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Post-data inference of coalescence times and segregating-site distribution in a two-island model with symmetric migration

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

Prakash Gorroochurn
Prakash Gorroochurn*
Affiliation:
University of Mauritius
*
∗ Postal address: Department of Mathematics, University of Mauritius, Réduit, Mauritius. Email address: [email protected]
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Abstract

In this paper, we present the distribution of the coalescence time of two DNA sequences (or genes) subject to symmetric migration between two islands, and conditional on the observed number of segregating sites in the sequences. The distribution for the segregating-site pattern is also obtained. Some surprising results emerge when both genes are initially on the same island. First, the post-data mean coalescence time is shown to be dependent on the migration parameter, as opposed to the pre-data mean. Second, both the post-data density and expectation for the coalescence time are shown to converge, in the weak-migration limit, to the corresponding panmictic results, as opposed to the pre-data situation where there is convergence in the density but not in the expectation. Finally, it is shown that there is convergence in the weak-migration limit in the distribution of the number of segregating sites but not in the expectation and variance. Numerical and graphical results for samples of size greater than two are also presented.

Keywords

Structured coalescentsegregating sitestwo-island modelsymmetric migrationmonomorphic

MSC classification

Primary: 92D25: Population dynamics (general)
Secondary: 60J85: Applications of branching processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 33 , Issue 3 , September 2001 , pp. 600 - 616
Copyright
Copyright © Applied Probability Trust 2001 

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References

Bahlo, M. and Griffiths, R. C. (2000). Inference from gene trees in a subdivided population. Theoret. Popn Biol. 57, 79–95.Google Scholar
Beerli, P. and Felsenstein, J. (1999). Maximum likelihood estimation of migration rates and effective population numbers in two populations using a coalescent approach. Genetics 152, 763–773.Google Scholar
Donnelly, P. and Tavaré, S. (1995). Coalescents and genealogical structure under neutrality. Annual Rev. Genet. 29, 401–421.Google Scholar
Donnelly, P., Tavaré, S., Balding, D. J. and Griffiths, R. C. (1996). Estimating the age of the common ancestor of men from the ZFY intron. Science 272, 1357–1359.Google Scholar
Dorit, R. L., Akashi, H. and Gilbert, W. (1995). Absence of polymorphism at the ZFY locus on the human Y chromosome. Science 268, 1183–1185.Google Scholar
Ewens, W. J. (1989). Population genetics—the past and the future. In Mathematical and Statistical Problems of Evolutionary Theory, ed. Lessard, S.. Kluwer, Dordrecht, pp. 177–227.Google Scholar
Fu, Y. X. and Li, W. H. (1998). Coalescent theory and its applications in population genetics. In Statistics in Genetics, ed. Halloran, E.. Springer, Berlin, pp. 45–79.Google Scholar
Fu, Y. X. and Li, W. H. (1999). Coalescing into the 21st century: an overview and prospects of coalescent theory. Theoret. Popn Biol. 56, 1–10.Google Scholar
Gorroochurn, P. (1999). Population genetics in subdivided populations. , Monash University.Google Scholar
Griffiths, R. C. and Tavaré, S. (1994). Ancestral inference in population genetics. Statist. Sci. 9, 307–319.Google Scholar
Herbots, H. M. (1997). The structured coalescent. In Progress in Population Genetics and Human Evolution (IMA Vols Math. Appl. 87), eds Donnelly, P. and Tavaré, S.. Springer, New York, pp. 231–255.Google Scholar
Hudson, R. R. (1991). Gene genealogies and the coalescent process. In Oxford Surveys in Evolutionary Biology 7, eds Futuyma, D. and Antonovics, J.. Oxford University Press, pp. 1–44.Google Scholar
Kingman, J. F. C. (1982a). The coalescent. Stoch. Proc. Appl. 13, 235–248.Google Scholar
Kingman, J. F. C. (1982b). Exchangeability and the evolution of large populations. In Exchangeability in Probability and Statistics, eds Koch, G. and Spizzichino, F.. North-Holland, Amsterdam, pp. 97–112.Google Scholar
Kingman, J. F. C. (1982c). On the genealogy of large populations. Adv. Appl. Prob. 19A, 27–43.Google Scholar
Marjoram, P. and Donnelly, P. (1994). Pairwise comparisons of mitochondrial DNA sequences in subdivided populations and implications for early human evolution. Genetics 136, 673–683.Google Scholar
Nagylaki, T. (1980). The strong-migration limit in geographically structured populations. J. Math. Biol. 9, 101–114.Google Scholar
Nagylaki, T. (2000). Geographical invariance and the strong-migration limit in subdivided populations. J. Math. Biol. 41, 123–142.Google Scholar
Nath, H. B. and Griffiths, R. C. (1993). The coalescent in two colonies with symmetric migration. J. Math. Biol. 31, 841–852.Google Scholar
Notohara, M. (1990). The coalescent and the genealogical process in geographically structured populations. J. Math. Biol. 29, 59–75.Google Scholar
Notohara, M. (1993). The strong-migration limit for the genealogical process in geographically structured populations. J. Math. Biol. 31, 115–122.Google Scholar
Notohara, M. (1997). The number of segregating sites in a sample of DNA sequences from a geographically structured population. J. Math. Biol. 36, 188–200.Google Scholar
Slatkin, M. (1987). The average number of sites separating DNA sequences drawn from a subdivided population. Theoret. Popn Biol. 32, 42–49.Google Scholar
Slatkin, M. and Hudson, R. R. (1991). Pairwise comparison of mitochondrial DNA sequences in stable and exponentially growing populations. Genetics 129, 555–562.Google Scholar
Tajima, F. (1983). Evolutionary relationships of DNA sequences in finite populations. Genetics 105, 437–460.Google Scholar
Takahata, N. (1988). The coalescent in two partially isolated diffusion populations. Genet. Res. Camb. 52, 213–222.Google Scholar
Takahata, N. (1991). Genealogy of neutral genes and spreading of selected mutations in a geographically structured population. Genetics 129, 585–595.CrossRefGoogle Scholar
Tavaré, S., (1984). Line-of-descent and genealogical processes and their applications in population genetics. Theoret. Popn Biol. 26, 119–164.Google Scholar
Tavaré, S., Balding, D. J., Griffiths, R. C. and Donnelly, P. (1997). Inferring coalescence times from DNA sequence data. Genetics 145, 505–518.CrossRefGoogle ScholarPubMed
Wakeley, J. (1998). Segregating sites in Wright's island model. Theoret. Popn Biol. 53, 166–174.Google Scholar
Watterson, G. A. (1975). On the number of segregating sites in genetic models without recombination. Theoret. Popn Biol. 7, 256–276.Google Scholar
Wilkinson-Herbots, H. M. (1998). Genealogy and subpopulation differentiation under various models of population structure. J. Math. Biol. 37, 535–585.Google Scholar
Wright, S. (1931). Evolution in Mendelian populations. Genetics 16, 97–159.Google Scholar