Chalmers Conferences, 9th European Conference on Mathematical and Theoretical Biology

Type distributions on partitions and their ancestry in the Moran model with recombination
Sebastian Probst

Last modified: 2014-04-02


We consider the Moran model with recombination, which describes the evolution of the genetic composition of a population under recombination and resampling. There are $n$ loci (or sites), a finite number of letters (or alleles) at every site, and we do not make any scaling assumptions. In particular, we do not assume a diffusion limit.

Due to the huge state space, it is notoriously difficult to find the full distribution of types in a sample. We therefore take an alternative route by concentrating on the joint probabilities of types at tuples of loci. Taking the usual genealogical approach, this is described via a process on the set of partitions of $\{1,\ldots,n\}$ (backward in time), which may be considered as a marginalised version of the ancestral recombination graph, and sheds new light on the work of A. Bobrowski, T. Wojdyla and M. Kimmel (2009). With the help of an inclusion-exclusion principle we show that the type distribution corresponding to a given partition may be represented in a systematic way, in terms of so-called recombinators. The same is true of linkage disequilibria of all orders.

Considering a graphical representation of the Moran model together with the ancestral process suggests a duality relation between the two. This will be taken up in the following talk.


Moran model; recombination; type distributions; joint probabilities; linkage disequilibira; duality