Asymptotic Behavior of a Moran Model with Mutations, Drift and Recombination Among Multiple Loci

In this paper, we extend the theoretical treatment of the Moran model of genetic drift with recombination and mutation, which was previously introduced by us for the case of two loci, to the case of n loci. Recombination, when considered in the Wright-Fisher model, makes it considerably less tractable. In the works of Griffiths, Hudson and Kaplan and their colleagues important properties were established using the coalescent approach. Other more recent approaches form a body of work to which we would like to contribute. The specific framework used in our paper allows finding close-form relationships, which however are limited to a set of distributions, which jointly characterize allelic states at a number of loci at the same or different chromosome(s) but which do not jointly characterize allelic states at a single locus on two or more chromosomes. However, the system is sufficiently rich to allow computing, albeit in general numerically, all possible multipoint linkage disequilibria under recombination, mutation and drift. We explore the algorithms enabling construction of the transition probability matrices of the Markov chain describing the process. We find that asymptotically the effects of recombination become indistinguishable, at least as characterized by the set of distributions we consider, from the effects of mutation and drift. Mathematically, the results are based on the foundations of the theory of semi-groups of operators. This approach allows generalization to any Markov-type mutation model. Based on these fundamental results, we explore the rates of convergence to the limit distribution, using Dobrushin's coefficient and spectral gap.