Bernoulli factories and duality in Wright-Fisher and Allen-Cahn models of population genetics.

Mathematical models of genetic evolution often come in pairs, connected by a so-called duality relation. The most seminal example are the Wright-Fisher diffusion and the Kingman coalescent, where the former describes the stochastic evolution of neutral allele frequencies in a large population forwards in time, and the latter describes the genetic ancestry of randomly sampled individuals from the population backwards in time. As well as providing a richer description than either model in isolation, duality often yields equations satisfied by quantities of interest. We employ the so-called Bernoulli factory - a celebrated tool in simulation-based computing - to derive duality relations for broad classes of genetics models. As concrete examples, we present Wright-Fisher diffusions with general drift functions, and Allen-Cahn equations with general, nonlinear forcing terms. The drift and forcing functions can be interpreted as the action of frequency-dependent selection. To our knowledge, this work is the first time a connection has been drawn between Bernoulli factories and duality in models of population genetics.

EWF: simulating exact paths of the Wright-Fisher diffusion.

MOTIVATION: The Wright-Fisher diffusion is important in population genetics in modelling the evolution of allele frequencies over time subject to the influence of biological phenomena such as selection, mutation and genetic drift. Simulating the paths of the process is challenging due to the form of the transition density. We present EWF, a robust and efficient sampler which returns exact draws for the diffusion and diffusion bridge processes, accounting for general models of selection including those with frequency dependence. RESULTS: Given a configuration of selection, mutation and endpoints, EWF returns draws at the requested sampling times from the law of the corresponding Wright-Fisher process. Output was validated by comparison to approximations of the transition density via the Kolmogorov-Smirnov test and QQ plots. AVAILABILITY AND IMPLEMENTATION: All softwares are available at https://github.com/JaroSant/EWF. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

Wright-Fisher diffusion bridges.

The trajectory of the frequency of an allele which begins at x at time 0 and is known to have frequency z at time T can be modelled by the bridge process of the Wright-Fisher diffusion. Bridges when x=z=0 are particularly interesting because they model the trajectory of the frequency of an allele which appears at a time, then is lost by random drift or mutation after a time T. The coalescent genealogy back in time of a population in a neutral Wright-Fisher diffusion process is well understood. In this paper we obtain a new interpretation of the coalescent genealogy of the population in a bridge from a time t∈(0,T). In a bridge with allele frequencies of 0 at times 0 and T the coalescence structure is that the population coalesces in two directions from t to 0 and t to T such that there is just one lineage of the allele under consideration at times 0 and T. The genealogy in Wright-Fisher diffusion bridges with selection is more complex than in the neutral model, but still with the property of the population branching and coalescing in two directions from time t∈(0,T). The density of the frequency of an allele at time t is expressed in a way that shows coalescence in the two directions. A new algorithm for exact simulation of a neutral Wright-Fisher bridge is derived. This follows from knowing the density of the frequency in a bridge and exact simulation from the Wright-Fisher diffusion. The genealogy of the neutral Wright-Fisher bridge is also modelled by branching Pólya urns, extending a representation in a Wright-Fisher diffusion. This is a new very interesting representation that relates Wright-Fisher bridges to classical urn models in a Bayesian setting.

Strong seed-bank effects in bacterial evolution.

Bacterial genomes are mosaics with fragments showing distinct phylogenetic origins or even being unrelated to any other genetic information (ORFan genes). Thus the analysis of bacterial population genetics is in large part a collection of explanations for anomalies in relation to classical population genetic models such as the Wright-Fisher model and the Kingman coalescent that do not adequately describe bacterial population genetics, genomics or evolution. The concept of "species" as an evolutionary coherent biological group that is genetically isolated and shares genetic information through recombination among its members cannot be applied to any bacterial group. Recently, a simple probabilistic model considering the role of strong seed-bank effects in population genetics has been proposed by Blath et al. This model suggests the existence of a genetic pool with high diversity that is not subject to classical selection and extinction. We reason that certain bacterial population genetics anomalies could be explained by the prevalence of strong seed-bank effects among bacteria. To address this possibility we analyzed the genome of the bacterium Azotobacter vinelandii and show that genes that code for functions that are essential for the bacterium biology do not have a relation of ancestry with closely related bacteria, or are ORFan genes. The existence of essential genes that are not inherited from the most recent ancestor cannot be explained by classical population genetics models and is irreconcilable with the current view of genes acquired by horizontal transfer as being accessory or adaptive.