Single-crossover recombination in discrete time.

*(English)*Zbl 1208.92050Summary: Modelling the process of recombination leads to a large coupled nonlinear dynamical system. 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 [M. Baake and E. Baake, Can. J. Math. 55, No. 1, 3–41 (2003; Zbl 1056.92040)], this no longer works for discrete time. A more general model (i.e., without the restriction to single crossovers) has been studied before [J. H. Bennett, Ann. Hum. Genet. 18, 311–317 (1954); K. J. Dawson, Theor. Popul. Biol. 58, No. 1, 1–20 (2000; Zbl 1011.92038); Linear Algebra Appl. 348, No. 1–3, 115–137 (2002; Zbl 1003.92023)], and was solved algorithmically by means of Haldane linearisation. Using the special formalism introduced by Baake and Baake, 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.

##### MSC:

92D10 | Genetics and epigenetics |

39A60 | Applications of difference equations |

37N25 | Dynamical systems in biology |

60J20 | Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) |

06A07 | Combinatorics of partially ordered sets |

##### Keywords:

Population genetics; Recombination dynamics; Möbius linearisation; Diagonalisation; Linkage disequilibria
PDF
BibTeX
XML
Cite

\textit{U. von Wangenheim} et al., J. Math. Biol. 60, No. 5, 727--760 (2010; Zbl 1208.92050)

**OpenURL**

##### References:

[1] | Aigner M (1979) Combinatorial theory. Springer, Berlin · Zbl 0415.05001 |

[2] | Baake M (2005) Recombination semigroups on measure spaces. Monatsh Math 146:267–278 (2005) and 150:83–84 (2007) (Addendum) |

[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) · Zbl 1056.92040 |

[4] | Baake E, Herms I (2008) Single-crossover dynamics: finite versus infinite populations. Bull Math Biol 70: 603–624 · Zbl 1139.92018 |

[5] | Bennett JH (1954) On the theory of random mating. Ann Hum Genet 18: 311–317 |

[6] | Bürger R (2000) The mathematical theory of selection, recombination and mutation. Wiley, Chichester · Zbl 0959.92018 |

[7] | Christiansen FB (1999) Population genetics of multiple loci. Wiley, Chichester · Zbl 0941.92019 |

[8] | Cohn DL (1980) Measure theory. Birkhäuser, Boston |

[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: 1–20 · Zbl 1011.92038 |

[10] | Dawson KJ (2002) The evolution of a population under recombination: how to linearise the dynamics. Linear Algebra Appl 348: 115–137 · Zbl 1003.92023 |

[11] | Geiringer H (1944) On the probability theory of linkage in Mendelian heredity. Ann Math Stat 15: 25–57 · Zbl 0063.01560 |

[12] | Hartl DL, Clark AG (1997) Principles of population genetic, 3rd edn. Sinauer, Sunderland |

[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: 97–154 |

[14] | Lyubich YI (1992) Mathematical structures in population genetics. Springer, Berlin · Zbl 0747.92019 |

[15] | McHale D, Ringwood G A (1983) Haldane linearisation of baric algebras. J Lond Math Soc (2) 28: 17–26 · Zbl 0515.17010 |

[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 308–319 |

[17] | Robbins RB (1918) Some applications of mathematics to breeding problems III. Genetics 3: 375–389 |

[18] | von Wangenheim U (2007) Diskrete Rekombinationsdynamik. Diplomarbeit, Universität Greifswald |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.