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.     

长期人工选择对群体遗传结构的影响——一项计算机模拟研究 Ⅱ.长期选择对群体遗传方差的作用

本文给出了显性与超显性模型下加性方差的分剖公式,为研究选择作用下基因间关系的变化提供了有力的方法。并模拟研究了群体大小、连锁强度与遗传力水平对遗传方差变化的影响。小群体中遗传方差在世代间波动很大;大群体中则稳定下降、波动较小。选择作用下平衡加性方差下降很快,特别是高遗传力性状。紧密连锁在小群体中一方面降低选择反应,一方面维持了更多的加性方差,从而使得预测长期选择反应甚为困难。     

Mutation Rate and Dominance of Genes Affecting Viability in DROSOPHILA MELANOGASTER

Spontaneous mutations were allowed to accumulate in a second chromosome that was transmitted only through heterozygous males for 40 generations. At 10-generation intervals the chromosomes were assayed for homozygous effects of the accumulated mutants. From the regression of homozygous viability on the number of generations of mutant accumulation and from the increase in genetic variance between replicate chromosomes it is possible to estimate the mutation rate and average effect of the individual mutants. Lethal mutations arose at a rate of 0.0060 per chromosome per generation. The mutants having small effects on viability are estimated to arise with a frequency at least 10 times as high as lethals, more likely 20 times as high, and possibly many more times as high if there is a large class of very nearly neutral mutations.-The dominance of such mutants was measured for chromosomes extracted from a natural population. This was determined from the regression of heterozygous viability on that of the sum of the two constituent homozygotes. The average dominance for minor viability genes in an equilibrium population was estimated to be 0.21. This is lower than the value for new mutants, as expected since those with the greatest heterozygous effect are most quickly eliminated from the population. That these mutants have a disproportionately large heterozygous effect on total fitness (as well as on the viability component thereof) is shown by the low ratio of the genetic load in equilibrium homozygotes to that of new mutant homozygotes.     

Linkage and the Maintenance of Heritable Variation by Mutation-Selection Balance

The role of linkage in influencing heritable variation maintained through a balance between mutation and stabilizing selection is investigated for two different models. In both cases one trait is considered and the interactions within and between loci are assumed to be additive. Contrary to most earlier investigations of this problem no a priori assumptions on the distribution of genotypic values are imposed. For a deterministic two-locus two-allele model with recombination and mutation, related to the symmetric viability model, a complete nonlinear analysis is performed. It is shown that, depending on the recombination rate, multiple stable equilibria may coexist. The equilibrium genetic and genic variances are calculated. For a polygenic trait in a finite population with a possible continuum of allelic effects a simulation study is performed. In both models the equilibrium genetic and genic variances are roughly equal to the house-of-cards prediction or its finite population counterpart as long as the recombination rate is not extremely low. However, negative linkage disequilibrium builds up. If the loci are very closely linked the equilibrium additive genetic variance is slightly lower than the house-of-cards prediction, but the genic variance is much higher. Depending on whether the parameters are in favor of the house-of-cards or the Gaussian approximation, different behavior of the genetic system occurs with respect to linkage.     

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.     

Epistasis in polygenic traits and the evolution of genetic architecture under stabilizing selection

We consider the effects of epistasis in a polygenic trait in the balance of mutation and stabilizing selection. The main issues are the genetic variation maintained in equilibrium and the evolution of the mutational effect distribution. The model assumes symmetric mutation and a continuum of alleles at all loci. Epistasis is modeled proportional to pairwise products of the single-locus effects. A general analytical formalism is developed. Assuming linkage equilibrium, we derive results for the equilibrium mutation load and the genetic and mutational variance in the house of cards and the Gaussian approximation. The additive genetic variation maintained in mutation-selection balance is reduced by any pattern of the epistatic interactions. The mutational variance, in contrast, is often increased. Large differences in mutational effects among loci emerge, and a negative correlation among (standard mean) locus mutation effects and mutation rates is predicted. Contrary to the common view since Waddington, we find that stabilizing selection in general does not lead to canalization of the trait. We propose that canalization as a target of selection instead occurs at the genic level. Here, primarily genes with a high mutation rate are buffered, often at the cost of decanalization of other genes. An intuitive interpretation of this view is given in the discussion.     

Modeling quantitative trait Loci and interpretation of models

A quantitative genetic model relates the genotypic value of an individual to the alleles at the loci that contribute to the variation in a population in terms of additive, dominance, and epistatic effects. This partition of genetic effects is related to the partition of genetic variance. A number of models have been proposed to describe this relationship: some are based on the orthogonal partition of genetic variance in an equilibrium population. We compare a few representative models and discuss their utility and potential problems for analyzing quantitative trait loci (QTL) in a segregating population. An orthogonal model implies that estimates of the genetic effects are consistent in a full or reduced model in an equilibrium population and are directly related to the partition of the genetic variance in the population. Linkage disequilibrium does not affect the estimation of genetic effects in a full model, but would in a reduced model. Certainly linkage disequilibrium would complicate the detection of QTL and epistasis. Using different models does not influence the detection of QTL and epistasis. However, it does influence the estimation and interpretation of genetic effects.     

Joint effect of selection, linkage and partial inbreeding on additive genetic variance in an infinite population

Under the inifinitesimal model of gene effects, selection reduces the additive genetic variance by inducing negative linkage disequilibrium among selected genes. If the selected genes are linked, the decay of linkage disequilibrium is delayed, and the reduction of additive genetic variance is enhanced. Inbreeding in an infinite population also alters the additive genetic variance through the generation of positive association among genes within a locus. In the present study, the joint effect of selection, linkage and partial inbreeding (partial selfing or partial full-sib mating) on the additive genetic variance was modeled. The recurrence relations of the additive genetic variance between successive generations and the prediction equation of the asymptotic additive genetic variance were derived. Numerical computation showed that although partially inbred populations initially maintain larger genetic variances, the accumulated effect of selection overrides the effect of inbreeding. Stochastic simulation was carried out to check the precision of prediction, showing that the obtained equations give a satisfactory prediction during initial generations. However, the predicted values always overestimate the simulated values, especially in later generations. Based on these results, possible extensions and perspectives of the assumed model were discussed.     

Quantitative genetic variance associated with chromosomal markers in segregating populations

Summary Use of chromosomal markers can accelerate genetic progress for quantitative traits in pedigree selection programs by providing early information on Mendelian segregation effects for individual progeny. Potential effectiveness of selection using markers is determined by the amount of additive genetic variance traced from parents to progeny by the markers. Theoretical equations for the amount of additive genetic variance associated with a marker were derived at the individual level and for a segregating population in joint linkage equilibrium. Factors considered were the number of quantitative trait loci linked to the marker, their individual effects, and recombination rates with the marker. Subsequently, the expected amount of genetic variance associated with a marker in a segregating population was derived. In pedigree selection programs in segregating populations, a considerable fraction of the genetic variance on a chromosome is expected to be associated with a marker located on that chromosome. For an average chromosome in the bovine, this fraction is approximately 40% of the Mendelian segregation variance contributed by the chromosome. The effects of interference and position of the marker on this expectation are relative small. Length of the chromosome has a large effect on the expected variance. Effectiveness of MAS is, however, greatly reduced by lack of polymorphism at the marker and inaccuracy of estimation of chromosome substitution effects. The size of the expected amount of genetic variance associated with a chromosomal marker indicates that, even when the marker is not the active locus, large chromosome substitution effects are not uncommon in segregating populations.     

Quantitative trait locus mapping using human pedigrees

In the past decade phenomenal progress has been made in molecular and statistical genetic methods for localizing quantitative trait loci. Because of these advances, we can anticipate a long period of active genetic research in which the genes influencing human quantitative variability will be mapped and their effects accurately evaluated. Here, we review the current state of the science in statistical genetic methods for quantitative trait linkage analysis. In particular, we detail a variance component-based framework for localizing quantitative trait loci and for accurately estimating their relative effect sizes. Attention is paid to the optimal design of human family studies for localizing genes of small to moderate effect. In addition, methods and strategies are described for dealing with the most important complications of quantitative variation, including the assessment of genotype x environment interaction and epistasis.     

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.     

Understanding the genetic architecture of a quantitative trait in gymnosperms by genotyping haploid megagametophytes

The gymnosperms are a group of plants characterized by a haploid female gametophyte (megagametophyte). With the function of bearing the female gametes and nourishing the developing embryo, the megagametophyte has provided a simple way to understand the genetics of gymnosperm species using biochemical or genetic markers. In this paper, a quantitative genetic approach is proposed to study the genetic architecture of a quantitative trait in gymnosperms by taking advantage of the megagametophyte and the concept of average effect of a gene. Average effect describes the value associated with an allele carried by an individual and transmitted to its offspring. Through the genetic dissection of the average effect and genetic variance associated with a gamete carrying candidate genes, this approach can provide estimates of basic population genetic parameters, such as additive, dominant and epistatic effects, allelic frequencies and linkage disequilibrium. The candidate genes, known through their major mutant phenotype, have been reported in gymnosperms. An example for a candidate gene affecting lignin biosynthesis was applied to demonstrate the statistical procedures of the approach and its advantage. The conditions upon which the approach can be effectively used are discussed. Received: 15 January 1999 / Accepted: 12 March 1999     

Power of different sampling strategies to detect quantitative trait loci variance effects

Summary Many studies have shown that segregating quantitative trait loci (QTL) can be detected via linkage to genetic markers. Power to detect a QTL effect on the trait mean as a function of the number of individuals genotyped for the marker is increased by selectively genotyping individuals with extreme values for the quantitative trait. Computer simulations were employed to study the effect of various sampling strategies on the statistical power to detect QTL variance effects. If only individuals with extreme phenotypes for the quantitative trait are selected for genotyping, then power to detect a variance effect is less than by random sampling. If 0.2 of the total number of individuals genotyped are selected from the center of the distribution, then power to detect a variance effect is equal to that obtained with random selection. Power to detect a variance effect was maximum when 0.2 to 0.5 of the individuals selected for genotyping were selected from the tails of the distribution and the remainder from the center.     

Pleiotropic model of maintenance of quantitative genetic variation at mutation-selection balance

A pleiotropic model of maintenance of quantitative genetic variation at mutation-selection balance is investigated. Mutations have effects on a metric trait and deleterious effects on fitness, for which a bivariate gamma distribution is assumed. Equations for calculating the strength of apparent stabilizing selection (V(s)) and the genetic variance maintained in segregating populations (V(G)) were derived. A large population can hold a high genetic variance but the apparent stabilizing selection may or may not be relatively strong, depending on other properties such as the distribution of mutation effects. If the distribution of mutation effects on fitness is continuous such that there are few nearly neutral mutants, or a minimum fitness effect is assumed if most mutations are nearly neutral, V(G) increases to an asymptote as the population size increases. Both V(G) and V(s) are strongly affected by the shape of the distribution of mutation effects. Compared with mutants of equal effect, allowing their effects on fitness to vary across loci can produce a much higher V(G) but also a high V(s) (V(s) in phenotypic standard deviation units, which is always larger than the ratio V(P)/V(m)), implying weak apparent stabilizing selection. If the mutational variance V(m) is approximately 10(-3)V(e) (V(e), environmental variance), the model can explain typical values of heritability and also apparent stabilizing selection, provided the latter is quite weak as suggested by a recent review.     

QUANTITATIVE GENETIC MODELING AND INFERENCE IN THE PRESENCE OF NONIGNORABLE MISSING DATA

Natural selection is typically exerted at some specific life stages. If natural selection takes place before a trait can be measured, using conventional models can cause wrong inference about population parameters. When the missing data process relates to the trait of interest, a valid inference requires explicit modeling of the missing process. We propose a joint modeling approach, a shared parameter model, to account for nonrandom missing data. It consists of an animal model for the phenotypic data and a logistic model for the missing process, linked by the additive genetic effects. A Bayesian approach is taken and inference is made using integrated nested Laplace approximations. From a simulation study we find that wrongly assuming that missing data are missing at random can result in severely biased estimates of additive genetic variance. Using real data from a wild population of Swiss barn owls Tyto alba, our model indicates that the missing individuals would display large black spots; and we conclude that genes affecting this trait are already under selection before it is expressed. Our model is a tool to correctly estimate the magnitude of both natural selection and additive genetic variance.     

Mutation load and mutation-selection-balance in quantitative genetic traits

Haldane (1937) showed that the reduction of equilibrium mean fitness in an infinite population due to recurrent deleterious mutations depends only on the mutation rate but not on the harmfulness of mutants. His analysis, as well as more recent ones (cf. Crow 1970), ignored back mutation. The purpose of the present paper is to extend these results to arbitrary mutation patterns among alleles and to quantitative genetic traits. We derive first-order approximations for the equilibrium mean fitness (and the mutation load) and determine the order of the error term. For a metric trait under mutation-stabilizing-selection balance our result differs qualitatively from that of Crow and Kimura (1964), whose analysis is based on a Gaussian assumption. Our general approach also yields a mathematical proof that the variance under the usual mutation-stabilizing-selection model is, to first order, µ/s (the house-of-cards approximation) as µ/s tends to zero. This holds for arbitrary mutant distributions and does not require that the population mean coincide with the optimum. We show how the mutant distribution determines the order of the error term, and thus the accuracy of the house-of-cards approximation. Upper and lower bounds to the equilibrium variance are derived that deviate only to second order as µ/s tends to zero. The multilocus case is treated under the assumption of global linkage equilibrium.     

Influence of dominance, leptokurtosis and pleiotropy of deleterious mutations on quantitative genetic variation at mutation-selection balance

In models of maintenance of genetic variance (V (G)) it has often been assumed that mutant alleles act additively. However, experimental data show that the dominance coefficient varies among mutant alleles and those of large effect tend to be recessive. On the basis of empirical knowledge of mutations, a joint-effect model of pleiotropic and real stabilizing selection that includes dominance is constructed and analyzed. It is shown that dominance can dramatically alter the prediction of equilibrium V (G). Analysis indicates that for the situations where mutations are more recessive for fitness than for a quantitative trait, as supported by the available data, the joint-effect model predicts a significantly higher V (G) than does an additive model. Importantly, for what seem to be realistic distributions of mutational effects (i.e., many mutants may not affect the quantitative trait substantially but are likely to affect fitness), the observed high levels of genetic variation in the quantitative trait under strong apparent stabilizing selection can be generated. This investigation supports the hypothesis that most V (G) comes from the alleles nearly neutral for fitness in heterozygotes while apparent stabilizing selection is contributed mainly by the alleles of large effect on the quantitative trait. Thus considerations of dominance coefficients of mutations lend further support to our previous conclusion that mutation-selection balance is a plausible mechanism of the maintenance of the genetic variance in natural populations.     

Predicting long-term response to selection

Lande's equation for predicting the response of trait means to a shift in optimal trait values is tested using a stochastic simulation model. The simulated population is finite, and each individual has a finite number of loci. Therefore, selection may cause allele frequencies and distributions to change over time. Since the equation assumes constant genetic parameters, the degree to which such allelic changes affect predictions can be examined. Predictions are based only on information available at generation zero of directional selection. The quality of the predictions depends on the nature of allelic distributions in the original population. If allelic effects are approximately normally distributed, as assumed in Lande's Gaussian approximation to the continuum-of-alleles model, the predictions are very accurate, despite small changes in the G matrix. If allelic effects have a leptokurtic distribution, as is likely in Turelli's 'house-of-cards' approximation, the equation underestimates the rate of response and correlated response, and overestimates the time required for the trait means to reach their equilibrium values. Models with biallelic loci have limits as to the amount of trait divergence possible, since only two allelic values are available at each of a finite set of loci. If the new optimal trait values lie within these limits, predictions are good, if not, singularity in the G matrix results in suboptimal equilibria, despite the presence of genetic variance for each individual trait.     

HOW MUCH HERITABLE VARIATION CAN BE MAINTAINED IN FINITE POPULATIONS BY MUTATION–SELECTION BALANCE?

The joint effects of stabilizing selection, mutation, recombination, and random drift on the genetic variability of a polygenic character in a finite population are investigated. A simulation study is performed to test the validity of various analytical predictions on the equilibrium genetic variance. A new formula for the expected equilibrium variance is derived that approximates the observed equilibrium variance very closely for all parameter combinations we have tested. The computer model simulates the continuum-of-alleles model of Crow and Kimura. However, it is completely stochastic in the sense that it models evolution as a Markov process and does not use any deterministic evolution equations. The theoretical results are compared with heritability estimates from laboratory and natural populations. Heritabilities ranging from 20% to 50%, as observed even in lab populations under a constant environment, can only be explained by a mutation-selection balance if the phenotypic character is neutral or the number of genes contributing to the trait is sufficiently high, typically several hundred, or if there are a few highly variable loci that influence quantitative traits.