The Maintenance of Genetic Variability in Two-Locus Models of Stabilizing Selection

The maintenance of genetic variability at two diallelic loci under stabilizing selection is investigated. Generations are discrete and nonoverlapping; mating is random; mutation and random genetic drift are absent; selection operates only through viability differences. The determination of the genotypic values is purely additive. The fitness function has its optimum at the value of the double heterozygote and decreases monotonically and symmetrically from its optimum, but is otherwise arbitrary. The resulting fitness scheme is identical to the symmetric viability model. Linkage disequilibrium is neglected, but the results are otherwise exact. Explicit formulas are found for all the equilibria, and explicit conditions are derived fro their existence and stability. A complete classification of the six possible global convergence patterns is presented. In addition to the symmetric equilibrium (with gene frequency 1/2 at both loci), a pair of unsymmetric equilibria may exist; the latter are usually, but not always, unstable. If the ratio of the effect of the major locus to that of the minor one exceeds a critical value, both loci will be stably polymorphic. If selection is weak at the minor locus, the more rapidly the fitness function decreases near the optimum, the lower is this critical value; for rapidly decreasing fitness functions, the critical value is close to one. If the fitness function is smooth at the optimum, then a stable polymorphism exists at both loci only if selection is strong at the major locus.     

Dynamics of Genetic Variability in Two-Locus Models of Stabilizing Selection

We study a two locus model, with additive contributions to the phenotype, to explore the dynamics of different phenotypic characteristics under stabilizing selection and recombination. We demonstrate that the interaction of selection and recombination results in constraints on the mode of phenotypic evolution. Let V(g) be the genic variance of the trait and C(L) be the contribution of linkage disequilibrium to the genotypic variance. We demonstrate that, independent of the initial conditions, the dynamics of the system on the plane (V(g), C(L)) are typically characterized by a quick approach to a straight line with slow evolution along this line afterward. We analyze how the mode and the rate of phenotypic evolution depend on the strength of selection relative to recombination, on the form of fitness function, and the difference in allelic effect. We argue that if selection is not extremely weak relative to recombination, linkage disequilibrium generated by stabilizing selection influences the dynamics significantly. We demonstrate that under these conditions, which are plausible in nature and certainly the case in artificial stabilizing selection experiments, the model can have a polymorphic equilibrium with positive linkage disequilibrium that is stable simultaneously with monomorphic equilibria.     

Additive Variation Maintained under Stabilizing Selection: A Two-Locus Model of Pleiotropy for Two Quantitative Characters

A model with two diallelic loci controlling two additive quantitative characters is suggested. One of the loci has a similar effect on both characters, whereas the second locus has an antagonistic effect on two characters. Both characters experience direct stabilizing selection. The model yields a stable polymorphic state, with both characters maintaining genetic variation. The genetic correlation between the characters at the equilibrium is zero, in spite of the pleiotropic effects of the loci controlling them.     

On Models of Quantitative Genetic Variability: A Stabilizing Selection-Balance Model

A model of stabilizing selection on a multilocus character is proposed that allows the maintenance of stable allelic polymorphism and linkage disequilibrium. The model is a generalization of Lerner's model of homeostasis in which heterozygotes are less susceptible to environmental variation and hence are superior to homozygotes under phenotypic stabilizing selection. The analysis is carried out for weak selection with a quadratic-deviation model for the stabilizing selection. The stationary state is characterized by unequal allele frequencies, unequal proportions of complementary gametes, and a reduction of the genetic (and phenotypic) variance by the linkage disequilibrium. The model is compared with Mather's polygenic balance theory, with models that include mutation-selection balance, and others that have been proposed to study the role of linkage disequilibrium in quantitative inheritance.     

Maintenance of Genetic Variability under the Joint Effect of Mutation, Selection and Random Drift

Formulae are developed for the distribution of allele frequencies (the frequency spectrum), the mean number of alleles in a sample, and the mean and variance of heterozygosity under mutation pressure and under either genic or recessive selection. Numerical computations are carried out by using these formulae and Watterson's (1977) formula for the distribution of allele frequencies under overdominant selection. The following properties are observed: (1) The effect of selection on the distribution of allele frequencies is slight when 4Ns 相似文献   

A Symmetric Two-Locus Fertility Model

A model in which selection is mediated by differential fertilities among the genotypes at two diallelic loci is proposed. Fertility depends only on the number of heterozygous loci participating in the mating. Classes analogous to symmetric equilibria in symmetric viability models are determined explicitly and shown to exhibit stability behavior very different from the viability results. Linkage equilibrium is shown to occur in a relatively asymmetric fashion and to overlap in stability with linkage disequilibrium. In many cases single-locus or two-locus polymorphism is shown to be stable simultaneously with chromosome fixation even under very tight linkage. It is suggested that historical effects may be of great significance in the evolution of systems in which fertility is the primary agent of natural selection.     

The Quantitative Genetic Consequences of Pleiotropy under Stabilizing and Directional Selection

The independence of two phenotypic characters affected by both pleiotropic and nonpleiotropic mutations is investigated using a generalization of M. Slatkin's stepwise mutation model of 1987. The model is used to determine whether predictions of either the multivariate normal model introduced in 1980 by R. Lande or the house-of-cards model introduced in 1985 by M. Turelli can be regarded as typical of models that are intermediate between them. We found that, under stabilizing selection, the variance of one character at equilibrium may depend on the strength of stabilizing selection on the other character (as in the house-of-cards model) or not (as in the multivariate normal model) depending on the types of mutations that can occur. Similarly, under directional selection, the genetic covariance between two characters may increase substantially (as in the house-of-cards model) or not (as in the multivariate normal model) depending on the kinds of mutations that are assumed to occur. Hence, even for the simple model we consider, neither the house-of-cards nor the multivariate normal model can be used to make predictions, making it unlikely that either could be used to draw general conclusions about more complex and realistic models.     

Pleiotropic Mutations Are Subject to Strong Stabilizing Selection

The assumption that pleiotropic mutations are more deleterious than mutations with more restricted phenotypic effects is an important premise in models of evolution. However, empirical evidence supporting this assumption is limited. Here, we estimated the strength of stabilizing selection on mutations affecting gene expression in male Drosophila serrata. We estimated the mutational variance (V_M) and the standing genetic variance (V_G) from two well-matched panels of inbred lines: a panel of mutation accumulation (MA) lines derived from a single inbred ancestral line and a panel of inbred lines derived from an outbred population. For 855 gene-expression traits, we estimated the strength of stabilizing selection as s = V_M/V_G. Selection was observed to be relatively strong, with 17% of traits having s > 0.02, a magnitude typically associated with life-history traits. Randomly assigning expression traits to five-trait sets, we used factor analytic mixed modeling in the MA data set to identify covarying traits that shared pleiotropic mutations. By assigning traits to the same trait sets in the outbred line data set, we then estimated s for the combination of traits affected by pleiotropic mutation. For these pleiotropic combinations, the median s was three times greater than s acting on the individual component traits, and 46% of the pleiotropic trait combinations had s > 0.02. Although our analytical approach was biased toward detecting mutations with relatively large effects, likely overestimating the average strength of selection, our results provide widespread support for the prediction that stronger selection can act against mutations with pleiotropic effects.THE extent to which new mutations have pleiotropic effects on multiple traits, and ultimately on fitness is central to our understanding of the maintenance of genetic variation and the process of adaptation (Kondrashov and Turelli 1992; Otto 2004; Johnson and Barton 2005; Zhang and Hill 2005). Analyses of Fisher’s (1930) geometric model of adaptation have shown that a mutation with effects on many traits will have a reduced probability of contributing to adaptive evolution (Orr 2000; Welch and Waxman 2003; see also Haygood 2006). For a population close to its optimum under mutation–selection balance, a direct corollary of this is that selection must act more strongly against mutations with wider pleiotropic effects (Zhang 2012).Evidence for the strength of selection increasing with the number of traits that are pleiotropically affected by a mutation is limited. At a phenotypic level, nonlinear (stabilizing) selection is much stronger on combinations of metric traits than on each individual trait contributing to the combination (Blows and Brooks 2003; Walsh and Blows 2009). Given that genetic correlations among such traits are expected to be a consequence of pleiotropic alleles (Lande 1980), stronger selection on trait combinations is consistent with stronger selection on pleiotropic mutations that are likely to underlie the genetic covariance among such traits. There is some evidence that per-trait allelic effects might be greater for alleles with more widespread pleiotropic effects (Wagner et al. 2008; Wang et al. 2010); as mutations with larger phenotypic effects might be more effectively targeted by selection, this also suggests stronger selection against more pleiotropic mutation.Mutation accumulation (MA) breeding designs, in which the opportunity for selection is reduced, allowing new mutations to drift to fixation, provide an opportunity to characterize the strength of selection acting directly against new mutations. Rice and Townsend (2012) proposed an approach for determining the strength of selection acting against mutations at individual loci, combining information from QTL mapping and MA studies. This approach could conceivably be extended to associate the strength of selection with the number of traits a QTL affects. More typically, estimates of selection from MA designs are focused on traits, rather than alleles. Under the assumption that most mutations are deleterious, an assumption supported by MA studies (Halligan and Keightley 2009), the strength of selection acting on mutations affecting quantitative traits can be measured as the ratio of the mutational to the standing genetic variance, s = V_M/V_G, where s is the selection coefficient of the mutation in heterozygous form (Barton 1990; Houle et al. 1996). While estimating s in this way provides a framework for estimating selection on pleiotropic combinations of traits, we are not aware of any studies adopting this approach to directly estimate the strength of selection acting on mutations affecting multiple traits.Within an MA framework, Estes and Phillips (2006) manipulated the opportunity for selection, providing rare direct evidence of stronger selection against mutations with pleiotropic effects. In a DNA repair-deficient strain of Caenorhabditis elegans, Estes and Phillips (2006) observed lower mutational covariance among life-history components when selection was allowed (larger populations) than when the opportunity for selection was limited (small populations). Similarly, McGuigan et al. (2011) compared Drosophila serrata MA lines accumulating mutations in the presence or absence of sexual selection on males, reporting reduced covariance between two fitness components in the selection treatment. These studies reveal that selection can eliminate nonlethal alleles with pleiotropic effects, but whether traits other than life-history components exhibit similar evidence of selection against pleiotropic alleles remains unknown.In parallel to the quantitative genetic predictions that pleiotropic alleles will be under stronger selection, molecular genetic theory predicts that the rate of gene evolution will be negatively correlated with pleiotropy (Pal et al. 2006; Salathe et al. 2006). More highly pleiotropic genes, as identified through the extent of connectivity (the number of interactions) in protein–protein interaction networks (Jeong et al. 2001), or the number of gene ontology (GO) terms (Jovelin and Phillips 2009) are more likely to be essential (i.e., knockout mutations result in lethality), suggesting that selection is stronger against large-effect (knockout) mutations in more highly pleiotropic genes. However, the selection acting against small-effect, nonlethal mutations in pleiotropic genes is less clear (Pal et al. 2006). Several studies have found an association between gene pleiotropy indices, such GO annotation of the number of biological processes or tissue specificity of expression, and the rate of sequence evolution (e.g., Pal et al. 2001; Salathe et al. 2006; Jovelin and Phillips 2009; Su et al. 2010). These pleiotropy indices typically explain little of the variation in sequence evolutionary rates, and it remains unclear whether more highly pleiotropic mutations are typically under stronger selection (Pal et al. 2006; Salathe et al. 2006).Here, we estimate the selection coefficients acting against naturally occurring mutations affecting gene-expression traits in male D. serrata to quantitatively test if selection is stronger on mutations that affect multiple traits. Gene-expression phenotypes are uniquely positioned to enable detailed investigations of pleiotropy: there are many of them, they represent a broad coverage of biological function, they can be analyzed to quantify developmental pleiotropy in the same way as traits traditionally considered in quantitative genetics, and GO information can be used to index molecular genetic pleiotropy. We use multivariate mixed-model analyses of expression traits in a set of inbred lines from a mutation accumulation experiment to estimate the mutational variance in individual expression traits, and the pleiotropic mutational covariance among random sets of five expression traits. Using a second panel of inbred lines, derived from a natural, outbred, population, we estimate the standing genetic variance in the same individual traits and five-trait combinations. From these estimates of mutational and standing genetic variance, we calculate s for each of the individual traits and trait combinations to determine whether selection has typically been stronger on mutations with pleiotropic effects than on other mutations affecting each trait. We complement this quantitative genetic analysis of developmental pleiotropy with an analysis of molecular genetic pleiotropy (Paaby and Rockman 2013), determining whether the strength of selection acting on individual expression traits can be predicted from the number of biological functions that the gene annotates to in the GO database or to the range of tissues in which the gene is expressed.     

Substitution Rates under Stabilizing Selection

Allelic substitutions under stabilizing phenotypic selection on quantitative traits are studied in Monte Carlo simulations of 8 and 16 loci. The results are compared and contrasted to analytical models based on work of M. Kimura for two and "infinite" loci. Selection strengths of S = 4Nes approximately four (which correspond to reasonable strengths of selection for quantitative characters) can retard substitution rates tenfold relative to rates under neutrality. An important finding is a strong dependence of per locus substitution rates on the number of loci.     

Stabilizing Selection and the Structural Variability of Flowers within Species

Zoophilous flowers often appear to be precisely formed for pollentransfer and exhibit relatively little variability in structurewithin species. Functional optimization by the seemingly exactingrequirements of pollen transfer may account for these observations.I used the results of a literature survey to examine the levelsof intraspecific variation in flowers across a wide range oftaxa. The least variable attributes were those potentially affectingthe mechanical fit between flower and pollinator, which arepotentially constrained by selection for pollination performance.I discuss six mechanisms by which plant-pollinator interactionscould generate stabilizing selection on flowers. In addition,I consider the stabilizing roles of limiting resources and alsotwo functionally-neutral mechanisms. Further work is requiredto identify the actual mechanisms by which selection stabilizesthe evolution of flowers.Copyright 1998 Annals of Botany Company stabilizing selection, flowers, pollination, variation     

Quasi Linkage Equilibrium under Weak Two-Locus Viability Selection: II. Loci with Multiple Alleles

Russian Journal of Genetics - A model of weak viability selection at two multiple-allele loci in a haploid population is considered. It is assumed that population is randomly mating, the...     

A Spatially Explicit Model of Stabilizing Selection for Improving Phylogenetic Inference

Ultraconserved elements (UCEs) are stretches of hundreds of nucleotides with highly conserved cores flanked by variable regions. Although the selective forces responsible for the preservation of UCEs are unknown, they are nonetheless believed to contain phylogenetically meaningful information from deep to shallow divergence events. Phylogenetic applications of UCEs assume the same degree of rate heterogeneity applies across the entire locus, including variable flanking regions. We present a Wright–Fisher model of selection on nucleotides (SelON) which includes the effects of mutation, drift, and spatially varying, stabilizing selection for an optimal nucleotide sequence. The SelON model assumes the strength of stabilizing selection follows a position-dependent Gaussian function whose exact shape can vary between UCEs. We evaluate SelON by comparing its performance to a simpler and spatially invariant GTR+Γ model using an empirical data set of 400 vertebrate UCEs used to determine the phylogenetic position of turtles. We observe much improvement in model fit of SelON over the GTR+Γ model, and support for turtles as sister to lepidosaurs. Overall, the UCE-specific parameters SelON estimates provide a compact way of quantifying the strength and variation in selection within and across UCEs. SelON can also be extended to include more realistic mapping functions between sequence and stabilizing selection as well as allow for greater levels of rate heterogeneity. By more explicitly modeling the nature of selection on UCEs, SelON and similar approaches can be used to better understand the biological mechanisms responsible for their preservation across highly divergent taxa and long evolutionary time scales.     

Maintenance of Genetic Variation with a Frequency-Dependent Selection Model as Compared to the Overdominant Model

A frequency-dependent selection model proposed by Huang, Singh and Kojima (1971) was found to be more effective at maintaining genetic variation in a finite population than the overdominant model. The fourth moment parameter of the distribution of unfixed states showed that there was a more platykurtic distribution for the frequency-dependent model. This agreed well with the expected gene frequency change found for an infinite population.     

A Two-Locus System with Strong Epistasis Underlies Rapid Parasite-Mediated Evolution of Host Resistance

Parasites are a major evolutionary force, driving adaptive responses in host populations. Although the link between phenotypic response to parasite-mediated natural selection and the underlying genetic architecture often remains obscure, this link is crucial for understanding the evolution of resistance and predicting associated allele frequency changes in the population. To close this gap, we monitored the response to selection during epidemics of a virulent bacterial pathogen, Pasteuria ramosa, in a natural host population of Daphnia magna. Across two epidemics, we observed a strong increase in the proportion of resistant phenotypes as the epidemics progressed. Field and laboratory experiments confirmed that this increase in resistance was caused by selection from the local parasite. Using a genome-wide association study, we built a genetic model in which two genomic regions with dominance and epistasis control resistance polymorphism in the host. We verified this model by selfing host genotypes with different resistance phenotypes and scoring their F1 for segregation of resistance and associated genetic markers. Such epistatic effects with strong fitness consequences in host–parasite coevolution are believed to be crucial in the Red Queen model for the evolution of genetic recombination.     

Deleterious Mutations, Apparent Stabilizing Selection and the Maintenance of Quantitative Variation

Apparent stabilizing selection on a quantitative trait that is not causally connected to fitness can result from the pleiotropic effects of unconditionally deleterious mutations, because as N. Barton noted, "...individuals with extreme values of the trait will tend to carry more deleterious alleles...." We use a simple model to investigate the dependence of this apparent selection on the genomic deleterious mutation rate, U; the equilibrium distribution of K, the number of deleterious mutations per genome; and the parameters describing directional selection against deleterious mutations. Unlike previous analyses, we allow for epistatic selection against deleterious alleles. For various selection functions and realistic parameter values, the distribution of K, the distribution of breeding values for a pleiotropically affected trait, and the apparent stabilizing selection function are all nearly Gaussian. The additive genetic variance for the quantitative trait is kQa2, where k is the average number of deleterious mutations per genome, Q is the proportion of deleterious mutations that affect the trait, and a2 is the variance of pleiotropic effects for individual mutations that do affect the trait. In contrast, when the trait is measured in units of its additive standard deviation, the apparent fitness function is essentially independent of Q and a2; and beta, the intensity of selection, measured as the ratio of additive genetic variance to the "variance" of the fitness curve, is very close to s = U/k, the selection coefficient against individual deleterious mutations at equilibrium. Therefore, this model predicts appreciable apparent stabilizing selection if s exceeds about 0.03, which is consistent with various data. However, the model also predicts that beta must equal Vm/VG, the ratio of new additive variance for the trait introduced each generation by mutation to the standing additive variance. Most, although not all, estimates of this ratio imply apparent stabilizing selection weaker than generally observed. A qualitative argument suggests that even when direct selection is responsible for most of the selection observed on a character, it may be essentially irrelevant to the maintenance of variation for the character by mutation-selection balance. Simple experiments can indicate the fraction of observed stabilizing selection attributable to the pleiotropic effects of deleterious mutations.     

Maintenance of Genetic Variability under the Pressure of Neutral and Deleterious Mutations in a Finite Population

In order to assess the effect of deleterious mutations on various measures of genic variation, approximate formulas have been developed for the frequency spectrum, the mean number of alleles in a sample, and the mean homozygosity; in some particular cases, exact formulas have been obtained. The assumptions made are that two classes of mutations exist, neutral and deleterious, and that selection is strong enough to keep deleterious alleles in low frequencies, the mode of selection being either genic or recessive. The main findings are: (1) If the expected value (q) of the sum of the frequencies of deleterious alleles is about 10% or less, then the presence of deleterious alleles causes only a minor reduction in the mean number of neutral alleles in a sample, as compared to the case of q = 0. Also, the low- and intermediate-frequency parts of the frequency spectrum of neutral alleles are little affected by the presence of deleterious alleles, though the high-frequency part may be changed drastically. (2) The contribution of deleterious mutations to the expected total number of alleles in a sample can be quite large even if q is only 1 or 2%. (3) The mean homozygosity is roughly equal to (1--2q)/(1 + theta 1), where theta 1 is twice the number of new neutral mutations occurring in each generation in the total population. Thus, deleterious mutations increase the mean heterozygosity by about 2q/(1 + theta 1). The present results have been applied to study the controversial problem of how deleterious mutations may affect the testing of the neutral mutation hypothesis.     

Genetic Variability and Rate of Gene Substitution in a Finite Population under Mutation and Fluctuating Selection

By using a numerical method of solving stochastic difference equations, the level of genetic variability maintained in a finite population and the rate of gene substitution under several models of fluctuating selection intensities were studied. It is shown that mutation and random genetic drift both play an important role in determining genetic variability and the rate of gene substitution. Compared with the case of neutral mutations, the fluctuation of selection intensity caused by temporal and spatial heterogeneity of environments generally increases the rate of gene substitution, but the level of genetic variability may be increased or decreased, depending upon the model and the parameters used. Although such a type of selection per se can not be ruled out, when mutation is taken into account, it is difficult to explain both the observed amount of genetic variability and the rough constancy of evolutionary rate within a framework of fluctuating selection models.