Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-23T05:10:21.915Z Has data issue: false hasContentIssue false
  • English
  • Français

The structured coalescent process with weak migration

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Morihiro Notohara
Morihiro Notohara*
Affiliation:
Kyushu University
*
∗ Postal address: Department of Biology, Faculty of Science, Kyushu University, Fukuoka 812-8581, Japan. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The aim of this paper is to study genealogical processes in a geographically structured population with weak migration. The coalescence time for sampled genes from different colonies diverges to infinity as the migration rates among colonies are close to zero. We investigate the moment generating functions of the coalescence time, the number of segregating sites and the number of allele types in sampled genes when there is low migration. Employing a perturbation method, we obtain a system of recurrence relations for the approximate solutions of these moment generating functions and solve them in some cases.

Keywords

Structured coalescentweak migrationperturbation

MSC classification

Primary: 92D10: Genetics
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 38 , Issue 1 , March 2001 , pp. 1 - 17
Copyright
Copyright © by the Applied Probability Trust 2001 

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

Donnelly, P. and Tavaré, S. (1995). Coalescents and genealogical structure under neutrality. Annual Rev. Genet. 29, 401–421.Google Scholar
Ewens, W. J. (1972). The sampling theory of selectively neutral alleles. Theoret. Popn Biol. 3, 87–112.Google Scholar
Fu, Y. X., and Li, W. H. (1993). Statistical tests of neutrality of mutations. Genetics 133, 693–709.Google Scholar
Herbots, H. M. (1997). The structured coalescent. In Progress in Population Genetics and Human Evolution, eds Donnelly, P. and Tavaré, S. Springer, New York, pp. 231–255.Google Scholar
Herbots, H. M. (1998). Genealogy and subpopulation differentiation under various models of population structure. J. Math. Biol. 37, 535–585.Google Scholar
Hudson, R. R. (1990). Gene genealogies and the coalescent process. In Oxford Surveys in Evolutionary Biology 7, eds Futuyma, D. J. and Antonovics, J. Oxford University Press, Oxford, pp. 1–44.Google Scholar
Kingman, J. F. C. (1982a). On the genealogy of large population. In Essays in Statistical Science (J. Appl. Prob. 19A), eds Gani, J. and Hannan, E. J. Applied Probability Trust, Sheffield, pp. 27–43.Google Scholar
Kingman, J. F. C. (1982b). The coalescent. Stoch. Proc. Appl. 13, 235–248.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 population. 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
Notohara, M. (2000). A perturbation method for the structured coalescent with strong migration. J. Appl. Prob. 37, 148–167.Google Scholar
Shiga, T. (1980). An interacting system in population genetics I and II. J. Math. Kyoto Univ. 20, 212–242 and 723–733.Google Scholar
Slatkin, M. (1991). Inbreeding coefficient and coalescent times. Genet. Res. 58, 167–175.Google Scholar
Tajima, F. (1983). Evolutionary relationship of DNA sequences in finite populations. Genetics 105, 437–460.Google Scholar
Tajima, F. (1989). DNA polymorphism in a subdivided population: the expected number of segregating sites in the two-subpopulation model. Genetics 123, 229–240.Google Scholar
Takahata, N. (1988). The coalescent in two partially isolated diffusion population. Genet. Res. 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.Google Scholar
Tavaré, S. (1984). Line-of-descent and genealogical processes, and their application in population genetics models. Theoret. Popn Biol. 26, 119–164.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. (1984). Lines of descent and the coalescent. Theoret. Popn Biol. 26, 77–92.Google Scholar