Single-crossover recombination in discrete time

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• 作者：Ute von Wangenheim ; Ellen Baake and Michael Baake
• 关键词：Population genetics ; Recombination dynamics ; M?bius linearisation ; Diagonalisation ; Linkage disequilibria
• 刊名：Journal of Mathematical Biology
• 出版年：2010
• 出版时间：May, 2010
• 年：2010
• 卷：60
• 期：5
• 页码：727-760
• 全文大小：393.8 KB

文摘

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.