PUB - Publikationen an der Universität Bielefeld

Modeling the process of recombination in the deterministic limit of an infinite population leads to a large coupled nonlinear dynamical system that is notoriously difficult to treat and solve . In this thesis, a particular case of recombination in discrete time, allowing only for single-crossovers, is studied extensively for the first time. We elaborate the underlying mathematical structure of the discrete-time process by providing a systematic approach that exploits the inherent (multi)linear and combinatorial structure of the problem. We then develop two different approaches to state an explicit solution to the dynamics. In a first approach, we construct a transformation of the equations to a solvable system in a two-step procedure: first linearisation followed by diagonalisation. Even though the coefficients of the second step must be determined in a recursive manner, once this is done for a given system, they allow for an explicit solution of the system valid for all times. The second approach aims to infer an explicit solution to the dynamics that does not employ recursions and contributes to a better understanding of the recombination dynamics. We here use the underlying stochastic process for finite populations, a Wright-Fisher model with single-crossovers, to trace recombination backwards in time, i.e. by backtracking the ancestry of single individuals. In the limit of infinite population size, this results in binary tree structures, the ancestral recombination trees. The ancestry is then formulated explicitly in terms of a (stochastic) segmentation process, which involves conditional independence between segments once they have occurred. As a consequence, the time evolution of the ancestral process can be calculated explicitly by assigning probabilities to the arising tree topologies. Taking into account all possible topologies, this finally yields an explicit solution to the recombination model.