Recombination dynamics and ancestral recombination trees
Ellen Baake (October 27, 2011)
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I will start with an overview over various models for the dynamics of the genetic composition of populations evolving under recombination. For the deterministic treatment that applies in the infinite-population limit, one has large, nonlinear dynamical systems; for the stochastic treatment required for finite populations, the Moran, or Wright-Fisher model is appropriate.
I will focus on models involving only single crossovers in every generation and contrast the situations in continuous and in discrete time. In continuous time, the deterministic model has a simple closed solution, which is due to the independence of the individual recombination events. In contrast, discrete time introduces dependencies between the links and leads to a much more complex situation. Nevertheless, the situation becomes tractable by looking backwards in time, starting from single individuals at present in a Wright-Fisher population with recombination and tracing back the ancestry of the various gene segments that result from recombination. In the limit of population size to infinity, these segments become independent. We identify the process that describes their history, together with the tree structures they define, which we like to call ancestral recombination trees. It turns out that the corresponding tree _topologies_ play a special role: Surprisingly, explicit probabilities may be assigned to them, which then leads to an explicit solution of the recombination dynamics.
This is joint work with Ute von Wangenheim.
 E. Baake, Deterministic and stochastic aspects of single-crossover recombination, Proceedings of the International Congress of Mathematicians, Hyderabad, India, 2010, Vol. VI, 3037-3053