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On the genealogy of nested subsamples from a haploid population

Published online by Cambridge University Press:  01 July 2016

Ian W. Saunders ,
Simon Tavaré  and
G. A. Watterson
Ian W. Saunders*
Affiliation:
CSIRO Division of Mathematics and Statistics
Simon Tavaré*
Affiliation:
Colorado State University
G. A. Watterson*
Affiliation:
Monash University
*
∗Postal address: CSIRO Division of Mathematics and Statistics, Private Bag 10, Clayton, VIC 3168, Australia.
∗∗ Postal address: Department of Statistics, Colorado State University, Fort Collins, CO 80532, USA.
∗∗∗Postal address: Department of Mathematics, Monash University, Clayton, VIC 3168, Australia.
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Abstract

For the haploid genetic model of Moran, the joint distribution of the numbers of distinct ancestors of a collection of nested subsamples is derived. These results are shown to apply to the diffusion approximations of a wide variety of other genetic models, including the Wright–Fisher process. The results allow us to relate the ancestries of populations sampled at different times. Analogous results for a line-of-descent process that incorporates the effect of mutation are given. Some results about the ages of alleles in an infinite-alleles model are described.

Keywords

MORAN MODELWRIGHT–FISHER MODELINFINITE ALLELESAGE OF ALLELESLINES OF DESCENTDIFFUSION APPROXIMATION
Type
Research Article
Information
Advances in Applied Probability , Volume 16 , Issue 3 , September 1984 , pp. 471 - 491
Copyright
Copyright © Applied Probability Trust 1984 

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References

Abramowitz, M. and Stegun, I. A. (1972) Handbook of Mathematical Functions. Dover, New York.Google Scholar
Cannings, C. (1974) The latest roots of certain Markov chains arising in genetics: a new approach. I. Haploid models. Adv. Appl. Prob. 6, 260–290.CrossRefGoogle Scholar
Ewens, W. J. (1972) The sampling theory of selectively neutral alleles. Theoret. Popn Biol. 3, 87–112.Google Scholar
Felsenstein, J. (1971) The rate of loss of multiple alleles in finite haploid populations. Theoret. Popn Biol. 2, 391–403.Google Scholar
Gladstien, K. (1978) The characteristic values and vectors for a class of stochastic matrices arising in genetics. SIAM J. Appl. Math. 34, 630–642.Google Scholar
Griffiths, R. C. (1980) Lines of descent in the diffusion approximation of neutral Wright-Fisher models. Theoret. Popn Biol. 17, 37–50.Google Scholar
Kelly, F. P. (1977) Exact results for the Moran neutral allele model. Adv. Appl. Prob. 9, 197–201.Google Scholar
Kelly, F. P. (1979) Reversibility and Stochastic Networks. Wiley, New York.Google Scholar
Kingman, J. F. C. (1980) The mathematics of genetic diversity. CBMS-NSF Regional Conf. Ser. Appl. Math. 34.Google 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
Moran, P. A. P. (1958) Random processes in genetics. Proc. Camb. Phil. Soc. 54, 60–71.Google Scholar
Tavare, S. (1984) Line-of-descent and genealogical processes and their applications in population genetics models. Theoret. Popn Biol. Google Scholar
Watterson, G. A. (1975) On the number of segregating sites in genetical models without recombination. Theoret. Popn Biol. 7, 256–276.CrossRefGoogle ScholarPubMed
Watterson, G. A. (1982a) Substitution times for mutant nucleotides. J. Appl. Prob. 19A, 59–70.Google Scholar
Watterson, G. A. (1982b) Mutant substitutions at linked nucleotide sites. Adv. Appl. Prob. 14, 206–224.Google Scholar
Watterson, G. A. and Guess, H. A. (1977) Is the most frequent allele the oldest? Theoret. Popn Biol. 11, 141–160.CrossRefGoogle ScholarPubMed