an:05815237
Zbl 1208.92050
von Wangenheim, Ute; Baake, Ellen; Baake, Michael
Single-crossover recombination in discrete time
EN
J. Math. Biol. 60, No. 5, 727-760 (2010).
0303-6812 1432-1416
2010
j
92D10 39A60 37N25 60J20 06A07
Population genetics; Recombination dynamics; MÃ¶bius linearisation; Diagonalisation; Linkage disequilibria
Summary: 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 [\textit{M. Baake} and \textit{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 [\textit{J. H. Bennett}, Ann. Hum. Genet. 18, 311--317 (1954); \textit{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.
1056.92040; 1011.92038; 1003.92023