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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 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]

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]

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.

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]

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 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]

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]

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]

The joint and accurate inference of selection and demography from genetic data is considered a particularly challenging question in population genetics, since both process may lead to very similar patterns of genetic diversity. However, additional information for disentangling these effects may be obtained by observing changes in allele frequencies over multiple time points. Such data are common in experimental evolution studies, as well as in the comparison of ancient and contemporary samples. Leveraging this information, however, has been computationally challenging, particularly when considering multilocus data sets. To overcome these issues, we introduce a novel, discrete approximation for diffusion processes, termed mean transition time approximation, which preserves the long-term behavior of the underlying continuous diffusion process. We then derive this approximation for the particular case of inferring selection and demography from time series data under the classic Wright-Fisher model and demonstrate that our approximation is well suited to describe allele trajectories through time, even when only a few states are used. We then develop a Bayesian inference approach to jointly infer the population size and locus-specific selection coefficients with high accuracy and further extend this model to also infer the rates of sequencing errors and mutations. We finally apply our approach to recent experimental data on the evolution of drug resistance in influenza virus, identifying likely targets of selection and finding evidence for much larger viral population sizes than previously reported. [ABSTRACT FROM AUTHOR]

Whales have 1000-fold more cells than humans and mice have 1000-fold fewer; however, cancer risk across species does not increase with the number of somatic cells and the lifespan of the organism. This observation is known as Peto's paradox. How much would evolution have to change the parameters of somatic evolution in order to equalize the cancer risk between species that differ by orders of magnitude in size? Analysis of previously published models of colorectal cancer suggests that a two- to three-fold decrease in the mutation rate or stem cell division rate is enough to reduce a whale's cancer risk to that of a human. Similarly, the addition of one to two required tumour-suppressor gene mutations would also be sufficient. We surveyed mammalian genomes and did not find a positive correlation of tumour-suppressor genes with increasing body mass and longevity. However, we found evidence of the amplification of TP53 in elephants, MAL in horses and FBXO31 in microbats, which might explain Peto's paradox in those species. Exploring parameters that evolution may have fine-tuned in large, long-lived organisms will help guide future experiments to reveal the underlying biology responsible for Peto's paradox and guide cancer prevention in humans. [ABSTRACT FROM AUTHOR]

We consider the classical Wright–Fisher model with mutation and selection. Mutations occur independently in each locus, and selection is performed according to the sharp peak landscape. In the asymptotic regime studied in Cerf (2014), a quasispecies is formed. We find explicitly the distribution of this quasispecies, which turns out to be the same distribution as for the Moran model. [ABSTRACT FROM AUTHOR]

In this paper we present a new framework of idealization in biology. We characterize idealizations as a network of counterfactual and hypothetical conditionals that can exhibit different 'degrees of contingency'. We use this idea to say that, in departing more or less from the actual world, idealizations can serve numerous epistemic, methodological or heuristic purposes within scientific research. We defend that, in part, this structure explains why idealizations, despite being deformations of reality, are so successful in scientific practice. For illustrative purposes, we provide an example from population genetics, the Wright-Fisher Model. [ABSTRACT FROM AUTHOR]

Abstract: The behaviour of ion channels within cardiac and neuronal cells is intrinsically stochastic in nature. When the number of channels is small this stochastic noise is large and can have an impact on the dynamics of the system which is potentially an issue when modelling small neurons and drug block in cardiac cells. While exact methods correctly capture the stochastic dynamics of a system they are computationally expensive, restricting their inclusion into tissue level models and so approximations to exact methods are often used instead. The other issue in modelling ion channel dynamics is that the transition rates are voltage dependent, adding a level of complexity as the channel dynamics are coupled to the membrane potential. By assuming that such transition rates are constant over each time step, it is possible to derive a stochastic differential equation (SDE), in the same manner as for biochemical reaction networks, that describes the stochastic dynamics of ion channels. While such a model is more computationally efficient than exact methods we show that there are analytical problems with the resulting SDE as well as issues in using current numerical schemes to solve such an equation. We therefore make two contributions: develop a different model to describe the stochastic ion channel dynamics that analytically behaves in the correct manner and also discuss numerical methods that preserve the analytical properties of the model. [ABSTRACT FROM AUTHOR]

One of the most important parameters in the application of genetic algorithms (GAs) is the population size N. In many cases, the choice of N determines the quality of the solutions obtained. The study of GAs with a finite population size requires a stochastic treatment of evolution. In this study, we examined the effects of genetic fluctuations on the performance of GA calculations. We considered the role of crossover by using the stochastic schema theory within the framework of the Wright-Fisher model of Markov chains. We also applied the diffusion approximation of the Wright-Fisher model. In numerical experiments, we studied effects of population size N and crossover rate pc on the success probability S. The success probability S is defined as the probability of obtaining the optimum solution within the limit of reaching the stationary state. We found that in a GA with pc, the diffusion equation can reproduce the success probability S. We also noted the role of crossover, which greatly increases S. [ABSTRACT FROM AUTHOR]

The goal of this paper is to provide formulae for the expectation and variance of the height and length of the ancestral recombination graph (ARG). While the formula for the expectation of the height is known (see, e.g. Krone and Neuhauser (1997)), the other formulae seem to be new. We obtain in particular (see Theorem 4.1) a very simple formula which expresses the expectation of the length of the ARG as a linear combination of the expectations of both the length of the coalescent tree and the height of the ARO. Finally, we study the speed at which the ARG comes down from infinity. [ABSTRACT FROM AUTHOR]

Computer simulation is an essential tool in the analysis of DNA sequence variation for mapping events of recent adaptive evolution in the genome. Various simulation methods are employed to predict the signature of selection in sequence variation. The most informative and efficient method currently in use is coalescent simulation. However, this method is limited to simple models of directional selection. Whole-population forward-in-time simulations are the alternative to coalescent simulations for more complex models. The notorious problem of excessive computational cost in forward-in-time simulations can be overcome by various simplifying amendments. Overall, the success of simulations depends on the creative application of some population genetic theory to the simulation algorithm. [ABSTRACT FROM AUTHOR]

An alternative version of the necessary and sufficient condition for almost sure fixation in the conditional branching process model is derived. This formulation provides an insight into why the examples considered in Buckley and Seneta (1983) all have the same condition for fixation. [ABSTRACT FROM AUTHOR]

Clonal interference, the competition between lineages arising from different beneficial mutations in an asexually reproducing population, is an important factor determining the tempo and mode of microbial adaptation. The standard theory of this phenomenon neglects the occurrence of multiple mutations as well as the correlation between loss by genetic drift and clonal competition, which is questionable in large populations. Working within the Wright-Fisher model with multiplicative fitness (no epistasis), we determine the rate of adaptation asymptotically for very large population sizes and show that the standard theory fails in this regime. Our study also explains the success of the standard theory in predicting the rate of adaptation for moderately large populations. Furthermore, we show that the nature of the substitution process changes qualitatively when multiple mutations are allowed for, because several mutations can be fixed in a single fixation event. As a consequence, the index of dispersion for counts of the fixation process displays a minimum as a function of population size, whereas the origination process of fixed mutations becomes completely regular for very large populations. We find that the number of mutations fixed in a single event is geometrically distributed as in the neutral case. These conclusions are based on extensive simulations combined with analytic results for the limit of infinite population size. [ABSTRACT FROM AUTHOR]

In this study, we investigated the influence of genetic drift on the performance of a genetic algorithm on the multiplicative landscape. We performed a theoretical investigation of the frequencies of the first-order schemata, and calculated their changes in time by using the Wright–Fisher model. We showed that this mathematical theory reasonably predicts various quantities, including the ultimate distribution of the first-order schemata. [ABSTRACT FROM AUTHOR]