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Partitioning, duality, and linkage disequilibria in the Moran model with recombination. (English) Zbl 1359.92080
This is a very comprehensive and unifying paper that looks at the genetic evolution of a population via a multilocus Moran model with recombination. Each unit in the population is a chromosome taken as an haploid individual and the total population remains constant; when an individual dies after an exponentially distributed lifetime, it is randomly replaced by a full copy of a single parent or by a recombination of two parents resulting from a single cross-over. The Moran model is a forward continuous-time Markov process that identifies the population at time $$t$$ with a random counting measure $$Z_t$$ on the type space (space of all possible chromosomal types).
The paper builds an important bridge between this forward approach and a genealogical (backward in time) approach using sampling formulae. Although the forward approach traditionally recurs to asymptotic approximations and the paper looks at them when appropriate, the bridge is made using the original Moran model by proving a duality between this model and a marginalized version (each locus being considered in one individual only) of the ancestral recombination process (ARP) (genealogical approach). This is done using what the authors call sampling functions as duality functions and is achieved by extending the recombinator formalism to the stochastic setting. Due to the above marginalization, this leads to an explicit closed ODE system for the expected sampling functions, which are building blocks for the linkage disequilibria that the paper also analyses.
Several tools are used to follow this program. One is the use of Möbius functions (and Möbius inversion) to turn sampling without replacement into sampling with replacement and to turn type frequencies into linkage disequilibria. Another is the partitioning process, a Markov process consisting of a mixture of splitting and coalescence events that describe how the sites are partitioned into different individuals backward in time. The paper studies its limiting behaviour as the population size $$N \rightarrow +\infty$$ (deterministic limit) and the limiting behaviour of a sequence of such processes with time sped up by a factor of $$N$$ and recombination probabilities also rescaled (diffusion limit). Finally, the extended recombinator formalism, together with Möbius functions, allows the introduction of the sampling functions.
At the end, there are applications to two or three sites; in particular, it is shown that linkage disequilibria decay exponentially. In the case of two sites, the time evolution of the expected composition of the population is obtained, as well as the fixation probabilities.

##### MSC:
 92D10 Genetics and epigenetics 92D15 Problems related to evolution 60J28 Applications of continuous-time Markov processes on discrete state spaces
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