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Register a new userStationary distribution of a 2-island 2-allele Wright-Fisher diffusion model with slow mutation and migration rates
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The stationary distribution of the diffusion limit of the 2-island, 2-allele Wright-Fisher with small but otherwise arbitrary mutation and migration rates is investigated. Following a method developed by Burden and Tang (2016, 2017) for approximating the forward Kolmogorov equation, the stationary distribution is obtained to leading order as a set of line densities on the edges of the sample space, corresponding to states for which one island is bi-allelic and the other island is non-segregating, and a set of point masses at the corners of the sample space, corresponding to states for which both islands are simultaneously non-segregating. Analytic results for the corner probabilities and line densities are verified independently using the backward generator and for the corner probabilities using the coalescent.
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The stationary distribution of a sample taken from a Wright-Fisher diffusion with general small mutation rates is found using a coalescent approach. The approximation is equivalent to having at most one mutation in the coalescent tree to the first order in the rates. The sample probabilities characterize an approximation for the stationary distribution from the Wright-Fisher diffusion. The approach is different from Burden and Tang (2016,2017) who use a probability flux argument to obtain the same results from a forward diffusion generator equation. The solution has interest because the solution is not known when rates are not small. An analogous solution is found for the configuration of alleles in a general exchangeable binary coalescent tree. In particular an explicit solution is found for a pure birth process tree when individuals reproduce at rate lambda.
The transition distribution of a sample taken from a Wright-Fisher diffusion with general small mutation rates is found using a coalescent approach. The approximation is equivalent to having at most one mutation in the coalescent tree of the sample up to the most recent common ancestor with additional mutations occurring on the lineage from the most recent common ancestor to the time origin if complete coalescence occurs before the origin. The sampling distribution leads to an approximation for the transition density in the diffusion with small mutation rates. This new solution has interest because the transition density in a Wright-Fisher diffusion with general mutation rates is not known.
We investigate the properties of a Wright-Fisher diffusion process started from frequency x at time 0 and conditioned to be at frequency y at time T. Such a process is called a bridge. Bridges arise naturally in the analysis of selection acting on standing variation and in the inference of selection from allele frequency time series. We establish a number of results about the distribution of neutral Wright-Fisher bridges and develop a novel rejection sampling scheme for bridges under selection that we use to study their behavior.
Consider a two-type Moran population of size $N$ subject to selection and mutation, which is immersed in a varying environment. The population is susceptible to exceptional changes in the environment, which accentuate the selective advantage of the fit individuals. In this setting, we show that the type-composition in the population is continuous with respect to the environment. This allows us to replace the deterministic environment by a random one, which is driven by a subordinator. Assuming that selection, mutation and the environment are weak in relation to $N$, we show that the type-frequency process, with time speed up by $N$, converges as $Ntoinfty$ to a Wright--Fisher-type SDE with a jump term modeling the effect of the environment. Next, we study the asymptotic behavior of the limiting model in the far future and in the distant past, both in the annealed and in the quenched setting. Our approach builds on the genealogical picture behind the model. The latter is described by means of an extension of the ancestral selection graph (ASG). The formal relation between forward and backward objects is given in the form of a moment duality between the type-frequency process and the line-counting process of a pruned version of the ASG. This relation yields characterizations of the annealed and the quenched moments of the asymptotic type distribution. A more involved pruning of the ASG allows us to obtain annealed and quenched results for the ancestral type distribution. In the absence of mutations, one of the types fixates and our results yield expressions for the fixation probabilities.
Duality plays an important role in population genetics. It can relate results from forwards-in-time models of allele frequency evolution with those of backwards-in-time genealogical models; a well known example is the duality between the Wright-Fisher diffusion for genetic drift and its genealogical counterpart, the coalescent. There have been a number of articles extending this relationship to include other evolutionary processes such as mutation and selection, but little has been explored for models also incorporating crossover recombination. Here, we derive from first principles a new genealogical process which is dual to a Wright-Fisher diffusion model of drift, mutation, and recombination. Our approach is based on expressing a putative duality relationship between two models via their infinitesimal generators, and then seeking an appropriate test function to ensure the validity of the duality equation. This approach is quite general, and we use it to find dualities for several important variants, including both a discrete L-locus model of a gene and a continuous model in which mutation and recombination events are scattered along the gene according to continuous distributions. As an application of our results, we derive a series expansion for the transition function of the diffusion. Finally, we study in further detail the case in which mutation is absent. Then the dual process describes the dispersal of ancestral genetic material across the ancestors of a sample. The stationary distribution of this process is of particular interest; we show how duality relates this distribution to haplotype fixation probabilities. We develop an efficient method for computing such probabilities in multilocus models.
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