Partitioning, duality, and linkage disequilibria in the Moran model with recombination
The multilocus Moran model with recombination is considered, which describes the evolution of the genetic composition of a population under recombination and resampling. We investigate a marginal ancestral recombination process, where each site is sampled only in one individual and we do not make any scaling assumptions in the first place. Following the ancestry of these loci backward in time yields a partition-valued Markov process, which experiences splitting and coalescence. In the diffusion limit, this process turns into a marginalised version of the multilocus ancestral recombination graph. With the help of an inclusion–exclusion principle and so-called recombinators we show that the type distribution corresponding to a given partition may be represented in a systematic way by a sampling function. The same is true of correlation functions (known as linkage disequilibria in genetics) of all orders. We prove that the partitioning process (backward in time) is dual to the Moran population process (forward in time), where the sampling function plays the role of the duality function. This sheds new light on the work of Bobrowski et al. (J Math Biol 61:455–473, 2010). The result also leads to a closed system of ordinary differential equations for the expectations of the sampling functions, which can be translated into expected type distributions and expected linkage disequilibria.
KeywordsMoran model with recombination Ancestral recombination process Linkage disequilibria Möbius inversion Duality
Mathematics Subject Classification92D10 60J28
It is our pleasure to thank Noemi Kurt, Cristian Giardina, and Frank Redig for a primer to duality theory, Fernando Cordero for helpful discussions, and Michael Baake for his help to improve the manuscript. The authors gratefully acknowledge the support from the Priority Programme Probabilistic Structures in Evolution (SPP 1590), which is funded by Deutsche Forschungsgemeinschaft (German Research Foundation, DFG).
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