Wright-Fisher Diffusion Model Research Articles

The stationary distribution of a sample from the Wright-Fisher diffusion model with general small mutation rates.

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 (Theor Popul Biol 112:22-32, 2016; Theor Popul Biol 113:23-33, 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 [Formula: see text].

Stationary distribution of a 2-island 2-allele Wright–Fisher diffusion model with slow mutation and migration rates

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.

Bayesian inference of selection in the Wright-Fisher diffusion model.

The increasing availability of population-level allele frequency data across one or more related populations necessitates the development of methods that can efficiently estimate population genetics parameters, such as the strength of selection acting on the population(s), from such data. Existing methods for this problem in the setting of the Wright-Fisher diffusion model are primarily likelihood-based, and rely on numerical approximation for likelihood computation and on bootstrapping for assessment of variability in the resulting estimates, requiring extensive computation. Recent work has provided a method for obtaining exact samples from general Wright-Fisher diffusion processes, enabling the development of methods for Bayesian estimation in this setting. We develop and implement a Bayesian method for estimating the strength of selection based on the Wright-Fisher diffusion for data sampled at a single time point. The method utilizes the latest algorithms for exact sampling to devise a Markov chain Monte Carlo procedure to draw samples from the joint posterior distribution of the selection coefficient and the allele frequencies. We demonstrate that when assumptions about the initial allele frequencies are accurate the method performs well for both simulated data and for an empirical data set on hypoxia in flies, where we find evidence for strong positive selection in a region of chromosome 2L previously identified. We discuss possible extensions of our method to the more general settings commonly encountered in practice, highlighting the advantages of Bayesian approaches to inference in this setting.

A coalescent dual process for a Wright–Fisher diffusion with recombination and its application to haplotype partitioning

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. The process is reminiscent of the ancestral recombination graph, a widely-used multilocus genealogical model, but here ancestral lineages are typed and transition rates are regarded as being conditioned on an observed configuration at the leaves of the genealogy. 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.

Duality Between the Two-Locus Wright–Fisher Diffusion Model and the Ancestral Process with Recombination

Known results on the moments of the distribution generated by the two-locus Wright–Fisher diffusion model, and the duality between the diffusion process and the ancestral process with recombination are briefly summarized. A numerical method for computing moments using a Markov chain Monte Carlo simulation and a method to compute closed-form expressions of the moments are presented. By applying the duality argument, the properties of the ancestral recombination graph are studied in terms of the moments.

Duality Between the Two-Locus Wright–Fisher Diffusion Model and the Ancestral Process with Recombination

Known results on the moments of the distribution generated by the two-locus Wright–Fisher diffusion model, and the duality between the diffusion process and the ancestral process with recombination are briefly summarized. A numerical method for computing moments using a Markov chain Monte Carlo simulation and a method to compute closed-form expressions of the moments are presented. By applying the duality argument, the properties of the ancestral recombination graph are studied in terms of the moments.

Analysis of market weights under volatility-stabilized market models

We derive the joint density of market weights, at fixed times and suitable stopping times, of the volatility-stabilized market models introduced by Fernholz and Karatzas in [Ann. Finan. 1 (2005) 149-177]. The argument rests on computing the exit density of a collection of independent Bessel-square processes of possibly different dimensions from the unit simplex. We show that the law of the market weights is the same as that of the multi-allele Wright-Fisher diffusion model, well known in population genetics. Thus, as a side result, we furnish a novel proof of the transition density function of the Wright-Fisher model which was originally derived by Griffiths by bi-orthogonal series expansion.

On the Karlin–Kimura approaches to the Wright–Fisher diffusion with fluctuatingselection

The goal of this work is a comparative study of two Wright–Fisher-like diffusionprocesses on the interval, one due to Karlin and the other one due to Kimura. Eachmodel accounts for the evolution of one two-locus colony undergoing randommating, under the additional action of selection in a random environment. In otherwords, we study the effect of disorder on the usual Wright–Fisher model withfixed (nonrandom) selection. There is a drastic qualitative difference betweenthe two models and between the random and nonrandom selection hypotheses.We first present a series of elementary stochastic models and tools that are needed toconduct this study in the context of diffusion process theory, including Kolmogorovbackward and forward equations, scale and speed functions, classification of boundaries,and Doob transformation of sample paths using additive functionals. In this spirit, webriefly revisit the neutral Wright–Fisher diffusion and the Wright–Fisher diffusion withnonrandom selection.With these tools at hand, we first deal with the Karlin approach to the Wright–Fisherdiffusion model with randomized selection differentials. The specificity of thismodel is that in the large population case, the boundaries of the state space arenatural and hence inaccessible, and so quasi-absorbing only. We supply somelimiting properties pertaining to times of hitting of points close to the boundaries.Next, we study the Kimura approach to the Wright–Fisher model with randomizedselection, which may be viewed as a modification of the Karlin model, using an appropriateDoob transform which we describe. This model also has natural boundaries, but they turnout to be much more attracting and sticky than in Karlin’s version. This leads to a fasterapproach to the quasi-absorbing states, to a larger time needed to move from the vicinity ofone boundary to the other and to a local critical behavior of the branching diffusionobtained after the relevant Doob transformation.

On Wright–Fisher diffusion and its relatives

We present a series of elementary stochastic models arising from population genetics.Various mathematical aspects are investigated at the intuitive level. Special emphasis is puton the diffusion method. The focus is on the Wright–Fisher diffusion model and itsvariations, describing the forward evolution of one colony undergoing random sampling,possibly under the additional forces arising from mutation and selection with or withoutdominance. Some aspects of their dual coalescents obtained while running the diffusionprocess backward in time are also investigated.