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The theory of selection fluctuation between generations has been a topic with much activities in population genetics and molecular evolution in 1970's. Most studies suggested that, as the result of fluctuating selection between generations, the frequency of an (on average) neutral mutation may fluctuate around 0.5 during the long-term evolution before it was ultimately fixed or lost. However, this pattern can only be derived from a specific type Wright-Fisher additive model, coined by the Nei-Yokoyama puzzle. In this commentary, I revisited this issue and figured out a theoretical assumption that has never been claimed explicitly, the notion of reference phenotype. Consider one locus with two-alleles: A is the wildtype allele and A' is the mutation. The fluctuating selection model actually requires a constraint that one of three genotypes (AA, AA', or A'A') must maintain a constant fitness without fluctuating between generations. It appears that the balancing selection at a frequency of 0.5 emerges only when the heterozygote (AA') is the reference genotype. Because it is difficult to determine which genotype could be the reference genotype in a real population, a desirable population genetics model should take all three possibilities into account. To this end, I propose a mixture model, where each genotype has a certain chance to be the reference genotype. My analysis showed that the emergence of balancing selection depends on the relative proportions of three different reference genotypes. [ABSTRACT FROM AUTHOR]

We study the response of a quantitative trait to exponential directional selection in a finite haploid population, both at the genetic and the phenotypic level. We assume an infinite sites model, in which the number of new mutations per generation in the population follows a Poisson distribution (with mean Θ) and each mutation occurs at a new, previously monomorphic site. Mutation effects are beneficial and drawn from a distribution. Sites are unlinked and contribute additively to the trait. Assuming that selection is stronger than random genetic drift, we model the initial phase of the dynamics by a supercritical Galton–Watson process. This enables us to obtain time-dependent results. We show that the copy-number distribution of the mutant in generation n , conditioned on non-extinction until n , is described accurately by the deterministic increase from an initial distribution with mean 1. This distribution is related to the absolutely continuous part W + of the random variable, typically denoted W , that characterizes the stochasticity accumulating during the mutant's sweep. A suitable transformation yields the approximate dynamics of the mutant frequency distribution in a Wright–Fisher population of size N. Our expression provides a very accurate approximation except when mutant frequencies are close to 1. On this basis, we derive explicitly the (approximate) time dependence of the expected mean and variance of the trait and of the expected number of segregating sites. Unexpectedly, we obtain highly accurate approximations for all times, even for the quasi-stationary phase when the expected per-generation response and the trait variance have equilibrated. The latter refine classical results. In addition, we find that Θ is the main determinant of the pattern of adaptation at the genetic level, i.e., whether the initial allele-frequency dynamics are best described by sweep-like patterns at few loci or small allele-frequency shifts at many. The number of segregating sites is an appropriate indicator for these patterns. The selection strength determines primarily the rate of adaptation. The accuracy of our results is tested by comprehensive simulations in a Wright–Fisher framework. We argue that our results apply to more complex forms of directional selection. • Accurate and tractable time-dependent results at genetic and phenotypic level. • Explicit approximations for the initial phase of polygenic adaptation. • Refined classical, quantitative genetics results for the stationary phase. • Population-wide mutation rate characterizes genetic pattern of adaptation. • Number of segregating sites is an indicator of the pattern of adaptation. [ABSTRACT FROM AUTHOR]

Population medical genetics aims at translating clinically relevant findings from recent studies of large cohorts into healthcare for individuals. Genetic counseling concerning reproductive risks and options is still mainly based on family history, and consanguinity is viewed to increase the risk for recessive diseases regardless of the demographics. However, in an increasingly multi‐ethnic society with diverse approaches to partner selection, healthcare professionals should also sharpen their intuition for the influence of different mating schemes in non‐equilibrium dynamics. We, therefore, revisited the so‐called out‐of‐Africa model and studied in forward simulations with discrete and not overlapping generations the effect of inbreeding on the average number of recessive lethals in the genome. We were able to reproduce in both frameworks the drop in the incidence of recessive disorders, which is a transient phenomenon during and after the growth phase of a population, and therefore showed their equivalence. With the simulation frameworks, we also provide the means to study and visualize the effect of different kin sizes and mating schemes on these parameters for educational purposes. [ABSTRACT FROM AUTHOR]

We construct a reliable estimation method for evolutionary parameters within the Wright-Fisher model, which describes changes in allele frequencies due to selection and genetic drift, from time-series data. Such data exist for biological populations, for example via artificial evolution experiments, and for the cultural evolution of behavior, such as linguistic corpora that document historical usage of different words with similar meanings. Our method of analysis builds on a Beta-with-Spikes approximation to the distribution of allele frequencies predicted by the Wright-Fisher model. We introduce a self-contained scheme for estimating parameters in the approximation, and demonstrate its robustness with synthetic data, especially in the strong-selection and near-extinction regimes where previous approaches fail. We further apply the method to allele frequency data for baker's yeast (Saccharomyces cerevisiae), finding a significant signal of selection in cases where independent evidence supports such a conclusion. We further demonstrate the possibility of detecting time points at which evolutionary parameters change in the context of a historical spelling reform in the Spanish language. [ABSTRACT FROM AUTHOR]

Growing evidence suggests that temporally fluctuating environments are important in maintaining variation both within and between species. To date, however, studies of genetic variation within a population have been largely conducted by evolutionary biologists (particularly population geneticists), while population and community ecologists have concentrated more on diversity at the species level. Despite considerable conceptual overlap, the commonalities and differences of these two alternative paradigms have yet to come under close scrutiny. Here, we review theoretical and empirical studies in population genetics and community ecology focusing on the 'temporal storage effect' and synthesise theories of diversity maintenance across different levels of biological organisation. Drawing on Chesson's coexistence theory, we explain how temporally fluctuating environments promote the maintenance of genetic variation and species diversity. We propose a further synthesis of the two disciplines by comparing models employing traditional frequency‐dependent dynamics and those adopting density‐dependent dynamics. We then address how temporal fluctuations promote genetic and species diversity simultaneously via rapid evolution and eco‐evolutionary dynamics. Comparing and synthesising ecological and evolutionary approaches will accelerate our understanding of diversity maintenance in nature. [ABSTRACT FROM AUTHOR]

Experimentalists observe allele frequency distributions and try to infer mutation rates and selection coefficients. How easy is this? We calculate limits to their ability in the context of the Wright-Fisher model by first finding the maximal amount of information that can be acquired using allele frequencies about the mutation rate and selection coefficient– at least 2 bits per allele– and then by finding how the organisms would have shaped their mutation rates and selection coefficients so as to maximize the information transfer. [ABSTRACT FROM AUTHOR]

We study the problem of the first passage time through a constant boundary for a jump diffusion process whose infinitesimal generator is a nonlocal Jacobi operator. Due to the lack of analytical results, we address the problem using a discretization scheme for simulating the trajectories of jump diffusion processes with state-dependent jumps in both frequency and amplitude. We obtain numerical approximations on their first passage time probability density functions and results for the qualitative behavior of other statistics of this random variable. Finally, we provide two examples of application of the method for different choices of the distribution involved in the mechanism of generation of the jumps. [ABSTRACT FROM AUTHOR]

1. The Wright-Fisher model, which directs how matings occur and how genes are transmitted across generations, has long been a lynchpin of evolutionary biology. This model is elegantly simple, analytically tractable and easy to implement, but it has one serious limitation: essentially no real species satisfies its many assumptions. With growing awareness of the importance of jointly considering both ecology and evolution in eco-evolutionary models, this limitation has become more apparent, causing many researchers to search for more realistic simulation models. 2. A recently described variation retains most of the Wright-Fisher simplicity but provides greater flexibility to accommodate departures from model assumptions. This generalized Wright-Fisher model relaxes the assumption that all individuals have identical expected reproductive success by introducing a vector of parental weights w that specifies relative probabilities different individuals have of producing offspring. With parental weights specified this way, expectations of key demographic parameters are simple functions of w. This allows researchers to quantitatively predict the consequences of non-Wright- Fisher features incorporated into their models. 3. An important limitation of the Wright-Fisher model is that it assumes discrete generations, whereas most real species are age structured. Here I show how an algorithm (TheWeight) that implements the generalized Wright-Fisher model can be used to model evolution in age-structured populations with overlapping generations. Worked examples illustrate simulation of seasonal and lifetime reproductive success and show how the user can pick vectors of weights expected to produce a desired level of reproductive skew or a desired Ne/N ratio. Alternatively, weights can be associated with heritable traits to provide a simple, quantitative way to model natural selection. Using TheWeight, it is easy to generate positive or negative correlations of individual reproductive success over time, thus allowing explicit modelling of common biological processes like skip breeding and persistent individual differences. 4. Code is provided to implement basic features of TheWeight and applications described here, including one scenario implemented in SLiM. However, required coding changes to the Wright-Fisher model are modest, so the real value of the new algorithm is to encourage users to adopt its features into their own or others' models. [ABSTRACT FROM AUTHOR]

We introduce a new model for plant metapopulations with a seed bank component, living in a fragmented environment in which local extinction events are frequent. This model is an intermediate between population dynamics models with a seed bank component, based on the classical Wright–Fisher model, and Stochastic Patch Occupancy Models (SPOMs) used in metapopulation ecology. Its main feature is the use of "ghost" individuals, which can reproduce but with a very strong selective disadvantage against "real" individuals, to artificially ensure a constant population size. We show the existence of an extinction threshold above which persistence of the subpopulation of "real" individuals is not possible, and investigate how the seed bank characteristics affect this extinction threshold. We also show the convergence of the model to a SPOM under an appropriate scaling, bridging the gap between individual-based models and occupancy models. [ABSTRACT FROM AUTHOR]

There are many problems in biology and related disciplines involving stochasticity, where a signal can only be detected when it lies above a threshold level, while signals lying below threshold are simply not detected. A consequence is that the detected signal is conditioned to lie above threshold, and is not representative of the actual signal. In this work, we present some general results for the conditioning that occurs due to the existence of such an observational threshold. We show that this conditioning is relevant, for example, to gene-frequency trajectories, where many loci in the genome are simultaneously measured in a given generation. Such a threshold can lead to severe biases of allele frequency estimates under purifying selection. In the analysis presented, within the context of Markov chains such as the Wright–Fisher model, we address two key questions: (1) "What is a natural measure of the strength of the conditioning associated with an observation threshold?" (2) "What is a principled way to correct for the effects of the conditioning?". We answer the first question in terms of a proportion. Starting with a large number of trajectories, the relevant quantity is the proportion of these trajectories that are above threshold at a later time and hence are detected. The smaller the value of this proportion, the stronger the effects of conditioning. We provide an approximate analytical answer to the second question, that corrects the bias produced by an observation threshold, and performs to reasonable accuracy in the Wright–Fisher model for biologically plausible parameter values. [ABSTRACT FROM AUTHOR]

Many organisms alternate between distinct haploid and diploid phases, which generates population structure according to ploidy level. In this research, we consider a haploid-diploid population using statistical approaches developed for spatially subdivided populations, where haploids represent one "patch" and diploids another "patch". In species with alternating generations, sexual reproduction causes movement from diploids to haploids (by meiosis with recombination) and from haploids to diploids (by syngamy). Thus, an allele in one ploidy phase can be said to "migrate" to the other ploidy phase by sexual reproduction and to "remain" in the same ploidy phase by asexual reproduction. By analyzing a coalescent model of the probability of identity by descent and by state for a haploid-diploid system, we define FST-like measures of differentiation between haploids and diploids and show that these measures can be simplified as a function of the extent of sexuality in each ploidy phase. We conduct simulations with an infinite-alleles model and discuss a method for estimating the degree of effective sexuality from genetic data sets that uses the observed FST measures of haploid-diploid species., (© The Author(s) 2024. Published by Oxford University Press on behalf of the European Society of Evolutionary Biology.)

The majority of population genetic theory assumes fully haploid or diploid organisms with obligate sexuality, despite complex life cycles with alternating generations being commonly observed. To reveal how natural selection and genetic drift shape the evolution of haploid–diploid populations, we analyze a stochastic genetic model for populations that consist of a mixture of haploid and diploid individuals, allowing for asexual reproduction and niche separation between haploid and diploid stages. Applying a diffusion approximation, we derive the fixation probability and describe its dependence on the reproductive values of haploid and diploid stages, which depend strongly on the extent of asexual reproduction in each phase and on the ecological differences between them. • Classical models consider fully haploid or diploid populations. • We model haploid–diploid life cycles allowing for asexual reproduction. • We obtain the fixation probability of alleles subject to selection and drift. • Reproductive values of haploid and diploid stages shape their evolution. [ABSTRACT FROM AUTHOR]

We construct weak approximations of the Wright-Fisher model and illustrate their accuracy by simulation examples.

We consider disease-causing mutations that are lethal when homozygous. Lethality involves the very strongest form of negative selection, with the selection coefficient against the disease-carrying homozygote having its maximum value of unity. We determine results for the behavior of the frequency of a lethal allele in an effectively infinite population. This includes an estimate of the time it takes to achieve equilibrium, and a description of transient behavior associated with a sudden change in the fitness of the heterozygote. We determine analogous results for a finite population, showing that a lethal disease-causing allele needs to be described by a modified Wright-Fisher model, which deviates from the standard model, where selection coefficients are assumed small compared with 1. We show that a by-product of the dynamics, resulting from the absence of the disease-allele carrying homozygote in adults, is the general constraint that the frequency of the disease-causing allele cannot exceed 12. The results presented in this work should prove useful to a number of areas including analysis of lethal/near lethal mutations in Mendelian disorders and, in particular, for exploring how mutation-selection-drift balance explains the current spectrum of mutation frequencies in humans. While the number of empirical examples of overdominance in lethal genetic disorders is not large, relatively high observed heterozygote frequencies may be a hint of transient heterozygous advantage in nature. For lethal disorders with anomalous frequencies, such as cystic fibrosis and Tay-Sachs, our analysis lends further support to the role that transitory episodes of weak overdominance may play in the evolution of lethal mutations.

The within‐host evolutionary dynamics of tuberculosis (TB) remain unclear, and underlying biological characteristics render standard population genetic approaches based upon the Wright‐Fisher model largely inappropriate. In addition, the compact genome combined with an absence of recombination is expected to result in strong purifying selection effects. Thus, it is imperative to establish a biologically relevant evolutionary framework incorporating these factors in order to enable an accurate study of this important human pathogen. Further, such a model is critical for inferring fundamental evolutionary parameters related to patient treatment, including mutation rates and the severity of infection bottlenecks. We here implement such a model and infer the underlying evolutionary parameters governing within‐patient evolutionary dynamics. Results demonstrate that the progeny skew associated with the clonal nature of TB severely reduces genetic diversity and that the neglect of this parameter in previous studies has led to significant mis‐inference of mutation rates. As such, our results suggest an underlying de novo mutation rate that is considerably faster than previously inferred, and a progeny distribution differing significantly from Wright‐Fisher assumptions. This inference represents a more appropriate evolutionary null model, against which the periodic effects of positive selection, associated with drug‐resistance for example, may be better assessed. [ABSTRACT FROM AUTHOR]

We consider disease-causing mutations that are lethal when homozygous. Lethality involves the very strongest form of negative selection, with the selection coefficient against the disease-carrying homozygote having its maximum value of unity. We determine results for the behavior of the frequency of a lethal allele in an effectively infinite population. This includes an estimate of the time it takes to achieve equilibrium, and a description of transient behavior associated with a sudden change in the fitness of the heterozygote. We determine analogous results for a finite population, showing that a lethal disease-causing allele needs to be described by a modified Wright-Fisher model, which deviates from the standard model, where selection coefficients are assumed small compared with 1. We show that a by-product of the dynamics, resulting from the absence of the disease-allele carrying homozygote in adults, is the general constraint that the frequency of the disease-causing allele cannot exceed 1 2 . The results presented in this work should prove useful to a number of areas including analysis of lethal/near lethal mutations in Mendelian disorders and, in particular, for exploring how mutation-selection-drift balance explains the current spectrum of mutation frequencies in humans. While the number of empirical examples of overdominance in lethal genetic disorders is not large, relatively high observed heterozygote frequencies may be a hint of transient heterozygous advantage in nature. For lethal disorders with anomalous frequencies, such as cystic fibrosis and Tay-Sachs, our analysis lends further support to the role that transitory episodes of weak overdominance may play in the evolution of lethal mutations. [ABSTRACT FROM AUTHOR]

Forward Wright–Fisher simulations are powerful in their ability to model complex demography and selection scenarios, but suffer from slow execution on the Central Processor Unit (CPU), thus limiting their usefulness. However, the single-locus Wright–Fisher forward algorithm is exceedingly parallelizable, with many steps that are so-called “embarrassingly parallel,” consisting of a vast number of individual computations that are all independent of each other and thus capable of being performed concurrently. The rise of modern Graphics Processing Units (GPUs) and programming languages designed to leverage the inherent parallel nature of these processors have allowed researchers to dramatically speed up many programs that have such high arithmetic intensity and intrinsic concurrency. The presented GPU Optimized Wright–Fisher simulation, or “GO Fish” for short, can be used to simulate arbitrary selection and demographic scenarios while running over 250-fold faster than its serial counterpart on the CPU. Even modest GPU hardware can achieve an impressive speedup of over two orders of magnitude. With simulations so accelerated, one can not only do quick parametric bootstrapping of previously estimated parameters, but also use simulated results to calculate the likelihoods and summary statistics of demographic and selection models against real polymorphism data, all without restricting the demographic and selection scenarios that can be modeled or requiring approximations to the single-locus forward algorithm for efficiency. Further, as many of the parallel programming techniques used in this simulation can be applied to other computationally intensive algorithms important in population genetics, GO Fish serves as an exciting template for future research into accelerating computation in evolution. GO Fish is part of the Parallel PopGen Package available at: http://dl42.github.io/ParallelPopGen/.

We explore the effect of different mechanisms of natural selection on the evolution of populations for one- and two-locus systems. We compare the effect of viability and fecundity selection in the context of the Wright-Fisher model with selection under the assumption of multiplicative fitness. We show that these two modes of natural selection correspond to different orderings of the processes of population regulation and natural selection in the Wright-Fisher model. We find that under the Wright-Fisher model these two different orderings can affect the distribution of trajectories of haplotype frequencies evolving with genetic recombination. However, the difference in the distribution of trajectories is only appreciable when the population is in significant linkage disequilibrium. We find that as linkage disequilibrium decays the trajectories for the two different models rapidly become indistinguishable. We discuss the significance of these findings in terms of biological examples of viability and fecundity selection, and speculate that the effect may be significant when factors such as gene migration maintain a degree of linkage disequilibrium.

We introduce a general solution concept for the Fokker–Planck (Kolmogorov) equation representing the diffusion limit of the Wright–Fisher model of random genetic drift for an arbitrary number of alleles at a single locus. This solution will continue beyond the transitions from the loss of alleles, that is, it will naturally extend to the boundary strata of the probability simplex on which the diffusion is defined. This also takes care of the degeneracy of the diffusion operator at the boundary. We shall then show the existence and uniqueness of a solution. From this solution, we can readily deduce information about the evolution of a Wright–Fisher population. [ABSTRACT FROM AUTHOR]

Abstract We consider two population models subject to the evolutionary forces of selection and mutation, the Moran model and the Λ-Wright–Fisher model. In such models the block counting process traces back the number of potential ancestors of a sample of the population at present. Under some conditions the block counting process is positive recurrent and its stationary distribution is described via a linear system of equations. In this work, we first characterise the measures Λ leading to a geometric stationary distribution, the Bolthausen–Sznitman model being the most prominent example having this feature. Next, we solve the linear system of equations corresponding to the Moran model. For the Λ-Wright–Fisher model we show that the probability generating function associated to the stationary distribution of the block counting process satisfies an integro differential equation. We solve the latter for the Kingman model and the star-shaped model. [ABSTRACT FROM AUTHOR]

For multivariant Wright–Fisher models in population genetics, we introduce equilibrium states, expressed by fluctuations of probability ratio, in contrast to the traditionally used fluctuations, expressed by the difference between the current value of the random process and its equilibrium value. Then the drift component of the dynamic process of gene frequencies, primarily expressed as a ratio of two quadratic forms, is transformed into a cubic parabola with a certain normalization factor. [ABSTRACT FROM AUTHOR]

During his well-known debate with Fisher regarding the phenotypic dataset of Panaxia dominula, Wright suggested fluctuating selection as a potential explanation for the observed change in allele frequencies. This model has since been invoked in a number of analyses, with the focus of discussion centering mainly on random or oscillatory fluctuations of selection intensities. Here, we present a novel method to consider nonrandom changes in selection intensities using Wright-Fisher approximate Bayesian (ABC)-based approaches, in order to detect and evaluate a change in selection strength from time-sampled data. This novel method jointly estimates the position of a change point as well as the strength of both corresponding selection coefficients (and dominance for diploid cases) from the allele trajectory. The simulation studies of this method reveal the combinations of parameter ranges and input values that optimize performance, thus indicating optimal experimental design strategies. We apply this approach to both the historical dataset of P. dominula in order to shed light on this historical debate, as well as to whole-genome time-serial data from influenza virus in order to identify sites with changing selection intensities in response to drug treatment.

Abstract We provide a general mathematical framework based on the theory of graphical models to study admixture graphs. Admixture graphs are used to describe the ancestral relationships between past and present populations, allowing for population merges and migration events, by means of gene flow. We give various mathematical properties of admixture graphs with particular focus on properties of the so-called F -statistics. Also the Wright–Fisher model is studied and a general expression for the loss of heterozygosity is derived. [ABSTRACT FROM AUTHOR]

Abstract It is well-known that for a one dimensional stochastic differential equation driven by Brownian noise, with coefficient functions satisfying the assumptions of the Yamada–Watanabe theorem (Yamada and Watanabe, 1971, [31,32]) and the Feller test for explosions (Feller, 1951, 1954), there exists a unique stationary distribution with respect to the Markov semigroup of transition probabilities. We consider systems on a restricted domain D of the phase space R and study the rate of convergence to the stationary distribution. Using a geometrical approach that uses the so called free energy function on the density function space, we prove that the density functions, which are solutions of the Fokker–Planck equation, converge to the stationary density function exponentially under the Kullback–Leibler divergence, thus also in the total variation norm. The results show that there is a relation between the Bakry–Émery curvature dimension condition and the dissipativity condition of the transformed system under the Fisher–Lamperti transformation. Several applications are discussed, including the Cox–Ingersoll–Ross model and the Ait-Sahalia model in finance and the Wright–Fisher model in population genetics. [ABSTRACT FROM AUTHOR]

The distribution of allele frequencies obtained from diffusion approximations to Wright–Fisher models is useful in developing intuition about the population level effects of evolutionary processes. The statistical properties of the stationary distributions of K -allele models have been extensively studied under neutrality or under selection. Here, we introduce a new family of Wright–Fisher models in which there are two hierarchical levels of genetic variability. The genotypes composed of alleles differing from each other at the selected level have fitness differences with respect to each other and evolve under selection. The genotypes composed of alleles differing from each other only at the neutral level have the same fitness and evolve under neutrality. We show that with an appropriate scaling of the mutation parameter with respect to the number of alleles at each level, the frequencies of alleles at the selected and the neutral level are conditionally independent of each other, conditional on knowing the number of alleles at all levels. This conditional independence allows us to simulate from the joint stationary distribution of the allele frequencies. We use these simulated frequencies to perform inference on parameters of the model with two levels of genetic variability using Approximate Bayesian Computation. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The Kingman coalescent is a commonly used model in genetics, which is often justified with reference to the Wright-Fisher (WF) model. Current proofs of convergence of WF and other models to the Kingman coalescent assume a constant sample size. However, sample sizes have become quite large in human genetics. Therefore, we develop a convergence theory that allows the sample size to increase with population size. If the haploid population size is N and the sample size is N 1 ∕ 3 − ϵ , ϵ > 0 , we prove that Wright-Fisher genealogies involve at most a single binary merger in each generation with probability converging to 1 in the limit of large N . Single binary merger or no merger in each generation of the genealogy implies that the Kingman partition distribution is obtained exactly. If the sample size is N 1 ∕ 2 − ϵ , Wright-Fisher genealogies may involve simultaneous binary mergers in a single generation but do not involve triple mergers in the large N limit. The asymptotic theory is verified using numerical calculations. Variable population sizes are handled algorithmically. It is found that even distant bottlenecks can increase the probability of triple mergers as well as simultaneous binary mergers in WF genealogies. [ABSTRACT FROM AUTHOR]

In this work we present a systematic mathematical approximation scheme that exposes the way that information, about the evolutionary forces of selection and random genetic drift, is encoded within gene-frequency trajectories. We determine approximate, time-dependent, gene-frequency trajectory statistics, assuming additive selection. We use the probability of fixation to test and illustrate the approximation scheme introduced. For the case where the strength of selection and the effective population size have constant values, we show how a standard diffusion approximation result, for the probability of fixation, systematically emerges when increasing numbers of approximate trajectory statistics are taken into account. We then provide examples of how time-dependent parameters influence gene-frequency statistics. [ABSTRACT FROM AUTHOR]

The Wright–Fisher model is the most popular population model for describing the behaviour of evolutionary systems with a finite population size. Approximations have commonly been used but the model itself has rarely been tested against time-resolved genomic data. Here, we evaluate the extent to which it can be inferred as the correct model under a likelihood framework. Given genome-wide data from an evolutionary experiment, we validate the Wright–Fisher drift model as the better option for describing evolutionary trajectories in a finite population. This was found by evaluating its performance against a Gaussian model of allele frequency propagation. However, we note a range of circumstances under which standard Wright–Fisher drift cannot be correctly identified. [ABSTRACT FROM AUTHOR]

We are interested in the numerical approximation of non-linear stochastic differential equations (SDEs) with solution in a certain domain. Our goal is to construct explicit numerical schemes that preserve that structure. We generalize the semi-discrete method (Halidias and Stamatiou, 2016), and propose a numerical scheme, for which we prove a strong convergence result, to a class of SDEs that appears in population dynamics and ion channel dynamics within cardiac and neuronal cells. We furthermore extend our scheme to a multidimensional case. [ABSTRACT FROM AUTHOR]

Random genetic drift, or stochastic change in gene frequency, is a fundamental evolutionary force that is usually defined within the ideal Wright-Fisher (WF) population. However, as the theory is increasingly applied to populations that deviate strongly from the ideal model, a paradox of random drift has emerged. When drift is defined by theWF model, it becomes stronger as the population size, N, decreases. However, the intensity of competition decreases when N decreases and, hence, drift might become weaker. To resolve the paradox, we propose that random drift be defined by the variance of "individual output", V(k) [k being the progeny number of each individual with the mean of E(k)], rather than by the WF sampling. If the distribution of k is known for any population, its strength of drift relative to a WF population of the same size, N, can be calculated. Generally, E(k) andV(k) should be density dependent but their relationships are different with or without competition, leading to opposite predictions on the efficiency of random drift as N changes. We apply the "individual output" model to asexual cell populations that are either unregulated (such as tumors) or negatively densitydependent (e.g., bacteria). In such populations, the efficiency of drift could be as low as<10% of that inWF populations. Interestingly, when N is below the carrying capacity, random drift could in fact increase as N increases. Growing asexual populations, especially tumors, may therefore be genetically even more heterogeneous than the high diversity estimated by some conventional models. [ABSTRACT FROM AUTHOR]

Recently, Burden and Tang (2016) provided an analytical expression for the stationary distribution of the multivariate neutral Wright–Fisher model with low mutation rates. In this paper we present a simple, alternative derivation that illustrates the approximation. Our proof is based on the discrete multivariate boundary mutation model which has three key ingredients. First, the decoupled Moran model is used to describe genetic drift. Second, low mutation rates are assumed by limiting mutations to monomorphic states. Third, the mutation rate matrix is separated into a time-reversible part and a flux part, as suggested by Burden and Tang (2016). An application of our result to data from several great apes reveals that the assumption of stationarity may be inadequate or that other evolutionary forces like selection or biased gene conversion are acting. Furthermore we find that the model with a reversible mutation rate matrix provides a reasonably good fit to the data compared to the one with a non-reversible mutation rate matrix. [ABSTRACT FROM AUTHOR]

A novel explicit time-stepping scheme, called Lamperti smooth sloping truncation (LSST) scheme, is devised in this paper to strongly approximate the Wright–Fisher model, whose coefficients violate the Lipschitz condition and whose solution process takes values in a bounded domain. The LSST scheme is constructed by combining the Lamperti-type transformation and the smooth sloping truncation. Under appropriate condition, it is proved that the convergence order of the LSST scheme can be up to one. Moreover, it is shown that the proposed scheme has a unique stationary distribution, which converges to that of the original model. Numerical examples are reported to confirm our theoretical findings. [ABSTRACT FROM AUTHOR]

Classical population genetic theory generally assumes either a fully haploid or fully diploid life cycle. However, many organisms exhibit more complex life cycles, with both free-living haploid and diploid stages. Here we ask what the probability of fixation is for selected alleles in organisms with haploid-diploid life cycles. We develop a genetic model that considers the population dynamics using both the Moran model and Wright-Fisher model. Applying a branching process approximation, we obtain an accurate fixation probability assuming that the population is large and the net effect of the mutation is beneficial. We also find the diffusion approximation for the fixation probability, which is accurate even in small populations and for deleterious alleles, as long as selection is weak. These fixation probabilities from branching process and diffusion approximations are similar when selection is weak for beneficial mutations that are not fully recessive. In many cases, particularly when one phase predominates, the fixation probability differs substantially for haploid-diploid organisms compared to either fully haploid or diploid species. [ABSTRACT FROM AUTHOR]