Application of support vector regression to genome-assisted prediction of quantitative traits

A byproduct of genome-wide association studies is the possibility of carrying out genome-enabled prediction of disease risk or of quantitative traits. This study is concerned with predicting two quantitative traits, milk yield in dairy cattle and grain yield in wheat, using dense molecular markers as predictors. Two support vector regression (SVR) models, ε-SVR and least-squares SVR, were explored and compared to a widely applied linear regression model, the Bayesian Lasso, the latter assuming additive marker effects. Predictive performance was measured using predictive correlation and mean squared error of prediction. Depending on the kernel function chosen, SVR can model either linear or nonlinear relationships between phenotypes and marker genotypes. For milk yield, where phenotypes were estimated breeding values of bulls (a linear combination of the data), SVR with a Gaussian radial basis function (RBF) kernel had a slightly better performance than with a linear kernel, and was similar to the Bayesian Lasso. For the wheat data, where phenotype was raw grain yield, the RBF kernel provided clear advantages over the linear kernel, e.g., a 17.5% increase in correlation when using the ε-SVR. SVR with a RBF kernel also compared favorably to the Bayesian Lasso in this case. It is concluded that a nonlinear RBF kernel may be an optimal choice for SVR, especially when phenotypes to be predicted have a nonlinear dependency on genotypes, as it might have been the case in the wheat data.     

Integrating Crop Growth Models with Whole Genome Prediction through Approximate Bayesian Computation

Genomic selection, enabled by whole genome prediction (WGP) methods, is revolutionizing plant breeding. Existing WGP methods have been shown to deliver accurate predictions in the most common settings, such as prediction of across environment performance for traits with additive gene effects. However, prediction of traits with non-additive gene effects and prediction of genotype by environment interaction (G×E), continues to be challenging. Previous attempts to increase prediction accuracy for these particularly difficult tasks employed prediction methods that are purely statistical in nature. Augmenting the statistical methods with biological knowledge has been largely overlooked thus far. Crop growth models (CGMs) attempt to represent the impact of functional relationships between plant physiology and the environment in the formation of yield and similar output traits of interest. Thus, they can explain the impact of G×E and certain types of non-additive gene effects on the expressed phenotype. Approximate Bayesian computation (ABC), a novel and powerful computational procedure, allows the incorporation of CGMs directly into the estimation of whole genome marker effects in WGP. Here we provide a proof of concept study for this novel approach and demonstrate its use with synthetic data sets. We show that this novel approach can be considerably more accurate than the benchmark WGP method GBLUP in predicting performance in environments represented in the estimation set as well as in previously unobserved environments for traits determined by non-additive gene effects. We conclude that this proof of concept demonstrates that using ABC for incorporating biological knowledge in the form of CGMs into WGP is a very promising and novel approach to improving prediction accuracy for some of the most challenging scenarios in plant breeding and applied genetics.     

Long-term impacts of genome-enabled selection

The objective was to evaluate the effects of directional selection based on estimated genomic breeding values (GEBVs) for a quantitative trait. Selection affects GEBV prediction accuracy as well as genetic architecture via changes in allelic frequencies and linkage disequilibrium (LD), and the resulting changes are different from those in the absence of selection. How marker density affects long-term GEBV accuracy and selection response needs to be understood as well. Simulations were used to characterize the impact of selection based on GEBVs over generations. Single-nucleotide polymorphism (SNP) marker effects were estimated with the Bayesian Lasso method in the base generation, and these estimates were used to calculate the GEBVs in subsequent generations. GEBV accuracy decreased over generations of selection, and it was lower than under random selection, where a decay took place as well. In the long term, selection response tended to reach a plateau, but, at higher marker density, both the magnitude and duration of the response were larger. Selection changed quantitative trait loci (QTL) allele frequencies and generated new but unfavorable LD for prediction. Family effects had a considerable contribution to GEBV accuracy in early generations of selection.     

Genome-enabled prediction of genetic values using radial basis function neural networks

The availability of high density panels of molecular markers has prompted the adoption of genomic selection (GS) methods in animal and plant breeding. In GS, parametric, semi-parametric and non-parametric regressions models are used for predicting quantitative traits. This article shows how to use neural networks with radial basis functions (RBFs) for prediction with dense molecular markers. We illustrate the use of the linear Bayesian LASSO regression model and of two non-linear regression models, reproducing kernel Hilbert spaces (RKHS) regression and radial basis function neural networks (RBFNN) on simulated data and real maize lines genotyped with 55,000 markers and evaluated for several trait-environment combinations. The empirical results of this study indicated that the three models showed similar overall prediction accuracy, with a slight and consistent superiority of RKHS and RBFNN over the additive Bayesian LASSO model. Results from the simulated data indicate that RKHS and RBFNN models captured epistatic effects; however, adding non-signal (redundant) predictors (interaction between markers) can adversely affect the predictive accuracy of the non-linear regression models.     

Genetic architecture of survival and fitness-related traits in two populations of Atlantic salmon

The additive genetic effects of traits can be used to predict evolutionary trajectories, such as responses to selection. Non-additive genetic and maternal environmental effects can also change evolutionary trajectories and influence phenotypes, but these effects have received less attention by researchers. We partitioned the phenotypic variance of survival and fitness-related traits into additive genetic, non-additive genetic and maternal environmental effects using a full-factorial breeding design within two allopatric populations of Atlantic salmon (Salmo salar). Maternal environmental effects were large at early life stages, but decreased during development, with non-additive genetic effects being most significant at later juvenile stages (alevin and fry). Non-additive genetic effects were also, on average, larger than additive genetic effects. The populations, generally, did not differ in the trait values or inferred genetic architecture of the traits. Any differences between the populations for trait values could be explained by maternal environmental effects. We discuss whether the similarities in architectures of these populations is the result of natural selection across a common juvenile environment.     

Additive and Non-additive Genetic Correlations. II. Application to Data Obtained from a 12 × 12 Diallel Cross

The proposed genetic correlation analysis, that involves the partitioning of the overall genetic correlation into an additive and a non-additive component, has been applied to data obtained from a diallel experiment involving 12 white modified opaque-2 maize inbred lines. The correlation analysis provided an insight into possible indirect selection strategies for the improvement of inferior kernel quality traits associated with the opaque-2 gene. Direct selection for high yield and low vitreousness rating would provide an efficient selection strategy for the development of high-yielding modified opaque-2 maize hybrids with desirable endosperm traits. It was concluded that it is not necessary to conduct the density, hardness and breakability determinations.     

Data and theory point to mainly additive genetic variance for complex traits

The relative proportion of additive and non-additive variation for complex traits is important in evolutionary biology, medicine, and agriculture. We address a long-standing controversy and paradox about the contribution of non-additive genetic variation, namely that knowledge about biological pathways and gene networks imply that epistasis is important. Yet empirical data across a range of traits and species imply that most genetic variance is additive. We evaluate the evidence from empirical studies of genetic variance components and find that additive variance typically accounts for over half, and often close to 100%, of the total genetic variance. We present new theoretical results, based upon the distribution of allele frequencies under neutral and other population genetic models, that show why this is the case even if there are non-additive effects at the level of gene action. We conclude that interactions at the level of genes are not likely to generate much interaction at the level of variance.     

Admixture mapping of quantitative traits in Populus hybrid zones: power and limitations

Uncovering the genetic architecture of species differences is of central importance for understanding the origin and maintenance of biological diversity. Admixture mapping can be used to identify the number and effect sizes of genes that contribute to the divergence of ecologically important traits, even in taxa that are not amenable to laboratory crosses because of their long generation time or other limitations. Here, we apply admixture mapping to naturally occurring hybrids between two ecologically divergent Populus species. We map quantitative trait loci for eight leaf morphological traits using 77 mapped microsatellite markers from all 19 chromosomes of Populus. We apply multivariate linear regression analysis allowing the modeling of additive and non-additive gene action and identify several candidate genomic regions associated with leaf morphology using an information-theoretic approach. We perform simulation studies to assess the power and limitations of admixture mapping of quantitative traits in natural hybrid populations for a variety of genetic architectures and modes of gene action. Our results indicate that (1) admixture mapping has considerable power to identify the genetic architecture of species differences if sample sizes and marker densities are sufficiently high, (2) modeling of non-additive gene action can help to elucidate the discrepancy between genotype and phenotype sometimes seen in interspecific hybrids, and (3) the genetic architecture of leaf morphological traits in the studied Populus species involves complementary and overdominant gene action, providing the basis for rapid adaptation of these ecologically important forest trees.     

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

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

Predicting expected progeny difference for marbling score in Angus cattle using artificial neural networks and Bayesian regression models

Background

Artificial neural networks (ANN) mimic the function of the human brain and are capable of performing massively parallel computations for data processing and knowledge representation. ANN can capture nonlinear relationships between predictors and responses and can adaptively learn complex functional forms, in particular, for situations where conventional regression models are ineffective. In a previous study, ANN with Bayesian regularization outperformed a benchmark linear model when predicting milk yield in dairy cattle or grain yield of wheat. Although breeding values rely on the assumption of additive inheritance, the predictive capabilities of ANN are of interest from the perspective of their potential to increase the accuracy of prediction of molecular breeding values used for genomic selection. This motivated the present study, in which the aim was to investigate the accuracy of ANN when predicting the expected progeny difference (EPD) of marbling score in Angus cattle. Various ANN architectures were explored, which involved two training algorithms, two types of activation functions, and from 1 to 4 neurons in hidden layers. For comparison, BayesCπ models were used to select a subset of optimal markers (referred to as feature selection), under the assumption of additive inheritance, and then the marker effects were estimated using BayesCπ with π set equal to zero. This procedure is referred to as BayesCpC and was implemented on a high-throughput computing cluster.

Results

The ANN with Bayesian regularization method performed equally well for prediction of EPD as BayesCpC, based on prediction accuracy and sum of squared errors. With the 3K-SNP panel, for example, prediction accuracy was 0.776 using BayesCpC, and ranged from 0.776 to 0.807 using BRANN. With the selected 700-SNP panel, prediction accuracy was 0.863 for BayesCpC and ranged from 0.842 to 0.858 for BRANN. However, prediction accuracy for the ANN with scaled conjugate gradient back-propagation was lower, ranging from 0.653 to 0.689 with the 3K-SNP panel, and from 0.743 to 0.793 with the selected 700-SNP panel.

Conclusions

ANN with Bayesian regularization performed as well as linear Bayesian regression models in predicting additive genetic values, supporting the idea that ANN are useful as universal approximators of functions of interest in breeding contexts.     

Estimating Additive and Non-Additive Genetic Variances and Predicting Genetic Merits Using Genome-Wide Dense Single Nucleotide Polymorphism Markers

Non-additive genetic variation is usually ignored when genome-wide markers are used to study the genetic architecture and genomic prediction of complex traits in human, wild life, model organisms or farm animals. However, non-additive genetic effects may have an important contribution to total genetic variation of complex traits. This study presented a genomic BLUP model including additive and non-additive genetic effects, in which additive and non-additive genetic relation matrices were constructed from information of genome-wide dense single nucleotide polymorphism (SNP) markers. In addition, this study for the first time proposed a method to construct dominance relationship matrix using SNP markers and demonstrated it in detail. The proposed model was implemented to investigate the amounts of additive genetic, dominance and epistatic variations, and assessed the accuracy and unbiasedness of genomic predictions for daily gain in pigs. In the analysis of daily gain, four linear models were used: 1) a simple additive genetic model (MA), 2) a model including both additive and additive by additive epistatic genetic effects (MAE), 3) a model including both additive and dominance genetic effects (MAD), and 4) a full model including all three genetic components (MAED). Estimates of narrow-sense heritability were 0.397, 0.373, 0.379 and 0.357 for models MA, MAE, MAD and MAED, respectively. Estimated dominance variance and additive by additive epistatic variance accounted for 5.6% and 9.5% of the total phenotypic variance, respectively. Based on model MAED, the estimate of broad-sense heritability was 0.506. Reliabilities of genomic predicted breeding values for the animals without performance records were 28.5%, 28.8%, 29.2% and 29.5% for models MA, MAE, MAD and MAED, respectively. In addition, models including non-additive genetic effects improved unbiasedness of genomic predictions.     

Allele frequency changes due to hitch-hiking in genomic selection programs

Background

Genomic selection makes it possible to reduce pedigree-based inbreeding over best linear unbiased prediction (BLUP) by increasing emphasis on own rather than family information. However, pedigree inbreeding might not accurately reflect loss of genetic variation and the true level of inbreeding due to changes in allele frequencies and hitch-hiking. This study aimed at understanding the impact of using long-term genomic selection on changes in allele frequencies, genetic variation and level of inbreeding.

Methods

Selection was performed in simulated scenarios with a population of 400 animals for 25 consecutive generations. Six genetic models were considered with different heritabilities and numbers of QTL (quantitative trait loci) affecting the trait. Four selection criteria were used, including selection on own phenotype and on estimated breeding values (EBV) derived using phenotype-BLUP, genomic BLUP and Bayesian Lasso. Changes in allele frequencies at QTL, markers and linked neutral loci were investigated for the different selection criteria and different scenarios, along with the loss of favourable alleles and the rate of inbreeding measured by pedigree and runs of homozygosity.

Results

For each selection criterion, hitch-hiking in the vicinity of the QTL appeared more extensive when accuracy of selection was higher and the number of QTL was lower. When inbreeding was measured by pedigree information, selection on genomic BLUP EBV resulted in lower levels of inbreeding than selection on phenotype BLUP EBV, but this did not always apply when inbreeding was measured by runs of homozygosity. Compared to genomic BLUP, selection on EBV from Bayesian Lasso led to less genetic drift, reduced loss of favourable alleles and more effectively controlled the rate of both pedigree and genomic inbreeding in all simulated scenarios. In addition, selection on EBV from Bayesian Lasso showed a higher selection differential for mendelian sampling terms than selection on genomic BLUP EBV.

Conclusions

Neutral variation can be shaped to a great extent by the hitch-hiking effects associated with selection, rather than just by genetic drift. When implementing long-term genomic selection, strategies for genomic control of inbreeding are essential, due to a considerable hitch-hiking effect, regardless of the method that is used for prediction of EBV.     

Combining ability analysis over F1-F5 generations in diallel crosses of bread wheat

Summary Combining ability studies for grain yield and its primary component traits in diallel crosses involving seven diverse wheat cultivars of bread wheat (Triticum aestivum L.) over generations F₁-F₅ are reported. The general and specific combining ability variances were significant in all generations for all the traits except specific combining ability variance for number of spikes per plant in the F₅. The ratio of general to specific combining ability variances was significant for all the traits except grain yield in all the generations. This indicated an equal role of additive and non-additive gene effects in the inheritance of grain yield, and the predominance of the former for its component traits. The presence of significant specific combining ability variances in even the advanced generations may be the result of an additive x additive type of epistasis or evolutionary divergence among progenies in the same parental array. The relative breeding values of the parental varieties, as indicated by their general combining ability effects, did not vary much over the generations. The cheap and reliable procedure observed for making the choice of parents, selecting hybrids and predicting advanced generation (F₅) bulk hybrid performance was the determination of breeding values of the parents on the relative performance of their F₂ progeny bulks.     

Improvement of Prediction Ability for Genomic Selection of Dairy Cattle by Including Dominance Effects

Dominance may be an important source of non-additive genetic variance for many traits of dairy cattle. However, nearly all prediction models for dairy cattle have included only additive effects because of the limited number of cows with both genotypes and phenotypes. The role of dominance in the Holstein and Jersey breeds was investigated for eight traits: milk, fat, and protein yields; productive life; daughter pregnancy rate; somatic cell score; fat percent and protein percent. Additive and dominance variance components were estimated and then used to estimate additive and dominance effects of single nucleotide polymorphisms (SNPs). The predictive abilities of three models with both additive and dominance effects and a model with additive effects only were assessed using ten-fold cross-validation. One procedure estimated dominance values, and another estimated dominance deviations; calculation of the dominance relationship matrix was different for the two methods. The third approach enlarged the dataset by including cows with genotype probabilities derived using genotyped ancestors. For yield traits, dominance variance accounted for 5 and 7% of total variance for Holsteins and Jerseys, respectively; using dominance deviations resulted in smaller dominance and larger additive variance estimates. For non-yield traits, dominance variances were very small for both breeds. For yield traits, including additive and dominance effects fit the data better than including only additive effects; average correlations between estimated genetic effects and phenotypes showed that prediction accuracy increased when both effects rather than just additive effects were included. No corresponding gains in prediction ability were found for non-yield traits. Including cows with derived genotype probabilities from genotyped ancestors did not improve prediction accuracy. The largest additive effects were located on chromosome 14 near DGAT1 for yield traits for both breeds; those SNPs also showed the largest dominance effects for fat yield (both breeds) as well as for Holstein milk yield.     

On marker-assisted prediction of genetic value: beyond the ridge

Marked-assisted genetic improvement of agricultural species exploits statistical dependencies in the joint distribution of marker genotypes and quantitative traits. An issue is how molecular (e.g., dense marker maps) and phenotypic information (e.g., some measure of yield in plants) is to be used for predicting the genetic value of candidates for selection. Multiple regression, selection index techniques, best linear unbiased prediction, and ridge regression of phenotypes on marker genotypes have been suggested, as well as more elaborate methods. Here, phenotype-marker associations are modeled hierarchically via multilevel models including chromosomal effects, a spatial covariance of marked effects within chromosomes, background genetic variability, and family heterogeneity. Lorenz curves and Gini coefficients are suggested for assessing the inequality of the contribution of different marked effects to genetic variability. Classical and Bayesian methods are presented. The Bayesian approach includes a Markov chain Monte Carlo implementation. The generality and flexibility of the Bayesian method is illustrated when a Lorenz curve is to be inferred.     

The Genetic Architecture of Quantitative Traits Cannot Be Inferred from Variance Component Analysis

Classical quantitative genetic analyses estimate additive and non-additive genetic and environmental components of variance from phenotypes of related individuals without knowing the identities of quantitative trait loci (QTLs). Many studies have found a large proportion of quantitative trait variation can be attributed to the additive genetic variance (V_A), providing the basis for claims that non-additive gene actions are unimportant. In this study, we show that arbitrarily defined parameterizations of genetic effects seemingly consistent with non-additive gene actions can also capture the majority of genetic variation. This reveals a logical flaw in using the relative magnitudes of variance components to indicate the relative importance of additive and non-additive gene actions. We discuss the implications and propose that variance component analyses should not be used to infer the genetic architecture of quantitative traits.     

Exploitation of heterosis for carotenoid content in African marigold (<Emphasis Type="Italic">Tagetes erecta</Emphasis> L.) and its correlation with esterase polymorphism

African marigold (Tagetes erecta L.), a major source of carotenoids, is also grown as a cut flower and a garden flower in addition to being grown for its medicinal values. We studied gene action, combining ability and heterosis, aiming at genetic improvement of T. erecta for enhanced carotenoid content in petals, and report for the first time that heterosis can be exploited for total carotenoids and its commercially important fractions. Total content of carotenoids and lutein appears to be governed by dominance (or non-additive) gene action, while content of xanthophyll esters is governed by both additive and dominance (or non-additive) gene actions. Specific combining ability variance was predominant for all the three traits. General and specific combining abilities and heterosis were highly significant. Heterobeltiosis was also positive. General combining ability (GCA) variances were not significantly correlated to performance per se. There was also no correlation between performance per se of normal petalled pollen parents and the performance of crosses made between male-sterile (female) and male-fertile (pollen) parents. These findings suggest that carotenoid content should not be the only criterion considered in the selection of parental lines. Studies on esterase in seeds and peroxidase in seedlings revealed a relatively high level of polymorphism in esterase with a total of 14 isoforms, whereas peroxidase showed low polymorphism. Similarity indices between different parental combinations, calculated based on seed esterase polymorphism, showed a significant negative correlation (r = -0.479, P = 0.05) with heterosis for carotenoid content. This indicates that the selection of parents with wider variation in their esterase profiles may possibly be exploited for genetic enhancement of carotenoids in T. erecta.     

Accuracies of genomic prediction for twenty economically important traits in Chinese Simmental beef cattle

Genomic prediction has been widely utilized to estimate genomic breeding values (GEBVs) in farm animals. In this study, we conducted genomic prediction for 20 economically important traits including growth, carcass and meat quality traits in Chinese Simmental beef cattle. Five approaches (GBLUP, BayesA, BayesB, BayesCπ and BayesR) were used to estimate the genomic breeding values. The predictive accuracies ranged from 0.159 (lean meat percentage estimated by BayesCπ) to 0.518 (striploin weight estimated by BayesR). Moreover, we found that the average predictive accuracies across 20 traits were 0.361, 0.361, 0.367, 0.367 and 0.378, and the averaged regression coefficients were 0.89, 0.86, 0.89, 0.94 and 0.95 for GBLUP, BayesA, BayesB, BayesCπ and BayesR respectively. The genomic prediction accuracies were mostly moderate and high for growth and carcass traits, whereas meat quality traits showed relatively low accuracies. We concluded that Bayesian regression approaches, especially for BayesR and BayesCπ, were slightly superior to GBLUP for most traits. Increasing with the sizes of reference population, these two approaches are feasible for future application of genomic selection in Chinese beef cattle.     

Accuracy of genomic selection methods in a standard data set of loblolly pine (Pinus taeda L.)

Genomic selection can increase genetic gain per generation through early selection. Genomic selection is expected to be particularly valuable for traits that are costly to phenotype and expressed late in the life cycle of long-lived species. Alternative approaches to genomic selection prediction models may perform differently for traits with distinct genetic properties. Here the performance of four different original methods of genomic selection that differ with respect to assumptions regarding distribution of marker effects, including (i) ridge regression-best linear unbiased prediction (RR-BLUP), (ii) Bayes A, (iii) Bayes Cπ, and (iv) Bayesian LASSO are presented. In addition, a modified RR-BLUP (RR-BLUP B) that utilizes a selected subset of markers was evaluated. The accuracy of these methods was compared across 17 traits with distinct heritabilities and genetic architectures, including growth, development, and disease-resistance properties, measured in a Pinus taeda (loblolly pine) training population of 951 individuals genotyped with 4853 SNPs. The predictive ability of the methods was evaluated using a 10-fold, cross-validation approach, and differed only marginally for most method/trait combinations. Interestingly, for fusiform rust disease-resistance traits, Bayes Cπ, Bayes A, and RR-BLUB B had higher predictive ability than RR-BLUP and Bayesian LASSO. Fusiform rust is controlled by few genes of large effect. A limitation of RR-BLUP is the assumption of equal contribution of all markers to the observed variation. However, RR-BLUP B performed equally well as the Bayesian approaches.The genotypic and phenotypic data used in this study are publically available for comparative analysis of genomic selection prediction models.     

Genomic-assisted prediction of genetic value with semiparametric procedures

Semiparametric procedures for prediction of total genetic value for quantitative traits, which make use of phenotypic and genomic data simultaneously, are presented. The methods focus on the treatment of massive information provided by, e.g., single-nucleotide polymorphisms. It is argued that standard parametric methods for quantitative genetic analysis cannot handle the multiplicity of potential interactions arising in models with, e.g., hundreds of thousands of markers, and that most of the assumptions required for an orthogonal decomposition of variance are violated in artificial and natural populations. This makes nonparametric procedures attractive. Kernel regression and reproducing kernel Hilbert spaces regression procedures are embedded into standard mixed-effects linear models, retaining additive genetic effects under multivariate normality for operational reasons. Inferential procedures are presented, and some extensions are suggested. An example is presented, illustrating the potential of the methodology. Implementations can be carried out after modification of standard software developed by animal breeders for likelihood-based or Bayesian analysis.