Multibreed analysis by splitting the breeding values

An equivalent model for multibreed variance covariance estimation is presented. It considers the additive case including or not the segregation variances. The model is based on splitting the additive genetic values in several independent parts depending on their genetic origin. For each part, it expresses the covariance between relatives as a partial numerator relationship matrix times the corresponding variance component. Estimation of fixed effects, random effects or variance components provided by the model are as simple as any model including several random factors. We present a small example describing the mixed model equations for genetic evaluations and two simulated examples to illustrate the Bayesian variance component estimation.     

Genetic evaluation by best linear unbiased prediction using marker and trait information in a multibreed population.

Genetic evaluation by best linear unbiased prediction (BLUP) requires modeling genetic means, variances, and covariances. This paper presents theory to model means, variances, and covariances in a multibreed population, given marker and breed information, in the presence of gametic disequilibrium between the marker locus (ML) and linked quantitative trait locus (MQTL). Theory and algorithms are presented to construct the matrix of conditional covariances between relatives (Gv) for the MQTL effects in a multibreed population and to obtain the inverse of Gv efficiently. Theory presented here accounts for heterogeneity of variances among pure breeds and for segregation variances between pure breeds. A numerical example was used to illustrate how the theory and algorithms can be used for genetic evaluation by BLUP using marker and trait information in a multibreed population.     

POPULATION STRUCTURE OF MORPHOLOGICAL TRAITS IN CLARKIA DUDLEYANA. II. CONSTANCY OF WITHIN-POPULATION GENETIC VARIANCE

Recent quantitative genetic studies have attempted to infer long-term selection responsible for differences in observed phenotypes. These analyses are greatly simplified by the assumption that the within-population genetic variance remains constant through time and over space, or for the multivariate case, that the matrix of additive genetic variances and covariances (G matrix) is constant. We examined differences in G matrices and the association of these differences with differences in multivariate means (Mahalanobis D²) among 11 populations of the California endemic annual plant, Clarkia dudleyana. Based on nine continuous morphological traits, the relationship between Mahalanobis D² and a distance measure summarizing differences in G matrices reflected no concomitant change in (co)variances with changes in means. Based on both broad- and narrow-sense analyses, we found little evidence that G matrices differed between populations. These results suggest that both the additive and nonadditive (co)variances for traits have remained relatively constant despite changes in means.     

Bottleneck Effects on Genetic Variance for Courtship Repertoire

Bottleneck effects on evolutionary potential in mating behavior were addressed through assays of additive genetic variances and resulting phenotypic responses to drift in the courtship repertoires of six two-pair founder-flush lines and two control populations of the housefly. A simulation addressed the complication that an estimate of the genetic variance for a courtship trait (e.g., male performance vigor or the female requirement for copulation) must involve assays against the background behavior of the mating partners. The additive ``environmental' effect of the mating partner's phenotype simply dilutes the net parent-offspring covariance for a trait. However, if there is an interaction with this ``environmental' component, negative parent-offspring covariances can result under conditions of high incompatibility between the population's distributions for male performance and female choice requirements, despite high levels of genetic variance. All six bottlenecked lines exhibited significant differentiation from the controls in at least one measure of the parent-offspring covariance for male performance or female choice (estimated by 50 parent-son and 50 parent-daughter covariances for 10 courtship traits per line) which translated to significant phenotypic drift. However, the average effect across traits or across lines did not yield a significant net increase in genetic variance due to bottlenecks. Concerted phenotypic differentiation due to the founder-flush event provided indirect evidence of directional dominance in a subset of traits. Furthermore, indirect evidence of genotype-environment interactions (potentially producing genotype-genotype effects) was found in the negative parent-offspring covariances predicted by the male-female interaction simulation and by the association of the magnitude of phenotypic drift with the absolute value of the parent-offspring covariance. Hence, nonadditive genetic effects on mating behavior may be important in structuring genetic variance for courtship, although most of the increases in genetic variance would be expected to reflect inbreeding depression with relatively rare situations representing the facilitation of speciation by bottlenecks.     

Inbreeding, coancestry, and covariance between relatives for X-chromosomal loci

Knowledge about the relationships between relatives for X-chromosomal loci is necessary to compute genetic variances and covariances for the genetic evaluation of individuals for economically important traits in livestock and poultry. Using a unified approach, we derived explicitly the coefficient of inbreeding for individuals and the coefficient of coancestry between collateral and lineal relatives of the same or different sex, assuming that the male is heterogametic and the female is homogametic. Collateral relatives include full sibs, paternal and maternal half-sibs, paternal and maternal single first cousins, and double first cousins. Lineal relatives include parent-offspring, paternal and maternal grandparent-grandoffspring, and aunt- or uncle-niece or -nephew. We also defined additive and dominance relationships to compute genetic covariance between relatives, assuming random mating equilibrium, and clarified misinterpretations and corrected errors in the literature. Our results are also applicable to organisms that have few autosomal loci, such as Drosophila, in which X-chromosomal loci can account for a large amount of genetic variance, and to haplodiploid organisms, such as the honeybee, in which the entire genome is equivalent to being X-chromosomal.     

The genetic structure of female life history in D. melanogaster: comparisons among populations

Two questions were addressed: (1) What is the genetic variance-covariance structure of a suite of four female life history traits in D. melanogaster? and (2) Does the genetic architecture of these traits differ among populations? Three populations of D. melanogaster were studied. Genetic variances and covariances were estimated by sib analysis three times for each population: immediately upon establishment of populations in the laboratory, and subsequently after approximately 6 months and 2 years of laboratory culture. Entire genetic variance-covariance matrices, as well as their individual components, were compared between populations by means of likelihood ratio tests. All traits studied were significantly heritable in at least one-half of estimates. Despite large sample sizes, additive genetic covariances were for the most part not statistically significant, and only two significant negative covariance estimates were obtained throughout the experiments. Therefore, these experiments provide little support for evolutionary life history theories that are based on negative genetic correlations among life history components. Neither do they support the idea that genetic variance for fitness components is maintained by trade-offs. Evidence suggests that the G matrix of one population was initially different from those of the other two populations. Those differences disappeared after 2 years of laboratory culture. At the level of individual (co)variance components, there were relatively few differences among populations, and the overall impression was that the three populations had generally similar genetic architectures for the traits studied.     

Dissecting total genetic variance into additive and dominance components of purebred and crossbred pig traits

The partition of the total genetic variance into its additive and non-additive components can differ from trait to trait, and between purebred and crossbred populations. A quantification of these genetic variance components will determine the extent to which it would be of interest to account for dominance in genomic evaluations or to establish mate allocation strategies along different populations and traits. This study aims at assessing the contribution of the additive and dominance genomic variances to the phenotype expression of several purebred Piétrain and crossbred (Piétrain × Large White) pig performances. A total of 636 purebred and 720 crossbred male piglets were phenotyped for 22 traits that can be classified into six groups of traits: growth rate and feed efficiency, carcass composition, meat quality, behaviour, boar taint and puberty. Additive and dominance variances estimated in univariate genotypic models, including additive and dominance genotypic effects, and a genomic inbreeding covariate allowed to retrieve the additive and dominance single nucleotide polymorphism variances for purebred and crossbred performances. These estimated variances were used, together with the allelic frequencies of the parental populations, to obtain additive and dominance variances in terms of genetic breeding values and dominance deviations. Estimates of the Piétrain and Large White allelic contributions to the crossbred variance were of about the same magnitude in all the traits. Estimates of additive genetic variances were similar regardless of the inclusion of dominance. Some traits showed relevant amount of dominance genetic variance with respect to phenotypic variance in both populations (i.e. growth rate 8%, feed conversion ratio 9% to 12%, backfat thickness 14% to 12%, purebreds-crossbreds). Other traits showed higher amount in crossbreds (i.e. ham cut 8% to 13%, loin 7% to 16%, pH semimembranosus 13% to 18%, pH longissimus dorsi 9% to 14%, androstenone 5% to 13% and estradiol 6% to 11%, purebreds-crossbreds). It was not encountered a clear common pattern of dominance expression between groups of analysed traits and between populations. These estimates give initial hints regarding which traits could benefit from accounting for dominance for example to improve genomic estimated breeding value accuracy in genetic evaluations or to boost the total genetic value of progeny by means of assortative mating.     

A MULTIVARIATE ANALYSIS OF GENETIC VARIATION IN THE ADVERTISEMENT CALL OF THE GRAY TREEFROG,HYLA VERSICOLOR

Genetic variation in sexual displays is crucial for an evolutionary response to sexual selection, but can be eroded by strong selection. Identifying the magnitude and sources of additive genetic variance underlying sexually selected traits is thus an important issue in evolutionary biology. We conducted a quantitative genetics experiment with gray treefrogs (Hyla versicolor) to investigate genetic variances and covariances among features of the male advertisement call. Two energetically expensive traits showed significant genetic variation: call duration, expressed as number of pulses per call, and call rate, represented by its inverse, call period. These two properties also showed significant genetic covariance, consistent with an energetic constraint to call production. Combining the genetic variance–covariance matrix with previous estimates of directional sexual selection imposed by female preferences predicts a limited increase in call duration but no change in call rate despite significant selection on both traits. In addition to constraints imposed by the genetic covariance structure, an evolutionary response to sexual selection may also be limited by high energetic costs of long‐duration calls and by preferences that act most strongly against very short‐duration calls. Meanwhile, the persistence of these preferences could be explained by costs of mating with males with especially unattractive calls.     

Developmental interactions and the constituents of quantitative variation

Development is the process by which genotypes are transformed into phenotypes. Consequently, development determines the relationship between allelic and phenotypic variation in a population and, therefore, the patterns of quantitative genetic variation and covariation of traits. Understanding the developmental basis of quantitative traits may lead to insights into the origin and evolution of quantitative genetic variation, the evolutionary fate of populations, and, more generally, the relationship between development and evolution. Herein, we assume a hierarchical, modular structure of trait development and consider how epigenetic interactions among modules during ontogeny affect patterns of phenotypic and genetic variation. We explore two developmental models, one in which the epigenetic interactions between modules result in additive effects on character expression and a second model in which these epigenetic interactions produce nonadditive effects. Using a phenotype landscape approach, we show how changes in the developmental processes underlying phenotypic expression can alter the magnitude and pattern of quantitative genetic variation. Additive epigenetic effects influence genetic variances and covariances, but allow trait means to evolve independently of the genetic variances and covariances, so that phenotypic evolution can proceed without changing the genetic covariance structure that determines future evolutionary response. Nonadditive epigenetic effects, however, can lead to evolution of genetic variances and covariances as the mean phenotype evolves. Our model suggests that an understanding of multivariate evolution can be considerably enriched by knowledge of the mechanistic basis of character development.     

THE EVOLUTIONARY GENETICS OF MALE LIFE-HISTORY CHARACTERS IN DROSOPHILA MELANOGASTER

Alternative models of the maintenance of genetic variability, theories of life-history evolution, and theories of sexual selection and mate choice can be tested by measuring additive and nonadditive genetic variances of components of fitness. A quantitative genetic breeding design was used to produce estimates of genetic variances for male life-history traits in Drosophila melanogaster. Additive genetic covariances and correlations between traits were also estimated. Flies from a large, outbred, laboratory population were assayed for age-specific competitive mating ability, age-specific survivorship, body mass, and fertility. Variance-component analysis then allowed the decomposition of phenotypic variation into components associated with additive genetic, nonadditive genetic, and environmental variability. A comparison of dominance and additive components of genetic variation provides little support for an important role for balancing selection in maintaining genetic variance in this suite of traits. The results provide support for the mutation-accumulation theory, but not the antagonistic-pleiotropy theory of senescence. No evidence is found for the positive genetic correlations between mating success and offspring quality or quantity that are predicted by “good genes” models of sexual selection. Additive genetic coefficients of variation for life-history characters are larger than those for body weight. Finally, this set of male life-history characters exhibits a very low correspondence between estimates of genetic and phenotypic correlations.     

AN ANALYSIS OF SELECTIONAL RESPONSE IN RELATION TO A POPULATION BOTTLENECK

Selection for increased morphometric shape (ratio of wing length to thorax width) was compared between control (nonbottlenecked) populations and bottlenecked populations founded with two male–female pairs of flies. Contrary to neutral expectation, selectional response was not reduced in bottlenecked populations, and the mean realized heritabilities and additive genetic variances were higher for the bottlenecked lines than for the nonbottlenecked lines. Additive genetic variances based on these realized heritabilities were consistent with independent estimates of genetic variances based on parent–offspring covariances. Joint scaling tests applied to the crosses between selected lines and their controls revealed significant nonadditive components of genetic variance in the ancestor, which were not detected in the crosses involving bottlenecked lines. The nonbottlenecked lines responded principally by changes in one trait or the other (wing length or thorax width) but not in both, and regardless of which trait responded, larger trait size was dominant and epistatic to smaller size. Stabilizing selection for morphometric shape in the ancestor likely molded the genetic architecture to include nonadditive genetic effects.     

Nonlinear selection and the evolution of variances and covariances for continuous characters in an anole

The pattern of genetic variances and covariances among characters, summarized in the additive genetic variance‐covariance matrix, G , determines how a population will respond to linear natural selection. However, G itself also evolves in response to selection. In particular, we expect that, over time, G will evolve correspondence with the pattern of multivariate nonlinear natural selection. In this study, we substitute the phenotypic variance‐covariance matrix ( P ) for G to determine if the pattern of multivariate nonlinear selection in a natural population of Anolis cristatellus, an arboreal lizard from Puerto Rico, has influenced the evolution of genetic variances and covariances in this species. Although results varied among our estimates of P and fitness, and among our analytic techniques, we find significant evidence for congruence between nonlinear selection and P , suggesting that natural selection may have influenced the evolution of genetic constraint in this species.     

Analysis of cytoplasmic and maternal effects I. A genetic model for diploid plant seeds and animals

A genetic model for modified diallel crosses is proposed for estimating variance and covariance components of cytoplasmic, maternal additive and dominance effects, as well as direct additive and dominance effects. Monte Carlo simulations were conducted to compare the efficiencies of minimum norm quadratic unbiased estimation (MINQUE) methods. For both balanced and unbalanced mating designs, MINQUE (0/1), which has 0 for all the prior covariances and 1 for all the prior variances, has similar efficiency to MINQUE(), which has parameter values for the prior values. Unbiased estimates of variance and covariance components and their sampling variances could be obtained with MINQUE(0/1) and jackknifing. A t-test following jackknifing is applicable to test hypotheses for zero variance and covariance components. The genetic model is robust for estimating variance and covariance components under several situations of no specific effects. A MINQUE(0/1) procedure is suggested for unbiased estimation of covariance components between two traits with equal design matrices. Methods of unbiased prediction for random genetic effects are discussed. A linear unbiased prediction (LUP) method is shown to be efficient for the genetic model. An example is given for a demonstration of estimating variance and covariance components and predicting genetic effects.     

Persistence of changes in the genetic covariance matrix after a bottleneck

Abstract.— Genetic variance, phenotypic variance, and the genetic covariance matrix ( G ) can change as a result of genetic drift. These changes will persist over time to some extent and will continue if population size remains relatively small. Nine populations founded by a single pair of Drosophila melanogaster were measured for a series of six morphological characteristics for a large number of parent-offspring families at both the third generation after the bottlenecks and after 20 generations. From these data, the phenotypic variance, additive genetic variance, and G were estimated for each line at each generation. Phenotypic and genetic variances were highly correlated over time, so that the measurements made at the third generation were predictive of the state of the population 17 generations later. Genetic covariances were also somewhat stable over time; however, the G matrices of some lines changed significantly over the intervening generations. This change did not return the populations toward their original state before the population bottlenecks. We conclude that the genetic covariance matrix can change as a result of mild genetic drift over a short span of time.     

EFFECT OF AN EXPERIMENTAL BOTTLENECK ON MORPHOLOGICAL INTEGRATION IN THE HOUSEFLY

Three measures of multivariate integration were derived from both additive genetic covariance and correlation matrices estimated from parent-offspring covariances to investigate the effect of bottlenecks of different sizes on genetic integration of morphological traits in the housefly, Musca domestica L. Bottleneck lines were initiated with one, four, or 16 pairs of flies sampled from a natural outbred (control) population. Bottlenecks of intermediate size significantly increased the average genetic correlation among traits, resulting in nearly isomorphic variation among all traits in these lines. Single-pair bottlenecks significantly disrupted the trait interrelationships, and the suites of traits identified by principal components of the additive genetic correlation and covariance matrices for the control population were no longer evident in these bottleneck lines. The alteration of the genetic relationships among traits as a result of a bottleneck suggests that nonadditive components of genetic variation affecting these traits were present in the control line. We discuss the implications of nonadditive gene action, particularly epistasis, for speciation via bottlenecks.     

Theory for modelling means and covariances in a two-breed population with dominance inheritance

This paper presents theory and methods to compute genotypic means and covariances in a two breed population under dominance inheritance, assuming multiple unlinked loci. It is shown that the genotypic mean is a linear function of five location parameters and that the genotypic covariance between relatives is a linear function of 25 dispersion parameters. Recursive procedures are given to compute the necessary identity coefficients. In the absence of inbreeding, the number of parameters for the mean is reduced from five to three and the number for the covariance is reduced from 25 to 12. In a two-breed population, for traits exhibiting dominance, the theory presented here can be used to obtain genetic evaluations by best linear unbiased prediction and to estimate genetic parameters by maximum likelihood.Supported in part by the Illinois Agricultural Experiment Station, Hatch Projects 35-0345 (R.L.F.) and 35-0367 (M.G.). A computer program implementing the methods described here is available upon request to R.L.F.     

The genetic variance but not the genetic covariance of life‐history traits changes towards the north in a time‐constrained insect

Seasonal time constraints are usually stronger at higher than lower latitudes and can exert strong selection on life‐history traits and the correlations among these traits. To predict the response of life‐history traits to environmental change along a latitudinal gradient, information must be obtained about genetic variance in traits and also genetic correlation between traits, that is the genetic variance‐covariance matrix, G . Here, we estimated G for key life‐history traits in an obligate univoltine damselfly that faces seasonal time constraints. We exposed populations to simulated native temperatures and photoperiods and common garden environmental conditions in a laboratory set‐up. Despite differences in genetic variance in these traits between populations (lower variance at northern latitudes), there was no evidence for latitude‐specific covariance of the life‐history traits. At simulated native conditions, all populations showed strong genetic and phenotypic correlations between traits that shaped growth and development. The variance–covariance matrix changed considerably when populations were exposed to common garden conditions compared with the simulated natural conditions, showing the importance of environmentally induced changes in multivariate genetic structure. Our results highlight the importance of estimating variance–covariance matrixes in environments that mimic selection pressures and not only trait variances or mean trait values in common garden conditions for understanding the trait evolution across populations and environments.     

RESISTANCE OF GENETIC CORRELATION STRUCTURE TO DIRECTIONAL SELECTION IN DROSOPHILA MELANOGASTER

The genetic covariance and correlation matrices for five morphological traits were estimated from four populations of fruit flies, Drosophila melanogaster, to measure the extent of change in genetic covariances as a result of directional selection. Two of the populations were derived from lines that had undergone selection for large or small thorax length over the preceding 23 generations. A third population was constituted using flies from control lines that were maintained with equivalent population sizes as the selected lines. The fourth population contained flies from the original cage population from which the selected and control lines had been started. Tests of the homogeneity of covariance matrices using maximum likelihood techniques revealed significant changes in covariance structure among the selected lines. Prediction of base population trait means from selected line means under the assumption of constant genetic covariances indicated that genetic covariances for the small population differed more from the base population than did the covariances for the large population. The predicted small population means diverged farther from the expected means because the additive genetic variance associated with several traits increased in value and most of the genetic covariances associated with one trait changed in sign. These results illustrate that genetic covariances may remain nearly constant in some situations while changing markedly in others. Possible developmental reasons for the genetic changes are discussed.     

Misspecification in Mixed-Model-Based Association Analysis

Additive genetic variance in natural populations is commonly estimated using mixed models, in which the covariance of the genetic effects is modeled by a genetic similarity matrix derived from a dense set of markers. An important but usually implicit assumption is that the presence of any nonadditive genetic effect increases only the residual variance and does not affect estimates of additive genetic variance. Here we show that this is true only for panels of unrelated individuals. In the case that there is genetic relatedness, the combination of population structure and epistatic interactions can lead to inflated estimates of additive genetic variance.     

The Contribution of Quantitative Trait Loci and Neutral Marker Loci to the Genetic Variances and Covariances among Quantitative Traits in Random Mating Populations

Using Cockerham's approach of orthogonal scales, we develop genetic models for the effect of an arbitrary number of multiallelic quantitative trait loci (QTLs) or neutral marker loci (NMLs) upon any number of quantitative traits. These models allow the unbiased estimation of the contributions of a set of marker loci to the additive and dominance variances and covariances among traits in a random mating population. The method has been applied to an analysis of allozyme and quantitative data from the European oyster. The contribution of a set of marker loci may either be real, when the markers are actually QTLs, or apparent, when they are NMLs that are in linkage disequilibrium with hidden QTLs. Our results show that the additive and dominance variances contributed by a set of NMLs are always minimum estimates of the corresponding variances contributed by the associated QTLs. In contrast, the apparent contribution of the NMLs to the additive and dominance covariances between two traits may be larger than, equal to or lower than the actual contributions of the QTLs. We also derive an expression for the expected variance explained by the correlation between a quantitative trait and multilocus heterozygosity. This correlation explains only a part of the genetic variance contributed by the markers, i.e., in general, a combination of additive and dominance variances and, thus, provides only very limited information relative to the method supplied here.