Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-05T02:01:02.318Z Has data issue: false hasContentIssue false
  • English
  • Français

Coalescence times and FST values in subdivided populations with symmetric structure

Part of: Graph theory Markov processes

Published online by Cambridge University Press:  01 July 2016

Hilde M. Wilkinson-Herbots
Hilde M. Wilkinson-Herbots*
Affiliation:
University College London
*
∗ Postal address: Department of Statistical Science, University College London, Gower Street, London WC1E 6BT, UK. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The structured coalescent is a continuous-time Markov chain which describes the genealogy of a sample of homologous genes from a subdivided population. Assuming this model, some results are proved relating to the genealogy of a pair of genes and the extent of subpopulation differentiation, which are valid under certain graph-theoretic symmetry and regularity conditions on the structure of the population. We first review and extend earlier results stating conditions under which the mean time since the most recent common ancestor of a pair of genes from any single subpopulation is independent of the migration rate and equal to that of two genes from an unstructured population of the same total size. Assuming the infinite alleles model of neutral mutation with a small mutation rate, we then prove a simple relationship between the migration rate and the value of Wright's coefficient FST for a pair of neighbouring subpopulations, which does not depend on the precise structure of the population provided that this is sufficiently symmetric.

Keywords

Structured coalescentgenealogysubpopulation differentiationF-statisticsgraph theoryMarkov chain

MSC classification

Primary: 92D10: Genetics
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces 05C90: Applications
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 35 , Issue 3 , September 2003 , pp. 665 - 690
Copyright
Copyright © Applied Probability Trust 2003 

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

Adel son-Velskii, G. M. (1969). Example of a graph without a transitive automorphism group. Soviet Math. Dokl. 10, 440–441.Google Scholar
Bahlo, M. and Griffiths, R. C. (2001). Coalescence time for two genes from a subdivided population. J. Math. Biol. 43, 397–410.CrossRefGoogle ScholarPubMed
Barton, N. H. and Wilson, I. (1995). Genealogies and geography. Phil. Trans. R. Soc. London B 349, 49–59.Google Scholar
Barton, N. H. and Wilson, I. (1996). Genealogies and geography. In New Uses for New Phylogenies, eds Harvey, P. H. et al., Oxford University Press, pp. 23–56.Google Scholar
Barton, N. H., Depaulis, F. and Etheridge, A. M. (2002). Neutral evolution in spatially continuous populations. Theoret. Pop. Biol. 61, 31–48.Google Scholar
Biggs, N. (1974). Algebraic Graph Theory. Cambridge University Press.CrossRefGoogle Scholar
Bollobás, B., (1979). Graph Theory. An Introductory Course (Graduate Texts Math. 63). Springer, New York.Google Scholar
Cannings, C. (1974). The latent roots of certain Markov chains arising in genetics: a new approach. I. Haploid models. 6, 260–290.Google Scholar
Crow, J. F. and Maruyama, T. (1971). The number of neutral alleles maintained in a finite, geographically structured population. Theoret. Pop. Biol. 2, 437–453.Google Scholar
Donnelly, P. and Tavaré, S. (1995). Coalescents and genealogical structure under neutrality. Annual Rev. Genet. 29, 401–421.CrossRefGoogle ScholarPubMed
Fu, Y. X. and Li, W. H. (1999). Minireview: coalescing into the 21st century: an overview and prospects of coalescent theory. Theoret. Pop. Biol. 56, 1–10.Google Scholar
Godsil, C. and Royle, G. (2001). Algebraic Graph Theory. Springer, New York.Google Scholar
Griffiths, R. C. and Tavaré, S. (1994a). Ancestral inference in population genetics. Statist. Sci. 9, 307–319.Google Scholar
Griffiths, R. C. and Tavaré, S. (1994b). Sampling theory for neutral alleles in a varying environment. Phil. Trans. R. Soc. London B 344, 403–410.Google Scholar
Griffiths, R. C. and Tavaré, S. (1994c). Simulating probability distributions in the coalescent. Theoret. Pop. Biol. 46, 131–159.Google Scholar
Herbots, H. M. (1994). Stochastic models in population genetics: genealogy and genetic differentiation in structured populations. Doctoral Thesis, University of London.Google Scholar
Herbots, H. M. (1997). The structured coalescent. In Progress in Population Genetics and Human Evolution (IMA Vol. Math. Appl. 87), eds Donnelly, P. and Tavaré, S., Springer, New York, pp. 231–255.Google Scholar
Hudson, R. R. (1990). Gene genealogies and the coalescent process. In Oxford Surveys in Evolutionary Biology, Vol. 7, eds Futuyma, D. J. and Antonovics, J., Oxford University Press, pp. 1–44.Google Scholar
Kingman, J. F. C. (1982a). 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. (1982b). On the genealogy of large populations. In Essays in Statistical Science (J. Appl. Prob. Spec. Vol. 19A), eds Gani, J. and Hannan, E. J., Applied Probability Trust, Sheffield, pp. 27–43.Google Scholar
Kingman, J. F. C. (1982c). The coalescent. Stoch. Process. Appl. 13, 235–248.Google Scholar
Li, W.-H. (1976). Distribution of nucleotide differences between two randomly chosen cistrons in a subdivided population: the finite island model. Theoret. Pop. Biol. 10, 303–308.Google Scholar
Nagylaki, T. (1980). The strong-migration limit in geographically structured populations. J. Math. Biol. 9, 101–114.CrossRefGoogle ScholarPubMed
Nagylaki, T. (1982). Geographical invariance in population genetics. J. Theoret. Biol. 99, 159–172.Google Scholar
Nagylaki, T. (1998). The expected number of heterozygous sites in a subdivided population. Genetics 149, 1599–1604.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.CrossRefGoogle ScholarPubMed
Nei, M. (1973). Analysis of gene diversity in subdivided populations. Proc. Nat. Acad. Sci. USA 70, 3321–3323.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
Rousset, F. (1996). Equilibrium values of measures of population subdivision for stepwise mutation processes. Genetics 142, 1357–1362.Google Scholar
Slatkin, M. (1985). Gene flow in natural populations. Annual Rev. Ecol. Systems 16, 393–430.CrossRefGoogle Scholar
Slatkin, M. (1987). The average number of sites separating DNA sequences drawn from a subdivided population. Theoret. Pop. Biol. 32, 42–49.Google Scholar
Slatkin, M. (1991). Inbreeding coefficients and coalescence times. Genetical Res. 58, 167–175.Google Scholar
Slatkin, M. (1993). Isolation by distance in equilibrium and non-equilibrium populations. Evolution 47, 264–279.Google Scholar
Slatkin, M. and Barton, N. H. (1989). A comparison of three indirect methods for estimating average levels of gene flow. Evolution 43, 1349–1368.CrossRefGoogle ScholarPubMed
Stephens, M. (2001). Inference under the coalescent. In Handbook of Statistical Genetics, eds Balding, D. J., Bishop, M. and Cannings, C., John Wiley, Chichester, pp. 213–238.Google Scholar
Strobeck, C. (1987). Average number of nucleotide differences in a sample from a single subpopulation: a test for population subdivision. Genetics 117, 149–153.Google Scholar
Takahata, N. (1988). The coalescent in two partially isolated diffusion populations. Genetical Res. 52, 213–222.CrossRefGoogle ScholarPubMed
Whitlock, M. C. and McCauley, D. E. (1999). Indirect measures of gene flow and migration: F_ ST neq 1 (4Nm + 1). Heredity 82, 117–125.Google Scholar
Wilkins, J. F. and Wakeley, J. (2002). The coalescent in a continuous, finite, linear population. Genetics 161, 873–888.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.CrossRefGoogle ScholarPubMed
Wright, S. (1951). The genetical structure of populations. Ann. Eugenics 15, 323–354.Google Scholar