Dissecting high-dimensional phenotypes with bayesian sparse factor analysis of genetic covariance matrices

High-dimensional multi-omics measured in controlled conditions are useful for maize platform and field trait predictions

KEY MESSAGE

Transcriptomics and proteomics information collected on a platform can predict additive and non-additive effects for platform traits and additive effects for field traits. The effects of climate change in the form of drought, heat stress, and irregular seasonal changes threaten global crop production. The ability of multi-omics data, such as transcripts and proteins, to reflect a plant's response to such climatic factors can be capitalized in prediction models to maximize crop improvement. Implementing multi-omics characterization in field evaluations is challenging due to high costs. It is, however, possible to do it on reference genotypes in controlled conditions. Using omics measured on a platform, we tested different multi-omics-based prediction approaches, using a high dimensional linear mixed model (MegaLMM) to predict genotypes for platform traits and agronomic field traits in a panel of 244 maize hybrids. We considered two prediction scenarios: in the first one, new hybrids are predicted (CV-NH), and in the second one, partially observed hybrids are predicted (CV-POH). For both scenarios, all hybrids were characterized for omics on the platform. We observed that omics can predict both additive and non-additive genetic effects for the platform traits, resulting in much higher predictive abilities than GBLUP. It highlights their efficiency in capturing regulatory processes in relation to growth conditions. For the field traits, we observed that the additive components of omics only slightly improved predictive abilities for predicting new hybrids (CV-NH, model MegaGAO) and for predicting partially observed hybrids (CV-POH, model GAOxW-BLUP) in comparison to GBLUP. We conclude that measuring the omics in the fields would be of considerable interest in predicting productivity if the costs of omics drop significantly.

Mega-scale Bayesian regression methods for genome-wide prediction and association studies with thousands of traits

Large-scale phenotype data are expected to increase the accuracy of genome-wide prediction and the power of genome-wide association analyses. However, genomic analyses of high-dimensional, highly correlated traits are challenging. We developed a method for implementing high-dimensional Bayesian multivariate regression to simultaneously analyze genetic variants underlying thousands of traits. As a demonstration, we implemented the BayesC prior in the R package MegaLMM. Applied to Genomic Prediction, MegaBayesC effectively integrated hyperspectral reflectance data from 620 hyperspectral wavelengths to improve the accuracy of genetic value prediction on grain yield in a wheat dataset. Applied to Genome-Wide Association Studies, we used simulations to show that MegaBayesC can accurately estimate the effect sizes of QTL across a range of genetic architectures and causes of correlations among traits. To apply MegaBayesC to a realistic scenario involving whole-genome marker data, we developed a 2-stage procedure involving a preliminary step of candidate marker selection prior to multivariate regression. We then used MegaBayesC to identify genetic associations with flowering time in Arabidopsis thaliana, leveraging expression data from 20,843 genes. MegaBayesC selected 15 single nucleotide polymorphisms as important for flowering time, with 13 located within 100 kb of known flowering-time related genes, a higher validation rate than achieved by a single-stage analysis using only the flowering time data itself. These results demonstrate that MegaBayesC can efficiently and effectively leverage high-dimensional phenotypes in genetic analyses.

Evolvability and constraint in the evolution of three-dimensional flower morphology

PREMISE

Flower phenotypes evolve to attract pollinators and to ensure efficient pollen transfer to and from the bodies of pollinators or, in self-compatible bisexual flowers, between anthers and stigmas. If functionally interacting traits are genetically correlated, response to selection may be subject to genetic constraints. Genetic constraints can be assessed by quantifying standing genetic variation in (multivariate) phenotypic traits and by asking how much the available variation is reduced under specific assumptions about phenotypic selection on functionally interacting and genetically correlated traits.

METHODS

We evaluated multivariate evolvability and potential genetic constraints underlying the evolution of the three-dimensional structure of Dalechampia blossoms. First, we used data from a greenhouse crossing design to estimate the G matrix for traits representing the relative positions of male and female sexual organs (anthers and stigmas) and used the G matrix to ask how genetic variation is distributed in multivariate space. To assess the evolutionary importance of genetic constraints, we related standing genetic variation across phenotypic space to evolutionary divergence of population and species in the same phenotypic directions.

RESULTS

Evolvabilities varied substantially across phenotype space, suggesting that certain traits or trait combinations may be subject to strong genetic constraint. Traits involved functionally in flower-pollinator fit and autonomous selfing exhibited considerable independent evolutionary potential, but population and species divergence tended to occur in phenotypic directions associated with greater-than-average evolvability.

CONCLUSIONS

These results are consistent with the hypothesis that genetic constraints can hamper joint trait evolution towards optimum flower-pollinator fit and optimum self-pollination rates.

Maintenance of quantitative genetic variance in complex, multi-trait phenotypes: The contribution of rare, large effect variants in two Drosophila species

The interaction of evolutionary processes to determine quantitative genetic variation has implications for contemporary and future phenotypic evolution, as well as for our ability to detect causal genetic variants. While theoretical studies have provided robust predictions to discriminate among competing models, empirical assessment of these has been limited. In particular, theory highlights the importance of pleiotropy in resolving observations of selection and mutation, but empirical investigations have typically been limited to few traits. Here, we applied high-dimensional Bayesian Sparse Factor Genetic modeling to gene expression datasets in 2 species, Drosophila melanogaster and Drosophila serrata, to explore the distributions of genetic variance across high-dimensional phenotypic space. Surprisingly, most of the heritable trait covariation was due to few lines (genotypes) with extreme [>3 interquartile ranges (IQR) from the median] values. Intriguingly, while genotypes extreme for a multivariate factor also tended to have a higher proportion of individual traits that were extreme, we also observed genotypes that were extreme for multivariate factors but not for any individual trait. We observed other consistent differences between heritable multivariate factors with outlier lines vs those factors without extreme values, including differences in gene functions. We use these observations to identify further data required to advance our understanding of the evolutionary dynamics and nature of standing genetic variation for quantitative traits.

MegaLMM: Mega-scale linear mixed models for genomic predictions with thousands of traits

Large-scale phenotype data can enhance the power of genomic prediction in plant and animal breeding, as well as human genetics. However, the statistical foundation of multi-trait genomic prediction is based on the multivariate linear mixed effect model, a tool notorious for its fragility when applied to more than a handful of traits. We present MegaLMM, a statistical framework and associated software package for mixed model analyses of a virtually unlimited number of traits. Using three examples with real plant data, we show that MegaLMM can leverage thousands of traits at once to significantly improve genetic value prediction accuracy.

Hamiltonian Monte Carlo method for estimating variance components

A Hamiltonian Monte Carlo algorithm is a Markov chain Monte Carlo method, and the method has a potential to improve estimating parameters effectively. Hamiltonian Monte Carlo is based on Hamiltonian dynamics, and it follows Hamilton's equations, which are expressed as two differential equations. In the sampling process of Hamiltonian Monte Carlo, a numerical integration method called leapfrog integration is used to approximately solve Hamilton's equations, and the integration is required to set the number of discrete time steps and the integration stepsize. These two parameters require some amount of tuning and calibration for effective sampling. In this study, we applied the Hamiltonian Monte Carlo method to animal breeding data and identified the optimal tunings of leapfrog integration for normal and inverse chi-square distributions. Then, using real pig data, we revealed the properties of the Hamiltonian Monte Carlo method with the optimal tuning by applying models including variance explained by pedigree information or genomic information. Compared with the Gibbs sampling method, the Hamiltonian Monte Carlo method had superior performance in both models. We have provided the source codes of this method written in the Fortran language at https://github.com/A-ARAKAWA/HMC.

Improving Genomic Prediction for Seed Quality Traits in Oat (Avena sativa L.) Using Trait-Specific Relationship Matrices

The observable phenotype is the manifestation of information that is passed along different organization levels (transcriptional, translational, and metabolic) of a biological system. The widespread use of various omic technologies (RNA-sequencing, metabolomics, etc.) has provided plant genetics and breeders with a wealth of information on pertinent intermediate molecular processes that may help explain variation in conventional traits such as yield, seed quality, and fitness, among others. A major challenge is effectively using these data to help predict the genetic merit of new, unobserved individuals for conventional agronomic traits. Trait-specific genomic relationship matrices (TGRMs) model the relationships between individuals using genome-wide markers (SNPs) and place greater emphasis on markers that most relevant to the trait compared to conventional genomic relationship matrices. Given that these approaches define relationships based on putative causal loci, it is expected that these approaches should improve predictions for related traits. In this study we evaluated the use of TGRMs to accommodate information on intermediate molecular phenotypes (referred to as endophenotypes) and to predict an agronomic trait, total lipid content, in oat seed. Nine fatty acids were quantified in a panel of 336 oat lines. Marker effects were estimated for each endophenotype, and were used to construct TGRMs. A multikernel TRGM model (MK-TRGM-BLUP) was used to predict total seed lipid content in an independent panel of 210 oat lines. The MK-TRGM-BLUP approach significantly improved predictions for total lipid content when compared to a conventional genomic BLUP (gBLUP) approach. Given that the MK-TGRM-BLUP approach leverages information on the nine fatty acids to predict genetic values for total lipid content in unobserved individuals, we compared the MK-TGRM-BLUP approach to a multi-trait gBLUP (MT-gBLUP) approach that jointly fits phenotypes for fatty acids and total lipid content. The MK-TGRM-BLUP approach significantly outperformed MT-gBLUP. Collectively, these results highlight the utility of using TGRM to accommodate information on endophenotypes and improve genomic prediction for a conventional agronomic trait.

Modeling multiple phenotypes in wheat using data-driven genomic exploratory factor analysis and Bayesian network learning

Inferring trait networks from a large volume of genetically correlated diverse phenotypes such as yield, architecture, and disease resistance can provide information on the manner in which complex phenotypes are interrelated. However, studies on statistical methods tailored to multidimensional phenotypes are limited, whereas numerous methods are available for evaluating the massive number of genetic markers. Factor analysis operates at the level of latent variables predicted to generate observed responses. The objectives of this study were to illustrate the manner in which data-driven exploratory factor analysis can map observed phenotypes into a smaller number of latent variables and infer a genomic latent factor network using 45 agro-morphological, disease, and grain mineral phenotypes measured in synthetic hexaploid wheat lines (Triticum aestivum L.). In total, eight latent factors including grain yield, architecture, flag leaf-related traits, grain minerals, yellow rust, two types of stem rust, and leaf rust were identified as common sources of the observed phenotypes. The genetic component of the factor scores for each latent variable was fed into a Bayesian network to obtain a trait structure reflecting the genetic interdependency among traits. Three directed paths were consistently identified by two Bayesian network algorithms. Flag leaf-related traits influenced leaf rust, and yellow rust and stem rust influenced grain yield. Additional paths that were identified included flag leaf-related traits to minerals and minerals to architecture. This study shows that data-driven exploratory factor analysis can reveal smaller dimensional common latent phenotypes that are likely to give rise to numerous observed field phenotypes without relying on prior biological knowledge. The inferred genomic latent factor structure from the Bayesian network provides insights for plant breeding to simultaneously improve multiple traits, as an intervention on one trait will affect the values of focal phenotypes in an interrelated complex trait system.

Facial shape and allometry quantitative trait locus intervals in the Diversity Outbred mouse are enriched for known skeletal and facial development genes

The biology of how faces are built and come to differ from one another is complex. Discovering normal variants that contribute to differences in facial morphology is one key to untangling this complexity, with important implications for medicine and evolutionary biology. This study maps quantitative trait loci (QTL) for skeletal facial shape using Diversity Outbred (DO) mice. The DO is a randomly outcrossed population with high heterozygosity that captures the allelic diversity of eight inbred mouse lines from three subspecies. The study uses a sample of 1147 DO animals (the largest sample yet employed for a shape QTL study in mouse), each characterized by 22 three-dimensional landmarks, 56,885 autosomal and X-chromosome markers, and sex and age classifiers. We identified 37 facial shape QTL across 20 shape principal components (PCs) using a mixed effects regression that accounts for kinship among observations. The QTL include some previously identified intervals as well as new regions that expand the list of potential targets for future experimental study. Three QTL characterized shape associations with size (allometry). Median support interval size was 3.5 Mb. Narrowing additional analysis to QTL for the five largest magnitude shape PCs, we found significant overrepresentation of genes with known roles in growth, skeletal and facial development, and sensory organ development. For most intervals, one or more of these genes lies within 0.25 Mb of the QTL's peak. QTL effect sizes were small, with none explaining more than 0.5% of facial shape variation. Thus, our results are consistent with a model of facial diversity that is influenced by key genes in skeletal and facial development and, simultaneously, is highly polygenic.

Gradients in richness and turnover of a forest passerine's diet prior to breeding: A mixed model approach applied to faecal metabarcoding data

Rather little is known about the dietary richness and variation of generalist insectivorous species, including birds, due primarily to difficulties in prey identification. Using faecal metabarcoding, we provide the most comprehensive analysis of a passerine's diet to date, identifying the relative magnitudes of biogeographic, habitat and temporal trends in the richness and turnover in diet of Cyanistes caeruleus (blue tit) along a 39 site and 2° latitudinal transect in Scotland. Faecal samples were collected in 2014-2015 from adult birds roosting in nestboxes prior to nest building. DNA was extracted from 793 samples and we amplified COI and 16S minibarcodes. We identified 432 molecular operational taxonomic units that correspond to putative dietary items. Most dietary items were rare, with Lepidoptera being the most abundant and taxon-rich prey order. Here, we present a statistical approach for estimation of gradients and intersample variation in taxonomic richness and turnover using a generalised linear mixed model. We discuss the merits of this approach over existing tools and present methods for model-based estimation of repeatability, taxon richness and Jaccard indices. We found that dietary richness increases significantly as spring advances, but changes little with elevation, latitude or local tree composition. In comparison, dietary composition exhibits significant turnover along temporal and spatial gradients and among sites. Our study shows the promise of faecal metabarcoding for inferring the macroecology of food webs, but we also highlight the challenge posed by contamination and make recommendations of laboratory and statistical practices to minimise its impact on inference.

Developmental structuring of phenotypic variation: A case study with a cellular automata model of ontogeny

Developmental mechanisms not only produce an organismal phenotype, but they also structure the way genetic variation maps to phenotypic variation. Here, we revisit a computational model for the evolution of ontogeny based on cellular automata, in which evolution regularly discovered two alternative mechanisms for achieving a selected phenotype, one showing high modularity, the other showing morphological integration. We measure a primary variational property of the systems, their distribution of fitness effects of mutation. We find that the modular ontogeny shows the evolution of mutational robustness and ontogenic simplification, while the integrated ontogeny does not. We discuss the wider use of this methodology on other computational models of development as well as real organisms.

Bayesian factor analytic model: An approach in multiple environment trials

One of the main challenges in plant breeding programs is the efficient quantification of the genotype-by-environment interaction (GEI). The presence of significant GEI may create difficulties for breeders in the selection and recommendation of superior genotypes for a wide environmental network. Among the diverse statistical procedures developed for this purpose, we highlight those based on mixed models and factor analysis that are called factor analytic (FA) models. However, some inferential issues are related to the factor analytic model, such as Heywood cases that make the model non-identifiable. Moreover, the representation of the loads and factors in the conventional biplot does not involve any measurement of uncertainty. In this work, we propose dealing with the FA model using the Bayesian framework with direct sampling of factor loadings via spectral decomposition; this guarantees identifiability in the estimation process and eliminates the need for the rotationality of factor loadings or imposition of any ad hoc constraints. We used simulated and real data to illustrate the method's application in multi-environment trials (MET) and to compare it with traditional FA mixed models on controlled unbalancing. In general, the Bayesian FA model was robust under different simulated unbalanced levels, presenting the superior predictive ability of missing data when compared to competing models, such as those based on FA mixed models. In addition, for some scenarios, the classical FA mixed model failed in estimating the full FA model, illustrating the parametric problems of convergence in these models. Our results suggest that Bayesian factorial models might be successfully used in plant breeding for MET analysis.

Genetic constraints on microevolutionary divergence of sex-biased gene expression

The evolution of sex-specific phenotypes is an important dimension of diversification and local adaptation. The sex-dependent regulation of gene expression is considered a key genomic mechanism facilitating sex-dependent adaptation. In many species, genes with male-biased expression evolve faster in DNA sequence and expression level than genes with female-biased or sexually monomorphic expression. While positive selection may be responsible for rapid DNA sequence evolution, why expression of male-biased genes also evolves rapidly remains unclear. Beyond sex differences in selection, some aspects of the genetic architecture of gene expression could contribute to the rapid evolution of male-biased gene expression. First, male-biased genes might simply have greater standing genetic variance than female-biased genes. Second, male-biased genes could be less constrained by pleiotropy, either within or between sexes. Here, we evaluate these alternative explanations on an intraspecific scale using a series of quantitative genetic experiments conducted on natural variation in male and female gene expression in the fly Drosophila serrata Male-biased genes had significantly higher genetic variance than female-biased genes and were generally more narrowly expressed across tissues, suggesting lower within-individual pleiotropy. However, consistent with stronger constraints due to between-sex pleiotropy, their between-sex genetic correlations, r_MF, were higher than for female-biased genes and more strongly negatively associated with sex bias. Using an extensive clinal dataset, we tested whether sex differences in gene expression divergence among populations have been shaped by pleiotropy. Here too, male-biased gene divergence was more strongly associated with between-sex pleiotropy than was female-biased gene divergence. Systematic differences in genetic variance and pleiotropy may be important factors influencing sex-specific adaptation arising through changes in gene expression.This article is part of the theme issue 'Linking local adaptation with the evolution of sex differences'.

Mouse Skull Mean Shape and Shape Robustness Rely on Different Genetic Architectures and Different Loci

The genetic architecture of skull shape has been extensively studied in mice and the results suggest a highly polygenic and additive basis. In contrast few studies have explored the genetic basis of the skull variability. Canalization and developmental stability are the two components of phenotypic robustness. They have been proposed to be emergent properties of the genetic networks underlying the development of the trait itself, but this hypothesis has been rarely tested empirically. Here we use outbred mice to investigate the genetic architecture of canalization of the skull shape by implementing a genome-wide marginal epistatic test on 3D geometric morphometric data. The same data set had been used previously to explore the genetic architecture of the skull mean shape and its developmental stability. Here, we address two questions: (1) Are changes in mean shape and changes in shape variance associated with the same genomic regions? and (2) Do canalization and developmental stability rely on the same loci and genetic architecture and do they involve the same patterns of shape variation? We found that unlike skull mean shape, among-individual shape variance and fluctuating asymmetry (FA) show a total lack of additive effects. They are both associated with complex networks of epistatic interactions involving many genes (protein-coding and regulatory elements). Remarkably, none of the genomic loci affecting mean shape contribute these networks despite their enrichment for genes involved in craniofacial variation and diseases. We also found that the patterns of shape FA and individual variation are largely similar and rely on similar multilocus epistatic genetic networks, suggesting that the processes channeling variation within and among individuals are largely common. However, the loci involved in these two networks are completely different. This in turn underlines the difference in the origin of the variation at these two levels, and points at buffering processes that may be specific to each level.

Fast and flexible linear mixed models for genome-wide genetics

Linear mixed effect models are powerful tools used to account for population structure in genome-wide association studies (GWASs) and estimate the genetic architecture of complex traits. However, fully-specified models are computationally demanding and common simplifications often lead to reduced power or biased inference. We describe Grid-LMM (https://github.com/deruncie/GridLMM), an extendable algorithm for repeatedly fitting complex linear models that account for multiple sources of heterogeneity, such as additive and non-additive genetic variance, spatial heterogeneity, and genotype-environment interactions. Grid-LMM can compute approximate (yet highly accurate) frequentist test statistics or Bayesian posterior summaries at a genome-wide scale in a fraction of the time compared to existing general-purpose methods. We apply Grid-LMM to two types of quantitative genetic analyses. The first is focused on accounting for spatial variability and non-additive genetic variance while scanning for QTL; and the second aims to identify gene expression traits affected by non-additive genetic variation. In both cases, modeling multiple sources of heterogeneity leads to new discoveries.

Uneven Distribution of Mutational Variance Across the Transcriptome of Drosophila serrata Revealed by High-Dimensional Analysis of Gene Expression

There are essentially an infinite number of traits that could be measured on any organism, and almost all individual traits display genetic variation, yet substantial genetic variance in a large number of independent traits is not plausible under basic models of selection and mutation. One mechanism that may be invoked to explain the observed levels of genetic variance in individual traits is that pleiotropy results in fewer dimensions of phenotypic space with substantial genetic variance. Multivariate genetic analyses of small sets of functionally related traits have shown that standing genetic variance is often concentrated in relatively few dimensions. It is unknown if a similar concentration of genetic variance occurs at a phenome-wide scale when many traits of disparate function are considered, or if the genetic variance generated by new mutations is also unevenly distributed across phenotypic space. Here, we used a Bayesian sparse factor model to characterize the distribution of mutational variance of 3385 gene expression traits of Drosophila serrata after 27 generations of mutation accumulation, and found that 46% of the estimated mutational variance was concentrated in just 21 dimensions with significant mutational heritability. We show that the extent of concentration of mutational variance into such a small subspace has the potential to substantially bias the response to selection of these traits.

Mutational Pleiotropy and the Strength of Stabilizing Selection Within and Between Functional Modules of Gene Expression

Variational modules, sets of pleiotropically covarying traits, affect phenotypic evolution, and therefore are predicted to reflect functional modules, such that traits within a variational module also share a common function. Such an alignment of function and pleiotropy is expected to facilitate adaptation by reducing the deleterious effects of mutations, and by allowing coordinated evolution of functionally related sets of traits. Here, we adopt a high-dimensional quantitative genetic approach using a large number of gene expression traits in Drosophila serrata to test whether functional grouping, defined by gene ontology (GO terms), predicts variational modules. Mutational or standing genetic covariance was significantly greater than among randomly grouped sets of genes for 38% of our functional groups, indicating that GO terms can predict variational modularity to some extent. We estimated stabilizing selection acting on mutational covariance to test the prediction that functional pleiotropy would result in reduced deleterious effects of mutations within functional modules. Stabilizing selection within functional modules was weaker than that acting on randomly grouped sets of genes in only 23% of functional groups, indicating that functional alignment can reduce deleterious effects of pleiotropic mutation but typically does not. Our analyses also revealed the presence of variational modules that spanned multiple functions.

Quantitative genetic methods depending on the nature of the phenotypic trait

A consequence of the assumptions of the infinitesimal model, one of the most important theoretical foundations of quantitative genetics, is that phenotypic traits are predicted to be most often normally distributed (so-called Gaussian traits). But phenotypic traits, especially those interesting for evolutionary biology, might be shaped according to very diverse distributions. Here, I show how quantitative genetics tools have been extended to account for a wider diversity of phenotypic traits using first the threshold model and then more recently using generalized linear mixed models. I explore the assumptions behind these models and how they can be used to study the genetics of non-Gaussian complex traits. I also comment on three recent methodological advances in quantitative genetics that widen our ability to study new kinds of traits: the use of "modular" hierarchical modeling (e.g., to study survival in the context of capture-recapture approaches for wild populations); the use of aster models to study a set of traits with conditional relationships (e.g., life-history traits); and, finally, the study of high-dimensional traits, such as gene expression.

The evolution of phenotypic integration: How directional selection reshapes covariation in mice

Variation is the basis for evolution, and understanding how variation can evolve is a central question in biology. In complex phenotypes, covariation plays an even more important role, as genetic associations between traits can bias and alter evolutionary change. Covariation can be shaped by complex interactions between loci, and this genetic architecture can also change during evolution. In this article, we analyzed mouse lines experimentally selected for changes in size to address the question of how multivariate covariation changes under directional selection, as well as to identify the consequences of these changes to evolution. Selected lines showed a clear restructuring of covariation in their cranium and, instead of depleting their size variation, these lines increased their magnitude of integration and the proportion of variation associated with the direction of selection. This result is compatible with recent theoretical works on the evolution of covariation that take the complexities of genetic architecture into account. This result also contradicts the traditional view of the effects of selection on available covariation and suggests a much more complex view of how populations respond to selection.

Structure and stability of genetic variance-covariance matrices: A Bayesian sparse factor analysis of transcriptional variation in the three-spined stickleback

The genetic variance-covariance matrix (G) is a quantity of central importance in evolutionary biology due to its influence on the rate and direction of multivariate evolution. However, the predictive power of empirically estimated G-matrices is limited for two reasons. First, phenotypes are high-dimensional, whereas traditional statistical methods are tuned to estimate and analyse low-dimensional matrices. Second, the stability of G to environmental effects and over time remains poorly understood. Using Bayesian sparse factor analysis (BSFG) designed to estimate high-dimensional G-matrices, we analysed levels variation and covariation in 10,527 expressed genes in a large (n = 563) half-sib breeding design of three-spined sticklebacks subject to two temperature treatments. We found significant differences in the structure of G between the treatments: heritabilities and evolvabilities were higher in the warm than in the low-temperature treatment, suggesting more and faster opportunity to evolve in warm (stressful) conditions. Furthermore, comparison of G and its phenotypic equivalent P revealed the latter is a poor substitute of the former. Most strikingly, the results suggest that the expected impact of G on evolvability-as well as the similarity among G-matrices-may depend strongly on the number of traits included into analyses. In our results, the inclusion of only few traits in the analyses leads to underestimation in the differences between the G-matrices and their predicted impacts on evolution. While the results highlight the challenges involved in estimating G, they also illustrate that by enabling the estimation of large G-matrices, the BSFG method can improve predicted evolutionary responses to selection.

Changes in human skull morphology across the agricultural transition are consistent with softer diets in preindustrial farming groups

Agricultural foods and technologies are thought to have eased the mechanical demands of diet-how often or how hard one had to chew-in human populations worldwide. Some evidence suggests correspondingly worldwide changes in skull shape and form across the agricultural transition, although these changes have proved difficult to characterize at a global scale. Here, adapting a quantitative genetics mixed model for complex phenotypes, we quantify the influence of diet on global human skull shape and form. We detect modest directional differences between foragers and farmers. The effects are consistent with softer diets in preindustrial farming groups and are most pronounced and reliably directional when the farming class is limited to dairying populations. Diet effect magnitudes are relatively small, affirming the primary role of neutral evolutionary processes-genetic drift, mutation, and gene flow structured by population history and migrations-in shaping diversity in the human skull. The results also bring an additional perspective to the paradox of why Homo sapiens, particularly agriculturalists, appear to be relatively well suited to efficient (high-leverage) chewing.

Accounting for Sampling Error in Genetic Eigenvalues Using Random Matrix Theory

The distribution of genetic variance in multivariate phenotypes is characterized by the empirical spectral distribution of the eigenvalues of the genetic covariance matrix. Empirical estimates of genetic eigenvalues from random effects linear models are known to be overdispersed by sampling error, where large eigenvalues are biased upward, and small eigenvalues are biased downward. The overdispersion of the leading eigenvalues of sample covariance matrices have been demonstrated to conform to the Tracy-Widom (TW) distribution. Here we show that genetic eigenvalues estimated using restricted maximum likelihood (REML) in a multivariate random effects model with an unconstrained genetic covariance structure will also conform to the TW distribution after empirical scaling and centering. However, where estimation procedures using either REML or MCMC impose boundary constraints, the resulting genetic eigenvalues tend not be TW distributed. We show how using confidence intervals from sampling distributions of genetic eigenvalues without reference to the TW distribution is insufficient protection against mistaking sampling error as genetic variance, particularly when eigenvalues are small. By scaling such sampling distributions to the appropriate TW distribution, the critical value of the TW statistic can be used to determine if the magnitude of a genetic eigenvalue exceeds the sampling error for each eigenvalue in the spectral distribution of a given genetic covariance matrix.

Genomic Characterization of the Evolutionary Potential of the Sea Urchin Strongylocentrotus droebachiensis Facing Ocean Acidification

Ocean acidification (OA) is increasing due to anthropogenic CO₂ emissions and poses a threat to marine species and communities worldwide. To better project the effects of acidification on organisms’ health and persistence, an understanding is needed of the 1) mechanisms underlying developmental and physiological tolerance and 2) potential populations have for rapid evolutionary adaptation. This is especially challenging in nonmodel species where targeted assays of metabolism and stress physiology may not be available or economical for large-scale assessments of genetic constraints. We used mRNA sequencing and a quantitative genetics breeding design to study mechanisms underlying genetic variability and tolerance to decreased seawater pH (-0.4 pH units) in larvae of the sea urchin Strongylocentrotus droebachiensis. We used a gene ontology-based approach to integrate expression profiles into indirect measures of cellular and biochemical traits underlying variation in larval performance (i.e., growth rates). Molecular responses to OA were complex, involving changes to several functions such as growth rates, cell division, metabolism, and immune activities. Surprisingly, the magnitude of pH effects on molecular traits tended to be small relative to variation attributable to segregating functional genetic variation in this species. We discuss how the application of transcriptomics and quantitative genetics approaches across diverse species can enrich our understanding of the biological impacts of climate change.

EvolQG - An R package for evolutionary quantitative genetics

We present an open source package for performing evolutionary quantitative genetics analyses in the R environment for statistical computing. Evolutionary theory shows that evolution depends critically on the available variation in a given population. When dealing with many quantitative traits this variation is expressed in the form of a covariance matrix, particularly the additive genetic covariance matrix or sometimes the phenotypic matrix, when the genetic matrix is unavailable and there is evidence the phenotypic matrix is sufficiently similar to the genetic matrix. Given this mathematical representation of available variation, the \textbf{EvolQG} package provides functions for calculation of relevant evolutionary statistics; estimation of sampling error; corrections for this error; matrix comparison via correlations, distances and matrix decomposition; analysis of modularity patterns; and functions for testing evolutionary hypotheses on taxa diversification.

EvolQG - An R package for evolutionary quantitative genetics

We present an open source package for performing evolutionary quantitative genetics analyses in the R environment for statistical computing. Evolutionary theory shows that evolution depends critically on the available variation in a given population. When dealing with many quantitative traits this variation is expressed in the form of a covariance matrix, particularly the additive genetic covariance matrix or sometimes the phenotypic matrix, when the genetic matrix is unavailable and there is evidence the phenotypic matrix is sufficiently similar to the genetic matrix. Given this mathematical representation of available variation, the \textbf{EvolQG} package provides functions for calculation of relevant evolutionary statistics; estimation of sampling error; corrections for this error; matrix comparison via correlations, distances and matrix decomposition; analysis of modularity patterns; and functions for testing evolutionary hypotheses on taxa diversification.

Rate of evolutionary change in cranial morphology of the marsupial genus Monodelphis is constrained by the availability of additive genetic variation

We tested the hypothesis that the rate of marsupial cranial evolution is dependent on the distribution of genetic variation in multivariate space. To do so, we carried out a genetic analysis of cranial morphological variation in laboratory strains of Monodelphis domestica and used estimates of genetic covariation to analyse the morphological diversification of the Monodelphis brevicaudata species group. We found that within-species genetic variation is concentrated in only a few axes of the morphospace and that this strong genetic covariation influenced the rate of morphological diversification of the brevicaudata group, with between-species divergence occurring fastest when occurring along the genetic line of least resistance. Accounting for the geometric distribution of genetic variation also increased our ability to detect the selective regimen underlying species diversification, with several instances of selection only being detected when genetic covariances were taken into account. Therefore, this work directly links patterns of genetic covariation among traits to macroevolutionary patterns of morphological divergence. Our findings also suggest that the limited distribution of Monodelphis species in morphospace is the result of a complex interplay between the limited dimensionality of available genetic variation and strong stabilizing selection along two major axes of genetic variation.

Evolutionary rates for multivariate traits: the role of selection and genetic variation

A fundamental question in evolutionary biology is the relative importance of selection and genetic architecture in determining evolutionary rates. Adaptive evolution can be described by the multivariate breeders' equation (Δz(-)=Gβ), which predicts evolutionary change for a suite of phenotypic traits (Δz(-)) as a product of directional selection acting on them (β) and the genetic variance-covariance matrix for those traits (G ). Despite being empirically challenging to estimate, there are enough published estimates of G and β to allow for synthesis of general patterns across species. We use published estimates to test the hypotheses that there are systematic differences in the rate of evolution among trait types, and that these differences are, in part, due to genetic architecture. We find some evidence that sexually selected traits exhibit faster rates of evolution compared with life-history or morphological traits. This difference does not appear to be related to stronger selection on sexually selected traits. Using numerous proposed approaches to quantifying the shape, size and structure of G, we examine how these parameters relate to one another, and how they vary among taxonomic and trait groupings. Despite considerable variation, they do not explain the observed differences in evolutionary rates.

Efficient multivariate linear mixed model algorithms for genome-wide association studies

Multivariate linear mixed models (mvLMMs) are powerful tools for testing associations between single-nucleotide polymorphisms and multiple correlated phenotypes while controlling for population stratification in genome-wide association studies. We present efficient algorithms in the genome-wide efficient mixed model association (GEMMA) software for fitting mvLMMs and computing likelihood ratio tests. These algorithms offer improved computation speed, power and P-value calibration over existing methods, and can deal with more than two phenotypes.

The nature and extent of mutational pleiotropy in gene expression of male Drosophila serrata

The nature and extent of mutational pleiotropy remain largely unknown, despite the central role that pleiotropy plays in many areas of biology, including human disease, agricultural production, and evolution. Here, we investigate the variation in 11,604 gene expression traits among 41 mutation accumulation (MA) lines of Drosophila serrata. We first confirmed that these expression phenotypes were heritable, detecting genetic variation in 96% of them in an outbred, natural population of D. serrata. Among the MA lines, 3385 (29%) of expression traits were variable, with a mean mutational heritability of 0.0005. In most traits, variation was generated by mutations of relatively small phenotypic effect; putative mutations with effects of greater than one phenotypic standard deviation were observed for only 8% of traits. With most (71%) traits unaffected by any mutation, our data provide no support for universal pleiotropy. We further characterized mutational pleiotropy in the 3385 variable traits, using sets of 5, randomly assigned, traits. Covariance among traits chosen at random with respect to their biological function is expected only if pleiotropy is extensive. Taking an analytical approach in which the variance unique to each trait in the random 5-trait sets was partitioned from variance shared among traits, we detected significant (at 5% false discovery rate) mutational covariance in 21% of sets. This frequency of statistically supported covariance implied that at least some mutations must pleiotropically affect a substantial number of traits (>70; 0.6% of all measured traits).