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Linked selected and neutral loci in heterogeneous environments

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Abstract

We analyze a system of ordinary differential equations modeling haplotype frequencies at a physically linked pair of loci, one selected and one neutral, in a population consisting of two demes with divergent selection regimes. The system is singularly perturbed, with the migration rate m between the demes serving as a small parameter. We use geometric singular perturbation theory to show that when m is sufficiently small, each solution not initially fixed for the same selected allele in both demes approaches one of a 1-dimensional continuum of equilibria. We then obtain asymptotic expansions of the solutions and show their validity on arbitrarily long finite time intervals. From these expansions we obtain formulas for the transient dynamics of F ST (a measure of population structure) at both loci, as well as for the rate of genotyping error if the allelic state at the selected locus is inferred from that at the neutral (marker) locus. We examine two cases in detail, one modeling two populations in secondary contact after a period of evolution in allopatry, and the other modeling the origination and spread of a resistance allele.

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References

  1. Aguilar A., Roemer G., Debenham S., Binns M., Garcelon D., Wayne R.K. (2004) High MHC diversity maintained by balancing selection in an otherwise genetically monomorphic mammal. Proc. Natl. Acad. Sci. USA 101, 3490–3494

    Article  Google Scholar 

  2. Aulbach B. (1984) Continuous and discrete dynamics near manifolds of equilibria. Springer, Berlin Heidelberg New York

    Book  MATH  Google Scholar 

  3. Barton N.H. (2000) Genetic hitchhiking. Phil. Trans. R. Soc. Lond B 355, 1553–1562

    Article  Google Scholar 

  4. Bürger R. (2000) The mathematical theory of selection, recombination, and mutation. Wiley, Chichester, UK

    MATH  Google Scholar 

  5. Christiansen F.B. (1975) Hard and soft selection in a subdivided population. Am. Nat. 109, 11–16

    Article  Google Scholar 

  6. Crow J.F., Aoki K. (1984) Group selection for a polygenic behavioral trait: estimating the degree of population subdivision. PNAS 81, 6073–6077

    Article  MATH  Google Scholar 

  7. Fenichel N. (1971) Persistence and smoothness of invariant manifolds for flows. Indiana Univ. Math. J. 21, 193–226

    Article  MathSciNet  MATH  Google Scholar 

  8. Fenichel N. (1974a) Asymptotic stability with rate conditions. Indiana Univ. Math. J. 23, 1109–1137

    Article  MathSciNet  MATH  Google Scholar 

  9. Fenichel N. (1974b) Asymptotic stability with rate conditions II. Indiana Univ. Math. J. 26, 81–93

    Article  MathSciNet  MATH  Google Scholar 

  10. Fenichel N. (1979) Geometric singular perturbation theory for ordinary differential equations. J. Diff. Eqs. 31, 53–98

    Article  MathSciNet  MATH  Google Scholar 

  11. Gillespie J.H. (2004) Population genetics: a concise guide, 2nd edn. Johns Hopkins University Press, Baltimore

    Book  Google Scholar 

  12. Hartl D.L., Clark A.G. (1997) Principles of population genetics. Sinauer Associates, Sunderland, MA, USA

    Google Scholar 

  13. Haubruge E., Arnaud L. (2001) Fitness consequences of malathion-specific resistance in red flour beetle (Coleoptera: Tenebrionidae) and selection for resistance in the absence of malathion. J. Econ. Entomol. 94, 552–557

    Article  Google Scholar 

  14. Hawthorne, D.J.: Personal communication, 19 June (2006)

  15. Hawthorne D.J., Via S. (2001) Genetic linkage of ecological specialization and reproductive isolation in pea aphids. Nature 412, 904–907

    Article  Google Scholar 

  16. Hirsch M., Smale S. (1974) Differential equations and dynamical systems, and linear algebra. Academic, San Diego

    MATH  Google Scholar 

  17. Hoppensteadt F.C. (1966) Singular perturbations on the infinite interval. Trans. Am. Math. Soc. 123, 521–535

    Article  MathSciNet  MATH  Google Scholar 

  18. Hoppensteadt F.C. (1971) Properties of solutions of ordinary differential equations with small parameters. Comm. Pure Appl. Math. 24, 807–40

    Article  MathSciNet  MATH  Google Scholar 

  19. Jones C.K.R.T. (1995) Geometric singular perturbation theory. In: Johnson R. (eds) Dynamical Systems: Montecatini Terme, 1994. Springer, Berlin Heidelberg New York, pp. 44–118

    Chapter  Google Scholar 

  20. Kaplan N.L., Hudson R.R., Langley C.H. (1989) The “hitchhiking effect” revisited. Genetics 123, 887–899

    Article  Google Scholar 

  21. Karlin S. (1975) General two-locus selection models: some objectives, results and interpretations. Theor. Pop. Biol. 7, 364–398

    Article  MathSciNet  MATH  Google Scholar 

  22. Kawecki T.J., Ebert D. (2004) Conceptual issues in local adaptation. Ecol. Lett. 7, 1225–1241

    Article  Google Scholar 

  23. Maynard Smith J., Haigh J. (1974) The hitch-hiking effect of a favorable gene. Genet. Res. 23, 23–35

    Article  Google Scholar 

  24. Miller J.R., Hawthorne D.J. (2005) Durability of marker-quantitative trait loci haplotypes in structured populations. Genetics 171, 1353–1364

    Article  Google Scholar 

  25. Nagylaki T. (1974) Continuous selective models with mutation and migration. Theor. Popul. Biol. 5, 284–295

    Article  MATH  Google Scholar 

  26. Nagylaki T. (1992) Introduction to theoretical population genetics. Springer, Berlin Heidelberg New York

    Book  MATH  Google Scholar 

  27. Nagylaki T. (1999) Convergence of multilocus systems under weak epistasis or weak selection. J. Math. Biol. 38, 103–133

    Article  MathSciNet  MATH  Google Scholar 

  28. Nagylaki T., Crow J.R. (1974) Continuous selective models. Theor. Popul. Biol. 5, 257–283

    Article  MATH  Google Scholar 

  29. Nilsson A.I., Kugelberg E., Berg O.G., Andersson D.I. (2004) Experimental adaptation of Salmonella typhimurium to mice. Genetics 168, 1119–1130

    Article  Google Scholar 

  30. O’Malley R.E. (1991) Singular perturbation methods for ordinary differential equations. Springer, Berlin Heidelberg New York

    Book  MATH  Google Scholar 

  31. Pannell J.R., Charlesworth B. (1999) Neutral genetic diversity in a metapopulation with recurrent local extinction and recolonization. Evolution 53, 664–676

    Article  Google Scholar 

  32. Pannell J.R., Charlesworth B. (2000) Effects of metapopulation processes on measures of genetic diversity. Phil. Trans. R. Soc. Lond B 355, 1851–1864

    Article  Google Scholar 

  33. Santiago E., Caballero A. (2005) Variation after a selective sweep in a subdivided population. Genetics 169, 475–483

    Article  Google Scholar 

  34. Slatkin M., Wiehe T. (1998) Genetic hitch-hiking in a subdivided population. Genet. Res. 71, 155–160

    Article  Google Scholar 

  35. Smith D.S. (1985) Singular-perturbation theory. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  36. Stephan W., Wiehe T.H.E., Lenz M.W. (1992) The effect of strongly selected substitutions on neutral polymorphism: analytical results based on diffusion theory. Theor. Popul. Biol. 41, 237–254

    Article  MATH  Google Scholar 

  37. Via S. (1999) Reproductive isolation between sympatric races of pea aphids. I. Gene flow restriction and habitat choice. Evolution 53, 1446–1457

    Article  Google Scholar 

  38. Via S., Bouck A.C., Skillman S. (2000) Reproductive isolation between divergent races of pea aphids on two hosts. II. Selection against migrants and hybrids in the parental environments. Evolution 54, 1626–1637

    Google Scholar 

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Author notes

  1. B. P. Wood

    Present address: Raytheon Solipsys, 6100 Chevy Chase Dr. Suite 200, Laurel, MD, 20707, USA

Authors and Affiliations

  1. Department of Mathematics, Georgetown University, Washington, DC, 20057, USA

    B. P. Wood & J. R. Miller

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  1. B. P. Wood
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  2. J. R. Miller
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Correspondence to J. R. Miller.

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Wood, B.P., Miller, J.R. Linked selected and neutral loci in heterogeneous environments. J. Math. Biol. 53, 939–975 (2006). https://doi.org/10.1007/s00285-006-0038-6

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  • DOI: https://doi.org/10.1007/s00285-006-0038-6

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