817–843.CrossRefGoogle Scholar
Freedman, D. (1971) Brownian Motion and Diffusion. Holden-Day, San Francisco.Google Scholar
Fukushima, M. and Stroock, D. (1986) Reversibility of solutions to martingale problems. Adv. Math. Suppl. Stud. 9, 107–123. Academic Press, Orlando, Fla. Google Scholar
Griffiths, R. C. (1979a) On the distribution of allele frequencies in a diffusion model. Theoret. Popn. Biol. 15, 140–158.Google Scholar
Griffiths, R. C. (1979b) A transition density expansion for a multi-allele diffusion model. Adv. Appl. Prob. 11, 310–325.Google Scholar
Hoppe, F. M. (1987) The sampling theory of neutral alleles and an urn model in population genetics. J. Math. Biol. 25, 123–159.Google Scholar
Kingman, J. F. C. (1977) The population structure associated with the Ewens sampling formula. Theoret. Popn. Biol. 11, 274–283.Google Scholar
Kurtz, T. G. (1981) Approximation of Population Processes. CBMS-NSF Regional Conference Series in Applied Mathematics, 36. SIAM, Philadelphia.CrossRefGoogle Scholar
Littler, R. A. and Good, A. J. (1978) Ages, extinction times, and first passage probabilities for a multiallele diffusion model with irreversible mutation. Theoret. Popn. Biol. 13, 214–225.Google Scholar
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.Google Scholar
Watterson, G. A. (1976b) The stationary distribution of the infinitely-many neutral alleles diffusion model. J. Appl. Prob. 13, 639–651.Google Scholar
Watterson, G. A. and Guess, H. A. (1977) Is the most frequent allele the oldest? Theoret. Popn. Biol. 11, 141–160.Google Scholar