title: Single-crossover recombination in discrete time.
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.
authors.affiliation: Technische Fakultät, Universität Bielefeld, Box 100131, 33501, Bielefeld, Germany. uvonwang@techfak.uni-bielefeld.de
authors.forename: Ute
authors.initials: U
authors.lastname: von Wangenheim
authors.forename: Ellen
authors.initials: E
authors.lastname: Baake
authors.forename: Michael
authors.initials: M
authors.lastname: Baake
mesh_terms: D003434:Crossing Over, Genetic; D005828:Genetics, Population; D006801:Humans; D008957:Models, Genetic; D011995:Recombination, Genetic