Single-crossover recombination and ancestral recombination trees |
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Authors: | Ellen Baake Ute von Wangenheim |
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Institution: | 1. Faculty of Technology, Bielefeld University, 33594, Bielefeld, Germany
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Abstract: | We consider the Wright–Fisher model for a population of $N$ individuals, each identified with a sequence of a finite number of sites, and single-crossover recombination between them. We trace back the ancestry of single individuals from the present population. In the $N \rightarrow \infty $ limit without rescaling of parameters or time, this ancestral process is described by a random tree, whose branching events correspond to the splitting of the sequence due to recombination. With the help of a decomposition of the trees into subtrees, we calculate the probabilities of the topologies of the ancestral trees. At the same time, these probabilities lead to a semi-explicit solution of the deterministic single-crossover equation. The latter is a discrete-time dynamical system that emerges from the Wright–Fisher model via a law of large numbers and has been waiting for a solution for many decades. |
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