Change in Genetic Variance under Selection in a Self-Fertilizing Population

In this study we show how the genetic variance of a quantitative trait changes in a self-fertilizing population under repeated cycles of truncation selection, with the analysis based on the infinitesimal model in which it is assumed that the trait is determined by an infinite number of unlinked loci without epistasis. The genetic variance is reduced not as a consequence of the genotypic frequency change but due to the build-up of linkage disequilibrium under truncation selection in this model. We assume that the order of the genotypic contribution from each locus is n(-1/2), where n is the number of loci involved, and investigate the change in linkage disequilibrium resulting from selection and self-fertilization using genotypic frequency dynamics in order to analyze the change in the genetic variance. Our analysis gives recurrence relations of genetic variance among the succeeding generations for the three cases of gene action, i.e., purely additive action, pure dominance without additive effect and the presence of both additive effect and dominance, respectively. Numerical examples are also given as a check on the recurrence formulas.     

The mutational load of a multigene family with uniform members

The mutational load of a multigene family with uniform members was studied by computer simulations. Two models of selection, truncation and exponential fitness, were examined, by using a simple model of gene conversion. It was found that the load is much smaller than the Haldane-Muller prediction under the truncation selection, and that it becomes approximately equal to the value calculated by the formula, nv(1-q)/(m-nq), where n is the copy number, v is the rate of detrimental mutation per gene copy, m is the truncation point in terms of the number of detrimental genes eliminated, and q is the equilibrium frequency of detrimental mutation. However the equilibrium frequency cannot be analytically obtained. For the exponential fitness model, the load is close to the Haldane-Muller value. When there is no gene conversion, the load becomes larger than the cases with conversion both for the truncation and the exponential fitness models. Thus, gene conversion or other mechanisms that are responsible for contraction-expansion of mutants on chromosomes helps eliminating deleterious mutations occurring in multigene families.     

Mutation and selection--a quantitative evaluation

A quantitative comparison of gene frequency change by mutation, selection, and mutation and selection combined, demonstrates that even when the mutation rate is quite high (1/10,000 per generation), and favorable selection almost imperceptibly small (.0025 per gene per generation) selection becomes the principal force changing gene frequency as soon as the new gene has reached the level of 4%.     

Inference of genome-wide mutation rates and distributions of mutation effects for fitness traits: a simulation study.

The properties and limitations of maximum likelihood (ML) inference of genome-wide mutation rates (U) and parameters of distributions of mutation effects are investigated. Mutation parameters are estimated from simulated experiments in which mutations randomly accumulate in inbred lines. ML produces more accurate estimates than the procedure of Bateman and Mukai and is more robust if the data do not conform to the model assumed. Unbiased ML estimates of the mutation effects distribution parameters can be obtained if a value for U can be assumed, but if U is estimated simultaneously with the distribution parameters, likelihood may increase monotonically as a function of U. If the distribution of mutation effects is leptokurtic, the number of mutation events per line is large, or if genotypic values are poorly estimated, only a lower limit for U, an upper limit for the mean mutation effect, and a lower limit for the kurtosis of the distribution can be given. It is argued that such lower (upper) limits are appropriate minima (maxima). Estimates of the mean mutational effect are unbiased but may convey little about the properties of the distribution if it is leptokurtic.     

Moments,cumulants, and polygenic dynamics

A new approach for describing the evolution of polygenic traits subject to selection and mutation is presented. Differential equations for the change of cumulants of the allelic frequency distribution at a particular locus and for the cumulants of the distributions of genotypic and phenotypic values are derived. The derivation is based on the assumptions of random mating, no sex differences, absence of random drift, additive gene action, linkage equilibrium, and Hardy-Weinberg proportions. Cumulants are a set of parameters that, like moments, describe the shape of a probability density. Compared with moments, however, they have properties that make them a much more convenient tool for investigating polygenic traits. Applications to directional and stabilizing selection are given.     

Tension versus ecological zones in a two-locus system

Previous theories show that tension and ecological zones are indistinguishable in terms of gene frequency clines. Here I analytically show that these two types of zones can be distinguished in terms of genetic statistics other than gene frequency. A two-locus cline model is examined with the assumptions of random mating, weak selection, no drift, no mutation, and multiplicative viabilities. The genetic statistics for distinguishing the two types of zones are the deviations of one- or two-locus genotypic frequencies from Hardy-Weinberg equilibrium (HWE) or from random association of gametes (RAG), and the deviations of additive and dominance variances from the values at HWE. These deviations have a discontinuous distribution in space and different extents of interruptions in the ecological zone with a sharp boundary, but exhibit a continuous distribution in the tension zone. Linkage disequilibrium enhances the difference between the deviations from HWE and from RAG for any two-locus genotypic frequency.     

The balance between pleiotropic mutation and selection,when alleles have discrete effects

The theory of pleiotropic mutation and selection is investigated and developed for a large population of asexual organisms. Members of the population are subject to stabilising selection on Omega phenotypic characters, which each independently affect fitness. Pleiotropy is incorporated into the model by allowing each mutation to simultaneously affect all characters. To expose differences with continuous-allele models, the characters are taken to originate from discrete-effect alleles and thus have discrete genotypic effects. Each character can take the values nxDelta where n=0,+/-1,+/-2, em leader, and the splitting in character effects, Delta, is a parameter of the model. When the distribution of mutant effects is normally distributed around the parental value, and Delta is large, a "stepwise" model of mutation arises, where only adjacent trait effects are accessible from a single mutation. The present work is primarily concerned with the opposite limit, where Delta is small and many different trait effects are accessible from a single mutation.In contrast to what has been established for continuous-effect models, discrete-effect models do not yield a singular equilibrium distribution of genotypic effects for any value of Omega. Instead, for different values of Omega, the equilibrium frequencies of trait values have very different dependencies on Delta. For Omega=1 and 2, decreasing Delta broadens the width of the frequency distribution and hence increases the equilibrium level of polymorphism. For all sufficiently large values of Omega, however, decreasing Delta decreases the width of the frequency distribution and the equilibrium level of polymorphism. The connection with continuous trait models follows when the limit Delta-->0 is considered, and a singular probability density of trait values is obtained for all sufficiently large Omega.     

The effect of positive assortative mating at one locus on a second linked locus

Summary Considerations proceed from a model of positive assortative mating based on genotype at one locus, with an arbitrary number of alleles, assuming no selection, mutation, or migration, hypothetically infinite population size, and discrete non-overlapping generations. From these conditions, inferences are made about the genotypic structure at a linked locus, as well as about the corresponding 2-locus gametic structure.The following main results are presented: in the course of the generations, the genotypic structure at the second locus and the 2-locus gametic structure always tend to a limit responsive to the initial conditions concerning the joint genotypic structure at the two loci and the degree of assortativity and linkage. A complete, analytical representation of the limits is given. In particular, if assortative mating is only partial and at the same time linkage is not complete, a population is not able to maintain a permanent deviation of the gametic structure from linkage equilibrium, and thus the genotypic structure at the second locus tends to Hardy-Weinberg proportions. On the other hand, if initial linkage disequilibrium is combined with partial assortative mating and complete linkage (or with complete assortative mating and unlinked loci) the population maintains this disequilibrium and thus the genotypic structure at the second locus need not tend to Hardy-Weinberg proportions. It turns out that the conditions not only of complete linkage, but also of unlinked loci together with complete assortativity, imply no change in gametic structure from the initial structure.In order to demonstrate the influence of several parameters on the speed of convergence to and the magnitude of the respective limits, several graphs are included.     

Second-order approximations for selection coefficients at polygenic loci

I determine the second-order approximation for the phenotypic distribution of an arbitrary number of quantitative traits, ignoring the effects of epistasis and linkage disequilibrium, conditioned on the presence of a specified genotype at one underlying locus of small effect. Using this approximation, I determine formulae for the effects of selection at a single locus with random mating under either Gaussian stabilizing selection, or correlated selection with truncation selection for one character. These formulae apply for arbitrary phenotypic distributions, yet even with multivariate Gaussian distributions of phenotypic effects the formula for correlated selection includes a correction to the standard formula in Falconer (1989). 1 demonstrate that this approximation has an error that is third order in the allelic or genotypic effects, independent of the form of the phenotypic distribution. I show also that the approximation of analogous form for the phenotypic distribution conditioned on the presence of a specified allele at a single locus is also correct to second order. Both approximations allow for dominance and are consistent in the sense that computing marginal fitnesses from approximations based on genotypic deviations and those based on average allelic effect yield the same answers.Supported by PHS Grant ROI GM 32130     

Effects of Mutation on Selection Limits in Finite Populations with Multiple Alleles

The ultimate response to directional selection (i.e., the selection limit) under recurrent mutation is analyzed by a diffusion approximation for a population in which there are k possible alleles at a locus. The limit mainly depends on two scaled parameters S (= 4Ns sigma a) and theta (= 4Nu) and k, the number of alleles, where N is the effective population size, u is the mutation rate, s is the selection coefficient, and sigma 2a is the variance of allelic effects. When the selection pressure is weak (S less than or equal to 0.5), the limit is given approximately by 2S sigma a[1 - (1 + c2)/k]/(theta + 1) for additive effects of alleles, where c is the coefficient of variation of the mutation rates among alleles. For strong selection, other approximations are devised to analyze the limit in different parameter regions. The effect of mutation on selection limits largely relies on the potential of mutation to introduce new and better alleles into the population. This effect is, however, bounded under the present model. Unequal mutation rates among alleles tend to reduce the selection limit, and can have a substantial effect only for small numbers of alleles and weak selection. The selection limit decreases as the mutation rate increases.     

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   

Definition and Estimation of Higher-Order Gene Fixation Indices

Fixation indices summarize the associations between genes that arise from the joint effects of inbreeding and selection. In this paper, fixation indices are derived for pairs, triplets and quadruplets of genes at a single multiallelic locus. The fixation indices are obtained by dividing cumulants by constants; the cumulants describe the statistical distribution of alleles and the constants are functions of gene frequency. The use of cumulants instead of moments is necessary only for four-gene indices, when the fourth cumulant is used. A second type of four-gene index is also required, and this index is based upon the covariation of second-order cumulants. At multiallelic loci, a large number of indices is possible. If alleles are selectively neutral, the number of indices is reduced and the relationship between gene identity and gene cumulants is shown.--Two-gene indices can always be estimated from genotypic frequency data at a single polymorphic locus. Three-gene indices are also estimable except when allele frequency equals one-half. Four-gene indices are not estimable unless selection is assumed to have an equal effect upon each allele (such as under selective neutrality) and the locus contains at least three alleles of unequal frequency. For diallelic or selected loci, an alternative four-gene fixation index is proposed. This index incorporates both types of four-gene associations but cannot be related to gene identity.     

The distribution of beneficial and fixed mutation fitness effects close to an optimum

The distribution of the selection coefficients of beneficial mutations is pivotal to the study of the adaptive process, both at the organismal level (theories of adaptation) and at the gene level (molecular evolution). A now famous result of extreme value theory states that this distribution is an exponential, at least when considering a well-adapted wild type. However, this prediction could be inaccurate under selection for an optimum (because fitness effect distributions have a finite right tail in this case). In this article, we derive the distribution of beneficial mutation effects under a general model of stabilizing selection, with arbitrary selective and mutational covariance between a finite set of traits. We assume a well-adapted wild type, thus taking advantage of the robustness of tail behaviors, as in extreme value theory. We show that, under these general conditions, both beneficial mutation effects and fixed effects (mutations escaping drift loss) are beta distributed. In both cases, the parameters have explicit biological meaning and are empirically measurable; their variation through time can also be predicted. We retrieve the classic exponential distribution as a subcase of the beta when there are a moderate to large number of weakly correlated traits under selection. In this case too, we provide an explicit biological interpretation of the parameters of the distribution. We show by simulations that these conclusions are fairly robust to a lower adaptation of the wild type and discuss the relevance of our findings in the context of adaptation theories and experimental evolution.     

The frequency of the perfect genotype in a population subject to pleiotropic mutation

We consider a large population of asexual organisms characterised by a number of quantitative traits that are subject to stabilising selection. Mutation is taken to act pleiotropically, with every mutation generally changing all of the traits under selection. We focus on the equilibrium distribution of the population, where mutation and selection are in balance. It has been previously established that the equilibrium distribution of genotypic effects may be anomalous, as it may contain a singular spike--a Dirac delta function--corresponding to a non-zero proportion of the population having exactly optimal genotypic values. In the present work, we present exact results for the case where three traits are under selection. These results give the equilibrium genetic variance of the population, and the proportion of the population that have the optimal genotype. This is achieved for two different spherically symmetric distributions of mutant effects. Additionally, a simple and robust numerical approach is also presented that allows the treatment of some other mutation distributions, where there are an arbitrary number of selected traits.     

Prospects for applying genotypic selection of somatic oncomutation to chemical risk assessment.

Genotypic selection methods detect rare sequence changes in populations of DNA molecules. These methods have been used to investigate the chemical induction of mutation and for the detection and diagnosis of cancer. The possible use of genotypic selection for improving current risk assessment practices is based on the premise that the frequency of somatic mutation is of critical importance in understanding and modeling carcinogenesis. If genotypic selection can measure the induction of specific mutations that disrupt normal cell/tissue homeostasis, then it could provide key mechanistic information for cancer risk assessment. For example, genotypic selection data might support a particular low-dose extrapolation method or characterize the relationship between rodent and human cancer risk. Strategies for evaluating the use of genotypic selection in cancer risk assessment include the concept of developing a battery of targets that detect a range of agent-specific effects. Ideal targets to examine by genotypic selection are the oncogene and tumor suppressor gene mutations frequently detected in human tumors because these are thought to represent tumor-initiating events. The most commonly occurring basepair (bp) substitutions within the ras and p53 genes are identified. Also, the battery of genotypic selection methods is defined in terms of the most important mutational specificities to include. In theory, the major basepair substitution mutations induced by 29 of 31 chemical carcinogens could be detected by analyzing three different mutations: G:C-->T:A, G:C-->A:T, and A:T-->T:A. Genotypic selection will have the greatest impact on risk assessment if measurement of spontaneous mutation is possible. Data from phenotypic selection assays suggest this corresponds to detection of mutant fractions of approximately 10(-7), and this would necessitate examining DNA samples containing >10(7) target molecules. Despite its apparent potential, considerable development and validation is needed before genotypic selection data can be applied to cancer risk assessment.     

Unpredictable selection in a structured population leads to local genetic differentiation in evolved reaction norms

Unpredictability during development of the optimum phenotype under future selection leads to a compromise reaction norm with a slope that is shallower than the slope of the optimum reaction norm. Unpredictability of selection can lead to an evolved curved reaction norm when genetic variation for curvature is available even if the optimum reaction norm is linear. This requires asymmetry in the frequency distribution of the habitats of selection; at small population size, stochasticity in the number of individuals per selection habitat is sufficient to generate such asymmetry. Unpredictability of selection in structured populations leads to local genetic differentiation of reaction norms. The mean habitat of a subpopulation is defined as the subpopulation's focal habitat. The evolved mean reaction norm of each subpopulation is anchored at the optimum genotypic value in its focal habitat. Linear reaction norms are parallel if the conditional distribution of adults around the focal habitats is the same for each subpopulation. Adult migration and absence of zygote dispersal represents the ultimate structured population, each habitat playing the role of focal habitat. Absence of zygote dispersal requires that the flow of individuals through the habitats is used instead of the habitats’ frequencies in the prediction of the evolved reaction norm. Adult migration in absence of zygote dispersal leads to an evolved pattern of locally differentiated reaction norms with optimum genotypic value anchored in the focal habitat and, for linear reaction norms, parallel slopes.     

The generation and maintenance of genetic variation by frequency-dependent selection: constructing polymorphisms under the pairwise interaction model

Frequency-dependent selection remains the most commonly invoked heuristic explanation for the maintenance of genetic variation. For polymorphism to exist, new alleles must be both generated and maintained in the population. Here we use a construction approach to model frequency-dependent selection with mutation under the pairwise interaction model. The pairwise interaction model is a general model of frequency-dependent selection at the genotypic level. We find that frequency-dependent selection is able to generate a large number of alleles at a single locus. The construction process generates multiallelic polymorphisms with a wide range of allele-frequency distributions and genotypic fitness relationships. Levels of polymorphism and mean fitness are uncoupled, so constructed polymorphisms remain permanently invasible to new mutants; thus the model never settles down to an equilibrium state. Analysis of constructed fitness sets reveals signatures of heterozygote advantage and positive frequency dependence.     

Frequency- and density-dependent selection on a quantitative character

The equilibrium distribution of a quantitative character subject to frequency- and density-dependent selection is found under different assumptions about the genetical basis of the character that lead to a normal distribution in a population. Three types of models are considered: (1) one-locus models, in which a single locus has an additive effect on the character, (2) continuous genotype models, in which one locus or several loci contribute additively to a character, and there is an effectively infinite range of values of the genotypic contributions from each locus, and (3) correlation models, in which the mean and variance of the character can change only through selection at modifier loci. It is shown that the second and third models lead to the same equilibrium values of the total population size and the mean and variance of the character. One-locus models lead to different equilibrium values because of constraints on the relationship between the mean and variance imposed by the assumptions of those models.——The main conclusion is that, at the equilibrium reached under frequency- and density-dependent selection, the distribution of a normally distributed quantitative character does not depend on the underlying genetic model as long as the model imposes no constraints on the mean and variance.     

Redistribution of gene frequency and changes of genetic variation following a bottleneck in population size

Although the distribution of frequencies of genes influencing quantitative traits is important to our understanding of their genetic basis and their evolution, direct information from laboratory experiments is very limited. In theory, different models of selection and mutation generate different predictions of frequency distributions. When a large population at mutation-selection balance passes through a rapid bottleneck in size, the frequency distribution of genes is dramatically altered, causing changes in observable quantities such as the mean and variance of quantitative traits. We investigate the gene frequency distribution of a population at mutation-selection balance under a joint-effect model of real stabilizing and pleiotropic selection and its redistribution and thus changes of the genetic properties of metric and fitness traits after the population passes a rapid bottleneck and expands in size. If all genes that affect the trait are neutral with respect to fitness, the additive genetic variance (VA) is always reduced by a bottleneck in population size, regardless of their degree of dominance. For genes that have been under selection, VA increases following a bottleneck if they are (partially) recessive, while the dominance variance increases substantially for any degree of dominance. With typical estimates of mutation parameters, the joint-effect model can explain data from laboratory experiments on the effect of bottlenecking on fitness and morphological traits, providing further support for it as a plausible mechanism for maintenance of quantitative genetic variation.     

Inbreeding effective population size under some artificial selection schemes

Summary It is well known that truncation selection is the most efficient form of directional selection in terms of changing gene frequency. In this paper we show circumstances where truncation selection followed by a balanced mating generates inbreeding effective population size smaller than that generated by a selection that assigns mating frequencies to individuals according to their breeding values, where both selection schemes give the same expected performance of selected individuals (selection differential). Breeding values of selected individuals and the weight used to determine mating frequencies are assumed to be linearly distributed on a performance scales, x. To assign mating frequencies to the individuals in the weighting system, the selected individuals are grouped using a constant , and ith group in the interval x_i, x_i + . With small number of groups, say 2 or 3, the weighting system in general generates inbreeding effective population size that is larger than that generated by a truncation selection. As the number of the groups increases, truncation selection generates larger effective numbers.