Additive Genetic Variance-covariance Matrix Research Articles

Population Divergence along a Genetic Line of Least Resistance in the Tree Species Eucalyptus globulus.

The evolutionary response to selection depends on the distribution of genetic variation in traits under selection within populations, as defined by the additive genetic variance-covariance matrix (G). The structure and evolutionary stability of G will thus influence the course of phenotypic evolution. However, there are few studies assessing the stability of G and its relationship with population divergence within foundation tree species. We compared the G-matrices of Mainland and Island population groups of the forest tree Eucalyptus globulus, and determined the extent to which population divergence aligned with within-population genetic (co)variation. Four key wood property traits exhibiting signals of divergent selection were studied—wood density, extractive content, and lignin content and composition. The comparison of G-matrices of the mainland and island populations indicated that the G-eigenstructure was relatively well preserved at an intra-specific level. Population divergence tended to occur along a major direction of genetic variation in G. The observed conservatism of G, the moderate evolutionary timescale, and close relationship between genetic architecture and population trajectories suggest that genetic constraints may have influenced the evolution and diversification of the E. globulus populations for the traits studied. However, alternative scenarios, including selection aligning genetic architecture and population divergence, are discussed.

Genetic covariances promote climatic adaptation in AustralianDrosophila*

Evolutionary potential for adaptation hinges upon the orientation of genetic variation for traits under selection, captured by the additive genetic variance-covariance matrix (G), as well as the evolutionary stability of G. Yet studies that assess both the stability of G and its alignment with selection are extraordinarily rare. We evaluated the stability of G in three Drosophila melanogaster populations that have adapted to local climatic conditions along a latitudinal cline. We estimated population- and sex-specific G matrices for wing size and three climatic stress-resistance traits that diverge adaptively along the cline. To determine how G affects evolutionary potential within these populations, we used simulations to quantify how well G aligns with the direction of trait divergence along the cline (as a proxy for the direction of local selection) and how genetic covariances between traits and sexes influence this alignment. We found that G was stable across the cline, showing no significant divergence overall, or in sex-specific subcomponents, among populations. G also aligned well with the direction of clinal divergence, with genetic covariances strongly elevating evolutionary potential for adaptation to climatic extremes. These results suggest that genetic covariances between both traits and sexes should significantly boost evolutionary responses to environmental change.

Evaluation criterion for response to selection with constraint.

The aims of the present study are to represent the concept of restricted breeding values algebraically and to propose a criterion for evaluating the genetic responses achieved by using a restricted selection procedure. An additive genetic mixed model characterized by multiple traits with constraints was assumed. If the random errors approach zero and the fixed effects can be completely estimated correctly in the model, the restricted best linear unbiased predictor of breeding values (u R) is equal to [Iq−G0C0(C0′G0C0)−1C0′]⊗Iu, where G 0, C 0, and u are the additive genetic variance‐covariance matrix for the q traits, the matrix for restriction, and the vector of breeding values, respectively. Therefore, if we want to evaluate the response to restricted selection, such as by a stochastic computer simulation study with known breeding values, we can use u R as only one criterion.

A case study of extant and extinct Xenarthra cranium covariance structure: implications and applications to paleontology

AbstractMost of the mammalian diversity is known only from fossils, and only a few of these fossils are well preserved or abundant. This undersampling poses serious problems for understanding mammalian phenotypic evolution under a quantitative genetics framework, since this framework requires estimation of a group’s additive genetic variance–covariance matrix (Gmatrix), which is impossible, and estimating a phenotypic variance–covariance matrix (Pmatrix) requires larger sample sizes than what is often available for extinct species. One alternative is to useGorPmatrices from extant taxa as surrogates for the extinct ones. Although there are reasons to believe this approach is usually safe, it has not been fully explored. By thoroughly determining the extant and some extinct Xenarthra (Mammalia) craniumPmatrices, this study aims to explore the feasibility of using extantGorPmatrices as surrogates for the extinct ones and to provide guidelines regarding the reliability of this strategy and the necessary sample sizes. Variance–covariance and correlationPmatrices for 35 cranium traits from 16 xenarthran genera (12 extant and 4 extinct) were estimated and compared between genera. Results show xenarthranP-matrix structures are usually very similar if sample sizes are reasonable. This study and others developed with extant therian mammals suggest, in general, that using extantGorPmatrices as an approximation to extinct ones is a valid approach. Nevertheless, the accuracy of this approach depends on sample size, selected traits, and the type of matrix being considered.

GENETIC CONSTRAINTS UNDERLYING HUMAN REPRODUCTIVE TIMING IN A PREMODERN SWISS VILLAGE

The trade-off between reproductive investment in early versus late life is central to life-history theory. Despite abundant empirical evidence supporting different versions of this trade-off, the specific trade-off between age at first reproduction (AFR) and age at last reproduction (ALR) has received little attention, especially in long-lived species with a pronounced reproductive senescence such as humans. Using genealogical data for a 19th-century Swiss village, we (i) quantify natural selection acting on reproductive timing, (ii) estimate the underlying additive genetic (co)variances, and (iii) use these to predict evolutionary responses. Selection gradients were computed using multiple linear regression, and the additive genetic variance-covariance matrix was estimated using a restricted maximum-likelihood animal model. We found strong selection for both an early AFR and a late ALR, which resulted from selection for an earlier and longer reproductive period (RP, i.e., ALR-AFR). Furthermore, postponing AFR shortened RP in both sexes, but twice as much in women. Finally, AFR and ALR were strongly and positively genetically correlated, which led to a considerable reduction in the predicted responses to selection, or even rendered them maladaptive. These results provide evidence for strong genetic constraints underlying reproductive timing in humans, which may have contributed to the evolution of menopause.

Metabolism, Growth, and the Energetic Definition of Fitness: A Quantitative Genetic Study in the Land SnailCornu aspersum

Life-history evolution-the way organisms allocate time and energy to reproduction, survival, and growth-is a central question in evolutionary biology. One of its main tenets, the allocation principle, predicts that selection will reduce energy costs of maintenance in order to divert energy to survival and reproduction. The empirical support for this principle is the existence of a negative relationship between fitness and metabolic rate, which has been observed in some ectotherms. In juvenile animals, a key function affecting fitness is growth rate, since fast growers will reproduce sooner and maximize survival. In principle, design constraints dictate that growth rate cannot be reduced without affecting maintenance costs. Hence, it is predicted that juveniles will show a positive relationship between fitness (growth rate) and metabolic rate, contrarily to what has been observed in adults. Here we explored this problem using land snails (Cornu aspersum). We estimated the additive genetic variance-covariance matrix for growth and standard metabolic rate (SMR; rate of CO2 production) using 34 half-sibling families. We measured eggs, hatchlings, and juveniles in 208 offspring that were isolated right after egg laying (i.e., minimizing maternal and common environmental variance). Surprisingly, our results showed that additive genetic effects (narrow-sense heritabilities, h(2)) and additive genetic correlations (rG) were small and nonsignificant. However, the nonadditive proportion of phenotypic variances and correlations (rC) were unexpectedly large and significant. In fact, nonadditive genetic effects were positive for growth rate and SMR ([Formula: see text]; [Formula: see text]), supporting the idea that fitness (growth rate) cannot be maximized without incurring maintenance costs. Large nonadditive genetic variances could result as a consequence of selection eroding the additive genetic component, which suggests that past selection could have produced nonadditive genetic correlation. It is predicted that this correlation is reduced when adulthood is attained and selection starts to promote the reduction in metabolic rate.

Comparing G: multivariate analysis of genetic variation in multiple populations

The additive genetic variance-covariance matrix (G) summarizes the multivariate genetic relationships among a set of traits. The geometry of G describes the distribution of multivariate genetic variance, and generates genetic constraints that bias the direction of evolution. Determining if and how the multivariate genetic variance evolves has been limited by a number of analytical challenges in comparing G-matrices. Current methods for the comparison of G typically share several drawbacks: metrics that lack a direct relationship to evolutionary theory, the inability to be applied in conjunction with complex experimental designs, difficulties with determining statistical confidence in inferred differences and an inherently pair-wise focus. Here, we present a cohesive and general analytical framework for the comparative analysis of G that addresses these issues, and that incorporates and extends current methods with a strong geometrical basis. We describe the application of random skewers, common subspace analysis, the 4th-order genetic covariance tensor and the decomposition of the multivariate breeders equation, all within a Bayesian framework. We illustrate these methods using data from an artificial selection experiment on eight traits in Drosophila serrata, where a multi-generational pedigree was available to estimate G in each of six populations. One method, the tensor, elegantly captures all of the variation in genetic variance among populations, and allows the identification of the trait combinations that differ most in genetic variance. The tensor approach is likely to be the most generally applicable method to the comparison of G-matrices from any sampling or experimental design.

TYPE I ERROR RATES FOR TESTING GENETIC DRIFT WITH PHENOTYPIC COVARIANCE MATRICES: A SIMULATION STUDY

Studies of evolutionary divergence using quantitative genetic methods are centered on the additive genetic variance-covariance matrix (G) of correlated traits. However, estimating G properly requires large samples and complicated experimental designs. Multivariate tests for neutral evolution commonly replace average G by the pooled phenotypic within-group variance-covariance matrix (W) for evolutionary inferences, but this approach has been criticized due to the lack of exact proportionality between genetic and phenotypic matrices. In this study, we examined the consequence, in terms of type I error rates, of replacing average G by W in a test of neutral evolution that measures the regression slope between among-population variances and within-population eigenvalues (the Ackermann and Cheverud [AC] test) using a simulation approach to generate random observations under genetic drift. Our results indicate that the type I error rates for the genetic drift test are acceptable when using W instead of average G when the matrix correlation between the ancestral G and P is higher than 0.6, the average character heritability is above 0.7, and the matrices share principal components. For less-similar G and P matrices, the type I error rates would still be acceptable if the ratio between the number of generations since divergence and the effective population size (t/N(e)) is smaller than 0.01 (large populations that diverged recently). When G is not known in real data, a simulation approach to estimate expected slopes for the AC test under genetic drift is discussed.

INDEPENDENT AXES OF GENETIC VARIATION AND PARALLEL EVOLUTIONARY DIVERGENCE OF OPERCLE BONE SHAPE IN THREESPINE STICKLEBACK

Evolution of similar phenotypes in independent populations is often taken as evidence of adaptation to the same fitness optimum. However, the genetic architecture of traits might cause evolution to proceed more often toward particular phenotypes, and less often toward others, independently of the adaptive value of the traits. Freshwater populations of Alaskan threespine stickleback have repeatedly evolved the same distinctive opercle shape after divergence from an oceanic ancestor. Here we demonstrate that this pattern of parallel evolution is widespread, distinguishing oceanic and freshwater populations across the Pacific Coast of North America and Iceland. We test whether this parallel evolution reflects genetic bias by estimating the additive genetic variance-covariance matrix (G) of opercle shape in an Alaskan oceanic (putative ancestral) population. We find significant additive genetic variance for opercle shape and that G has the potential to be biasing, because of the existence of regions of phenotypic space with low additive genetic variation. However, evolution did not occur along major eigenvectors of G, rather it occurred repeatedly in the same directions of high evolvability. We conclude that the parallel opercle evolution is most likely due to selection during adaptation to freshwater habitats, rather than due to biasing effects of opercle genetic architecture.

EVIDENCE FOR STRONG INTRALOCUS SEXUAL CONFLICT IN THE INDIAN MEAL MOTH, PLODIA INTERPUNCTELLA

Males and females share a genome and express many shared phenotypic traits, which are often selected in opposite directions. This generates intralocus sexual conflict that may constrain trait evolution by preventing the sexes from reaching their optimal phenotype. Furthermore, if present across multiple loci, intralocus sexual conflict can result in a gender load that may diminish the benefits of sexual selection and help maintain genetic variation for fitness. Despite the importance of intralocus sexual conflict, surprisingly few empirical studies conclusively demonstrate its operation. We show that the pattern of multivariate selection acting on three sexually dimorphic life-history traits (development time, body size, and longevity) in the Indian meal moth, Plodia interpunctella, is opposing for the sexes. Moreover, we combined our estimates of selection with the additive genetic variance-covariance matrix (G) to predict the evolutionary response of the life-history traits in the sexes and showed that the angle between the vector of responses and the vector of sexually antagonistic selection was almost orthogonal at 84.70°. Thus, G biases the predicted response of life-history traits in the sexes away from the direction of sexually antagonistic selection, confirming the presence of strong intralocus sexual conflict in this species. Despite this, sexual dimorphism has evolved in all of the life-history traits examined suggesting that mechanism(s) have evolved to resolve this conflict and allow the sexes to reach their life-history optima. We argue that intralocus sexual conflict is likely to play an important role in the evolution of divergent life-history strategies between the sexes in this species.

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.

Multivariate Quantitative Genetics of Anthropometric Traits from the Boas Data

The use of multivariate quantitative trait information to address questions of population relationships and evolutionary issues has a longstanding history in human anthropometry. Previous analyses have usually rested on a number of explicit or implicit assumptions that allow phenotypic information to be used as a proxy for quantitative genetic information. One (usually implicit) assumption is that the additive genetic variance-covariance matrix (G) among traits is proportional to the phenotypic variance-covariance matrix (P). In this study we discuss the implications of this assumption, demonstrating that if it is true that G = h 2 P, where h 2 is some constant of proportionality, then (1) the biological (phenotypic) Mahalanobis distance will be proportional to genetic distance, (2) phenotypic and genetic allometry coefficients will be equal, and (3) evolutionary models will become simplified. We then use a multivariate quantitative genetic analysis of 12 anthropometric traits in 5 tribes to demonstrate that G = h 2 P for at least a portion of the Boas data.

Multivariate Quantitative Genetics of Anthropometric Traits from the Boas Data

The use of multivariate quantitative trait information to address questions of population relationships and evolutionary issues has a long-standing history in human anthropometry. Previous analyses have usually rested on a number of explicit or implicit assumptions that allow phenotypic information to be used as a proxy for quantitative genetic information. One (usually implicit) assumption is that the additive genetic variance-covariance matrix (G) among traits is proportional to the phenotypic variance-covariance matrix (P). In this study we discuss the implications of this assumption, demonstrating that if it is true that G = h2P, where h2 is some constant of proportionality, then (1) the biological (phenotypic) Mahalanobis distance will be proportional to genetic distance, (2) phenotypic and genetic allometry coefficients will be equal, and (3) evolutionary models will become simplified. We then use a multivariate quantitative genetic analysis of 12 anthropometric traits in 5 tribes to demonstrate that G = h2P for at least a portion of the Boas data.

EMPIRICAL COMPARISON OF G MATRIX TEST STATISTICS: FINDING BIOLOGICALLY RELEVANT CHANGE

A central assumption of quantitative genetic theory is that the breeder's equation (R=GP(-1)S) accurately predicts the evolutionary response to selection. Recent studies highlight the fact that the additive genetic variance-covariance matrix (G) may change over time, rendering the breeder's equation incapable of predicting evolutionary change over more than a few generations. Although some consensus on whether G changes over time has been reached, multiple, often-incompatible methods for comparing G matrices are currently used. A major challenge of G matrix comparison is determining the biological relevance of observed change. Here, we develop a "selection skewers"G matrix comparison statistic that uses the breeder's equation to compare the response to selection given two G matrices while holding selection intensity constant. We present a bootstrap algorithm that determines the significance of G matrix differences using the selection skewers method, random skewers, Mantel's and Bartlett's tests, and eigenanalysis. We then compare these methods by applying the bootstrap to a dataset of laboratory populations of Tribolium castaneum. We find that the results of matrix comparison statistics are inconsistent based on differing a priori goals of each test, and that the selection skewers method is useful for identifying biologically relevant G matrix differences.

THE PHENOTYPIC AND GENETIC COVARIANCE STRUCTURE OF DROSPHILID WINGS

Evolutionary constraint results from the interaction between the distribution of available genetic variation and the position of selective optima. The availability of genetic variance in multitrait systems, as described by the additive genetic variance-covariance matrix (G), has been the subject of recent attempts to assess the prevalence of genetic constraints. However, evolutionary constraints have not yet been considered from the perspective of the phenotypes available to multivariate selection, and whether genetic variance is present in all phenotypes potentially under selection. Determining the rank of the phenotypic variance-covariance matrix (P) to characterize the phenotypes available to selection, and contrasting it with the rank of G, may provide a general approach to determining the prevalence of genetic constraints. In a study of a laboratory population of Drosophila bunnanda from northern Australia we applied factor-analytic modeling to repeated measures of individual wing phenotypes to determine the dimensionality of the phenotypic space described by P. The phenotypic space spanned by the 10 wing traits had 10 statistically supported dimensions. In contrast, factor-analytic modeling of G estimated for the same 10 traits from a paternal half-sibling breeding design suggested G had fewer dimensions than traits. Statistical support was found for only five and two genetic dimensions, describing a total of 99% and 72% of genetic variance in wing morphology in females and males, respectively. The observed mismatch in dimensionality between P and G suggests that although selection might act to shift the intragenerational population mean toward any trait combination, evolution may be restricted to fewer dimensions.

SEXUAL DIMORPHISM IN THE QUANTITATIVE-GENETIC ARCHITECTURE OF FLORAL, LEAF, AND ALLOCATION TRAITS IN SILENE LATIFOLIA

Many species exhibit sexual dimorphism in a variety of characters, and the underlying genetic architecture of dimorphism potentially involves sex-specific differences in the additive-genetic variance-covariance matrix (G) of dimorphic traits. We investigated the quantitative-genetic structure of dimorphic traits in the dioecious plant Silene latifolia by estimating G (including within-sex matrices, G(m), G(f), and the between-sex variance-covariance matrix, B), and the phenotypic variance-covariance matrix (P) for seven traits. Flower number was the most sexually dimorphic trait, and was significantly genetically correlated with all traits within each sex. Negative genetic correlations between flower size and number suggested a genetic trade-off in investment, but positive environmental correlations between the same traits resulted in no physical evidence for a trade-off in the phenotype. Between-sex genetic covariances for homologous traits were always greater than 0 but smaller than 1, showing that some, but not all, of the variation in traits is caused by genes or alleles with sex-limited expression. Using common principal-components analysis (CPCA), a maximum-likelihood (ML) estimation approach, and element-by-element comparison to compare matrices, we found that G(m) and G(f) differed significantly in eigenstructure because of dissimilarity in covariances involving leaf traits, suggesting the presence of variation in sex-limited genes with pleiotropic effects and/or linkage between sex-limited loci. The sex-specific structure of G is expected to cause differences in the correlated responses to selection within each sex, promoting the further evolution and maintenance of dimorphism.

VARIATION IN PLEIOTROPY AND THE MUTATIONAL UNDERPINNINGS OF THE G-MATRIX

Abstract The pattern and extent of pleiotropic gene action can contribute substantially to the internal structure and shape of the additive genetic variance-covariance matrix (G)–a key determinant of evolutionary trajectories. We use data from our study (Estes et al. 2004) on the univariate effects of mutation in a mismatch-repair-defective strain, msh-2, of Caenorhabditis elegans to address the impact of increasing levels of selection on the magnitude and pattern of genetic covariance due to new mutations. Mutational covariances between three life-history traits are shown to exhibit a weak pattern of decline with increasing population size (increasing selection), while the orientation of mutational matrices remains reasonably constant. This suggests that mutations with smaller effects on fitness may tend to be slightly more confined in their influence than large-effect mutations (i.e., small-effect mutations reduce the magnitude of covariation between characters), but do not change the direction of this covariation.

VARIATION IN PLEIOTROPY AND THE MUTATIONAL UNDERPINNINGS OF THE G-MATRIX

The pattern and extent of pleiotropic gene action can contribute substantially to the internal structure and shape of the additive genetic variance-covariance matrix (G)--a key determinant of evolutionary trajectories. We use data from our study (Estes et al. 2004) on the univariate effects of mutation in a mismatch-repair-defective strain, msh-2, of Caenorhabditis elegans to address the impact of increasing levels of selection on the magnitude and pattern of genetic covariance due to new mutations. Mutational covariances between three life-history traits are shown to exhibit a weak pattern of decline with increasing population size (increasing selection), while the orientation of mutational matrices remains reasonably constant. This suggests that mutations with smaller effects on fitness may tend to be slightly more confined in their influence than large-effect mutations (i.e., small-effect mutations reduce the magnitude of covariation between characters), but do not change the direction of this covariation.

THE ULTIMATE CAUSES OF PHENOTYPIC INTEGRATION: LOST IN TRANSLATION1

Since Darwin's time we have come to recognize two major proximate causes of genetic correlation within populations (pleiotropy and linkage disequilibrium). Today's challenge, taken up by Phenotypic Integration, is to understand the ultimate causes of phenotypic and genetic correlation. A major advance on this front was made by Olson and Miller (1958) who proposed that phenotypic correlations between morphological structures may often reflect functional interactions. Olson and Miller' s insight reverberates today as the first clear proposal for an ultimate cause of correlation. Olson and Miller coined the term morphological integration to describe character correlations that are shaped by selection. The first genetic model of morphological integration was constructed by Lande (1980). Lande was unable to derive a general expression for how genetic correlations evolve from generation to generation. Nevertheless, he was able to characterize the additive genetic variance-covariance matrix (G-matrix) at equilibrium as it pulsates and wobbles under the opposing forces of mutation, recombination, and selection. Among other things, the resulting expression identifies correlational selection (selection that directly changes the covariance between two traits) as the major selective force shaping the evolution of genetic correlation and hence morphological integration. In an important early implementation of Lande's (1980) results, Cheverud (1982) showed how morphological integration could be pursued in a quantitative genetic framework. For many years, Cheverud and his colleagues seemed to be the only modern champions of morphological integration. Thus, Pigliucci and Preston's book, Phenotypic Integration, brings welcome attention to empirical laws of correlation that remain, in many ways, mysterious. Phenotypic Integration brings together contributions from empiricists and theoreticians, botanists and zoologists, and carries a strong multivariate theme throughout its 19 chapters. The term phenotypic integration is a welcome broadening of morphological integration to include behavior and other kinds of nonmorphological traits. Does this volume introduce a new theory of phenotypic integration? Not only is the answer No!, in several chapters this volume steps backward more than two decades, as the authors create new conceptual muddles and needlessly fret over issues that have long enjoyed

POPULATION DIFFERENTIATION IN G MATRIX STRUCTURE DUE TO NATURAL SELECTION IN RANA TEMPORARIA

The additive genetic variance-covariance matrix (G) is a concept central to discussions about evolutionary change over time in a suite of traits. However, at the moment we do not know how fast G itself changes as a consequence of selection or how sensitive it is to environmental influences. We investigated possible evolutionary divergence and environmental influences on G using data from a factorial common-garden experiment where common frog (Rana temporaria) tadpoles from two divergent populations were exposed to three different environmental treatments. G-matrices were estimated using an animal model approach applied to data from a NCII breeding design. Matrix comparisons using both Flury and multivariate analysis of variance methods revealed significant differences in G matrices both between populations and between treatments within populations, the former being generally larger than the latter. Comparison of levels of population differentiation in trait means using QST indices with that observed in microsatellite markers (FST) revealed that the former values generally exceeded the neutral expectation set by FST. Hence, the results suggest that intraspecific divergence in G matrix structure has occurred mainly due to natural selection.