Wright-Fisher Process Research Articles

The Effect of Copy Number Hemiplasy on Gene Family Evolution.

The evolution of gene families is complex, involving gene-level evolutionary events such as gene duplication, horizontal gene transfer, and gene loss, and other processes such as incomplete lineage sorting (ILS). Because of this, topological differences often exist between gene trees and species trees. A number of models have been recently developed to explain these discrepancies, the most realistic of which attempts to consider both gene-level events and ILS. When unified in a single model, the interaction between ILS and gene-level events can cause polymorphism in gene copy number, which we refer to as copy number hemiplasy (CNH). In this paper, we extend the Wright-Fisher process to include duplications and losses over several species, and show that the probability of CNH for this process can be significant. We study how well two unified models-multilocus multispecies coalescent (MLMSC), which models CNH, and duplication, loss, and coalescence (DLCoal), which does not-approximate the Wright-Fisher process with duplication and loss. We then study the effect of CNH on gene family evolution by comparing MLMSC and DLCoal. We generate comparable gene trees under both models, showing significant differences in various summary statistics; most importantly, CNH reduces the number of gene copies greatly. If this is not taken into account, the traditional method of estimating duplication rates (by counting the number of gene copies) becomes inaccurate. The simulated gene trees are also used for species tree inference with the summary methods ASTRAL and ASTRAL-Pro, demonstrating that their accuracy, based on CNH-unaware simulations calibrated on real data, may have been overestimated.

Extinction scenarios in evolutionary processes: a multinomial Wright-Fisher approach.

We study a discrete-time multi-type Wright-Fisher population process. The mean-field dynamics of the stochastic process is induced by a general replicator difference equation. We prove several results regarding the asymptotic behavior of the model, focusing on the impact of the mean-field dynamics on it. One of the results is a limit theorem that describes sufficient conditions for an almost certain path to extinction, first eliminating the type which is the least fit at the mean-field equilibrium. The effect is explained by the metastability of the stochastic system, which under the conditions of the theorem spends almost all time before the extinction event in a neighborhood of the equilibrium. In addition to the limit theorems, we propose a maximization principle for a general deterministic replicator dynamics and study its implications for the stochastic model.

The path integral formula for the stochastic evolutionary game dynamics

Although the long-term behavior of stochastic evolutionary game dynamics in finite populations has been fully investigated, its evolutionary characteristics in a limited period of time is still unclear. In order to answer this question, we introduce the formulation of the path integral approach for evolutionary game theory. In this framework, the transition probability is the sum of all the evolutionary paths. The path integral formula of the transition probability is expected to be a new mathematical tool to explore the stochastic game evolutionary dynamics. As an example, we derive the transition probability for stochastic evolutionary game dynamics by the path integral in a limited period of time with the updating rule of the Wright-Fisher process.

Half a century of evolutionary games: a synthesis of theory, application and future directions.

Half a century of evolutionary games: a synthesis of theory, application and future directions.

EWF: simulating exact paths of the Wright–Fisher diffusion

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. 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. All softwares are available at https://github.com/JaroSant/EWF. Supplementary data are available at Bioinformatics online.

Evolutionary dynamics of multi-player snowdrift games based on the Wright-Fisher process

Although cooperative behavior is ubiquitous in biological and social systems, the causes and mechanisms of cooperation are a basic problem in evolutionary theory. The snowdrift game is considered as an effective evolutionary game model to describe cooperative behavior in a competitive situation. Thus, this paper studies the evolutionary dynamics of cooperative behavior in multi-player snowdrift games. This work establishes a stochastic two-strategy multi-player snowdrift game based on the Wright-Fisher (W-F) update process. Next, a specific analytical expression for fixation probabilities of cooperation and defection is considered, and the conditions under which cooperative strategies take root in a population and become an evolutionarily stable strategy are given. Finally, the relationships between the fixation probability of cooperation and each parameter involved in the game are obtained via simulation analysis. A simulation analysis reveals that the fixation probability of cooperation decreases with selection intensity, the number of players playing in multi-player snowdrift games, and population size but increases with the benefit-cost ratio. The present work promotes an understanding of the evolutionary dynamics of cooperative behavior and the theory of multi-player snowdrift games with the W-F update process.

Weak Approximations of the Wright–Fisher Process

In this paper, we construct first- and second-order weak split-step approximations for the solutions of the Wright–Fisher equation. The discretization schemes use the generation of, respectively, two- and three-valued random variables at each discretization step. The accuracy of constructed approximations is illustrated by several simulation examples.

Comparison principles and applications to mathematical modelling of vegetal meta-communities

<abstract><p>This article partakes of the PEGASE project the goal of which is a better understanding of the mechanisms explaining the behaviour of species living in a network of forest patches linked by ecological corridors (hedges for instance). Actually we plan to study the effect of the <italic>fragmentation</italic> of the habitat on biodiversity. A simple neutral model for the evolution of abundances in a vegetal metacommunity is introduced. Migration between the communities is explicitely modelized in a deterministic way, while the reproduction process is dealt with using Wright-Fisher models, independently within each community. The large population limit of the model is considered. The hydrodynamic limit of this split-step method is proved to be the solution of a partial differential equation with a deterministic part coming from the migration process and a diffusion part due to the Wright-Fisher process. Finally, the diversity of the metacommunity is adressed through one of its indicators, the mean extinction time of a species. At the limit, using classical comparison principles, the exchange process between the communities is proved to slow down extinction. This shows that the existence of corridors seems to be good for the biodiversity.</p></abstract>

The Rescaled Pólya Urn and the Wright—Fisher Process with Mutation

In recent papers the authors introduce, study and apply a variant of the Eggenberger—Pólya urn, called the “rescaled” Pólya urn, which, for a suitable choice of the model parameters, exhibits a reinforcement mechanism mainly based on the last observations, a random persistent fluctuation of the predictive mean and the almost sure convergence of the empirical mean to a deterministic limit. In this work, motivated by some empirical evidence, we show that the multidimensional Wright—Fisher diffusion with mutation can be obtained as a suitable limit of the predictive means associated to a family of rescaled Pólya urns.

Multiple Loci Selection with Multi-Way Epistasis in Coalescence with Recombination

As studies move into deeper characterization of the impact of selection through non-neutral mutations in whole genome population genetics, modeling for selection becomes crucial. Moreover, epistasis has long been recognized as a significant component in understanding the evolution of complex genetic systems. We present a backward coalescent model, EpiSimRA, that accommodates multiple loci selection, with multi-way (k-way) epistasis for any arbitrary k. Starting from arbitrary extant populations with epistatic sites, we trace the Ancestral Recombination Graph (ARG), sampling relevant recombination and coalescent events. Our framework allows for studying different complex evolutionary scenarios in the presence of selective sweeps, positive and negative selection with multiway epistasis. We also present a forward counterpart of the coalescent model based on a Wright-Fisher (WF) process, which we use as a validation framework, comparing the hallmarks of the ARG between the two. We provide the first framework that allows a nose-to-nose comparison of multiway epistasis in a coalescent simulator with its forward counterpart with respect to the hallmarks of the ARG. We demonstrate, through extensive experiments, that EpiSimRA is consistently superior in terms of performance (seconds vs. hours) in comparison to the forward model without compromising on its accuracy.

Replicator equations induced by microscopic processes in nonoverlapping population playing bimatrix games

This paper is concerned with exploring the microscopic basis for the discrete versions of the standard replicator equation and the adjusted replicator equation. To this end, we introduce frequency-dependent selection-as a result of competition fashioned by game-theoretic consideration-into the Wright-Fisher process, a stochastic birth-death process. The process is further considered to be active in a generation-wise nonoverlapping finite population where individuals play a two-strategy bimatrix population game. Subsequently, connections among the corresponding master equation, the Fokker-Planck equation, and the Langevin equation are exploited to arrive at the deterministic discrete replicator maps in the limit of infinite population size.

Moran process and Wright-Fisher process favor low variability

We study evolutionary dynamics in finite populations. We assume the individuals are one of two competing genotypes, \begin{document}$ A $\end{document} or \begin{document}$ B $\end{document} . The genotypes have the same average fitness but different variances and/or third central moments. We focus on two frequency-independent stochastic processes: (1) Wright-Fisher process and (2) Moran process. Both processes have two absorbing states corresponding to homogeneous populations of all \begin{document}$ A $\end{document} or all \begin{document}$ B $\end{document} . Despite the fact that types \begin{document}$ A $\end{document} and \begin{document}$ B $\end{document} have the same average fitness, both stochastic dynamics differ from a random drift. In both processes, the selection favors \begin{document}$ A $\end{document} replacing \begin{document}$ B $\end{document} and opposes \begin{document}$ B $\end{document} replacing \begin{document}$ A $\end{document} if the fitness variance for \begin{document}$ A $\end{document} is smaller than the fitness variance for \begin{document}$ B $\end{document} . In the case the variances are equal, the selection favors \begin{document}$ A $\end{document} replacing \begin{document}$ B $\end{document} and opposes \begin{document}$ B $\end{document} replacing \begin{document}$ A $\end{document} if the third central moment of \begin{document}$ A $\end{document} is larger than the third central moment of \begin{document}$ B $\end{document} . We show that these results extend to structured populations and other dynamics where the selection acts at birth. We also demonstrate that the selection favors a larger variance in fitness if the selection acts at death.

On the Simpson index for the Wright–Fisher process with random selection and immigration

Moran or Wright–Fisher processes are probably the most well known models to study the evolution of a population under environmental various effects. Our object of study will be the Simpson index which measures the level of diversity of the population, one of the key parameters for ecologists who study for example, forest dynamics. Following ecological motivations, we will consider, here, the case, where there are various species with fitness and immigration parameters being random processes (and thus time evolving). The Simpson index is difficult to evaluate when the population is large, except in the neutral (no selection) case, because it has no closed formula. Our approach relies on the large population limit in the “weak” selection case, and thus to give a procedure which enables us to approximate, with controlled rate, the expectation of the Simpson index at fixed time. We will also study the long time behavior (invariant measure and convergence speed towards equilibrium) of the Wright–Fisher process in a simplified setting, allowing us to get a full picture for the approximation of the expectation of the Simpson index.

Evolutionary dynamics of Chinese tourism market extremely low-priced strategies based on wright-fisher processes

Extremely low-priced of package tours is an important cause of Chinese tourism market price disorder. Regulating the price of package tours in Chinese tourism market, can solve the problem of Chinese tourism market price disorder. In order to research the reason of extremely low package tours price, consider the stochastic dynamics of N homogeneous mixed travel agency pricing strategies, and each travel agency could choose Extremely low-pricing or regular pricing. Based on Wright-Fisher processes, we obtain the dominant strategy of Extremely low-pricing and regular pricing under the weak selectivity in the package tours price setting, when the travel agency groups are finite. And then further verify these conditions by numerical simulation. The simulation shows that the Price elasticity of tourism demand, the transparency of the tourism market and the degree of government supervision strength have important effects on the stochastic evolution of package tours price setting. Reducing the demand elasticity of tourism market, increasing the transparency of the tourism market and the degree of government supervision strength can make travel agency choose regular pricing and solve chaotic of Chinese tourism market further. Therefore, this paper puts the suggestion of “two increases and one decrease” for the development of Chinese tourism market.

Effects of update rules on networked N-player trust game dynamics

We investigate the effects of update rules on the dynamics of an evolutionary game-theoretic model – the N-player evolutionary trust game – consisting of three types of players: investors (trusters), trustworthy trustees, and untrustworthy trustees. Interactions among players are constrained by local neighborhoods predefined by spatial or social network topologies. We compare different evolutionary update rules with behaviors that rely on the level of payoffs obtained by neighbors. In particular, we study the dynamics resulting from players using a deterministic rule (i.e., unconditional imitation with and without using a noise process induced by a voter model), a stochastic pairwise payoff-based strategy (i.e., proportional imitation), and stochastic local Wright-Fisher processes. We study these dynamics with different social network structures and varying levels of game difficulty. We see that there are significant differences on the level of promoted trust and global net wealth depending on the update rule. Under ‘harder’ game settings, rules based on unconditional imitation achieve the highest global net wealth in the population. We observe that there are key spatio-temporal correlations in the system for all rules. The update rules lead to the formation of fractal structures on a lattice, and low frequencies in the output signal of the system (i.e., long-term memory) when the rules are stochastic.

Computation and Simulation of Evolutionary Game Dynamics in Finite Populations

The study of evolutionary dynamics increasingly relies on computational methods, as more and more cases outside the range of analytical tractability are explored. The computational methods for simulation and numerical approximation of the relevant quantities are diverging without being compared for accuracy and performance. We thoroughly investigate these algorithms in order to propose a reliable standard. For expositional clarity we focus on symmetric 2 × 2 games leading to one-dimensional processes, noting that extensions can be straightforward and lessons will often carry over to more complex cases. We provide time-complexity analysis and systematically compare three families of methods to compute fixation probabilities, fixation times and long-term stationary distributions for the popular Moran process. We provide efficient implementations that substantially improve wall times over naive or immediate implementations. Implications are also discussed for the Wright-Fisher process, as well as structured populations and multiple types.

A general framework for moment-based analysis of genetic data.

In population genetics, the Dirichlet (also called the Balding-Nichols) model has for 20years been considered the key model to approximate the distribution of allele fractions within populations in a multi-allelic setting. It has often been noted that the Dirichlet assumption is approximate because positive correlations among alleles cannot be accommodated under the Dirichlet model. However, the validity of the Dirichlet distribution has never been systematically investigated in a general framework. This paper attempts to address this problem by providing a general overview of how allele fraction data under the most common multi-allelic mutational structures should be modeled. The Dirichlet and alternative models are investigated by simulating allele fractions from a diffusion approximation of the multi-allelic Wright-Fisher process with mutation, and applying a moment-based analysis method. The study shows that the optimal modeling strategy for the distribution of allele fractions depends on the specific mutation process. The Dirichlet model is only an exceptionally good approximation for the pure drift, Jukes-Cantor and parent-independent mutation processes with small mutation rates. Alternative models are required and proposed for the other mutation processes, such as a Beta-Dirichlet model for the infinite alleles mutation process, and a Hierarchical Beta model for the Kimura, Hasegawa-Kishino-Yano and Tamura-Nei processes. Finally, a novel Hierarchical Beta approximation is developed, a Pyramidal Hierarchical Beta model, for the generalized time-reversible and single-step mutation processes.

Evolutionary game dynamics of the Wright-Fisher process with different selection intensities

Evolutionary game dynamics in finite populations can be described by a frequency-dependent, stochastic Wright-Fisher process. The fitness of individuals in a population is not only linked to environmental conditions but also tightly coupled to the types and frequencies of competitors, leading to different types of individuals with different selection intensities. We studied a 2 × 2 symmetric game in a finite population and established a dynamic model of the Wright-Fisher process by introducing different selection intensities for different strategies. Thus, we provided another effective way to study the evolutionary dynamics of a finite population and obtained the analytical expressions of fixation probabilities under weak selection. The fixation probability of a strategy is not only related to a game matrix but also to different selection intensities. The conditions required for natural selection to favor one strategy and for that strategy to be an evolutionary stable strategy (ESSN) are specified in our model. We compared our results with those of a Moran dynamic process with different selection intensities to explore these two processes better. In the two processes, the conditions conducive to the strategy's taking fixation are the same. By simulation analysis, the dynamic relationships between the fixation probabilities and selection intensities were intuitively observed in the prisoner's dilemma, coordination, and coexistence games. The fixation probability of the cooperative strategy in the prisoner's dilemma decreases with the increase of its own selection intensity. In the coexistence and coordination games, the fixation probability of the cooperative strategy increases with its own selection intensity. For the three types of games, the fixation probability of the cooperative strategy decreases with the increase of the selection intensity of the defection strategy.

Stochastic dynamics in the fitness-based process which can be on behalf of the standard Moran, local and Wright-Fisher processes

In this paper, we proposed a frequency dependent fitness-based process, which is an extension of both the standard Moran process and the Wright-Fisher process. Some individuals are selected into a parent's pool and reproduce. Then the offspring is selected to replace individuals in the entire parent generation. We explored the influence of the size of parent pool and the number of offspring on a single cooperator's fixation. The less offspring leads to higher fixation probability of s single cooperator. Meanwhile, the fixation probability decreases with the growth of the local level. In other words, the direction of the number of offspring's impact on fixation probability is in accordance with that of the local level's impact. The less offspring in one generation or the smaller parent's pool contributes to promoting cooperation with the fitness-based updating rule.

Impact of demography on extinction/fixation events.

In this article we consider diffusion processes modeling the dynamics of multiple allelic proportions (with fixed and varying population size). We are interested in the way alleles extinctions and fixations occur. We first prove that for the Wright-Fisher diffusion process with selection, alleles get extinct successively (and not simultaneously), until the fixation of one last allele. Then we introduce a very general model with selection, competition and Mendelian reproduction, derived from the rescaling of a discrete individual-based dynamics. This multi-dimensional diffusion process describes the dynamics of the population size as well as the proportion of each type in the population. We prove first that alleles extinctions occur successively and second that depending on population size dynamics near extinction, fixation can occur either before extinction almost surely, or not. The proofs of these different results rely on stochastic time changes, integrability of one-dimensional diffusion processes paths and multi-dimensional Girsanov's tranform.