Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-27T20:03:26.722Z Has data issue: false hasContentIssue false
  • English
  • Français

A transition density expansion for a multi-allele diffusion model

Published online by Cambridge University Press:  01 July 2016

R. C. Griffiths
R. C. Griffiths*
Affiliation:
Monash University
*
∗Postal address: Department of Mathematics, Monash University, Clayton, Victoria 3168, Australia.
Get access Rights & Permissions [Opens in a new window]

Abstract

An expansion in orthogonal polynomials is found for the transition density in a neutral multi-allele diffusion model where the mutation rates of allele types Ai → Aj are assumed to be uj(≥ O). The density is found when the mutation rate is positive for all allele types, and when some or all have zero mutation. The asymptotic conditional density is found for a mixture of positive and zero mutation rates.

The infinite alleles limit with equal mutation is studied. Eigenfunctions of the process are derived and the frequency spectrum found. An important result is that the first eigenfunction depends only on the homozygosity.

A density for the time to fixation with zero mutation is found for the K allele, and infinite alleles model.

Keywords

ALLELE FREQUENCYDIFFUSIONDIRICHLET DISTRIBUTIONORTHOGONAL POLYNOMIALSTRANSITION DENSITY
Type
Research Article
Information
Advances in Applied Probability , Volume 11 , Issue 2 , June 1979 , pp. 310 - 325
Copyright
Copyright © Applied Probability Trust 1979 

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, I. A. (1965) Handbook of Mathematical Functions. Dover, New York.Google Scholar
Crow, J. F. and Kimura, M. (1970) An Introduction to Population Genetics Theory. Harper and Row, New York.Google Scholar
Ewens, W. J. and Kirby, K. (1975) The eigenvalues of the neutral alleles process. Theoret. Popn Biol. 7, 212–220.Google Scholar
Fackerell, E. D. and Littler, R. A. (1974) Polynomials biorthogonal to Appell's polynomials. Bull. Austral Math. Soc. 11, 181–195.CrossRefGoogle Scholar
Griffiths, R. C. (1979a) On the distribution of allele frequencies in a diffusion model. Theoret. Popn Biol. 15.Google Scholar
Griffiths, R. C. (1979b) Exact sampling distributions from the infinite neutral alleles model. Adv. Appl. Prob. 11, 326–354.Google Scholar
Karlin, S. and Avni, H. (1975) Derivation of the eigenvalues of the configuration process induced by a labelled direct product branching process. Theoret. Popn Biol. 7, 221–228.CrossRefGoogle ScholarPubMed
Karlin, S. and McGregor, J. (1964) On some stochastic models in genetics. In Stochastic Models in Medicine and Biology, ed. Gurland, J. University of Wisconsin Press, Madison.Google Scholar
Keilson, J. (1965) A review of transient behavior in regular diffusion and birth-death processes II. J. Appl. Prob. 2, 405–428.Google Scholar
Kimura, M. (1956) Random genetic drift in a tri-allelic locus; exact solution with a continuous model. Biometrics 12, 57–66.CrossRefGoogle Scholar
Kimura, M. (1964) Diffusion models in population genetics. J. Appl. Prob. 1, 177–232.CrossRefGoogle Scholar
Kingman, J. F. C. (1975) Random discrete distributions. J. R. Statist. Soc. B37, 1–15.Google Scholar
Kingman, J. F. C. (1977) The population structure associated with the Ewens sampling formula. Theoret. Popn Biol. 11, 274–283.Google Scholar
Li, W. (1977) Maintenance of genetic variability under mutation and selection pressures in a finite population. Proc. Natn. Acad. Sci. U.S.A. 74, 2509–2513.Google Scholar
Littler, R. A. (1975) Loss of variability in a finite population. Math. Biosci. 25, 151–163.CrossRefGoogle Scholar
Littler, R. A. and Fackerell, E. D. (1975) Transition densities for neutral multi-allele diffusion models. Biometrics 31, 117–123.CrossRefGoogle ScholarPubMed
Nei, M. and Li, W. (1976) The transient distribution of allele frequencies under mutation pressure. Genet. Res. Cambridge 28, 205–214.CrossRefGoogle ScholarPubMed
Watterson, G. A. (1976) The stationary distribution of the infinitely-many neutral alleles diffusion model. J. Appl. Prob. 13, 639–651.CrossRefGoogle Scholar
Watterson, G. A. (1977) Heterosis or neutrality? Genetics 85, 789–814.CrossRefGoogle ScholarPubMed