BACKGROUND
whole-genome prediction  is expected to reshape plant breeding fundametally in the near future  <cit> .
whereas the approach has been initially proposed  <cit>  and rapidly implemented in animal breeding  <cit> , recent empirical studies demonstrated also its potential in hybrid maize breeding  <cit> .
recently, we showed that wgp allows a reliable screening of large germplasm collections of diverse maize inbred lines for their potential to create superior hybrids  <cit> .
however, these studies exclusively focused on predicting highly polygenic traits such as grain yield or biomass accumulation with genetic architectures close to the infinitesimal genetic model  <cit> .
in maize, several economically important traits are genetically less complex with few quantitative trait loci  explaining a large proportion of the genetic variance.
examples include pest and disease resistances or nutrifunctional compounds such as bioavailable minerals  <cit>  or β-carotene  <cit> .
in addition, disease resistances are often found to be controlled by a combination of race-specific resistance loci with large effects involved in pathogen recognition, and a large number of loci with small effects involved in basal resistance.
such a mixed qtl effect distribution can be found in maize e.g.
for rust  <cit> , giberella ear rot  <cit>  or to a lesser extent for northern corn leaf blight  <cit> .
for such traits, the assumption of normally distributed snp effects underlying ridge regression, the most commonly applied wgp model, is severely violated.
heslot et al.
<cit>  found for polygenic traits in several plant species only minor differences between ridge regression and models with different assumptions of the underlying distribution of snp effects.
however, these differences are expected to be much larger for traits controlled by only a few qtl.
recently, clark et al.
<cit>  simulated this situation under the assumption of the historical population structure of holstein cattle.
they found that under the assumption of either few common or few rare quantitative trait loci, a bayesian variable selection model  outperforms ridge regression by far.
for holstein cattle, hayes et al.
<cit>  found also the bayesa model to be superior to ridge regression in the case of coat color or milk-fat percentage.
cattle differs greatly in its population structure and ld level from elite maize germplasm, which has faced severe genetic bottlenecks during domestication and the creation of genetically distinct heterotic pools to maximize exploitation of heterosis in hybrid breeding  <cit> .
hence, results from cattle might not be directly transferable to maize, for which little is known about wgp for traits with a simpler genetic architecture.
moreover, the genetic architecture of a trait is often unclear in crops.
especially if the trait has not yet been extensively dissected by linkage or association mapping, which might be the case for traits which gained only recently in importance such as nutritional properties, nutrient acquisition traits or traits related to climate change adaptation.
to fill this apparent gap of knowledge, we investigated wgp in a diverse collection of  <dig> maize inbred lines with traits which largely vary in their genetic architecture.
to let the genetic basis differ as much as possible, we chose three highly polgygenic agronomic traits and three metabolites, each one controlled by a different major qtl explaining about 22% to 30% of genetic variance.
with this empirical set-up, we asked the following questions:
● to what extent do distinct wgp models differ in their prediction accuracies for a diversity panel of maize inbred lines if the genetic architecture of the trait changes dramatically?
● are there guidelines for plants concerning the choice of the most promising wgp model?
methods
genetic material
the genetic material consisted of  <dig> maize inbred lines which were previously described in great detail  <cit> .
the population constituted a global sample of elite breeding material from worldwide sources with a focus on north america and europe and encompassed  <dig> lines from the dent heterotic pool  and  <dig> from the european flint pool, which served as check genotypes.
genotyping
the population was genotyped with the illumina snp chip maizesnp <dig> containing  <dig>  snps  <cit> .
quality control preprocessing of snps was performed by eliminating snps that did not match the following criteria:  less than 10% missing values,  minor allele frequency of greater than  <dig> %,  no more than three heterozygous genotypes, and  unique allele assignment for the  <dig> replicated checks of genotype b <dig>  a total of  <dig>  snps remained and were used for further analysis.
linkage disequilibrium  declined to r2= <dig>  at approximately  <dig> kb with a mean ld between adjacent snps of  <dig>   <cit> .
field trials
the population was phenotyped in six environments  in germany  <cit> .
briefly, the population was split into three maturity groups based on prior knowledge of their flowering time.
in the trials of each of the three maturity groups,  <dig> genotypes, including five common check genotypes, were randomized in a  <dig> ×  <dig> α-lattice design with two replications and were planted in 2-row plots.
plots were thinned to a final plant density of  <dig>  plants/ha.
the common check genotypes were used to adjust for potential differences in the soil fertility among trials in each environment.
metabolites
leaf samples were collected in one location  <dig> d after sowing and processed using an established gc-ms method  <cit> .
genotypic means of box-cox transformed metabolite concentrations were obtained using a linear mixed model analysis including effects for field trial, replication, block, and batch.
the whole metabolic profiling procedure including statistical analysis has been described in detail previously  <cit> .
from the measured metabolite concentrations, we chose three highly heritable substances as model traits: dopamine, ribitol, and an unknown metabolite .
for each metabolite, we found in a genome-wide association  study a major metabolite qtl  on different chromosomes after correcting for population structure and kinship  <cit> .
for dopamine, the major mqtl was found on chromosome  <dig> and explained  <dig> % of the genetic variance.
for ribitol, the major mqtl was found on chromosome  <dig> and explained  <dig> % oft the genetic variance.
for the unknown metabolite, the major mqtl was found on chromosome  <dig> and explained  <dig> % of the genetic variance.
the metabolites were uncorrelated with each other  and only weakly  correlated with agronomic traits .
agronomic traits
dry matter yield of whole-plant biomass  and plant height  were measured per field plot of the inbred lines.
lignin content was measured as acid detergent lignin  in the harvested plant material of the inbred lines using calibrated near-infrared spectroscopy .
the nirs calibration model was built using phenotypic data from  <dig> inbred lines,  <dig> testcrosses and  <dig> hybrids grown in the same environments as the population of inbred lines analyzed in this study  <cit> .
heritability estimates and genotypic means were obtained using a one-step linear mixed model analysis as described previously  <cit> .
using a 1% bonferroni corrected significance threshold, we could not find any significant snp-trait association signal using the the same gwa model as for metabolites.
since population size, marker density, and heritabilites were sufficiently high for detecting qtl with large effects, the absence of any significant trait-snp associations suggest a highly polygenic genetic architecture for the agronomic traits with no major qtl.
genome partitioning of the genetic variance
to further characterize the genetic architectures of the investigated traits irrespective of the significance thresholds for snp-trait associations, we compared how the ten chromosomes contributed to the total genetic variance.
later on, we will use these results as a guideline for model selection based on trait architecture.
we adopted the approach of yang et al.
<cit>  to simultaneously estimate the genetic variance explained by each chromosome.
in order to derive a guideline which is purely based on trait architecture and not on population structure artefacts, we additionally corrected for population structure by regressing the trait values on the first ten principal components.
this linear model can be written as
y=1μ+qβ+∑c=110+e
where y is a vector with n trait values, 1is a vector of 1’s, qis a matrix of size n× <dig> containing the first  <dig> principal components calculated from snp data with β containing the corresponding regression coefficients, s is an incidence matrix allocating components of yto components of gc, which is a vector of length n with random genotypic effects attributable to chromosome c with gc∼n and gc=zczct/pc where zc is a matrix of size n×pcwith standardized levels of snp alleles on chromosome c. vector e contains normally distributed residuals with e∼n.
the genetic variance contributed by chromosome c was then estimated as σgc2/+σe2).
variance components were estimated by restricted maximum likelihood  using asreml-r  <dig>  <cit> .
since matrices gcwere often found to be singular, we used the algorithm of higham  <cit>  implemented in the function nearpd of the r-package matrix  <cit> , to approximate the nearest positive definite matrices.
wgp models
we investigated five wgp models that have been recently advocated in the literature for this purpose  <cit> .
all based upon the classical regression set-up
y=1μ+zu+e
where y is a vector with n trait values, μ is the overall mean,  <dig> is a vector with 1’s, z is the n×p matrix of standardized values of snp alleles, u is a vector with snp effects, and e is a vector of residuals with e∼n.
depending on the trait, a combination of genotypic and phenotypic information was available for  <dig> to  <dig> genotypes which were used for wgp.
rr-blup
ridge regression  tackles the p≫nproblem in wgp by minimizing the residual sum of squares t) by bounding the euclidean  norm of u to a constraint: ||u||2=∑i=1pui2<crr which leads to a homogeneous shrinkage of all snp effects towards zero .
the rr estimator is given by
u^rr=argminu.
the lagrangian multiplier λrris a regularization parameter which controls the amount of shrinkage.
it can be estimated as λrr=σe2/σu <dig> by regarding u as random effects with u∼n with σu <dig> being the snp effect variance estimated by reml.
in this setting, u^rr is equivalent to the best linear unbiased predictor  of u <cit> .
for computational convenience, rr-blup can be transformed to a mathematically equivalent model
y=1μ+sg+e
with g being a vector of random genotype effects with var = gσg <dig> and whole-genome relationship matrix g=zzt/p.
the solution vector of snp effects can then be obtained as u^rr=ztg−1g^ <cit> .
here, g is an inner-product kernel which allows to perform all computations in the space of n genotypes instead of p snps, a shortcut which is well established in the field of kernel-based machine learning  <cit> .
lasso
as an alternative to ridge regression, it was suggested to use an l <dig> penalty to tackle the p≫nproblem  <cit> .
this estimator was termed least absolute shrinkage and selection operator  and has recently been suggested for whole-genome prediction  <cit> .
the estimator is given by
u^l=argminu
which bounds the manhattan  norm of u to a constraint: ||u||1=∑i=1p|ui|<cl.
the lasso penalty is a diamond shaped constraint which allows not only to shrink coefficients towards zero but to set some coefficients to exactly zero .
unlike rr, lasso, cannot be ’kernelized’, i.e., it is not possible to transform the lasso estimator into an equivalent kernel regression problem in the space of n genotypes  <cit> .
hence, lasso regression has to be carried out with the full set of snps.
here, we used the r package glmnet, a fast implementation using cyclic coordinate descent to compute the complete lasso path solution  <cit> .
elastic net
the lasso penalty is known to be somewhat indifferent to the choice among a set of strong but correlated variables.
the rr penalty, on the other side, tends to shrink the coefficients of correlated variables toward each other  <cit> .
the elastic net  estimator is a compromise which can be written as
u^en=argminu||u||2)),
and is a weighted mixture between the rr penalty  and the lasso penalty   <cit> .
while the rr penalty encourages highly correlated variables to be averaged, the lasso penalty encourages a sparse solution  <cit> .
we again used the implementation in glmnet and performed a grid search to find the combination of α and λen, which yielded the lowest mean squared error in the training population.
reproducing kernel hilbert space  regression
the theory of rkhs regression is rooted in the field of kernel-based machine-learning  <cit>  and has recently been advocated for whole-genome prediction  <cit> .
the approach uses equation  <dig> but replaces the inner-product matrix g with a kernel matrix k. the motivation behind rkhs regression lies in the ability to effectively perform non-linear regression in a higher-dimensional feature space so it might capture non-additive genetic effects, if present.
here, we used a gaussian kernel on genetic distances with kij=exp, where gdij is the modified rogers’ genetic distance  between genotype i and j, and θ is a smoothing parameter which controls the rate of decay of kij with increasing genetic distance.
the optimum value for θwas chosen from a sequence from  <dig>  to  <dig> at which the maximum likelihood was obtained.
bayesb
as a bayesian approach, we used a modified version of bayesb, which has a prior assumption that the snp effects are t-distributed with a point-mass at zero  <cit> .
following the suggestions of yang and tempelman  <cit> , we modeled several hyperparameters as uncertain too.
details of the priors used can be found in table  <dig>  to fit the model, we ran the gibbs-sampler for  <dig>  iterations.
the first  <dig>  iterations were discarded as burn-in and only samples from every 10th post burn-in iteration were stored.
for computational convenience, we reduced the number of markers to  <dig>  snps for which we did not observe any decline in prediction accuracy up to the numerical precision reported in this study.
validation
a five-fold cross-validation scheme was applied and repeated  <dig> times.
in each repetition, the dataset was divided into  <dig> disjoint subsets of genotypes whereas one subset served as the validation set and the other four subsets served as the training population to estimate the model parameters for predicting the left-out genotypes in the validation set.
in each of the five rounds, the pearson correlation between the observed and predicted phenotypic values was calculated.
the procedure was repeated twenty times to yield  <dig> cross-validation runs.
the predictive ability was then calculated as the pearson correlation ) between the observed  and predicted  phenotypic values.
the ’prediction accuracy’ estimates the correlation ) between the predicted  and unobserved true genetic values  and was calculated by r=r/h where h is the square root of the heritability on a line-mean basis for the agronomic traits.
for metabolites, the square root of the estimated repeatability was used.
RESULTS
the contribution of the individual chromosomes to the genetic variance differed largely between metabolites and agronomic traits .
for the metabolites, the chromosomes containing the major mqtl  captured by far the largest portion of the genetic variance, leaving the remaining genetic variance equally distributed over the remaining chromosomes.
for the agronomic traits, the total genetic variance was largely uniformly distributed over all chromosomes.
using the same gwa model as for the metabolites  <cit> , we found that for dry matter yield and plant height, the chromosomes which captured the largest portion of genetic variance contain the strongest gwa signals.
however, in no instance was the 1% bonferroni corrected significance threshold surpassed .
when excluding chromosomes containing either these two association signals or major mqtl, we observed a tendency that longer chromosomes captured more genetic variance than shorter ones .
this trend was significant  for lignin content  and dry matter yield .
the grey area in figure 2b was therefore regarded as the range in chromosomal genetic variance explainable by the length of the chromosome.
on the other side, the genetic variance contributed by the left-out chromosomes was highly correlated  with the explained genetic variance of the individual snps found by gwa mapping .
the total genetic variance summed over all chromosomes amounted to  <dig>  for dry matter yield,  <dig>  for plant height,  <dig>  for lignin content,  <dig>  for dopamine,  <dig>  for ribitol, and  <dig>  for the unknown metabolite.
these values were close to the heritabilities and repeatabilities obtained from the phenotypic analysis .
results are averaged over all  <dig> cross-validation runs.
for the agronomic traits, h2is the heritability on a line-mean basis and for the metabolites, the repeatability is shown.
prediction accuracies of wgp ranged between  <dig>  and  <dig>  with standard deviations between  <dig>  to  <dig>  across traits and models .
the largest differences in accuracies between models ranged from  <dig>  to  <dig>  for the same trait.
between rr-blup and rkhs, we found no difference in the prediction accuracies above  <dig>  for any trait.
for agronomic traits, prediction accuracies were highest for rr-blup with a drop of  <dig>  to  <dig>  if lasso or elastic net was used and with a drop of  <dig>  to  <dig>  if bayesb was used.
the ranking of the prediction accuracies for the wgp models was reverse for metabolites.
here, prediction accuracies were highest for lasso or elastic net with a drop of  <dig>  to  <dig>  when using rr-blup.
for metabolites, no differences in the prediction accuracies above  <dig>  were observed between rr-blup and bayesb.
for dopamine and the unknown metabolites, the mqtl were precisely found with lasso, elastic net and also rr-blup .
for all three models, their largest absolute snp effect matched exactly with the snp identified by gwa mapping.
however, the three models differed drastically in their sparsity in snp effects, and the distance over which the mqtl effect was distributed.
whereas the mqtl effects declined sharply with lasso or elastic net, they were diluted over a much longer distance with rr-blup.
discussion
we found in a diverse panel of elite maize inbred lines that prediction accuracies obtained with five different wgp models were remarkable similar, even for traits with drastically deviating genetic architecture.
our results suggest that small gains in accuracies  can be gained if the wgp model is selected according to the genetic architecture underlying the trait.
recently, heslot et al.
<cit>  reported similar small differences for seven parametric wgp models when comparing them for different presumable highly polygenic agronomic traits over eight datasets of barley, arabidopsis thaliana, maize, and wheat.
for the metabolites, however, our results differ from those obtained from clark et al.
<cit> , who investigated the influence of genetic architecture on prediction accuracies achieved by rr-blup or bayesb.
whereas these authors found only slight differences for simulated traits with a genetic architecture close to the infinitesimal genetic model, bayesb outperformed rr-blup by an increase in prediction accuracy of ≈ <dig>  if the trait is controlled by either a few common or a few rare qtl.
simulation also predicted a drop in prediction accuracy in case of rr-blup for traits controlled by a small number of qtl  <cit> .
although lasso, elastic net, and bayesb showed higher accuracies compared to rr-blup for metabolites, we found the differences to be remarkable small in case of lasso or elastic net and negliable in the case of bayesb.
one major reason of the minor differences in prediction accuracies among the different models lies in the high level of ld found in elite breeding germplasm of maize.
our results suggest that with this level of ld , accuracies are quite similar irrespective whether the effect of large qtl are precisely captured  or spread over a larger region .
since our population was highly diverse for elite maize germplasm in europe, it is unlikely that breeders are confronted with lower levels of ld unless they work with highly exotic germplasm for which ld has been reported to decline within 5- <dig> kb  <cit> .
moreover, the high similarity of rkhs and rr-blup suggest that either  non-additive, epistatic genetic effects are not present,  these are so small that they are negligible in wgp for the investigated traits, or  rkhs regression is unable to capture them.
in either case, for prediction purposes rkhs does not seem to yield any advancements over rr-blup for situations comparable to our germplasm and traits.
dominance, as another source of non-additive genetic, effects cannot be present in the inbred lines investigated in this study.
for predicting heterozygeous f <dig> maize hybrids, however, it has been shown that modeling dominance effects can result in higher prediction accuracies  <cit> .
although bayesb reached for  <dig> of the  <dig> traits a higher prediction accuracy than the worst model, we cannot recommend it because of the excessively larger computation time and the negliable differences in prediction accuracies compared with rr-blup in case of the metabolites as the result of probably only sampling error.
we found the approach to partition genetic variance over chromosomes useful for guiding the breeder which wgp model to prefer in the case of little or no prior knowledge on the genetic architecture.
whereas for the agronomic traits an approximately linear increase of cumulative explained genetic variance matched with a superiority of the l <dig> penalty , the l1penalty  or a mixture of both penalties  performed better in the case of the metabolites with a strong convex curve curvature .
although for dry matter yield and plant height, barely significant association signals with a proportion of explained genetic variance <9% led to a chromosomal genetic variance slightly above the range expected from length of the chromosome , these effects were too small to justify the use of the elastic net or lasso.
as an alternative to this approach, hayes et al.
<cit>  estimated successively the genetic variance explained by each chromosome segment and compared it with the genetic variance captured by the remaining part of the genome.
to correct for the non-independence of neighbouring segments, they applied a bias correction using an expectation maximization  algorithm.
such a correction is not necessary if the variance components for all chromosomes are estimated simultaneously as applied in this study; this is a further advantage besides its straightforward implementation using standard mixed model software packages such as asreml.
CONCLUSIONS
our empirical data of wgp in a large panel of diverse maize inbred lines suggest that  different wgp models differ only slightly in their prediction accuracies, irrespective of the number and effects of qtl found in association analysis,  small gains in prediction accuracies can be obtained if the wgp model is selected according to the genetic architecture of the trait,  genome partitioning of genetic variance offers a straightforward approach for model selection if the genetic architecture is unknown.
the question of which wgp model to choose is therefore not expected to hamper implementation of wgp in maize breeding.
competing interests
the authors declare that they have no competing interests.
authors’ contributions
cr analyzed the data and wrote the manuscript.
aem supervised the research.
ft wrote the c code for implementing bayesb.
all authors read and approved the final manuscript.

 whole-genome prediction is expected to reshape plant breeding fundametally in the near future. whereas the approach has been initially proposed and rapidly implemented in animal breeding, recent empirical studies demonstrated also its potential in hybrid maize breeding. recently, we showed that wgp allows a reliable screening of large germplasm collections of diverse maize inbred lines for their potential to create superior hybrids. however, these studies exclusively focused on predicting highly polygenic traits such as grain yield or biomass accumulation with genetic architectures close to the infinitesimal genetic model. in maize, several economically important traits are genetically less complex with few quantitative trait loci explaining a large proportion of the genetic variance. examples include pest and disease resistances or nutrifunctional compounds such as bioavailable minerals or β-carotene. in addition, disease resistances are often found to be controlled by a combination of race-specific resistance loci with large effects involved in pathogen recognition, and a large number of loci with small effects involved in basal resistance. such a mixed qtl effect distribution can be found in maize e.g. for rust, giberella ear rot or to a lesser extent for northern corn leaf blight. for such traits, the assumption of normally distributed snp effects underlying ridge regression, the most commonly applied wgp model, is severely violated. heslot et al. found for polygenic traits in several plant species only minor differences between ridge regression and models with different assumptions of the underlying distribution of snp effects. however, these differences are expected to be much larger for traits controlled by only a few qtl. recently, clark et al. simulated this situation under the assumption of the historical population structure of holstein cattle. they found that under the assumption of either few common or few rare quantitative trait loci, a bayesian variable selection model outperforms ridge regression by far. for holstein cattle, hayes et al. found also the bayesa model to be superior to ridge regression in the case of coat color or milk-fat percentage. cattle differs greatly in its population structure and ld level from elite maize germplasm, which has faced severe genetic bottlenecks during domestication and the creation of genetically distinct heterotic pools to maximize exploitation of heterosis in hybrid breeding. hence, results from cattle might not be directly transferable to maize, for which little is known about wgp for traits with a simpler genetic architecture. moreover, the genetic architecture of a trait is often unclear in crops. especially if the trait has not yet been extensively dissected by linkage or association mapping, which might be the case for traits which gained only recently in importance such as nutritional properties, nutrient acquisition traits or traits related to climate change adaptation. to fill this apparent gap of knowledge, we investigated wgp in a diverse collection of maize inbred lines with traits which largely vary in their genetic architecture. to let the genetic basis differ as much as possible, we chose three highly polgygenic agronomic traits and three metabolites, each one controlled by a different major qtl explaining about 22% to 30% of genetic variance. with this empirical set-up, we asked the following questions: ● to what extent do distinct wgp models differ in their prediction accuracies for a diversity panel of maize inbred lines if the genetic architecture of the trait changes dramatically? ● are there guidelines for plants concerning the choice of the most promising wgp model? methods genetic material the genetic material consisted of maize inbred lines which were previously described in great detail. the population constituted a global sample of elite breeding material from worldwide sources with a focus on north america and europe and encompassed lines from the dent heterotic pool and from the european flint pool, which served as check genotypes. genotyping the population was genotyped with the illumina snp chip maizesnp containing snps. quality control preprocessing of snps was performed by eliminating snps that did not match the following criteria: less than 10% missing values, minor allele frequency of greater than %, no more than three heterozygous genotypes, and unique allele assignment for the replicated checks of genotype b a total of snps remained and were used for further analysis. linkage disequilibrium declined to r2= at approximately kb with a mean ld between adjacent snps of. field trials the population was phenotyped in six environments in germany. briefly, the population was split into three maturity groups based on prior knowledge of their flowering time. in the trials of each of the three maturity groups, genotypes, including five common check genotypes, were randomized in a × α-lattice design with two replications and were planted in 2-row plots. plots were thinned to a final plant density of plants/ha. the common check genotypes were used to adjust for potential differences in the soil fertility among trials in each environment. metabolites leaf samples were collected in one location d after sowing and processed using an established gc-ms method. genotypic means of box-cox transformed metabolite concentrations were obtained using a linear mixed model analysis including effects for field trial, replication, block, and batch. the whole metabolic profiling procedure including statistical analysis has been described in detail previously. from the measured metabolite concentrations, we chose three highly heritable substances as model traits: dopamine, ribitol, and an unknown metabolite. for each metabolite, we found in a genome-wide association study a major metabolite qtl on different chromosomes after correcting for population structure and kinship. for dopamine, the major mqtl was found on chromosome and explained % of the genetic variance. for ribitol, the major mqtl was found on chromosome and explained % oft the genetic variance. for the unknown metabolite, the major mqtl was found on chromosome and explained % of the genetic variance. the metabolites were uncorrelated with each other and only weakly correlated with agronomic traits. agronomic traits dry matter yield of whole-plant biomass and plant height were measured per field plot of the inbred lines. lignin content was measured as acid detergent lignin in the harvested plant material of the inbred lines using calibrated near-infrared spectroscopy. the nirs calibration model was built using phenotypic data from inbred lines, testcrosses and hybrids grown in the same environments as the population of inbred lines analyzed in this study. heritability estimates and genotypic means were obtained using a one-step linear mixed model analysis as described previously. using a 1% bonferroni corrected significance threshold, we could not find any significant snp-trait association signal using the the same gwa model as for metabolites. since population size, marker density, and heritabilites were sufficiently high for detecting qtl with large effects, the absence of any significant trait-snp associations suggest a highly polygenic genetic architecture for the agronomic traits with no major qtl. genome partitioning of the genetic variance to further characterize the genetic architectures of the investigated traits irrespective of the significance thresholds for snp-trait associations, we compared how the ten chromosomes contributed to the total genetic variance. later on, we will use these results as a guideline for model selection based on trait architecture. we adopted the approach of yang et al. to simultaneously estimate the genetic variance explained by each chromosome. in order to derive a guideline which is purely based on trait architecture and not on population structure artefacts, we additionally corrected for population structure by regressing the trait values on the first ten principal components. this linear model can be written as y=1μ+qβ+∑c=110+e where y is a vector with n trait values, 1is a vector of 1’s, qis a matrix of size n× containing the first principal components calculated from snp data with β containing the corresponding regression coefficients, s is an incidence matrix allocating components of yto components of gc, which is a vector of length n with random genotypic effects attributable to chromosome c with gc∼n and gc=zczct/pc where zc is a matrix of size n×pcwith standardized levels of snp alleles on chromosome c. vector e contains normally distributed residuals with e∼n. the genetic variance contributed by chromosome c was then estimated as σgc2/+σe2). variance components were estimated by restricted maximum likelihood using asreml-r. since matrices gcwere often found to be singular, we used the algorithm of higham implemented in the function nearpd of the r-package matrix, to approximate the nearest positive definite matrices. wgp models we investigated five wgp models that have been recently advocated in the literature for this purpose. all based upon the classical regression set-up y=1μ+zu+e where y is a vector with n trait values, μ is the overall mean, is a vector with 1’s, z is the n×p matrix of standardized values of snp alleles, u is a vector with snp effects, and e is a vector of residuals with e∼n. depending on the trait, a combination of genotypic and phenotypic information was available for to genotypes which were used for wgp. rr-blup ridge regression tackles the p≫nproblem in wgp by minimizing the residual sum of squares t) by bounding the euclidean norm of u to a constraint: ||u||2=∑i=1pui2<crr which leads to a homogeneous shrinkage of all snp effects towards zero. the rr estimator is given by u^rr=argminu. the lagrangian multiplier λrris a regularization parameter which controls the amount of shrinkage. it can be estimated as λrr=σe2/σu by regarding u as random effects with u∼n with σu being the snp effect variance estimated by reml. in this setting, u^rr is equivalent to the best linear unbiased predictor of u. for computational convenience, rr-blup can be transformed to a mathematically equivalent model y=1μ+sg+e with g being a vector of random genotype effects with var = gσg and whole-genome relationship matrix g=zzt/p. the solution vector of snp effects can then be obtained as u^rr=ztg−1g^. here, g is an inner-product kernel which allows to perform all computations in the space of n genotypes instead of p snps, a shortcut which is well established in the field of kernel-based machine learning. lasso as an alternative to ridge regression, it was suggested to use an l penalty to tackle the p≫nproblem. this estimator was termed least absolute shrinkage and selection operator and has recently been suggested for whole-genome prediction. the estimator is given by u^l=argminu which bounds the manhattan norm of u to a constraint: ||u||1=∑i=1p|ui|<cl. the lasso penalty is a diamond shaped constraint which allows not only to shrink coefficients towards zero but to set some coefficients to exactly zero. unlike rr, lasso, cannot be ’kernelized’, i.e., it is not possible to transform the lasso estimator into an equivalent kernel regression problem in the space of n genotypes. hence, lasso regression has to be carried out with the full set of snps. here, we used the r package glmnet, a fast implementation using cyclic coordinate descent to compute the complete lasso path solution. elastic net the lasso penalty is known to be somewhat indifferent to the choice among a set of strong but correlated variables. the rr penalty, on the other side, tends to shrink the coefficients of correlated variables toward each other. the elastic net estimator is a compromise which can be written as u^en=argminu||u||2)), and is a weighted mixture between the rr penalty and the lasso penalty. while the rr penalty encourages highly correlated variables to be averaged, the lasso penalty encourages a sparse solution. we again used the implementation in glmnet and performed a grid search to find the combination of α and λen, which yielded the lowest mean squared error in the training population. reproducing kernel hilbert space regression the theory of rkhs regression is rooted in the field of kernel-based machine-learning and has recently been advocated for whole-genome prediction. the approach uses equation but replaces the inner-product matrix g with a kernel matrix k. the motivation behind rkhs regression lies in the ability to effectively perform non-linear regression in a higher-dimensional feature space so it might capture non-additive genetic effects, if present. here, we used a gaussian kernel on genetic distances with kij=exp, where gdij is the modified rogers’ genetic distance between genotype i and j, and θ is a smoothing parameter which controls the rate of decay of kij with increasing genetic distance. the optimum value for θwas chosen from a sequence from to at which the maximum likelihood was obtained. bayesb as a bayesian approach, we used a modified version of bayesb, which has a prior assumption that the snp effects are t-distributed with a point-mass at zero. following the suggestions of yang and tempelman, we modeled several hyperparameters as uncertain too. details of the priors used can be found in table to fit the model, we ran the gibbs-sampler for iterations. the first iterations were discarded as burn-in and only samples from every 10th post burn-in iteration were stored. for computational convenience, we reduced the number of markers to snps for which we did not observe any decline in prediction accuracy up to the numerical precision reported in this study. validation a five-fold cross-validation scheme was applied and repeated times. in each repetition, the dataset was divided into disjoint subsets of genotypes whereas one subset served as the validation set and the other four subsets served as the training population to estimate the model parameters for predicting the left-out genotypes in the validation set. in each of the five rounds, the pearson correlation between the observed and predicted phenotypic values was calculated. the procedure was repeated twenty times to yield cross-validation runs. the predictive ability was then calculated as the pearson correlation ) between the observed and predicted phenotypic values. the ’prediction accuracy’ estimates the correlation ) between the predicted and unobserved true genetic values and was calculated by r=r/h where h is the square root of the heritability on a line-mean basis for the agronomic traits. for metabolites, the square root of the estimated repeatability was used.  the contribution of the individual chromosomes to the genetic variance differed largely between metabolites and agronomic traits. for the metabolites, the chromosomes containing the major mqtl captured by far the largest portion of the genetic variance, leaving the remaining genetic variance equally distributed over the remaining chromosomes. for the agronomic traits, the total genetic variance was largely uniformly distributed over all chromosomes. using the same gwa model as for the metabolites, we found that for dry matter yield and plant height, the chromosomes which captured the largest portion of genetic variance contain the strongest gwa signals. however, in no instance was the 1% bonferroni corrected significance threshold surpassed. when excluding chromosomes containing either these two association signals or major mqtl, we observed a tendency that longer chromosomes captured more genetic variance than shorter ones. this trend was significant for lignin content and dry matter yield. the grey area in figure 2b was therefore regarded as the range in chromosomal genetic variance explainable by the length of the chromosome. on the other side, the genetic variance contributed by the left-out chromosomes was highly correlated with the explained genetic variance of the individual snps found by gwa mapping. the total genetic variance summed over all chromosomes amounted to for dry matter yield, for plant height, for lignin content, for dopamine, for ribitol, and for the unknown metabolite. these values were close to the heritabilities and repeatabilities obtained from the phenotypic analysis. results are averaged over all cross-validation runs. for the agronomic traits, h2is the heritability on a line-mean basis and for the metabolites, the repeatability is shown. prediction accuracies of wgp ranged between and with standard deviations between to across traits and models. the largest differences in accuracies between models ranged from to for the same trait. between rr-blup and rkhs, we found no difference in the prediction accuracies above for any trait. for agronomic traits, prediction accuracies were highest for rr-blup with a drop of to if lasso or elastic net was used and with a drop of to if bayesb was used. the ranking of the prediction accuracies for the wgp models was reverse for metabolites. here, prediction accuracies were highest for lasso or elastic net with a drop of to when using rr-blup. for metabolites, no differences in the prediction accuracies above were observed between rr-blup and bayesb. for dopamine and the unknown metabolites, the mqtl were precisely found with lasso, elastic net and also rr-blup. for all three models, their largest absolute snp effect matched exactly with the snp identified by gwa mapping. however, the three models differed drastically in their sparsity in snp effects, and the distance over which the mqtl effect was distributed. whereas the mqtl effects declined sharply with lasso or elastic net, they were diluted over a much longer distance with rr-blup. discussion we found in a diverse panel of elite maize inbred lines that prediction accuracies obtained with five different wgp models were remarkable similar, even for traits with drastically deviating genetic architecture. our results suggest that small gains in accuracies can be gained if the wgp model is selected according to the genetic architecture underlying the trait. recently, heslot et al. reported similar small differences for seven parametric wgp models when comparing them for different presumable highly polygenic agronomic traits over eight datasets of barley, arabidopsis thaliana, maize, and wheat. for the metabolites, however, our results differ from those obtained from clark et al., who investigated the influence of genetic architecture on prediction accuracies achieved by rr-blup or bayesb. whereas these authors found only slight differences for simulated traits with a genetic architecture close to the infinitesimal genetic model, bayesb outperformed rr-blup by an increase in prediction accuracy of ≈ if the trait is controlled by either a few common or a few rare qtl. simulation also predicted a drop in prediction accuracy in case of rr-blup for traits controlled by a small number of qtl. although lasso, elastic net, and bayesb showed higher accuracies compared to rr-blup for metabolites, we found the differences to be remarkable small in case of lasso or elastic net and negliable in the case of bayesb. one major reason of the minor differences in prediction accuracies among the different models lies in the high level of ld found in elite breeding germplasm of maize. our results suggest that with this level of ld, accuracies are quite similar irrespective whether the effect of large qtl are precisely captured or spread over a larger region. since our population was highly diverse for elite maize germplasm in europe, it is unlikely that breeders are confronted with lower levels of ld unless they work with highly exotic germplasm for which ld has been reported to decline within 5- kb. moreover, the high similarity of rkhs and rr-blup suggest that either non-additive, epistatic genetic effects are not present, these are so small that they are negligible in wgp for the investigated traits, or rkhs regression is unable to capture them. in either case, for prediction purposes rkhs does not seem to yield any advancements over rr-blup for situations comparable to our germplasm and traits. dominance, as another source of non-additive genetic, effects cannot be present in the inbred lines investigated in this study. for predicting heterozygeous f maize hybrids, however, it has been shown that modeling dominance effects can result in higher prediction accuracies. although bayesb reached for of the traits a higher prediction accuracy than the worst model, we cannot recommend it because of the excessively larger computation time and the negliable differences in prediction accuracies compared with rr-blup in case of the metabolites as the result of probably only sampling error. we found the approach to partition genetic variance over chromosomes useful for guiding the breeder which wgp model to prefer in the case of little or no prior knowledge on the genetic architecture. whereas for the agronomic traits an approximately linear increase of cumulative explained genetic variance matched with a superiority of the l penalty, the l1penalty or a mixture of both penalties performed better in the case of the metabolites with a strong convex curve curvature. although for dry matter yield and plant height, barely significant association signals with a proportion of explained genetic variance <9% led to a chromosomal genetic variance slightly above the range expected from length of the chromosome, these effects were too small to justify the use of the elastic net or lasso. as an alternative to this approach, hayes et al. estimated successively the genetic variance explained by each chromosome segment and compared it with the genetic variance captured by the remaining part of the genome. to correct for the non-independence of neighbouring segments, they applied a bias correction using an expectation maximization algorithm. such a correction is not necessary if the variance components for all chromosomes are estimated simultaneously as applied in this study; this is a further advantage besides its straightforward implementation using standard mixed model software packages such as asreml.  our empirical data of wgp in a large panel of diverse maize inbred lines suggest that different wgp models differ only slightly in their prediction accuracies, irrespective of the number and effects of qtl found in association analysis, small gains in prediction accuracies can be obtained if the wgp model is selected according to the genetic architecture of the trait, genome partitioning of genetic variance offers a straightforward approach for model selection if the genetic architecture is unknown. the question of which wgp model to choose is therefore not expected to hamper implementation of wgp in maize breeding. competing interests the authors declare that they have no competing interests. authors’ contributions cr analyzed the data and wrote the manuscript. aem supervised the research. ft wrote the c code for implementing bayesb. all authors read and approved the final manuscript. 
here, we compared five wgp models with different assumptions about the distribution of the underlying genetic effects. our results suggest that in diverse germplasm of elite maize inbred lines with a high level of ld, wgp models differ only slightly in their accuracies, irrespective of the number and effects of qtl found in previous linkage or association mapping studies. whereas rr-blup performed best for the agronomic traits, it was inferior to lasso or elastic net for the three metabolites. as contrasting model traits, we chose three highly polygenic agronomic traits and three metabolites each with a major qtl explaining to 30% of the genetic variance in a panel of diverse maize inbred lines genotyped with snps. we found the approach of genome partitioning of genetic variance, first applied in human genetics, as useful in guiding the breeder which model to choose, if prior knowledge of the trait architecture is lacking. for novel traits which only recently received attention in breeding, we suggest to inspect the distribution of the genetic variance explained by each chromosome for guiding model selection in wgp. we found the five wgp models to be remarkable robust towards trait architecture with the largest differences in prediction accuracies ranging between and for the same trait, most likely as the result of the high level of linkage disequilibrium prevailing in elite maize germplasm. however, there is a lack of knowledge about the sensitivity of wgp models towards the genetic architecture of the trait. whereas previous studies exclusively focused on highly polygenic traits, important agronomic traits such as disease resistances, nutrifunctional or climate adaptational traits have a genetic architecture which is either much less complex or unknown. however, small gains in prediction accuracies can be achieved if the wgp model is selected according to the genetic architecture of the trait. if the trait architecture is unknown e.g. there is increasing empirical evidence that whole-genome prediction is a powerful tool for predicting line and hybrid performance in maize. genomic selectionwhole-genome predictiongenetic architecturecomplex traitszea mays for such cases, information about model robustness and guidelines for model selection are lacking.
 there is increasing empirical evidence that whole-genome prediction is a powerful tool for predicting line and hybrid performance in maize. however, there is a lack of knowledge about the sensitivity of wgp models towards the genetic architecture of the trait. whereas previous studies exclusively focused on highly polygenic traits, important agronomic traits such as disease resistances, nutrifunctional or climate adaptational traits have a genetic architecture which is either much less complex or unknown. for such cases, information about model robustness and guidelines for model selection are lacking. here, we compared five wgp models with different assumptions about the distribution of the underlying genetic effects. as contrasting model traits, we chose three highly polygenic agronomic traits and three metabolites each with a major qtl explaining to 30% of the genetic variance in a panel of diverse maize inbred lines genotyped with snps.  we found the five wgp models to be remarkable robust towards trait architecture with the largest differences in prediction accuracies ranging between and for the same trait, most likely as the result of the high level of linkage disequilibrium prevailing in elite maize germplasm. whereas rr-blup performed best for the agronomic traits, it was inferior to lasso or elastic net for the three metabolites. we found the approach of genome partitioning of genetic variance, first applied in human genetics, as useful in guiding the breeder which model to choose, if prior knowledge of the trait architecture is lacking.  our results suggest that in diverse germplasm of elite maize inbred lines with a high level of ld, wgp models differ only slightly in their accuracies, irrespective of the number and effects of qtl found in previous linkage or association mapping studies. however, small gains in prediction accuracies can be achieved if the wgp model is selected according to the genetic architecture of the trait. if the trait architecture is unknown e.g. for novel traits which only recently received attention in breeding, we suggest to inspect the distribution of the genetic variance explained by each chromosome for guiding model selection in wgp. genomic selectionwhole-genome predictiongenetic architecturecomplex traitszea mays 