Linkage and the maintenance of variation for quantitative 
traits by mutation–selection balance: an infinitesimal model

The infinitesimal model is extended to cover linkage in finite populations. General equations to predict the dynamics of the genetic variation under the joint effects of mutation, selection and drift are derived. Under truncation and stabilizing selection, the quadratic equations for the asymptotic genetic variance (VG) are respectivelyV2G(1+kS)+VG (Ve−2NeVm) −2NeVmVe=0andV2G(1+S)+VG (Ve+γ−2NeVm) −2NeVm(Ve+γ)=0,where Ne is the effective population size, Vm is the mutational variance, Ve is the environmental variance, γ is the parameter that measures the spread of fitness around the optimum under stabilizing selection, k is equal to i(i−x) where i is the selection intensity and x is the cut-off point under truncation selection. The term S is a function of the number of chromosomes (v) and the average chromosome length (l):formula hereThese predictions are accurate when compared with results of simulations of small populations unless the number of genes is small. The infinitesimal model reduces to the continuum of alleles model if there is no recombination between homologous chromosomes.

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Pleiotropic Model of Maintenance of Quantitative Genetic Variation at Mutation-Selection Balance

Genetics ◽

10.1093/genetics/161.1.419 ◽

2002 ◽

Vol 161 (1) ◽

pp. 419-433 ◽

Cited By ~ 6

Author(s):

Xu-Sheng Zhang ◽

Jinliang Wang ◽

William G Hill

Keyword(s):

Genetic Variation ◽

Genetic Variance ◽

Large Population ◽

Stabilizing Selection ◽

Fitness Effect ◽

Environmental Variance ◽

Quantitative Genetic ◽

Phenotypic Standard Deviation ◽

Bivariate Gamma Distribution ◽

Mutational Variance

AbstractA 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 (Vs) and the genetic variance maintained in segregating populations (VG) 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, VG increases to an asymptote as the population size increases. Both VG and Vs 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 VG but also a high Vs (Vs in phenotypic standard deviation units, which is always larger than the ratio VP/Vm), implying weak apparent stabilizing selection. If the mutational variance Vm is ∼10-3 Ve (Ve, 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.

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Quantitative genetic variability maintained by mutation-stabilizing selection balance in finite populations

Genetics Research ◽

10.1017/s0016672300027282 ◽

1988 ◽

Vol 52 (1) ◽

pp. 33-43 ◽

Cited By ~ 42

Author(s):

Peter D. Keightley ◽

William G. Hill

Keyword(s):

Standard Deviation ◽

Quantitative Traits ◽

Genetic Variance ◽

Finite Population ◽

Stabilizing Selection ◽

Infinite Population ◽

Quantitative Genetic ◽

Optimum Model ◽

Mutational Variance ◽

Allele Model

SummaryModels of variability in quantitative traits maintained by a balance between mutation and stabilizing selection are investigated. The effects of mutant alleles are assumed to be additive and to be randomly sampled from a stationary distribution. With a two-allele model the equilibrium genetic variance in an infinite population is independent of the distribution of mutant effects, and dependent only on the total number of mutants appearing per generation. In a finite population, however, both the shape and standard deviation of the distribution of mutant effects are important. The equilibrium variance is lower when most of the mutational variance is contributed by few genes of large effect. Genes of small effect can eventually contribute substantially to the variance with increasing population size (N.) The equilibrium variance can be higher in a finite than an infinite population since near-neutral alleles can drift to intermediate frequencies where selection is weakest. Linkage leads to a reduction in the maintained variance which is small unless linkage is very tight and selection is strong, but the reduction becomes greater with increasing N since more mutants segregate. A multi-allele model is simulated and it is concluded that the two-allele model gives a good approximation of its behaviour. It is argued that the total number of loci capable of influencing most quantitative traits is large, and that the distribution of mutant effects is highly leptokurtic with the effects of most mutants very small, and such mutants are important in contributing to the maintained variance since selection against them is slight. The weakness of the simple optimum model is discussed in relation to the likely consequences of pleiotropy.

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Quantitative genetic variability maintained by mutation-stabilizing selection balance: sampling variation and response to subsequent directional selection

Genetics Research ◽

10.1017/s0016672300028366 ◽

1989 ◽

Vol 54 (1) ◽

pp. 45-58 ◽

Cited By ~ 17

Author(s):

Peter D. Keightley ◽

William G. Hill

Keyword(s):

Genetic Variance ◽

Directional Selection ◽

Quantitative Character ◽

Stabilizing Selection ◽

Effective Population ◽

Selection Responses ◽

Quantitative Genetic ◽

Before And After ◽

Cage Population ◽

Selection Of

SummaryA model of genetic variation of a quantitative character subject to the simultaneous effects of mutation, selection and drift is investigated. Predictions are obtained for the variance of the genetic variance among independent lines at equilibrium with stabilizing selection. These indicate that the coefficient of variation of the genetic variance among lines is relatively insensitive to the strength of stabilizing selection on the character. The effects on the genetic variance of a change of mode of selection from stabilizing to directional selection are investigated. This is intended to model directional selection of a character in a sample of individuals from a natural or long-established cage population. The pattern of change of variance from directional selection is strongly influenced by the strengths of selection at individual loci in relation to effective population size before and after the change of regime. Patterns of change of variance and selection responses from Monte Carlo simulation are compared to selection responses observed in experiments. These indicate that changes in variance with directional selection are not very different from those due to drift alone in the experiments, and do not necessarily give information on the presence of stabilizing selection or its strength.

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On the distribution of the mean and variance of a quantitative trait under mutation-selection-drift balance.

Genetics ◽

10.1093/genetics/138.3.901 ◽

1994 ◽

Vol 138 (3) ◽

pp. 901-912 ◽

Cited By ~ 4

Author(s):

R Bürger ◽

R Lande

Keyword(s):

Genetic Drift ◽

Quantitative Trait ◽

Quantitative Traits ◽

Genetic Variance ◽

Stochastic Simulations ◽

Stabilizing Selection ◽

Phenotypic Evolution ◽

Theoretical Predictions ◽

Mean And Variance ◽

The Mean

Abstract The distributions of the mean phenotype and of the genetic variance of a polygenic trait under a balance between mutation, stabilizing selection and genetic drift are investigated. This is done by stochastic simulations in which each individual and each gene are represented. The results are compared with theoretical predictions. Some aspects of the existing theories for the evolution of quantitative traits are discussed. The maintenance of genetic variance and the average dynamics of phenotypic evolution in finite populations (with Ne < 1000) are generally simpler than those suggested by some recent deterministic theories for infinite populations.

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Polygenic adaptation after a sudden change in environment

10.1101/792952 ◽

2019 ◽

Cited By ~ 10

Author(s):

Laura K. Hayward ◽

Guy Sella

Keyword(s):

Genetic Variance ◽

Sudden Change ◽

Directional Selection ◽

Stabilizing Selection ◽

Effective Population ◽

Short Term ◽

Exponential Process ◽

Optimal Value ◽

Polygenic Adaptation ◽

The Mean

AbstractPolygenic adaptation in response to selection on quantitative traits is thought to be ubiquitous in humans and other species, yet this mode of adaptation remains poorly understood. We investigate the dynamics of this process, assuming that a sudden change in environment shifts the optimal value of a highly polygenic quantitative trait. We find that when the shift is not too large relative to the genetic variance in the trait and this variance arises from segregating loci with small to moderate effect sizes (defined in terms of the selection acting on them before the shift), the mean phenotype’s approach to the new optimum is well approximated by a rapid exponential process first described by Lande (1976). In contrast, when the shift is larger or large effect loci contribute substantially to genetic variance, the initially rapid approach is succeeded by a much slower one. In either case, the underlying changes to allele frequencies exhibit different behaviors short and long-term. Over the short term, strong directional selection on the trait introduces small differences between the frequencies of minor alleles whose effects are aligned with the shift in optimum versus those with effects in the opposite direction. The phenotypic effects of these differences are dominated by contributions from alleles with moderate and large effects, and cumulatively, these effects push the mean phenotype close to the new optimum. Over the longer term, weak directional selection on the trait can amplify the expected frequency differences between opposite alleles; however, since the mean phenotype is close to the new optimum, alleles are mainly affected by stabilizing selection on the trait. Consequently, the frequency differences between opposite alleles translate into small differences in their probabilities of fixation, and the short-term phenotypic contributions of large effect alleles are largely supplanted by contributions of fixed, moderate ones. This process takes on the order of ~4Ne generations (where Ne is the effective population size), after which the steady state architecture of genetic variation around the new optimum is restored.

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The response to artificial selection from new mutations in Drosophila melanogaster.

Genetics ◽

10.1093/genetics/128.1.89 ◽

1991 ◽

Vol 128 (1) ◽

pp. 89-102 ◽

Cited By ~ 5

Author(s):

A Caballero ◽

M A Toro ◽

C López-Fanjul

Keyword(s):

Drosophila Melanogaster ◽

Single Mutation ◽

Effective Population ◽

Environmental Variance ◽

Epistatic Interactions ◽

Bristle Number ◽

Population Sizes ◽

Selection For ◽

Mutational Variance ◽

New Mutations

Abstract Twenty generations of divergent selection for abdominal bristle number were carried out starting from a completely homozygous population of Drosophila melanogaster. All lines were selected with the same proportion (20%) but at two different numbers of selected parents of each sex (5 or 25). A significant response to selection was detected in eight lines (out of 40) and, in most cases, it could be wholly attributed to a single mutation of relatively large effect (0.5-2 phenotypic standard deviations). The ratio of new mutational variance to environmental variance was estimated to be (0.33 +/- 0.11) X 10(-3). The distribution of mutant effects was asymmetrical, both with respect to bristle number (85% of it was negative) and to fitness (most detected bristle mutations were lethal or semilethal). Moreover, this distribution was leptokurtic, due to the presence of major genes. Gene action on bristles ranged from additive to completely recessive, no epistatic interactions being found. In agreement with theory, larger responses in each direction were achieved by those lines selected at greater effective population sizes. Furthermore, the observed divergence between lines selected in opposite directions was proportional to their effective size, as predicted for mutations of large effect.

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Transposable element-induced polygenic mutations in Drosophila melanogaster

Genetics Research ◽

10.1017/s0016672300027117 ◽

1987 ◽

Vol 49 (3) ◽

pp. 225-233 ◽

Cited By ~ 19

Author(s):

Trudy F. C. Mackay

Keyword(s):

Drosophila Melanogaster ◽

Transposable Element ◽

Quantitative Traits ◽

Natural Populations ◽

P Element ◽

Environmental Variance ◽

X Ray ◽

P Elements ◽

Very High ◽

Mutational Variance

SummaryP-element mutagenesis was used to contaminate M-strain second chromosomes with P elements. The effect of P-element transposition on abdominal and sternopleural bristle scores and on female productivity was deduced by comparing the distributions of these quantitative traits among the contaminated second-chromosome lines with a control population of M-strain second-chromosome lines free of P elements. Estimates of P-element-induced mutational variance, Vm, for these characters are very high, and mutational ‘heritabilities’ (Vm/Ve, the ratio of mutational variance to environmental variance) are of the same order as heritabilities of these traits from natural populations. P-element-induced mutational variance of abdominal bristle score is roughly two orders of magnitude greater than spontaneous and X-ray-induced Vm/Ve for this trait.

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Is Function of the Drosophila Homeotic Gene Ultrabithorax Canalized?

Genetics ◽

10.1093/genetics/147.3.1155 ◽

1997 ◽

Vol 147 (3) ◽

pp. 1155-1168 ◽

Cited By ~ 3

Author(s):

Greg Gibson ◽

Sylvie van Helden

Keyword(s):

Genetic Variance ◽

Population Level ◽

Homeotic Gene ◽

Stabilizing Selection ◽

Homeotic Genes ◽

Wild Type ◽

Isofemale Lines ◽

Environmental Variance ◽

Quantitative Genetic ◽

Genetic Contributions

Genetic variation affecting the expressivity of an amorphic allele of the homeotic gene Ultrabithorax, (Ubx1) was characterized after 11 generations of introgression into 29 different isofemale lines. Heterozygotes display a range of haploinsufficient phenotypes, from overlap with wild-type halteres to dramatic transformations such as a 50% increase in area and the presence of over 20 bristles on the anterior margin of each haltere. In both the wild-type and mutant genetic backgrounds, there is moderate genetic variance and low environmental variance/developmental asymmetry, as expected of a trait under stabilizing selection pressure. Surprisingly, there is little evidence that mutant halteres are more variable than wild-type ones, so it is unclear that haltere development is also canalized. The correlation between wild-type and Ubx haltere size is very low, indicating that interactions among modifiers of Ubx are complex, and in some cases sex-specific. The potential quantitative genetic contributions of homeotic genes to appendage morphology are discussed, noting that population-level effects of variation in key regulatory genes may be prevalent and complex but cannot be readily extrapolated to macroevolutionary diversification.

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Neutral additive genetic variance in a metapopulation

Genetics Research ◽

10.1017/s0016672399004127 ◽

1999 ◽

Vol 74 (3) ◽

pp. 215-221 ◽

Cited By ~ 133

Author(s):

MICHAEL C. WHITLOCK

Keyword(s):

Genetic Variance ◽

Additive Genetic Variance ◽

Additive Model ◽

Divergent Evolution ◽

Fixation Index ◽

Effective Population ◽

Quantitative Genetic ◽

Population Structures ◽

Diploid Individual ◽

Mutational Variance

For neutral, additive quantitative characters, the amount of additive genetic variance within and among populations is predictable from Wright's FST, the effective population size and the mutational variance. The structure of quantitative genetic variance in a subdivided metapopulation can be predicted from results from coalescent theory, thereby allowing single-locus results to predict quantitative genetic processes. The expected total amount of additive genetic variance in a metapopulation of diploid individual is given by 2Neσ2m (1 + FST), where FST is Wright's among-population fixation index, Ne is the eigenvalue effective size of the metapopulation, and σ2m is the mutational variance. The expected additive genetic variance within populations is given by 2Neσ2e(1 − FST), and the variance among demes is given by 4FSTNeσ2m. These results are general with respect to the types of population structure involved. Furthermore, the dimensionless measure of the quantitative genetic variance among populations, QST, is shown to be generally equal to FST for the neutral additive model. Thus, for all population structures, a value of QST greater than FST for neutral loci is evidence for spatially divergent evolution by natural selection.

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The contribution of mutation and selection to multivariate quantitative genetic variance in an outbred population of Drosophila serrata

Proceedings of the National Academy of Sciences ◽

10.1073/pnas.2026217118 ◽

2021 ◽

Vol 118 (31) ◽

pp. e2026217118

Author(s):

Robert J. Dugand ◽

J. David Aguirre ◽

Emma Hine ◽

Mark W. Blows ◽

Katrina McGuigan

Keyword(s):

Genetic Variance ◽

Genetic Correlations ◽

Stabilizing Selection ◽

Outbred Population ◽

Drosophila Serrata ◽

Slow Evolution ◽

Multivariate Traits ◽

Mutational Variance ◽

Mutation And Selection ◽

Multivariate Phenotypes

Genetic variance is not equal for all multivariate combinations of traits. This inequality, in which some combinations of traits have abundant genetic variation while others have very little, biases the rate and direction of multivariate phenotypic evolution. However, we still understand little about what causes genetic variance to differ among trait combinations. Here, we investigate the relative roles of mutation and selection in determining the genetic variance of multivariate phenotypes. We accumulated mutations in an outbred population of Drosophila serrata and analyzed wing shape and size traits for over 35,000 flies to simultaneously estimate the additive genetic and additive mutational (co)variances. This experimental design allowed us to gain insight into the phenotypic effects of mutation as they arise and come under selection in naturally outbred populations. Multivariate phenotypes associated with more (less) genetic variance were also associated with more (less) mutational variance, suggesting that differences in mutational input contribute to differences in genetic variance. However, mutational correlations between traits were stronger than genetic correlations, and most mutational variance was associated with only one multivariate trait combination, while genetic variance was relatively more equal across multivariate traits. Therefore, selection is implicated in breaking down trait covariance and resulting in a different pattern of genetic variance among multivariate combinations of traits than that predicted by mutation and drift. Overall, while low mutational input might slow evolution of some multivariate phenotypes, stabilizing selection appears to reduce the strength of evolutionary bias introduced by pleiotropic mutation.

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