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Exact sampling distributions from the infinite neutral alleles 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.
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Abstract

The exact transient sampling distribution is found for the infinite neutral alleles model with mutation, thus extending the stationary distribution found by Ewens (1972). The first eigenfunction in the sampling distribution depends only on the homozygosity. In a large sample the expected number of allele types is close to the expected number in a stationary distribution and depends little on the initial frequencies or the time of sampling. The expected number of types in a sample for a non-stationary population is tabulated.

The sampling distribution from a population with no mutation is found. The probability that the population is monomorphic given that the sample contains only one type is tabulated, where initially the population has a large number of alleles with equal frequencies.

A study is made of the joint distribution of the allele frequencies in two samples taken time t apart from a stationary population. Of particular interest is the number of allele types in common in two such samples. The distribution of the number of types in common in a population viewed at time t apart is also found and tabulated for different mutation rates and time. The expected frequency of the allele types in common is tabulated.

The distribution of allele frequencies in two divergent populations of common ancestry is shown to be the same as the distribution of frequencies in a single population at two time points 2t apart. A sufficient statistic for the time of divergence is shown to be the paired frequencies of the allele types in common. The expected waiting time until the populations have no allele types in common is tabulated.

Keywords

ALLELE FREQUENCIESDIFFUSIONGENETIC DISTANCENEUTRAL INFINITE ALLELES MODELSAMPLING DISTRIBUTIONTRANSITION DENSITY
Type
Research Article
Information
Advances in Applied Probability , Volume 11 , Issue 2 , June 1979 , pp. 326 - 354
Copyright
Copyright © Applied Probability Trust 1979 

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References

Crow, J. F. and Kimura, M. (1956) Some genetic problems in natural populations. Proc. 3rd Berkeley Symp. Math. Statist. Prob., 1–22.CrossRefGoogle Scholar
Crow, J. F. and Kimura, M. (1970) An Introduction to Population Genetics Theory. Haper and Row, New York.Google Scholar
Ewens, W. J. (1972) The sampling theory of selectively neutral allelecs. Theoret. Popn Biol 3, 87–112.Google Scholar
Griffiths, R. C. (1979a) On the distribution of allele frequencies in a diffusion model. Theoret. Popn Biol. 15.CrossRefGoogle Scholar
Griffiths, R. C. (1979b) A transition density expansion for a multi-allele diffusion model. Adv. Appl. Prob. 11, 310–325.Google Scholar
Karlin, S. and McGregor, J. (1967) The number of mutant forms maintained in a population. Proc. 5th Berkeley Symp. Math. Statist. Prob. 4, 415–438.Google Scholar
Karlin, S. and McGregor, J. (1972) Addendum to a paper of W. Ewens. Theoret. Popn Biol. 3, 113–116.CrossRefGoogle ScholarPubMed
Kelly, F. P. (1976) On stochastic population models in genetics. J. Appl. Prob. 13, 127–131.Google Scholar
Kelly, F. P. (1977) Exact results for the Moran neutral allele model (abstract). Adv. Appl. Prob. 9, 197–201.Google Scholar
Kimura, M. (1955) Random genetic drift in a multi-allelic locus. Evolution 9, 419–435.CrossRefGoogle Scholar
Kimura, M. (1956) Random genetic drift in a tri-allelic locus; exact solution with a continuous model. Biometrics 12, 57–66.Google 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. H. and Nei, M. (1977) Persistence of common alleles in two related populations or species. Genetics 86, 901–914.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.Google Scholar
Nei, M. (1976) Mathematical models of speciation and genetic distance. In Population Genetics and Ecology, ed. Karlin, S. and Nevo, E. Academic Press, New York.Google Scholar
Trajstman, A. C. (1974) On a conjective of G. A. Watterson. Adv. Appl. Prob. 6, 489–493.Google Scholar
Watterson, G. A. (1974) The sampling theory of selectively neutral alleles. Adv. Appl. Prob. 6, 463–488.Google Scholar
Watterson, G. A. (1975) On the number of segregating sites in genetical models without recombination. Theoret. Popn Biol. 7, 256–276.Google Scholar
Watterson, G. A. (1976) The stationary distribution of the infinitely many 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