Partitioning, duality, and linkage disequilibria in the Moran model with recombination

The Moran model with recombination is considered, which describes the evolution of the genetic composition of a population under recombination and resampling. There are $n$ sites (or loci), 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. We consider the following marginal ancestral recombination process. Let $S = {1,...,n}$ and $mathcal A={A_1, ..., A_m}$ be a partition of $S$. We concentrate on the joint probability of the letters at the sites in $A_1$ in individual $1$, $...$, and at the sites in $A_m$ in individual $m$, where the individuals are sampled from the current population without replacement. Following the ancestry of these sites backwards in time yields a process on the set of partitions of $S$, which, in the diffusion limit, turns into a marginalised version of the $n$-locus ancestral recombination graph. 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, with the help of so-called recombinators and sampling functions. 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, Wojdyla, and Kimmel (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.

Looking down in the ancestral selection graph: A probabilistic approach to the common ancestor type distribution

In a (two-type) Wright-Fisher diffusion with directional selection and two-way mutation, let $x$ denote todays frequency of the beneficial type, and given $x$, let $h(x)$ be the probability that, among all individuals of todays population, the individual whose progeny will eventually take over in the population is of the beneficial type. Fearnhead [Fearnhead, P., 2002. The common ancestor at a nonneutral locus. J. Appl. Probab. 39, 38-54] and Taylor [Taylor, J. E., 2007. The common ancestor process for a Wright-Fisher diffusion. Electron. J. Probab. 12, 808-847] obtained a series representation for $h(x)$. We develop a construction that contains elements of both the ancestral selection graph and the lookdown construction and includes pruning of certain lines upon mutation. Besides being interesting in its own right, this construction allows a transparent derivation of the series coefficients of $h(x)$ and gives them a probabilistic meaning.

The Moran model with selection: Fixation probabilities, ancestral lines, and an alternative particle representation

We reconsider the Moran model in continuous time with population size $N$, two allelic types, and selection. We introduce a new particle representation, which we call the labelled Moran model, and which has the same distribution of type frequencies as the original Moran model, provided the initial values are chosen appropriately. In the new model, individuals are labelled $1,2, dots, N$; neutral resampling events may take place between arbitrary labels, whereas selective events only occur in the direction of increasing labels. With the help of elementary methods only, we not only recover fixation probabilities, but also obtain detailed insight into the number and nature of the selective events that play a role in the fixation process forward in time.

The common ancestor process revisited

We consider the Moran model in continuous time with two types, mutation, and selection. We concentrate on the ancestral line and its stationary type distribution. Building on work by Fearnhead (J. Appl. Prob. 39 (2002), 38-54) and Taylor (Electron. J. Probab. 12 (2007), 808-847), we characterise this distribution via the fixation probability of the offspring of all individuals of favourable type (regardless of the offsprings types). We concentrate on a finite population and stay with the resulting discrete setting all the way through. This way, we extend previous results and gain new insight into the underlying particle picture.