Wright-Fisher Process Research Articles

Transition probabilities for degenerate diffusions arising in population genetics

We provide a detailed description of the structure of the transition probabilities and of the hitting distributions on boundary components of a manifold with corners for a degenerate strong Markov process arising in population genetics. The Markov processes that we study are a generalization of the classical Wright–Fisher process. The main ingredients in our proofs are based on the analysis of the regularity properties of solutions to a forward Kolmogorov equation defined on a compact manifold with corners, which is degenerate in the sense that it is not strictly elliptic and the coefficients of the first order drift term have mild logarithmic singularities.

Public goods games in populations with fluctuating size

Many mathematical frameworks of evolutionary game dynamics assume that the total population size is constant and that selection affects only the relative frequency of strategies. Here, we consider evolutionary game dynamics in an extended Wright–Fisher process with variable population size. In such a scenario, it is possible that the entire population becomes extinct. Survival of the population may depend on which strategy prevails in the game dynamics. Studying cooperative dilemmas, it is a natural feature of such a model that cooperators enable survival, while defectors drive extinction. Although defectors are favored for any mixed population, random drift could lead to their elimination and the resulting pure-cooperator population could survive. On the other hand, if the defectors remain, then the population will quickly go extinct because the frequency of cooperators steadily declines and defectors alone cannot survive. In a mutation–selection model, we find that (i) a steady supply of cooperators can enable long-term population survival, provided selection is sufficiently strong, and (ii) selection can increase the abundance of cooperators but reduce their relative frequency. Thus, evolutionary game dynamics in populations with variable size generate a multifaceted notion of what constitutes a trait’s long-term success.

Duality and fixation in $\Xi$-Wright–Fisher processes with frequency-dependent selection

A two-types, discrete-time population model with finite, constant size is constructed, allowing for a general form of frequency-dependent selection and skewed offspring distribution. Selection is defined based on the idea that individuals first choose a (random) number of \emph{potential} parents from the previous generation and then, from the selected pool, they inherit the type of the fittest parent. The probability distribution function of the number of potential parents per individual thus parametrises entirely the selection mechanism. Using sampling- and moment-duality, weak convergence is then proved both for the allele frequency process of the selectively weak type and for the population's ancestral process. The scaling limits are, respectively, a two-types $\Xi$-Fleming-Viot jump-diffusion process with frequency-dependent selection, and a branching-coalescing process with general branching and simultaneous multiple collisions. Duality also leads to a characterisation of the probability of extinction of the selectively weak allele, in terms of the ancestral process' ergodic properties.

The effect of the stake size on the evolution of fairness

The stake-size effect on the evolution of fairness has attracted increasing attention from experiments for a long time, yet there has not been a clear and consistent conclusion. Budgetary restrictions make it necessary to construct theoretical models. Here, we investigate the stake-size effect, which has been neglected by almost all previous models, in an evolutionary model of the Ultimatum Game. Our study proceeds in group-structured populations of finite size. For the Moran process and the Wright-Fisher process, we calculate the precise expressions of the average offer and the average demand under weak selection. The increase of the stake size causes populations to be closer to the rational selfish strategy, and we analyze how three key parameters influence the approach speed. We divide all possible stakes according to the evolution of generosity and tolerance, and find that the majority of all possible stakes promote generosity and tolerance for small group numbers. The largest stake promotes tolerance to an extreme. The intermediate stake or the largest stake promotes generosity to an extreme, and we analyze the effect of three key parameters on it. Our results can be used to help design more reasonable and more efficient experiments under restricted budgets.

Dimension reduction for stochastic dynamical systems forced onto a manifold by large drift: a constructive approach with examples from theoretical biology

Systems composed of large numbers of interacting agents often admit an effective coarse-grained description in terms of a multidimensional stochastic dynamical system, driven by small-amplitude intrinsic noise. In applications to biological, ecological, chemical and social dynamics it is common for these models to posses quantities that are approximately conserved on short timescales, in which case system trajectories are observed to remain close to some lower-dimensional subspace. Here, we derive explicit and general formulae for a reduced-dimension description of such processes that is exact in the limit of small noise and well-separated slow and fast dynamics. The Michaelis–Menten law of enzyme-catalysed reactions, and the link between the Lotka–Volterra and Wright–Fisher processes are explored as a simple worked examples. Extensions of the method are presented for infinite dimensional systems and processes coupled to non-Gaussian noise sources.

Evaluating the maintenance of disease-associated variation at the blood group-related gene B4galnt2 in house mice

BackgroundB4galnt2 is a blood group-related glycosyltransferase that displays cis-regulatory variation for its tissue-specific expression patterns in house mice. The wild type allele, found e.g. in the C57BL/6 J strain, directs intestinal expression of B4galnt2, which is the pattern observed among vertebrates, including humans. An alternative allele class found in the RIIIS/J strain and other mice instead drives expression in blood vessels, which leads to a phenotype similar to type 1 von Willebrand disease (VWD), a common human bleeding disorder. We previously showed that alternative B4galnt2 alleles are subject to long-term balancing selection in mice and that variation in B4galnt2 expression influences host-microbe interactions in the intestine. This suggests that the costs of prolonged bleeding in RIIIS/J allele-bearing mice might be outweighed by benefits associated with resistance against gastrointestinal pathogens. However, the conditions under which such trade-offs could lead to the long-term maintenance of disease-associated variation at B4galnt2 are unclear.ResultsTo explore the persistence of B4galnt2 alleles in wild populations of house mice, we combined B4galnt2 haplotype frequency data together with a mathematical model based on an evolutionary game framework with a modified Wright-Fisher process. In particular, given the potential for a heterozygote advantage as a possible explanation for balancing selection, we focused on heterozygous mice, which express B4galnt2 in both blood vessels and the gastrointestinal tract. We show that B4galnt2 displays an interesting spatial allelic distribution in Western Europe, likely due to the recent action of natural selection. Moreover, we found that the genotype frequencies observed in nature can be produced by pathogen-driven selection when both heterozygotes and RIIIS/J homozygotes are protected against infection and the fitness cost of bleeding is roughly half that of infection.ConclusionBy comparing the results of our models to the patterns of polymorphism at B4galnt2 in natural populations, we are able to recognize the long-term maintenance of the RIIIS/J allele through host-pathogen interactions as a viable hypothesis. Further, our models identify that a putative dominant-, yet unknown protective function of the RIIIS/J allele appears to be more likely than a protective loss of intestinal B4galnt2 expression in RIIIS/J homozygotes.

On selection in finite populations.

Two major forces shaping evolution are drift and selection. The standard models of neutral drift-the Wright-Fisher (WF) and Moran processes-can be extended to include selection. However, these standard models are not always applicable in practice, and-even without selection-many other drift models make very different predictions. For example, "generalised Wright-Fisher" models (so-called because their first two conditional moments agree with those of the WF process) can yield wildly different absorption times from WF. Additionally, evolutionary stability in finite populations depends only on fixation probabilities, which can be evaluated under less restrictive assumptions than those required to estimate fixation times or more complex population-genetic quantities. We therefore distill the notion of a selection process into a broad class of finite-population, mutationless models of drift and selection (including the WF and Moran processes). We characterize when selection favours fixation of one strategy over another, for any selection process, which allows us to derive finite-population conditions for evolutionary stability independent of the selection process. In applications, the precise details of the selection process are seldom known, yet by exploiting these new theoretical results it is now possible to make rigorously justifiable inferences about fixation of traits.

Exact simulation of the Wright–Fisher diffusion

The Wright-Fisher family of diffusion processes is a widely used class of evolutionary models. However, simulation is difficult because there is no known closed-form formula for its transition function. In this article we demonstrate that it is in fact possible to simulate exactly from a broad class of Wright-Fisher diffusion processes and their bridges. For those diffusions corresponding to reversible, neutral evolution, our key idea is to exploit an eigenfunction expansion of the transition function; this approach even applies to its infinite-dimensional analogue, the Fleming-Viot process. We then develop an exact rejection algorithm for processes with more general drift functions, including those modelling natural selection, using ideas from retrospective simulation. Our approach also yields methods for exact simulation of the moment dual of the Wright-Fisher diffusion, the ancestral process of an infinite-leaf Kingman coalescent tree. We believe our new perspective on diffusion simulation holds promise for other models admitting a transition eigenfunction expansion.

On the stochastic evolution of finite populations.

This work is a systematic study of discrete Markov chains that are used to describe the evolution of a two-types population. Motivated by results valid for the well-known Moran (M) and Wright-Fisher (WF) processes, we define a general class of Markov chains models which we term the Kimura class. It comprises the majority of the models used in population genetics, and we show that many well-known results valid for M and WF processes are still valid in this class. In all Kimura processes, a mutant gene will either fixate or become extinct, and we present a necessary and sufficient condition for such processes to have the probability of fixation strictly increasing in the initial frequency of mutants. This condition implies that there are WF processes with decreasing fixation probability-in contradistinction to M processes which always have strictly increasing fixation probability. As a by-product, we show that an increasing fixation probability defines uniquely an M or WF process which realises it, and that any fixation probability with no state having trivial fixation can be realised by at least some WF process. These results are extended to a subclass of processes that are suitable for describing time-inhomogeneous dynamics. We also discuss the traditional identification of frequency dependent fitnesses and pay-offs, extensively used in evolutionary game theory, the role of weak selection when the population is finite, and the relations between jumps in evolutionary processes and frequency dependent fitnesses.

Impact of migration on the multi-strategy selection in finite group-structured populations.

For large quantities of spatial models, the multi-strategy selection under weak selection is the sum of two competition terms: the pairwise competition and the competition of multiple strategies with equal frequency. Two parameters σ1 and σ2 quantify the dependence of the multi-strategy selection on these two terms, respectively. Unlike previous studies, we here do not require large populations for calculating σ1 and σ2, and perform the first quantitative analysis of the effect of migration on them in group-structured populations of any finite sizes. The Moran and the Wright-Fisher process have the following common findings. Compared with well-mixed populations, migration causes σ1 to change with the mutation probability from a decreasing curve to an inverted U-shaped curve and maintains the increase of σ2. Migration (probability and range) leads to a significant change of σ1 but a negligible one of σ2. The way that migration changes σ1 is qualitatively similar to its influence on the single parameter characterizing the two-strategy selection. The Moran process is more effective in increasing σ1 for most migration probabilities and the Wright-Fisher process is always more effective in increasing σ2. Finally, our findings are used to study the evolution of cooperation under direct reciprocity.

Stationary Stability for Evolutionary Dynamics in Finite Populations

We demonstrate a vast expansion of the theory of evolutionary stability to finite populations with mutation, connecting the theory of the stationary distribution of the Moran process with the Lyapunov theory of evolutionary stability. We define the notion of stationary stability for the Moran process with mutation and generalizations, as well as a generalized notion of evolutionary stability that includes mutation called an incentive stable state (ISS) candidate. For sufficiently large populations, extrema of the stationary distribution are ISS candidates and we give a family of Lyapunov quantities that are locally minimized at the stationary extrema and at ISS candidates. In various examples, including for the Moran and Wright–Fisher processes, we show that the local maxima of the stationary distribution capture the traditionally-defined evolutionarily stable states. The classical stability theory of the replicator dynamic is recovered in the large population limit. Finally we include descriptions of possible extensions to populations of variable size and populations evolving on graphs.

ARGON: fast, whole-genome simulation of the discrete time Wright-fisher process.

Simulation under the coalescent model is ubiquitous in the analysis of genetic data. The rapid growth of real data sets from multiple human populations led to increasing interest in simulating very large sample sizes at whole-chromosome scales. When the sample size is large, the coalescent model becomes an increasingly inaccurate approximation of the discrete time Wright-Fisher model (DTWF). Analytical and computational treatment of the DTWF, however, is generally harder. We present a simulator (ARGON) for the DTWF process that scales up to hundreds of thousands of samples and whole-chromosome lengths, with a time/memory performance comparable or superior to currently available methods for coalescent simulation. The simulator supports arbitrary demographic history, migration, Newick tree output, variable mutation/recombination rates and gene conversion, and efficiently outputs pairwise identical-by-descent sharing data. ARGON (version 0.1) is written in Java, open source, and freely available at https://github.com/pierpal/ARGON CONTACT: ppalama@hsph.harvard.edu Supplementary data are available at Bioinformatics online.

The common ancestor type distribution of a $\Lambda$-Wright-Fisher process with selection and mutation

Using graphical methods based on a `lookdown' and pruned version of the {\em ancestral selection graph}, we obtain a representation of the type distribution of the ancestor in a two-type Wright-Fisher population with mutation and selection, conditional on the overall type frequency in the old population. This extends results from Lenz, Kluth, Baake, and Wakolbinger (Theor. Pop. Biol., 103 (2015), 27-37) to the case of heavy-tailed offspring, directed by a reproduction measure $\Lambda$. The representation is in terms of the equilibrium tail probabilities of the line-counting process $L$ of the graph. We identify a strong pathwise Siegmund dual of $L$, and characterise the equilibrium tail probabilities of $L$ in terms of hitting probabilities of the dual process.

The multivariate Wright–Fisher process with mutation: Moment-based analysis and inference using a hierarchical Beta model

We consider the diffusion approximation of the multivariate Wright–Fisher process with mutation. Analytically tractable formulas for the first-and second-order moments of the allele frequency distribution are derived, and the moments are subsequently used to better understand key population genetics parameters and modeling frameworks. In particular we investigate the behavior of the expected homozygosity (the probability that two randomly sampled genes are identical) in the transient and stationary phases, and how appropriate the Dirichlet distribution is for modeling the allele frequency distribution at different evolutionary time scales. We find that the Dirichlet distribution is adequate for the pure drift model (no mutations allowed), but the distribution is not sufficiently flexible for more general mutation models. We suggest a new hierarchical Beta distribution for the allele frequencies in the Wright–Fisher process with a mutation model on the nucleotide level that distinguishes between transitions and transversions.

Harnack Inequality and Applications for Infinite-Dimensional GEM Processes

The dimension-free Harnack inequality and uniform heat kernel upper/lower bounds are derived for a class of infinite-dimensional GEM processes, which was introduced in \cite{FW} to simulate the two-parameter GEM distributions. In particular, the associated Dirichlet form satisfies the super log-Sobolev inequality which strengthens the log-Sobolev inequality derived in \cite{FW}. To prove the main results, explicit Harnack inequality and super Poincar\'e inequality are established for the one-dimensional Wright-Fisher diffusion processes. The main tool of the study is the coupling by change of measures.

Indirect Reciprocity with Optional Interactions and Private Information

We consider indirect reciprocity with optional interactions and private information. A game is offered between two players and accepted unless it is known that the other person is a defector. Whenever a defector manages to exploit a cooperator, his or her reputation is revealed to others in the population with some probability. Therefore, people have different private information about the reputation of others, which is a setting that is difficult to analyze in the theory of indirect reciprocity. Since a defector loses a fraction of his social ties each time he exploits a cooperator, he is less efficient at exploiting cooperators in subsequent rounds. We analytically calculate the critical benefit-to-cost ratio above which cooperation is successful in various settings. We demonstrate quantitative agreement with simulation results of a corresponding Wright–Fisher process with optional interactions and private information. We also deduce a simple necessary condition for the critical benefit-to-cost ratio.

Multi-locus analysis of genomic time series data from experimental evolution.

Genomic time series data generated by evolve-and-resequence (E&R) experiments offer a powerful window into the mechanisms that drive evolution. However, standard population genetic inference procedures do not account for sampling serially over time, and new methods are needed to make full use of modern experimental evolution data. To address this problem, we develop a Gaussian process approximation to the multi-locus Wright-Fisher process with selection over a time course of tens of generations. The mean and covariance structure of the Gaussian process are obtained by computing the corresponding moments in discrete-time Wright-Fisher models conditioned on the presence of a linked selected site. This enables our method to account for the effects of linkage and selection, both along the genome and across sampled time points, in an approximate but principled manner. We first use simulated data to demonstrate the power of our method to correctly detect, locate and estimate the fitness of a selected allele from among several linked sites. We study how this power changes for different values of selection strength, initial haplotypic diversity, population size, sampling frequency, experimental duration, number of replicates, and sequencing coverage depth. In addition to providing quantitative estimates of selection parameters from experimental evolution data, our model can be used by practitioners to design E&R experiments with requisite power. We also explore how our likelihood-based approach can be used to infer other model parameters, including effective population size and recombination rate. Then, we apply our method to analyze genome-wide data from a real E&R experiment designed to study the adaptation of D. melanogaster to a new laboratory environment with alternating cold and hot temperatures.

Towards A Theoretical Framework for Analysis and Intervention of Random Drift on General Networks

It is well known that evolution is a fundamental phenomenon driving nature. And random drift is a basic force for population evolution. This technical note aims at constructing a unified theoretical framework for analyzing and intervening in random drift of binary states on general networks. In detail, a general methodology is developed for calculating the fixation probability with different dynamics, including the Wright-Fisher (WF), birth-death (BD), and death-birth (DB) processes. In particular, we prove that the fixation probability of a mutant at node k corresponds to the k-th element of stationary distribution of a stochastic matrix deduced from the weight matrix of the networks. Intuitively, it provides an effective way to discover the invasion hubs of structured population and further to intervene in random drift on networks. Finally, a typical example is then given to validate the above theoretical results.

Formal properties of the probability of fixation: Identities, inequalities and approximations

The formula for the probability of fixation of a new mutation is widely used in theoretical population genetics and molecular evolution. Here we derive a series of identities, inequalities and approximations for the exact probability of fixation of a new mutation under the Moran process (equivalent results hold for the approximate probability of fixation under the Wright–Fisher process, after an appropriate change of variables). We show that the logarithm of the fixation probability has particularly simple behavior when the selection coefficient is measured as a difference of Malthusian fitnesses, and we exploit this simplicity to derive inequalities and approximations. We also present a comprehensive comparison of both existing and new approximations for the fixation probability, highlighting those approximations that induce a reversible Markov chain when used to describe the dynamics of evolution under weak mutation. To demonstrate the power of these results, we consider the classical problem of determining the total substitution rate across an ensemble of biallelic loci and prove that, at equilibrium, a strict majority of substitutions are due to drift rather than selection.

Optimal filtering and the dual process

We link optimal filtering for hidden Markov models to the notion of duality for Markov processes. We show that when the signal is dual to a process that has two components, one deterministic and one a pure death process, and with respect to functions that define changes of measure conjugate to the emission density, the filtering distributions evolve in the family of finite mixtures of such measures and the filter can be computed at a cost that is polynomial in the number of observations. Special cases of our framework include the Kalman filter, and computable filters for the Cox-Ingersoll-Ross process and the one-dimensional Wright-Fisher process, which have been investigated before. The dual we obtain for the Cox-Ingersoll-Ross process appears to be new in the literature.