Journal of Mathematical Biology

, Volume 48, Issue 3, pp 275–292 | Cite as

The two-locus ancestral graph in a subdivided population: convergence as the number of demes grows in the island model



We study the ancestral recombination graph for a pair of sites in a geographically structured population. In particular, we consider the limiting behavior of the graph, under Wright’s island model, as the number of subpopulations, or demes, goes to infinity. After an instantaneous sample-size adjustment, the graph becomes identical to the two-locus graph in an unstructured population, but with a time scale that depends on the migration rate and the deme size. Interestingly, when migration is gametic, this rescaling of time increases the population mutation rate but does not affect the population recombination rate. We compare this to the case of a partially-selfing population, in which both mutation and recombination depend on the selfing rate. Our result for gametic migration holds both for finite-sized demes, and in the limit as the deme size goes to infinity. However, when migration occurs during the diploid phase of the life cycle and demes are finite in size, the population recombination rate does depend on the migration rate, in a way that is reminiscent of partial selfing. Simulations imply that convergence to a rescaled panmictic ancestral recombination graph occurs for any number of sites as the number of demes approaches infinity.


Coalescent Island model Migration Recombination 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Begun and Aquadro1992 BegunAndAquadro92Begun, D.J., Aquadro, C.F.: Levels of naturally occurring polymorphism correlate with recombination rate in Drosophila melanogaster. Nature 356, 519–520 (1992)PubMedGoogle Scholar
  2. 2.
    Fearnhead, P., Donnelly, P.: Estimating recombination rates from population genetic data. Genetics 159, 1299–1318 (2001)Google Scholar
  3. 3.
    Fisher, R.A.: The Genetical Theory of Natural Selection. Clarendon, Oxford, 1930Google Scholar
  4. 4.
    Golding, G.B., Strobeck, C.: Linkage disequilibrium in a finite population that is partially selfing. Genetics 94, 777–789 (1980)MathSciNetGoogle Scholar
  5. 5.
    Griffiths, R.C., Marjoram, P.: Ancestral inference from samples of DNA sequences with recombination. J. Comp. Biol. 3, 479–502 (1996)Google Scholar
  6. 6.
    Hey, J., Wakeley, J.: A coalescent estimator of the population recombination rate. Genetics 145, 833–846 (1997)PubMedGoogle Scholar
  7. 7.
    Hill, W.G., Robertson, A.R.: Linkage disequilibrium in finite populations. Theoret. Appl. Genet. 38, 226–231 (1968)Google Scholar
  8. 8.
    Hudson, R.R.: Properties of a neutral allele model with intragenic recombination. Theoret. Pop. Biol. 23, 183–201 (1983)MATHGoogle Scholar
  9. 9.
    Hudson, R.R.: Estimating the recombination parameter of a finite population model without selection. Genet. Res. Camb. 50, 245–250 (1987)Google Scholar
  10. 10.
    Hudson, R.R.: Two-locus sampling distributions and their application. Genetics 159, 1805–1817 (2001)Google Scholar
  11. 11.
    Hudson, R.R., Kaplan, N.L.: Statistical properties of the number of recombination events in the history of a sample of DNA sequences. Genetics 111, 147–164 (1985)PubMedGoogle Scholar
  12. 12.
    Hudson, R.R., Kaplan, N.L.: The coalescent process in models with selection and recombination. Genetics 120, 831–840 (1988)PubMedGoogle Scholar
  13. 13.
    Hudson, R.R., Slatkin, M., Maddison, W.P.: Estimation of levels of gene flow from DNA sequence data. Genetics 132, 583–589 (1992)Google Scholar
  14. 14.
    Jorde, L.B.: Linkage disequilibrium as a gene mapping tool. Am. J. Hum. Genet. 56, 11–14 (1995)Google Scholar
  15. 15.
    Kaplan, N.L., Darden, T., Hudson, R.R.: Coalescent process in models with selection. Genetics 120, 819–829 (1988)Google Scholar
  16. 16.
    Kaplan, N.L., Hudson, R.R., Iizuka, M.: Coalescent processes in models with selection, recombination and geographic subdivision. Genet. Res. Camb. 57, 83–91 (1991)Google Scholar
  17. 17.
    Kingman, J.F.C.: The coalescent. Stochastic Process. Appl. 13, 235–248 (1982)CrossRefMATHGoogle Scholar
  18. 18.
    Kruglyak, L.: Prospects for whole genome linkage disequilibrium mapping of common diseases. Nature Genetics 22, 139–144 (1999)CrossRefGoogle Scholar
  19. 19.
    Lander, E.S.: The new genomics: global views of biology. Sci. 274, 536–539 (1996)CrossRefGoogle Scholar
  20. 20.
    Latter, B.D.H.: The island model of population differentiation: a general solution. Genetics 73, 147–157 (1973)Google Scholar
  21. 21.
    Lewontin, R.C., Kojima, K.: The evolutionary dynamics of complex polymorphisms. Evolution 14, 450–472 (1960)Google Scholar
  22. 22.
    Maruyama, T.: Effective number of alleles in a subdivided population. Theoret. Pop. Biol. 1, 273–306 (1970)MATHGoogle Scholar
  23. 23.
    Möhle, M.: A convergence theorem for Markov chains arising in population genetics and the coalescent with partial selfing. Adv. Appl. Prob. 30, 493–512 (1998)CrossRefGoogle Scholar
  24. 24.
    Moran, P.A.P.: The theory of some genetical effects of population subdivision. Austr. J. Biol. Sci. 12, 109–116 (1959)MATHGoogle Scholar
  25. 25.
    Nachman, M.W.: Single nucleotide polymorphisms and recombination rate in humans. Trends in Genetics 17, 481–485 (2001)CrossRefGoogle Scholar
  26. 26.
    Nordborg, M.: Linkage disequilibrium, gene trees and selfing: an ancestral recombination graph with partial selfing. Genetics 154, 923–929 (2000)PubMedGoogle Scholar
  27. 27.
    Nordborg, M.: Coalescent theory. In: Handbook of Statistical Genetics, D. J. Balding, M. J. Bishop, C. Cannings, (eds.), John Wiley & Sons, Chichester, England, 2001Google Scholar
  28. 28.
    Nordborg, M., Donnelly, P.: The coalescent process with selfing. Genetics 146, 1185–1195 (1997)Google Scholar
  29. 29.
    Nordborg, M., Krone, S.M.: Separation of time scales and convergence to the coalescent in structured populations. In: Modern Developments in Theoretical Population Genetics, M. Slatkin, M. Veuille, (eds.), Oxford University Press, Oxford, UK, 2002Google Scholar
  30. 30.
    Pollack, E.: On the theory of partially inbreeding finite populations. I. Partial selfing. Genetics 117, 353–360 (1987)Google Scholar
  31. 31.
    Reich, D.E., Cargill, M., Bolk, S., Ireland, J., Sabeti, P.C., Richter, D.J., Lavery, T., Farhadian, S.F., Ward, R., Lander, E.S.: Linkage disequilibrium in the human genome. Nature 411, 199–204 (2001)CrossRefPubMedGoogle Scholar
  32. 32.
    Risch, N., Merikangas, K.: The future of genetic studies of complex human diseases. Sci. 273, 1516–1517 (1996)Google Scholar
  33. 33.
    Wakeley, J.: Segregating sites in Wright’s island model. Theoret. Pop. Biol. 53, 166–175 (1998)CrossRefMATHGoogle Scholar
  34. 34.
    Wakeley, J.: Non-equilibrium migration in human history. Genetics 153, 1863–1871 (1999)Google Scholar
  35. 35.
    Wakeley, J.: The coalescent in an island model of population subdivision with variation among demes. Theoret. Pop. Biol. 59, 133–144 (2001)CrossRefGoogle Scholar
  36. 36.
    Wakeley, J., Lessard, S.: Theory of the effects of population structure and sampling on patterns of linkage disequilibrium applied to genomic data from humans. Genetics 164, 1043–1053 (2003)Google Scholar
  37. 37.
    Wall, J.D.: A comparison of estimators of the population recombination rate. Mol. Biol. Evol. 17, 156–163 (1999)Google Scholar
  38. 38.
    Wilkinson-Herbots, H.M.: Genealogy and subpopulation differentiation under various models of population structure. J. Math. Biol. 37, 535–585 (1998)CrossRefMATHGoogle Scholar
  39. 39.
    Wright, S.: Evolution in Mendelian populations. Genetics 16, 97–159 (1931)Google Scholar
  40. 40.
    Wright, S.: Isolation by distance. Genetics 28, 114–138 (1943)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  1. 1.Département de mathématiques et de statistique
  2. 2.Department of Organismic and Evolutionary Biology

Personalised recommendations