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Journal of Mathematical Biology

, Volume 60, Issue 5, pp 727–760 | Cite as

Single-crossover recombination in discrete time

  • Ute von Wangenheim
  • Ellen BaakeEmail author
  • Michael Baake
Article

Abstract

Modelling the process of recombination leads to a large coupled nonlinear dynamical system. Here, we consider a particular case of recombination in discrete time, allowing only for single crossovers. While the analogous dynamics in continuous time admits a closed solution (Baake and Baake in Can J Math 55:3–41, 2003), this no longer works for discrete time. A more general model (i.e. without the restriction to single crossovers) has been studied before (Bennett in Ann Hum Genet 18:311–317, 1954; Dawson in Theor Popul Biol 58:1–20, 2000; Linear Algebra Appl 348:115–137, 2002) and was solved algorithmically by means of Haldane linearisation. Using the special formalism introduced by Baake and Baake (Can J Math 55:3–41, 2003), we obtain further insight into the single-crossover dynamics and the particular difficulties that arise in discrete time. We then transform the equations to a solvable system in a two-step procedure: linearisation followed by diagonalisation. Still, the coefficients of the second step must be determined in a recursive manner, but once this is done for a given system, they allow for an explicit solution valid for all times.

Keywords

Population genetics Recombination dynamics Möbius linearisation Diagonalisation Linkage disequilibria 

Mathematics Subject Classification (2000)

92D10 37N30 06A07 60J05 

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References

  1. Aigner M (1979) Combinatorial theory. Springer, BerlinzbMATHGoogle Scholar
  2. Baake M (2005) Recombination semigroups on measure spaces. Monatsh Math 146:267–278 (2005) and 150:83–84 (2007) (Addendum)Google Scholar
  3. Baake M, Baake E (2003) An exactly solved model for mutation, recombination and selection. Can J Math 55:3–41 (2003) and 60:264–265 (2008) (Erratum)Google Scholar
  4. Baake E, Herms I (2008) Single-crossover dynamics: finite versus infinite populations. Bull Math Biol 70: 603–624zbMATHCrossRefMathSciNetGoogle Scholar
  5. Bennett JH (1954) On the theory of random mating. Ann Hum Genet 18: 311–317Google Scholar
  6. Bürger R (2000) The mathematical theory of selection, recombination and mutation. Wiley, ChichesterzbMATHGoogle Scholar
  7. Christiansen FB (1999) Population genetics of multiple loci. Wiley, ChichesterzbMATHGoogle Scholar
  8. Cohn DL (1980) Measure theory. Birkhäuser, BostonzbMATHGoogle Scholar
  9. Dawson KJ (2000) The decay of linkage disequilibria under random union of gametes: How to calculate Bennett’s principal components. Theor Popul Biol 58: 1–20zbMATHCrossRefGoogle Scholar
  10. Dawson KJ (2002) The evolution of a population under recombination: how to linearise the dynamics. Linear Algebra Appl 348: 115–137zbMATHCrossRefMathSciNetGoogle Scholar
  11. Geiringer H (1944) On the probability theory of linkage in Mendelian heredity. Ann Math Stat 15: 25–57zbMATHCrossRefMathSciNetGoogle Scholar
  12. Hartl DL, Clark AG (1997) Principles of population genetic, 3rd edn. Sinauer, SunderlandGoogle Scholar
  13. Jennings HS (1917) The numerical results of diverse systems of breeding, with respect to two pairs of characters, linked or independent, with special relation to the effects of linkage. Genetics 2: 97–154Google Scholar
  14. Lyubich YI (1992) Mathematical structures in population genetics. Springer, BerlinzbMATHGoogle Scholar
  15. McHale D, Ringwood G A (1983) Haldane linearisation of baric algebras. J Lond Math Soc (2) 28: 17–26zbMATHCrossRefGoogle Scholar
  16. Popa E (2007) Some remarks on a nonlinear semigroup acting on positive measures. In: Carja O, Vrabie II (eds) Applied analysis and differential equations. World Scientific, Singapore, pp 308–319Google Scholar
  17. Robbins RB (1918) Some applications of mathematics to breeding problems III. Genetics 3: 375–389Google Scholar
  18. von Wangenheim U (2007) Diskrete Rekombinationsdynamik. Diplomarbeit, Universität GreifswaldGoogle Scholar

Copyright information

© The Authors 2009

Authors and Affiliations

  • Ute von Wangenheim
    • 1
  • Ellen Baake
    • 1
    Email author
  • Michael Baake
    • 2
  1. 1.Technische FakultätUniversität BielefeldBielefeldGermany
  2. 2.Fakultät für MathematikUniversität BielefeldBielefeldGermany

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