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The mean number of alleles in multigene families

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

G. A. Watterson
G. A. Watterson*
Affiliation:
Monash University
*
∗Postal address: Department of Mathematics, Monash University, VIC 3168, Australia.
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Abstract

The paper considers a random sample of r chromosomes, each having n genes subject to intrachromosomal gene conversion, and mutation. The probability distribution and moments for the number of alleles present is investigated, when the number, k, of possible alleles at each locus, is either finite or infinite. Explicit formulas are given for the mean numbers of alleles on r = 1, 2, or 3 chromosomes, which simplify previously known results. For fixed r, in the infinitely-many-alleles case, the mean number increases asymptotically like r θ log (n) as n→∞, where θ is a mutation parameter. But results for large samples remain elusive.

Keywords

GENE CONVERSIONREVERSIBLE MARKOV PROCESS

MSC classification

Primary: 92D15: Problems related to evolution
Secondary: 60J85: Applications of branching processes
Type
Research Article
Information
Advances in Applied Probability , Volume 24 , Issue 1 , March 1992 , pp. 1 - 19
Copyright
Copyright © Applied Probability Trust 1992 

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References

Abramowitz, M. and Stegun, I. A. (1972) Handbook of Mathematical Functions. Dover, New York.Google Scholar
Donnelly, P. (1984) The transient behaviour of the Moran model in population genetics. Math. Proc. Camb. Phil. Soc. 95, 349–358.CrossRefGoogle Scholar
Ewens, W. J. (1972) The sampling theory of selectively neutral alleles. Theoret. Popn Biol. 3, 87–112.Google Scholar
Felsenstein, J. (1981) Evolutionary trees from DNA sequences: a maximum likelihood approach. J. Mol. Evol. 17, 368–376.Google Scholar
Griffiths, R. C. (1991) Two chromosomes with multigene families. Theoret. Popn Biol. 39, 263–272.Google Scholar
Griffiths, R. C. and Watterson, G. A. (1990) The number of alleles in multigene families. Theoret. Popn Biol. 37, 110–123.Google Scholar
Karlin, S. and Mcgregor, J. (1962) On a genetics model of Moran. Proc. Camb. Phil. Soc. 58, 299–311.Google Scholar
Kingman, J. F. C. (1982) The coalescent. Stoch. Proc. Appl. 13, 235–248.Google Scholar
Moran, P. A. P. (1958) Random processes in genetics. Proc. Camb. Phil. Soc. 54, 60–71.Google Scholar
Ohta, T. (1986) Actual number of alleles contained in a multigene family. Genet. Res. 48, 119–123.Google Scholar
Padmadisastra, S. (1990) Allele frequencies in multigene families. Theoret. Popn Biol. To appear.Google Scholar
Shimizu, A. (1987) Stationary distribution of a diffusion process taking values in probability distributions on the partitions. In Stochastic Models in Biology , ed. Kimura, M., Kallianpur, G. and Hida, T. Springer-Verlag, Berlin.Google Scholar
Watterson, G. A. (1989a) Allele frequencies in multigene families. I. Diffusion equation approach. Theoret. Popn Biol. 35, 142–160.CrossRefGoogle ScholarPubMed
Watterson, G. A. (1989b) Allele frequencies in multigene families. II. Coalescent approach. Theoret. Popn Biol. 35, 161–180.Google Scholar
Watterson, G. A. (1990) Alleles in multigene families. In Population Biology of Genes and Molecules , ed. Takahata, N. and Crow, J. F. Baifukan, Tokyo.Google Scholar