Theoretical Study of near Neutrality. I. Heterozygosity and Rate of Mutant Substitution

In order to clarify the nature of "near neutrality" in molecular evolution and polymorphism, extensive simulation studies were performed. Selection coefficients of new mutations are assumed to be small so that both random genetic drift and selection contribute to determining the behavior of mutants. The model also incorporates normally distributed spatial fluctuation of selection coefficients. If the system starts from "average neutrality," it will move to a better adapted state, and most new mutations will become "slightly deleterious." Monte Carlo simulations have indicated that such adaptation is attained, but that the rate of such "progress" is very low for weak selection. In general, the larger the population size, the more effective the selection becomes. Also, as selection becomes weaker, the behavior of the mutants approaches that of completely neutral genes. Thus, the weaker the selection, the smaller is the effect of population size on mutant dynamics. Increase of heterozygosity with population size is very pronounced for subdivided populations. The significance of these results is discussed in relation to various observed facts on molecular evolution and polymorphism, such as generation-time dependency and overdispersion of the molecular clock, or contrasting patterns of DNA and protein polymorphism among some closely related species.     

The Effect of Deleterious Mutations on Neutral Molecular Variation

Selection against deleterious alleles maintained by mutation may cause a reduction in the amount of genetic variability at linked neutral sites. This is because a new neutral variant can only remain in a large population for a long period of time if it is maintained in gametes that are free of deleterious alleles, and hence are not destined for rapid elimination from the population by selection. Approximate formulas are derived for the reduction below classical neutral values resulting from such background selection against deleterious mutations, for the mean times to fixation and loss of new mutations, nucleotide site diversity, and number of segregating sites. These formulas apply to random-mating populations with no genetic recombination, and to populations reproducing exclusively asexually or by self-fertilization. For a given selection regime and mating system, the reduction is an exponential function of the total mutation rate to deleterious mutations for the section of the genome involved. Simulations show that the effect decreases rapidly with increasing recombination frequency or rate of outcrossing. The mean time to loss of new neutral mutations and the total number of segregating neutral sites are less sensitive to background selection than the other statistics, unless the population size is of the order of a hundred thousand or more. The stationary distribution of allele frequencies at the neutral sites is correspondingly skewed in favor of rare alleles, compared with the classical neutral result. Observed reductions in molecular variation in low recombination genomic regions of sufficiently large size, for instance in the centromere-proximal regions of Drosophila autosomes or in highly selfing plant populations, may be partly due to background selection against deleterious mutations.     

The nearly neutral and selection theories of molecular evolution under the fisher geometrical framework: substitution rate, population size, and complexity

The general theories of molecular evolution depend on relatively arbitrary assumptions about the relative distribution and rate of advantageous, deleterious, neutral, and nearly neutral mutations. The Fisher geometrical model (FGM) has been used to make distributions of mutations biologically interpretable. We explored an FGM-based molecular model to represent molecular evolutionary processes typically studied by nearly neutral and selection models, but in which distributions and relative rates of mutations with different selection coefficients are a consequence of biologically interpretable parameters, such as the average size of the phenotypic effect of mutations and the number of traits (complexity) of organisms. A variant of the FGM-based model that we called the static regime (SR) represents evolution as a nearly neutral process in which substitution rates are determined by a dynamic substitution process in which the population's phenotype remains around a suboptimum equilibrium fitness produced by a balance between slightly deleterious and slightly advantageous compensatory substitutions. As in previous nearly neutral models, the SR predicts a negative relationship between molecular evolutionary rate and population size; however, SR does not have the unrealistic properties of previous nearly neutral models such as the narrow window of selection strengths in which they work. In addition, the SR suggests that compensatory mutations cannot explain the high rate of fixations driven by positive selection currently found in DNA sequences, contrary to what has been previously suggested. We also developed a generalization of SR in which the optimum phenotype can change stochastically due to environmental or physiological shifts, which we called the variable regime (VR). VR models evolution as an interplay between adaptive processes and nearly neutral steady-state processes. When strong environmental fluctuations are incorporated, the process becomes a selection model in which evolutionary rate does not depend on population size, but is critically dependent on the complexity of organisms and mutation size. For SR as well as VR we found that key parameters of molecular evolution are linked by biological factors, and we showed that they cannot be fixed independently by arbitrary criteria, as has usually been assumed in previous molecular evolutionary models.     

A Study on a Nearly Neutral Mutation Model in Finite Populations

As a nearly neutral mutation model, the house-of-cards model is studied in finite populations using computer simulations. The distribution of the mutant effect is assumed to be normal. The behavior is mainly determined by the product of the population size, N, and the standard deviation, sigma, of the distribution of the mutant effect. If 4N sigma is large compared to one, a few advantageous mutants are quickly fixed in early generations. Then most mutation becomes deleterious and very slow increase of the average selection coefficient follows. It takes very long for the population to reach the equilibrium state. Substitutions of alleles occur very infrequently in the later stage. If 4N sigma is the order of one or less, the behavior is qualitatively similar to that of the strict neutral case. Gradual increase of the average selection coefficient occurs and in generations of several times the inverse of the mutation rate the population almost reaches the equilibrium state. Both advantageous and neutral (including slightly deleterious) mutations are fixed. Except in the early stage, an increase of the standard deviation of the distribution of the mutant effect decreases the average heterozygosity. The substitution rate is reduced as 4N sigma is increased. Three tests of neutrality, one using the relationship between the average and the variance of heterozygosity, another using the relationship between the average heterozygosity and the average number of substitutions and Watterson's homozygosity test are applied to the consequences of the present model. It is found that deviation from the neutral expectation becomes apparent only when 4N sigma is more than two. Also a simple approximation for the model is developed which works well when the mutation rate is very small.     

Large number of replacement polymorphisms in rapidly evolving genes of Drosophila. Implications for genome-wide surveys of DNA polymorphism

We present a survey of nucleotide polymorphism of three novel, rapidly evolving genes in populations of Drosophila melanogaster and D. simulans. Levels of silent polymorphism are comparable to other loci, but the number of replacement polymorphisms is higher than that in most other genes surveyed in D. melanogaster and D. simulans. Tests of neutrality fail to reject neutral evolution with one exception. This concerns a gene located in a region of high recombination rate in D. simulans and in a region of low recombination rate in D. melanogaster, due to an inversion. In the latter case it shows a very low number of polymorphisms, presumably due to selective sweeps in the region. Patterns of nucleotide polymorphism suggest that most substitutions are neutral or nearly neutral and that weak (positive and purifying) selection plays a significant role in the evolution of these genes. At all three loci, purifying selection of slightly deleterious replacement mutations appears to be more efficient in D. simulans than in D. melanogaster, presumably due to different effective population sizes. Our analysis suggests that current knowledge about genome-wide patterns of nucleotide polymorphism is far from complete with respect to the types and range of nucleotide substitutions and that further analysis of differences between local populations will be required to understand the forces more completely. We note that rapidly diverging and nearly neutrally evolving genes cannot be expected only in the genome of Drosophila, but are likely to occur in large numbers also in other organisms and that their function and evolution are little understood so far.     

The efficiency of purifying selection in Mammals vs. Drosophila for metabolic genes

The nearly neutral theory of molecular evolution states that the efficiency of natural selection depends on the effective population size. By using a wide range of multispecies data on nucleotide polymorphism, we have tried to ascertain whether there are any differences in the level of selective constraints of metabolic process genes between Mammals and Drosophila species. The results are consistent with a higher selective constraint in Drosophila than in Mammals, according to the expected under the nearly neutral model: purifying selection seems to be more efficient in species with a larger effective population size.     

Studying models of balancing selection using phase-type theory

Balancing selection (BLS) is the evolutionary force that maintains high levels of genetic variability in many important genes. To further our understanding of its evolutionary significance, we analyze models with BLS acting on a biallelic locus: an equilibrium model with long-term BLS, a model with long-term BLS and recent changes in population size, and a model of recent BLS. Using phase-type theory, a mathematical tool for analyzing continuous time Markov chains with an absorbing state, we examine how BLS affects polymorphism patterns in linked neutral regions, as summarized by nucleotide diversity, the expected number of segregating sites, the site frequency spectrum, and the level of linkage disequilibrium (LD). Long-term BLS affects polymorphism patterns in a relatively small genomic neighborhood, and such selection targets are easier to detect when the equilibrium frequencies of the selected variants are close to 50%, or when there has been a population size reduction. For a new mutation subject to BLS, its initial increase in frequency in the population causes linked neutral regions to have reduced diversity, an excess of both high and low frequency derived variants, and elevated LD with the selected locus. These patterns are similar to those produced by selective sweeps, but the effects of recent BLS are weaker. Nonetheless, compared to selective sweeps, nonequilibrium polymorphism and LD patterns persist for a much longer period under recent BLS, which may increase the chance of detecting such selection targets. An R package for analyzing these models, among others (e.g., isolation with migration), is available.     

Fixation probability in spatially changing environments.

The fixation probability of a mutant in a subdivided population with spatially varying environments is investigated using a finite island model. This probability is different from that in a panmictic population if selection is intermediate to strong and migration is weak. An approximation is used to compute the fixation probability when migration among subpopulations is very weak. By numerically solving the two-dimensional partial differential equation for the fixation probability in the two subpopulation case, the approximation was shown to give fairly accurate values. With this approximation, we show in the case of two subpopulations that the fixation probability in subdivided populations is greater than that in panmictic populations mostly. The increase is most pronounced when the mutant is selected for in one subpopulation and is selected against in the other subpopulation. Also it is shown that when there are two types of environments, further subdivision of subpopulations does not cause much change of the fixation probability in the no dominance case unless the product of the selection coefficient and the local population size is less than one. With dominance, the effect of subdivision becomes more complex.     

Models of nearly neutral mutations with particular implications for nonrandom usage of synonymous codons

Summary The population dynamics of nearly neutral mutations are studied using a single-site and a multisite model. In the latter model, the nucleotides in a sequence are completely linked and the selection schemes employed are additive, multiplicative, and additive with a threshold. Although the third selection scheme is very different from the first two, the three schemes produce identical results for a wide range of parameter values. Thus the present study provides a general theory for the population dynamics of nearly neutral mutations because the results can also be used to draw inferences about other selection schemes such as stabilizing selection and synergistic selection. It is shown that the number of slightly deleterious mutations accumulated in a sequence can be considerably larger under the multisite model than under the single-site model, particularly if the sequence is long or if the mutation rate per site is high. The results show that even a very slight selective difference between synonymous codons can produce a strong bias in codon usage. Three alternative explanations for the strong bias in codon usage in bacterial and yeast genes are considered. The implications of the present results for molecular evolution are discussed.     

Transition between stochastic evolution and deterministic evolution in the presence of selection: general theory and application to virology.

We present here a self-contained analytic review of the role of stochastic factors acting on a virus population. We develop a simple one-locus, two-allele model of a haploid population of constant size including the factors of random drift, purifying selection, and random mutation. We consider different virological experiments: accumulation and reversion of deleterious mutations, competition between mutant and wild-type viruses, gene fixation, mutation frequencies at the steady state, divergence of two populations split from one population, and genetic turnover within a single population. In the first part of the review, we present all principal results in qualitative terms and illustrate them with examples obtained by computer simulation. In the second part, we derive the results formally from a diffusion equation of the Wright-Fisher type and boundary conditions, all derived from the first principles for the virus population model. We show that the leading factors and observable behavior of evolution differ significantly in three broad intervals of population size, N. The "neutral limit" is reached when N is smaller than the inverse selection coefficient. When N is larger than the inverse mutation rate per base, selection dominates and evolution is "almost" deterministic. If the selection coefficient is much larger than the mutation rate, there exists a broad interval of population sizes, in which weakly diverse populations are almost neutral while highly diverse populations are controlled by selection pressure. We discuss in detail the application of our results to human immunodeficiency virus population in vivo, sampling effects, and limitations of the model.     

The spatial scale of population fluctuations and quasi-extinction risk

Using a spatially homogeneous population model with migration (random individual dispersal) and spatially autocorrelated environmental noise, we show how migration and local density regulation affect the spatial scale of fluctuations in the log of population sizes as well as the 1-yr differences in these. The difference between the squares of these two spatial scales of population fluctuations does not depend on the spatial scale of the noise but only on migration rate and strength of local density regulation. We also show how migration, local density regulation, and spatially correlated environmental noise affect the realized population process at a specific location. As the migration increases, the realized local density regulation and the expected population size increase, while the realized environmental noise decreases. This approach also enables us to analyze the dynamics of the total population size within quadrats of different sizes. The risk of local quasi extinction is strongly reduced by increasing quadrat size or migration rate, while an increase in environmental stochasticity or spatial correlation in the environmental noise increases the risk of quasi extinction.     

An exact steady state solution of Fisher's geometric model and other models

Because nearly neutral substitutions are thought to contribute substantially to molecular evolution, and much of our insight about the workings of nearly neutral evolution relies on theory, solvable models of this process are of particular interest. Here, I present an analytical method for solving models of nearly neutral evolution at steady state. The steady state solution applies to any constant fitness landscape under a dynamic of successive fixations, each of which occurs on the background of the population's most recent common ancestor. Because this dynamic neglects the effects of polymorphism in the population beyond the mutant allele under consideration, the steady state solution provides a decent approximation of evolutionary dynamics when the population mutation rate is low (Nu<1). To demonstrate the method, I apply it to two examples: Fisher's geometric model (FGM), and a simple model of molecular evolution. Since recent papers have studied the steady state behavior of FGM under this dynamic, I analyze its behavior in detail and compare the results with previous work.     

Fixation probabilities in a spatially heterogeneous environment

 We consider a simple model of a one-locus, two-allele population inhibiting a two-patch system and experiencing spatially heterogeneous viability selection. The populaton size is finite. We use a diffusion approximation and singular perturbation techniques to find the probability of fixation of a mutant allele. We focus on situations in which each allele is advantageous in one patch and deleterious in the other patch. Our theoretical results support the previous conclusions that, under certain conditions, small populations respond faster to selection than do large populations. We emphasize that knowledge of the dependence of migration rates on population size is crucial in evaluating the effects of population size on the rate of evolution.     

Dynamics of inbreeding depression due to deleterious mutations in small populations: mutation parameters and inbreeding rate

A multilocus stochastic model is developed to simulate the dynamics of mutational load in small populations of various sizes. Old mutations sampled from a large ancestral population at mutation-selection balance and new mutations arising each generation are considered jointly, using biologically plausible lethal and deleterious mutation parameters. The results show that inbreeding depression and the number of lethal equivalents due to partially recessive mutations can be partly purged from the population by inbreeding, and that this purging mainly involves lethals or detrimentals of large effect. However, fitness decreases continuously with inbreeding, due to increased fixation and homozygosity of mildly deleterious mutants, resulting in extinctions of very small populations with low reproductive rates. No optimum inbreeding rate or population size exists for purging with respect to fitness (viability) changes, but there is an optimum inbreeding rate at a given final level of inbreeding for reducing inbreeding depression or the number of lethal equivalents. The interaction between selection against partially recessive mutations and genetic drift in small populations also influences the rate of decay of neutral variation. Weak selection against mutants relative to genetic drift results in apparent overdominance and thus an increase in effective size (Ne) at neutral loci, and strong selection relative to drift leads to a decrease in Ne due to the increased variance in family size. The simulation results and their implications are discussed in the context of biological conservation and tests for purging.     

Genealogy of Neutral Genes and Spreading of Selected Mutations in a Geographically Structured Population

In a geographically structured population, the interplay among gene migration, genetic drift and natural selection raises intriguing evolutionary problems, but the rigorous mathematical treatment is often very difficult. Therefore several approximate formulas were developed concerning the coalescence process of neutral genes and the fixation process of selected mutations in an island model, and their accuracy was examined by computer simulation. When migration is limited, the coalescence (or divergence) time for sampled neutral genes can be described by the convolution of exponential functions, as in a panmictic population, but it is determined mainly by migration rate and the number of demes from which the sample is taken. This time can be much longer than that in a panmictic population with the same number of breeding individuals. For a selected mutation, the spreading over the entire population was formulated as a birth and death process, in which the fixation probability within a deme plays a key role. With limited amounts of migration, even advantageous mutations take a large number of generations to spread. Furthermore, it is likely that these mutations which are temporarily fixed in some demes may be swamped out again by non-mutant immigrants from other demes unless selection is strong enough. These results are potentially useful for testing quantitatively various hypotheses that have been proposed for the origin of modern human populations.     

The effect of subdivision on variation at multi-allelic loci under balancing selection

Simulations are used to investigate the expected pattern of variation at loci under different forms of multi-allelic balancing selection in a finite island model of a subdivided population. The objective is to evaluate the effect of restricted migration among demes on the distribution of polymorphism at the selected loci at equilibrium, and to compare the results with those expected for a neutral locus. The results show that the expected number of alleles maintained, and numbers of nucleotide differences between alleles, are relatively insensitive to the migration rate, and differentiation remains low even under very restricted migration. However, nucleotide divergence between copies of functionally identical alleles increases sharply when migration decreases. These results are discussed in relation to published surveys of allelic diversity in MHC and plant self-incompatibility systems, and to the possibility of inferring ancient population genetic events and processes. In addition, it is shown that, for sporophytic self-incompatibility systems, it is not necessarily true in a subdivided population that recessive alleles are more frequent than dominant ones.     

Environmental Fluctuations and Level of Density-Compensation Strongly Affects the Probability of Fixation and Fixation Times

The probability of, and time to, fixation of a mutation in a population has traditionally been studied by the classic Wright–Fisher model where population size is constant. Recent theoretical expansions have covered fluctuating populations in various ways but have not incorporated models of how the environment fluctuates in combination with different levels of density-compensation affecting fecundity. We tested the hypothesis that the probability of, and time to, fixation of neutral, advantageous and deleterious mutations is dependent on how the environment fluctuates over time, and on the level of density-compensation. We found that fixation probabilities and times were dependent on the pattern of autocorrelation of carrying capacity over time and interacted with density-compensation. The pattern found was most pronounced at small population sizes. The patterns differed greatly depending on whether the mutation was neutral, advantageous, or disadvantageous. The results indicate that the degree of mismatch between carrying capacity and population size is a key factor, rather than population size per se, and that effective population sizes can be very low also when the census population size is far above the carrying capacity. This study highlights the need for explicit population dynamic models and models for environmental fluctuations for the understanding of the dynamics of genes in populations.     

Fixation probability for a beneficial allele and a mutant strategy in a linear game under weak selection in a finite island model

The effect of population structure on the probability of fixation of a newly introduced mutant under weak selection is studied using a coalescent approach. Wright's island model in a framework of a finite number of demes is assumed and two selection regimes are considered: a beneficial allele model and a linear game among offspring. A first-order approximation of the fixation probability for a single mutant with respect to the intensity of selection is deduced. The approximation requires the calculation of expected coalescence times, under neutrality, for lineages starting from two or three sampled individuals. The results are obtained in a general setting without assumptions on the number of demes, the deme size or the migration rate, which allows for simultaneous coalescence or migration events in the genealogy of the sampled individuals. Comparisons are made with limit cases as the deme size or the number of demes goes to infinity or the migration rate goes to zero for which a diffusion approximation approach is possible. Conditions for selection to favor a mutant strategy replacing a resident strategy in the context of a linear game in a finite island population are addressed.     

A Pleiotropic Nonadditive Model of Variation in Quantitative Traits

A model of mutation-selection-drift balance incorporating pleiotropic and dominance effects of new mutations on quantitative traits and fitness is investigated and used to predict the amount and nature of genetic variation maintained in segregating populations. The model is based on recent information on the joint distribution of mutant effects on bristle traits and fitness in Drosophila melanogaster from experiments on the accumulation of spontaneous and P element-induced mutations. These experiments suggest a leptokurtic distribution of effects with an intermediate correlation between effects on the trait and fitness. Mutants of large effect tend to be partially recessive while those with smaller effect are on average additive, but apparently with very variable gene action. The model is parameterized with two different sets of information derived from P element insertion and spontaneous mutation data, though the latter are not fully known. They differ in the number of mutations per generation which is assumed to affect the trait. Predictions of the variance maintained for bristle number assuming parameters derived from effects of P element insertions, in which the proportion of mutations with an effect on the trait is small, fit reasonably well with experimental observations. The equilibrium genetic variance is nearly independent of the degree of dominance of new mutations. Heritabilities of between 0.4 and 0.6 are predicted with population sizes from 10(4) to 10(6), and most of the variance for the metric trait in segregating populations is due to a small proportion of mutations (about 1% of the total number) with neutral or nearly neutral effects on fitness and intermediate effects on the trait (0.1-0.5σ(P)). Much of the genetic variance is contributed by recessive or partially recessive mutants, but only a small proportion (about 10%) of the genetic variance is dominance variance. The amount of apparent selection on the trait itself generated by the model is very small. If a model is assumed in which all mutation events have an effect on the quantitative trait, the majority of the genetic variance is contributed by deleterious mutations with tiny effects on the trait. If such a model is assumed for viability, the heritability is about 0.1, independent of the population size.     

The Probability of Fixation in Populations of Changing Size

The rate of adaptive evolution of a population ultimately depends on the rate of incorporation of beneficial mutations. Even beneficial mutations may, however, be lost from a population since mutant individuals may, by chance, fail to reproduce. In this paper, we calculate the probability of fixation of beneficial mutations that occur in populations of changing size. We examine a number of demographic models, including a population whose size changes once, a population experiencing exponential growth or decline, one that is experiencing logistic growth or decline, and a population that fluctuates in size. The results are based on a branching process model but are shown to be approximate solutions to the diffusion equation describing changes in the probability of fixation over time. Using the diffusion equation, the probability of fixation of deleterious alleles can also be determined for populations that are changing in size. The results developed in this paper can be used to estimate the fixation flux, defined as the rate at which beneficial alleles fix within a population. The fixation flux measures the rate of adaptive evolution of a population and, as we shall see, depends strongly on changes that occur in population size.