Random forests (RF) was used to correlate spectral responses to known wet chemistry carotenoid concentrations including total carotenoid content (TCC), all-trans β-carotene (ATBC), violaxanthin (VIO), lutein (LUT), 15-cis beta-carotene (15CBC), 13-cis beta-carotene (13CBC), alpha-carotene (AC), 9-cis beta-carotene (9CBC), and phytoene (PHY) from laboratory analysis of 173 cassava root samples in Columbia. The cross-validated correlations between the actual and estimated carotenoid values using RF ranged from 0.62 in PHY to 0.97 in ATBC. The developed models were used to evaluate the carotenoids of 594 cassava clones with spectral information collected across three locations in a national breeding program , Nigeria. Both populations contained cassava clones characterized as white and yellow. The NRCRI evaluated phenotypes were used to assess the genetic correlations, conduct genome-wide association studies (GWAS), and genomic predictions. Estimates of genetic correlation showed various levels of the relationship among the carotenoids. The associations between TCC and the individual carotenoids were all significant (P < 0.001) with high positive values . The GWAS revealed significant genomic regions on chromosomes 1, 2, 4, 13, 14, and 15 associated with variation in at least one of the carotenoids. One of the identified candidate genes, phytoene synthase (PSY) has been widely reported for variation in TCC in cassava. On average, genomic prediction accuracies from the single-trait genomic best linear unbiased prediction (GBLUP) and RF as well as from a multiple-trait GBLUP model ranged from ∼0.2 in LUT and PHY to 0.52 in TCC. The multiple-trait GBLUP model gave slightly higher accuracies than the single trait GBLUP and RF models. This study is one of the initial attempts in understanding the genetic basis of individual carotenoids and demonstrates the usefulness of NIRS in cassava improvement. Carotenoids are well known for their nutritional and health benefits, particularly in the prevention of a number of human cancers and eye diseases. Most important is the vitamin A activity of the provitamin A carotenoids (PVAC), especially, beta-carotene, alpha-carotene, beta-cryptoxanthin, and gamma-carotene . VitaminCassava is the fourth most important basic food after rice, wheat, and maize worldwide and provides food for many people, particularly in sub-Saharan Africa, where over 600 million people depend on it to meet their energy requirements . GeneralIn cassava phenotyping, the conventional use of color intensity in quantifying carotenoid content in cassava roots is challenging and restricted to a qualitative classification of clones into white, cream and yellow categories . Other aLinear regression models have been widely used in developing NIRS calibrations—correlating spectral response of each sample at individual wavelengths to known chemical concentrations from laboratory analysis , but theThe lack of adequate phenotyping tools especially in dissecting total carotenoid content (TCC) into its individual components is a limiting factor in the genetic studies of PVAC in cassava. Genome-wide association studies (GWAS), which leverage available marker polymorphisms distributed throughout the cassava genome, have been useful in identifying the genomic regions associated mainly with TCC variation in cassava . GWAS coNaturally, carotenoids are present in various configurations and isomerization . In addrUnlike GWAS, genomic selection (GS) is a breeding technology that is used to predict the genetic potential of individuals in a breeding program without necessarily uncovering the underlying genes and quantitative trait loci (QTL) behind the traits of interest . It promLaboratory facilities to assay the full suite of carotenoids are not readily available in Nigeria. In order to assess the full spectrum of carotenoids in Nigerian cassava germplasm, we leveraged a calibration population developed at the International Center for Tropical Agriculture (CIAT) in Cali-Palmira, Colombia, to predict content of carotenoids in a cassava population of the National Root Crops Research Institute (NRCRI) in Umudike, Nigeria. We validated the predictions by assessing their genotype to phenotype relationships in terms of heritability, genomic prediction accuracy, and the identification of significant GWAS hits. We used RF, a nonlinear method for NIRs prediction for TCC, ATBC, VIO, LUT, 15CBC, 13CBC, AC, 9CBC, and PHY in cassava and employed the calibration models in analyzing the spectral information of a training population from NRCRI in Nigeria. We estimated the genetic correlations, identified the underlying genomic regions associated with the variation, and demonstrated the potential of using GS for the rapid improvement of these traits, comparing linear with nonlinear prediction models. While many GS predictions are performed on a single trait basis, the use of multiple-trait models has shown prediction improvements in various cases . TherefoNRCRI has a training population currently used for the implementation of GS in cassava which has been fully described . The gerSpectral data on the TP were collected using a full range (350 – 2500 nm wavelength in 1-nm increments) portable visible and infrared spectrometer (Vis/NIRS) . Root samples from two to three sizeable roots were randomly selected from a plot and the selected roots were peeled with knives, washed, and homogenized into a paste-like mash using a portable power-operated grater. Spectral data were collected from homogenized mashed samples in quartz sampling cups placed against the window of the portable Vis/NIRS device. Each final spectral output was a mean of fifty scans .caret package  developed from the breeding population of CIAT using RF. Usually, the use of NIR instruments for analyses require the training, also known as the calibration of the instruments for the evaluation of traits of interest. Calibration establishes a mathematical relationship between the absorption spectra from the NIR instruments and the factor of interest . Develop package .Prior to building calibration models, standard normal variate and detrending (SNVD) spectra pretreatment  was applied to correct for external interferences on the spectral data, where D indicates the derivative order number , G indicates the gap (the number of data points over which derivation is computed), S1 indicates the number of data points in the first smoothing (1 means no smoothing), and S2 indicates the number of data points in the second smoothing (1 means no smoothing) .The cross-validation of the NIRS calibration models was done by dividing the calibration set into training and testing sets in a ratio of 3:1 which was repeated 10 times. After assessing the performance of the initial calibration models on the testing set, a final model was fitted on the full calibration set for each trait in order to maximize the number of calibration samples. These final RF models were used in predicting the carotenoids of over 4,000 spectra from 594 clones from NRCRI across Umudike, Otobi, and Kano locations.The genotype data used in this study have been previously described . The datThe estimate of genetic correlations set  = random effect of set nested in trial and location: where y = the NIRS predicted phenotypes; µ = population mean; loc = fixed effect of location; clone = random effect of clone: age in R .H2) was calculated using the estimated variance components from the mixed models according to  and unequal shrinking of the BLUPS for clones, BLUPs were deregressed on the basis of PEV . The derrding to  aswhere, Genomic estimated breeding values (GEBVs) for the clones were extracted using the genomic BLUP (GBLUP) model which is defined asy is the vector of dBLUPs for each carotenoid; 1 is a vector of ones; µ is the mean for the dBLUP values; u is the vector of random additive genomic effects (GEBVs) with the corresponding design matrix Z; and K is the additive genomic relationship matrix calculated from SNPs using method 1 of ; U is a random matrix of additive genomic effects vector (GEBVs) with the design matrix Z and E is an independent residual matrix; U and E are assumed to have independent matrix variate normal distributions given as N and N respectively. The multiple-trait GEBVs were derived using the EMMREML package in R  ranged from 0.66 in PHY to 0.97 in ATBC Table 1.-1 to 15.39 µgg-1 with an average of 4.72 µgg-1 (fresh weight basis). Variation in ATBC ranged from 0.53 µgg-1 to 10.18 µgg-1 with a mean of 1.58 µgg-1 , we used a generalized extreme studentized deviate outlier test to identify and remove the extreme points and recalculated the correlations Figure 1-7) and the remaining two markers tagged another peak around 21.85 Mbp on chromosome 13  (-7) and significant for variation in both 13CBC and VIO. On the other hand, five hits were associated with variation in LUT on chromosome 1, tagged by four significant markers of which three were localized between 4.81 Mbp and 4.86 Mbp and the remaining marker around 17.48 Mbp, five markers tagged a peak around 22.54 Mbp to 23.69 Mbp on chromosome 4 and a marker on each of chromosome 13 , chromosome 14 , and chromosome 15 .We identified a total of 42 unique markers significantly associated with variation in TCC and individual carotenoids . Most of the significant markers were associated with variation in more than one trait Table S1Manes.01G124200, a phytoene synthase (PSY) gene known for increasing the accumulation of carotenoid in cassava roots   on Phytova roots  and Maneynthesis , locatedThe result of the genomic predictions for the studied carotenoids showed a slight increase in prediction accuracies using ST-RF compared to the linear ST-GBLUP models Figure 32) have been used in assessing calibration performance  were encouraging and suggested that these traits could be improved concurrently. Because of its health benefits, the positive and significant genetic association between LUT and the PVAC Figure 1Manes.01G124200) has been reported as a single genomic region associated with quantitative variation in TCC using both a panel of partial S1 and S2 inbreds  as part of the Next Generation Cassava Breeding project and the PhD degree of Ugochukwu N Ikeogu.Funding for this work was provided by the Bill and Melinda Gates Foundation and UKAID (Grant 1048542, The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


N = 6110 pairs and 6110 unrelated individuals, respectively) were compared to those for individual measures. Twin heritability increased from 45% on average for individual measures to 76% (se = 0.023) by focusing on stable trait variance. SNP heritability rose from 5% on average (n.s.) to 14% . Heritability was also higher for stable within-rater composites. Polygenic scores for both adult anxiety and depression significantly explained variance in stable emotional problems . The variance explained was more than in most individual measures. Stable emotional problems also showed significant genetic correlation with adult depression and anxiety (average = 52%). These results demonstrate the value of examining stable emotional problems in gene-finding and prediction studies.Twin studies have shown that emotional problems (anxiety and depression) in childhood and adolescence are moderately heritable (~20–50%). In contrast, DNA-based ‘SNP heritability’ estimates are generally <15% and non-significant. One notable feature of emotional problems is that they can be somewhat transient, but the moderate stability seen across time and across raters is predominantly influenced by stable genetic influences. This suggests that by capturing what is in common across time and across raters, we might be more likely to tap into any underlying genetic vulnerability. We therefore hypothesised that a phenotype capturing the pervasive stability of emotional problems would show higher heritability. We fitted single-factor latent trait models using 12 emotional problems measures across ages 7, 12 and 16, rated by parents, teachers and children themselves in the Twins Early Development Study sample. Twin and SNP heritability estimates for stable emotional problems ( The average age of onset for anxiety is strikingly young (between 11 and 14), and 50% of anxiety disorders begin before age 112. Similarly, 25% of depression cases are diagnosed before age 192. Emotional problems often take a chronic course throughout life3, and predict numerous difficulties in adulthood: anxiety4, bipolar disorder5, disruptive disorders and schizophrenia2. Given that emotional problems are common and disabling, it is essential to unravel their origins in childhood to identify risk factors and develop preventative methods and treatments.Anxiety and depression, known collectively as emotional problems, are highly prevalent in childhood and adolescence 10. In contrast, DNA-based ‘SNP heritability’ estimates for childhood emotional problems are generally low and non-significant ). Association studies seeking to identify genetic variants influencing the heritability of childhood anxiety and depression have so far been underpowered and unsuccessful14. Also, polygenic scores  for a range of traits explain <1% of the variance in emotional problems in childhood and adolescence17.Twin studies have shown that emotional problems in childhood and adolescence are moderately heritable (~20–50%)23. Moreover, longitudinally assessed stable anxiety traits are more heritable than anxiety at a single time-point26. In one study, the heritability of adolescents’ stable trait anxiety sensitivity across ages 14, 15 and 17 was 61%, whereas age-specific heritability was zero and non-significant apart from at one age24.A potential way to improve power to identify significant SNP heritability is to tap into the genetic core of emotional problems by assessing problems that are stable across time. Longitudinal twin studies have shown that emotional problems are only moderately stable across childhood and adolescence, but that stability is predominantly influenced by stable genetic influences27. For example, two studies found higher twin heritability for rater-common than rater-specific parts of variance in childhood anxiety (~46% vs ~17%)29. Further, the substantial covariation between anxiety and depression in childhood and adolescence is largely genetically influenced30.Aggregation across reporters and across measures also yields a more heritable core phenotype. Twin research suggests that variation in the part of children’s behaviour that raters see in common captures more of the genetic action than rater-specific variation32 and Item Response Theory33 and obtained more reliable psychopathology measures, from which more accurate heritability estimates can be derived, for example, a latent ‘general childhood psychopathology’ factor had a high SNP heritability of 38%, reflecting pervasiveness across domains and raters, as well as lower error32. However, no research thus far has applied latent modelling approaches to longitudinal multi-rater emotional problems data, and estimated both twin and SNP heritability.Analysing phenotypic measurements can also increase heritability through improved reliability. Latent trait modelling allows us to account for measurement error, and more robustly summarise measurements of the same latent characteristics across time, measures and raters on one scale. In genetic research, reducing measurement error variance by definition reduces environmental variance. This increases the proportion of phenotypic variance explained by genetic variance—heritability (both twin and SNP). Previous studies have used latent trait modelling including factor analysisOur primary hypothesis was that a phenotype capturing the stability of emotional problems across ages, across measures, and across raters would yield higher heritability than individual anxiety and depression measures. To test our hypothesis, we estimated twin and SNP heritabilities of stable emotional problems phenotypes derived from latent modelling. The use of measures assessing both anxiety and depression across multiple raters increases the likelihood of capturing stability in the emotional problems trait, free of rater- and measure-specific views and bias. By ‘stability’, we therefore refer to trait variance in childhood emotional problems that is shared across time, situations (raters), and measures.33. To test this, we compared the twin and SNP heritabilities of factor scores derived from two complementary methods: scale-level Confirmatory Factor Analysis (CFA) and Item Response Theory (IRT). In supplementary analyses, we assessed whether heritability was also higher for a crude composite constructed without latent modelling. We compared the independent contributions of stability across age and across raters by analysing cross-age composites for each rater and cross-rater composites for each age. We also investigated the extent that the heritability of stable emotional problems is inflated by individuals with persistent, severe symptoms.Our secondary hypothesis was that the heritability of a stable factor based on all questionnaire items would yield higher heritability than a scale-level approach. Items have unique properties, so heritability can be assessed more accurately by optimally weighting items and reducing the proportion of variance accounted for by item-specific measurement error34 and major depression (PGC)35. Additionally, we report results from genome-wide genetic correlation analyses. Our research therefore sheds light on both the genetic architecture of early emotional problems, and links with adult emotional problems.In sum, the present study estimates twin and SNP heritability of a stable emotional problems phenotype constructed from 12 measures from three ages and three raters. These are compared to the twin and SNP heritabilities of the individual anxiety and depression measures. We also extend previous work by assessing the prediction of our stable emotional problems phenotype by polygenic scores for adult anxiety (UK Biobank)36 Analyses were conducted on a sub-sample of unrelated individuals with available emotional problem data and genome-wide genotyping, plus their co-twins (6110 pairs). Informed consent was obtained from all subjects.The sample is from the Twins Early Development Study (TEDS), a multivariate, longitudinal study of >10,000 twin pairs representative of England and Wales, recruited 1994–1996N = 3665) and HumanOmniExpressExome-8v1.2 arrays (N = 4649). Typical quality control procedures were followed . Genotypes from the two platforms were separately imputed and then merged.Full details are in the All anxiety and depression variables from ages 7, 12 and 16, and from self-, parent- and teacher ratings, were included. Across these ages, anxiety and depression were measured with 12 scales. See Supplementary Table 37 at ages 7, 12 and 16 was completed by parents12, teachers12 and children themselves16.The emotional problems subscale of the Strengths and Difficulties Questionnaire (SDQ)38 (self- and parent-rated) assessed depressive symptoms at ages 12 and 16. Note that for all MFQ measures except self-rated at age 16, 11 rather than 13 items were collected, due to similarity to SDQ emotional problems items.The Moods and Feelings Questionnaire (MFQ)6 was administered at age 16 to assess parent ratings of anxiety symptoms, behaviour, emotions and cognition.The Anxiety-Related Behaviours Questionnaire (ARBQ)39 was used at age 16 to measure self-rated anxiety sensitivity .The Childhood Anxiety Sensitivity Index (CASI)Total scores were derived for each scale by taking a mean of the items, requiring at least half the items to be present (e.g. at least 9 for the 18-item CASI). All items had three response categories, taking integer values 0–2. Hence each scale has a range of values from 0 to (2 × number of items).N = 6110). Stage three involved four analyses using the stable emotional problems factor scores: (i) estimating twin heritability; (ii) estimating SNP heritability; (iii) predicting stable emotional problems with polygenic scores for adult emotional problems; and (iv) genome-wide association analysis of stable emotional problems. In the fourth stage, we used summary statistics from GWA to explore genetic correlations between stable emotional problems and adult psychopathology. Each stage is explained further below.Figure To test whether heritability results held with a simpler, non-latent modelling approach, and without combining raters, we conducted sensitivity analyses using simple mean composites across all 12 variables, across age for each variable, and across variables at each age . Individuals’ factor scores were extracted for estimation of twin and SNP heritability.The model was estimated in Mplus, taking into account within-twin-pair correlations at the factor level and at the scale level  into genetic and environmental components. In Supplementary Table IRT models describe the relationship between individuals’ responses to specific questionnaire items and their level of the ‘latent variable’ being measured. In the standard two-parameter IRT model, the parameters (item difficulty and discrimination) are equivalent to thresholds and factor loadings in CFA. However, IRT involves categorical observed variables and logistic regressions (rather than continuous and linear). We used all 112 items from the 12 longitudinal cross-rater measures of anxiety and depression in an IRT model. A diagram would look the same as the CFA model in Fig. Factor scores obtained from the above models for 6110 twin pairs were used to estimate twin and SNP heritabilities. For comparison, we estimated twin and SNP heritabilities for the 12 individual measures.40.In the twin design, differences in within-pair correlations for MZ and DZ twins are used to estimate genetic, shared environmental and non-shared environmental effects on traits. Greater MZ than DZ similarity indicates genetic influence. Within-pair similarity that is not due to genetic factors is attributed to shared environmental influences. Non-shared environment accounts for individual-specific factors that influence differences among siblings from the same family, plus measurement error. Twin model fitting analysis using full-information maximum likelihood was carried out with structural equation modelling software OpenMx in RSNP heritability analyses were conducted using one twin of each pair. Analyses were restricted to individuals with available genotyping and factor scores. During estimation of SNP heritability, individuals with missing covariates and excessive genetic relatedness were removed  of >0.025 (third degree relatives)), reducing the final samples to 6002 and 6001, respectively.41. For this method, we calculate genetic similarity for each pair of unrelated individuals across all genotyped SNPs. To decompose trait variance, genetic similarity is used to predict phenotypic similarity. GCTA only detects additive genetic effects tagged by common SNPs  in our DNA arrays; the residual component includes any other source of variance, including non-additive genetic effects, rare variants, environment, gene–environment interaction and error. We used sex and the first 10 principal components as covariates. See SNP heritability was estimated using genomic relatedness matrix restricted maximum likelihood (GREML), implemented in the Genome-wide Complex Trait Analysis (GCTA) program42 to obtain the most predictive polygenic score  for each phenotype. Our analyses included 10,000 permutations to obtain more stringent empirical p-values.Polygenic scores aggregate the effects of thousands of SNPs from genome-wide association studies, including variants that do not achieve genome-wide significance, to provide individual-specific ‘genetic propensity’ estimates. An individual’s polygenic score is the sum of their allele count weighted by the effect size for each SNP, as derived from GWAS. We used the high-resolution approach in PRSice 235 and UK Biobank case–control anxiety 34 genome-wide association studies. We predicted phenotypic variance in the CFA- and IRT-derived stable emotional problems scores with the polygenic scores for depression and anxiety. To compare the prediction of the stable scores with that of individual measures, the 12 individual measures were also regressed on the two polygenic scores. We used sex and the first 10 principal components as covariates.In this study, we generated polygenic scores using summary statistics from PGC Depression  using PLINK v1.90b3.3143.Genome-wide association analysis was performed on the CFA-derived stable emotional problems phenotype  and with a range of phenotypes from publicly available external GWAS using LD HubComputer code used in our analyses is available from the authors upon request.p < 0.0005). As expected given the low stability of anxiety and depression across childhood . For most individual anxiety and depression measures, point estimates are low and the intervals based on standard errors cross zero. None of these individual measures had a significant SNP heritability (p < 0.05), except teacher-rated SDQ at age 7.Results support our hypothesis that extracting stable variance increases the heritability of childhood emotional problems. Figure Scale- and item-level approaches increased SNP heritability equally. Promisingly, IRT factor scores are more normally distributed .Polygenic scores for anxiety and depression significantly explain variance in stable emotional problems Table . More vaNo genome-wide significant SNP associations were identified. See Our stable emotional problems score showed significant genetic correlations with adult depression (case–control (0.48) and symptoms (0.64)), case–control anxiety (0.45), and wellbeing (−0.36). The latter did not remain significant after multiple testing correction, and genetic correlations with nine other phenotypes were non-significant (see Supplementary Table p = 0.002) by capturing common variance across ages and raters. These findings were consistent for a common factor based on scale-level and item-level data for the 12 measures. The findings also held for a simple, non-latent variable approach. This simple approach is easier to calculate but does not account for measurement error. Additionally, polygenic scores for adult anxiety and depression significantly explained variance in stable emotional problems, and the variance explained (0.4% (p = 0.0001) was higher than in most individual measures. Stable emotional problems showed significant genetic correlation with adult depression and anxiety (average = 52%), mirroring previous findings in adults46. Together, the polygenic score and genetic correlation results demonstrate that a significant proportion of common SNPs influencing stable emotional problems in childhood also influence adult anxiety and depression.This study found that a phenotype capturing the pervasive stability of emotional problems across childhood and adolescence showed higher heritability than individual measures. Twin heritability increased from 45% on average for individual measures to 76%  for stable emotional problems. SNP heritability rose from 5% on average to 14% (se = 0.05; 95% CI = 0.04–0.24; 47 can be used to explicitly separate these effects. However, our inclusion of multi-rater data is an advantage of this study. The common variance that has been extracted is free of rater bias or rater-specific views. Our factor scores therefore reflect a core emotional problems trait: longitudinally stable and agreed upon by three raters.This research has several limitations. Our hypothesis was concerned with capturing stability in emotional problems, and not with finding the best-fitting model explaining stability and change across age, and the structure of age-, scale- and rater-specific influences. Consequently, we did not explicitly separate out transient and enduring factors, but acknowledge that the presence of genes driving stable emotional problems does not preclude age effects. We also note the contribution of stability in measurement, not only of longitudinal stability, to increased heritability, although heritability remains higher for stable within-rater composites (Supplementary Figure 18. Further research could also investigate whether extracting a measure of stability increases the heritability of other less developmentally stable and less heritable traits.Our findings suggest that common genetic variants have stable influences on emotional problems throughout childhood and adolescence, which extend into later life. Additional research from large longitudinal studies is needed to determine whether the SNP heritability of emotional problems is more accurately estimated by capturing pervasive stability beyond age 16, or even throughout the life course. This is likely, given the evidence from twin research that genetic influences on emotional problems at age 3 remain influential well into adulthoodFuture genomic studies of emotional problems could benefit from adopting a lifelong approach, using measures of adult case/control status as well as childhood dimensions. Our results indicate that stable emotional problems offer a more useful phenotype than individual measures for: finding variants predisposing to early emotional problems; creating polygenic scores for emotional problems in childhood and adolescence; and for using as a target phenotype for polygenic prediction. Subsequent research should examine how polygenic risk for stable child and adolescent emotional problems is expressed throughout development and into adulthood, as well as the mediators, moderators and multivariate outcomes of this polygenic risk.The young age of onset of emotional problems, and their persistent, wide-ranging negative outcomes, mean that prediction and prevention should be prioritised. The key contribution of genomic research into early emotional problems is likely to be the predictive value of polygenic scores. The predictive accuracy of polygenic scores is increasing with gains in power for genome-wide association, thanks to collaborative consortia and large national projects with homogenous phenotyping such as the UK Biobank. Researchers are working towards the much-needed large-scale genomic study of emotional problems in childhood itself. The present research underlines the utility of extracting a more stable emotional problems phenotype. This phenotype, theoretically grounded in evidence from decades of twin research, is more reliable, heritable and useful for prediction studies.Supplementary Information


Incorporating measurements on correlated traits into genomic prediction models can increase prediction accuracy and selection gain. However, multi-trait genomic prediction models are complex and prone to overfitting which may result in a loss of prediction accuracy relative to single-trait genomic prediction. Cross-validation is considered the gold standard method for selecting and tuning models for genomic prediction in both plant and animal breeding. When used appropriately, cross-validation gives an accurate estimate of the prediction accuracy of a genomic prediction model, and can effectively choose among disparate models based on their expected performance in real data. However, we show that a naive cross-validation strategy applied to the multi-trait prediction problem can be severely biased and lead to sub-optimal choices between single and multi-trait models when secondary traits are used to aid in the prediction of focal traits and these secondary traits are measured on the individuals to be tested. We use simulations to demonstrate the extent of the problem and propose three partial solutions: 1) a parametric solution from selection index theory, 2) a semi-parametric method for correcting the cross-validation estimates of prediction accuracy, and 3) a fully non-parametric method which we call CV2*: validating model predictions against focal trait measurements from genetically related individuals. The current excitement over high-throughput phenotyping suggests that more comprehensive phenotype measurements will be useful for accelerating breeding programs. Using an appropriate cross-validation strategy should more reliably determine if and when combining information across multiple traits is useful. Genomic Selection (GS) aims to increase the speed and accuracy of selection in breeding programs by predicting the genetic worth of candidate individuals or lines earlier in the selection process, or for individuals that cannot be directly phenotyped   relatedness to the training population as individuals in the remaining target population  for model training, then the testing and training datasets are overlapping, even if the testing individuals themselves are excluded from model training  based on the full dataset before cross-validationPre-selection of features  should be done using only the training partition of phenotypes and the selection itself should be repeated each replicate of the cross-validation on each new training dataset. If the feature selection is only done once on the whole dataset before cross-validation begins, this can lead to biased estimates of model accuracy . In this case, standard cross-validation approaches lead to biased results. As we discuss below, the source of bias is not data leakage between the training and testing data per se, but correlated errors with respect to the true genetic merit between the secondary traits in the training data and the focal train in the testing data. Note that this issue only occurs when the multiple traits are measured on the same individuals, and the traits share non-genetic covariance. When traits are measured on different individuals, the standard cross-validation approach is appropriate.If these mistakes are avoided, cross-validation generally works well for comparing among single-trait methods, and in some cases for multi-trait methods. However, our goal in this paper is to highlight a challenge with using cross-validation to choose between single-trait methods and multi-trait methods; specifically multi-trait methods that use information from “secondary” traits In the following sections, we first describe the opportunity offered by multi-trait genomic prediction models in this setting, and the challenge in evaluating them. We then develop a simulation study that highlights the extent of the problem. Next, we propose three partial solutions that lead to fairly consistent model selections between single and multi-trait models under certain situations. Finally, we draw conclusions on when this issue is likely to arise and when it can be safely ignored.cause of the correlations , and iii) the relative expenses of collecting data on each trait.Multi-trait genomic prediction is useful in two general settings: 1) When the overall value of an individual depends on each trait simultaneously (ex. fruit number and fruit size) and these traits are correlated, and 2) When a focal trait is difficult or expensive to measure on every individual, but other correlated traits are more readily available  and phenotypic information on other traits. Even within this context, there are also two distinct prediction settings: 1) Predicting the focal trait value for new individuals that are yet to be phenotyped for any of the traits, and 2) Predicting the focal trait value for individuals that have been partially phenotyped; phenotypic values for the secondary traits are known and we wish to predict the individual’s genetic value for the focal trait. These settings were described by . However, it comes with a cost for the evaluation of prediction accuracy by cross-validation. Since we do not know the true genetic values for the testing individuals, we must either use a model to estimate the genetic values or simply use their phenotypic value as a proxy. Unfortunately, if we use our genetic model to estimate these values, we are breaking the independence between the testing and training data, and therefore have biased estimates of cross-validation accuracy. On the other hand, if we simply use the phenotypic values of the focal trait as our predictand, these may be biased toward or away from the true genetic values depending on the non-genetic correlation between the focal and secondary traits. This leads to either over- or under-estimation of the prediction accuracy of our multi-trait models. In realistic scenarios, this can lead users to select worse models.The key difference between CV1 and CV2-style multi-trait prediction is that in the former, the secondary traits help refine estimates of the genetic values of We used a simulation study to explore conditions when naive cross-validation experiments as described above lead to sub-optimal choices between single and multi-trait genomic prediction methods. Our simulations were designed to mimic the process of using cross-validation to compare single and multi-trait models based on their prediction accuracies. We repeated this simulation across scenarios with different genetic architectures for two traits: a single “focal” trait and a single “secondary” trait. Specifically, we modified the heritability and correlation structure of the two traits. These are the most important parameters for determining the relative efficiencies of single- and multi-trait prediction models . Sample n individuals, To make our simulations realistic, we based them on genomic marker data from 803 lines from a real wheat breeding program  and the subscript o refers to the training partition . We use the hat symbol (^) to denote parameter estimates or predictions.After creating the 803 simulated individuals, we randomly divided them into a training partition and a testing partition. We arranged the rows of We then fit single- and multi-trait linear mixed models to the training data and used these model fits to predict the genetic values for the focal trait (trait 1) in the testing partition.Specifically, for the single-trait method we fit a univariate linear mixed model to the training data To make predictions of the genetic values for the focal trait in the testing partition in the CV1 case without use of To make predictions of the genetic values for the focal trait in the testing partition in the CV2 case, using the phenotypic observations of the secondary trait We measured the accuracy of these three predictions by calculating the correlation between the prediction For the second accuracy measure that uses phenotypic values as predictands, we “corrected“ the correlations by dividing by the true value of As described below, we also simulated phenotypes for an additional set of individuals For each combination of genetic parameters, we declared the “best” prediction method to be the one with the highest average correlation with the true genetic values across the 500 simulations. Then we counted the proportion of the simulations in which this “best” method actually had the highest estimated accuracy when scored against https://github.com/deruncie/multiTrait_crossValidation_scripts. Supplemental material available at FigShare: https://doi.org/10.25387/g3.9762899.Scripts for running all simulations and analyses described here are available at Although we ran simulations for two levels of heritability for the focal trait  that used phenotypic information only on the training partition slightly out-performed the single-trait model in some settings, more-so when the genetic and non-genetic correlations between traits were large and opposing and when the genetic determinacy of the secondary trait was high  when i.e., accuracy is underestimated) when these covariances are in opposite directions. Given this result, we can correct In principle, we can correct for the bias in the non-parametric accuracy estimate  are tested, and each line is grown in several plots in a field First, we considered the situation common in plant breeding where inbred lines (i) and held this additional set of 10% of the lines as Second, we considered the situation more common in animal breeding where clones are not available. In this case, the best option for CV2* would be to select pairs of closely related individuals to include in the training set; we use the first individual of the pair as are used , and so Our study highlights a potential pitfall in using cross-validation to estimate the accuracy of multi-trait genomic prediction methods. When secondary traits are used to aid in the prediction of focal traits and these secondary traits are measured on the individuals to be tested, cross-validation evaluated against phenotypic observations can be severely biased and result in poor model choices. Unfortunately, we rarely know the true genetic value of any individual and therefore can only evaluate our models with phenotypic data . We cani.e., the secondary traits) and our best predictand  – non-genetic correlations between two traits measured on the same individual are expected. However, in some cases this correlation is zero by construction, and standard cross-validation approaches can be valid. For example, in the original description of the CV2 cross-validation method by . For example, in medical genetics the aim is to predict whether or not a person will get a disease or not, not her genetic propensity to get a disease had she been raised in a different environment  data, does not get around the problem identified here. Using estimated genetic values instead of phenotypic values will tend to increase the genetic repeatability of the training and validation values, and therefore increase the overall prediction accuracy of all methods. But these estimated genetic values will still be biased by the non-genetic variation, and the biases across traits will still be correlated by the non-genetic correlations. Therefore the same issue will arise.i.e., any method that is evaluated with the CV2 cross-validation method on multiple traits measured on the same individual  as the prediction model, it loses some of the guarantees of reliability that completely non-parametric cross-validation methods can claim.The parametric solution relies on fitting a new multi-trait mixed model to the predicted values and the predictand, with the accuracy estimated as the genetic correlation scaled by the heritability of the prediction. This solution is always available as long as the individuals in the validation partition have non-zero genomic relatedness and the full dataset is large enough to estimate genetic correlations in both training and validation partitions. However it generally worked poorly in our simulations because i.e., GBLUP and RKHS-style methods with fixed kernels), although the approximation The semi-parametric solution aims to correct the non-parametric correlation estimate for the bias caused by the non-null residual correlation among traits. This correction factor is only needed for CV2-style multi-trait prediction approaches, and is similar to the approach of  for singi.e., not a joint average over all replicates of the line as would be typical for single-trait cross-validation). However, because of the separation of the replicates, each replicate will have higher residual variance, which reduces the accuracy of this method. Clones are less common outside of plant breeding, so more distant relatives need to be used instead. In this case, the estimated prediction accuracies of CV2-style methods will be downwardly biased. In our simulations, despite relatively close relatives for each validation line being available, this approach was not successful.As a third alternative, we proposed the CV2* cross-validation method, a fully non-parametric approach for assessing CV2-style multi-trait prediction accuracy. CV2* uses phenotypic values of the focal trait from relatives of the testing individuals in place of the phenotypic values of that trait from the testing individuals themselves. If the close relatives are raised independently, they will not share non-genetic variation, removing the source of bias in the cross-validation estimate . The CV2In our simulations, the semi-parametric approach was the most reliable, and the fully parametric approach the least reliable. However the fully parametric approach is always possible to implement while our semi-parametric and non-parametric approaches may not be possible depending on the prediction model used and the structure of the experimental design.We expect that multi-trait methods for genomic prediction carry great promise to accelerate both plant and animal breeding. However there is a need to design better methods to evaluate and train the prediction methods to ensure that models can be accurately compared. We have presented and compared three contrasting methods to evaluating multi-trait methods. Each of these methods is preferred to naive cross-validation when secondary traits of the target individuals are used to predict their focal traits. However, the methods can give contrasting answers for different datasets, so careful consideration of which evaluation method to use is critical when choosing among prediction methods.


The capacity of organisms to tune their development in response to environmental cues is pervasive in nature. This phenotypic plasticity is particularly striking in plants, enabled by their modular and continuous development. A good example is the activation of lateral shoot branches in Arabidopsis, which develop from axillary meristems at the base of leaves. The activity and elongation of lateral shoots depends on the integration of many signals both external  and internal . Here, we characterise natural variation in plasticity of shoot branching in response to nitrate supply using two diverse panels of Arabidopsis lines. We find extensive variation in nitrate sensitivity across these lines, suggesting a genetic basis for variation in branching plasticity. High plasticity is associated with extreme branching phenotypes such that lines with the most branches on high nitrate have the fewest under nitrate deficient conditions. Conversely, low plasticity is associated with a constitutively moderate level of branching. Furthermore, variation in plasticity is associated with alternative life histories with the low plasticity lines flowering significantly earlier than high plasticity lines. In Arabidopsis, branching is highly correlated with fruit yield, and thus low plasticity lines produce more fruit than high plasticity lines under nitrate deficient conditions, whereas highly plastic lines produce more fruit under high nitrate conditions. Low and high plasticity, associated with early and late flowering respectively, can therefore be interpreted alternative escape vs mitigate strategies to low N environments. The genetic architecture of these traits appears to be highly complex, with only a small proportion of the estimated genetic variance detected in association mapping. Arabidopsis thaliana lines for their branching response to the availability of a key nutrient, nitrogen. We found that some lines adjusted their degree of branching according to the level of nitrogen available, whereas others did not. These latter low plasticity genotypes constitutively produce an intermediate number of branches and also flower earlier than the more plastic genotypes that responded to the nutrient treatment. One interpretation of these results is that flowering time and branch number are traded off in some way, reflecting at the extremes alternative strategies to cope with low nitrogen availability: an escape strategy of early flowering, and a mitigation strategy involving additional nutrient foraging. Using quantitative genetics methods, we found that the genetic basis for this response is likely to be complex, although not intractable when analysis methods that take into account genetic variability at the relevant candidate loci are used.Many organisms adjust their development depending on environmental conditions. This is particularly striking in plants, with development constantly tuned throughout their lives. A good example is the modulation of shoot branching in response to external environmental cues. We characterised hundreds of genetically distinct  Most organisms experience environmental heterogeneity. For mobile organisms, adverse environments can be avoided by migration and habitat selection. Immobile organisms, such as higher plants, can avoid adverse environments by early reproduction and survival as seeds, or they may mitigate negative environmental impacts by physiological and/or developmental adjustments. It is therefore not surprising that plant development is so remarkably plastic, with a single plant genotype able to give rise to a wide range of phenotypes, depending on the prevailing environmental conditions .There is a substantial body of theory, with some experimental support, concerning the circumstances under which plasticity is adaptive ,3. Key fArabidopsis thaliana (Brassicaceae) as a system to dissect the genetics of shoot branching plasticity in response to nitrate. Arabidopsis is an ideal system for the study of plasticity. Firstly, the availability of many inbred lines from a wide geographic range provides ample genetic material for quantitative genetic studies [Here, we use the annual model species  studies . Secondl studies ,14. Fina studies –23.We focus on the plasticity of shoot branching and its relationship with other morphological and life-history traits. The major determinant of branch number, and particularly its plasticity, is the degree of activity of axillary shoot apical meristems, laid down in the axil of each leaf as they form on the primary shoot apical meristem. Shoot apical meristems can remain dormant, restricting the shoot system to a single axis of growth, or they can activate to produce a branch, reiterating the development of the primary axis, and allowing the possibility of higher order branches.Axillary meristem activity is regulated by diverse inputs, including environmental factors such as nutrient availability, shading, and damage to the primary shoot apical meristem . In addiThus, plasticity in response to nitrate supply depends on the hormone network, with constitutively extreme phenotypes associated with either low cytokinin or low strigolactone. This would suggest a mechanism for variation in branching plasticity in nature in which plants with low plasticity adopt these extreme phenotypes due to compromised strigolactone or cytokinin biology. However, the roles of these hormones in shoot branching have largely been elucidated using null alleles, which simultaneously affect both the activities of the genes in question and any ability to modulate these activities dynamically in response to environmental cues, making it difficult to assess their likely roles in variation in plasticity in nature.To address whether and how plasticity varies in natural genotypes, we have analysed shoot branching responses to nitrate supply in two populations of Arabidopsis: a set of recombinant inbred lines from a mapping population derived from 19 accessions (the MAGIC lines ) and a sTo investigate whether there is natural genetic variation for shoot branching plasticity in response to nitrate supply, we analysed a set of 374 Multi-parent Advanced Generation Inter-Cross (MAGIC) Arabidopsis lines grown under nitrate sufficient (high N—HN) and nitrate deficient (low N—LN) conditions ,28. PlanAIC = -419.32; likelihood ratio test p-value < 10−6); The effect of N treatment on the different traits was variable . Most ofGxE interactions affect the overall variance of a trait in such a way that the mean trait value for a genotype in one environment is a bad predictor of the mean for that trait in another environment. For shoot branching, this can be readily seen as a low correlation between the number of branches for each line under the two nitrate treatments , with anTo assess whether natural accessions of Arabidopsis showed the same trends, 278 accessions were grown as described for the MAGIC lines. The same phenotypic traits were scored, but in addition the total number of secondary shoots was re-counted when the first siliques started to senesce, and the total number of siliques was scored as a proxy for reproductive fitness ,32. As eAIC = -237.95; likelihood ratio test p-value < 10−6); Similarly to the MAGIC lines, there was virtually no plasticity for flowering time, with the variation in this trait being primarily due to genotype . This laTo analyse shoot branching plasticity directly, we calculated the shoot branching response to N supply in each line as the difference between the mean number of secondary shoots formed on HN vs LN. Under HN conditions, for both the MAGIC lines  and the As expected given the correlation between branch number and fruit number, when the natural accessions were grown on HN, there was a positive correlation between branching plasticity and silique number, whereas on LN the correlation was negative . Thus thWe investigated the relationship between shoot branching plasticity and flowering time. Overall, there was a low or no significant linear correlation between shoot branching plasticity and flowering time in the two populations, although there is a clear non-linear trend in the data Figs  and 4D. These results suggest alternative strategies for growth under N limitation. At one extreme there is a rapid exit, escape strategy where plants flower early and branch regardless of N supply. This strategy results in higher fruit numbers on constitutively LN. At the other extreme are the later flowering lines that adjust their branching according to N supply. This strategy results in higher fruit numbers on constitutively HN. The lines we assessed form a continuum between these extremes.The lack of response to N by lines at the low plasticity end of the spectrum could be due to an inability to sense nitrate. To test this hypothesis, we selected three lower plasticity lines (Sha and Hi-0 accessions and MAGIC.11) and three higher plasticity lines (Rsch-4 and Tsu-0 accessions and MAGIC.345) and assessed their primary nitrate response using a panel of seven nitrate responsive genes. Of these, six are up-regulated and one is down-regulated in response to nitrate resupply after a period of starvation .Rapid local transcriptional responses to nitrate are known to be distinct from plant N status responses –42. We hWe next investigated the genetic architecture of shoot branching and its plasticity. Compared with the other traits we measured, shoot branching has generally lower broad-sense heritability, ranging from ~25–35% depending on the nitrate treatment and population analysed . The herERECTA gene carried by one of the founding accessions, Ler-0 [To identify QTL associated with each trait in the MAGIC lines, we focused on the early-flowering lines and used methods suitable for the analysis of multi-parent populations . These ms, Ler-0 ,44,45. WVIP5 and FT on chromosome 1 and, among others, FLC, FY and CO on chromosome 5 [We also detected significant QTL for flowering time , despitemosome 5 –48. TogeFRI . Shoot branching h2GWAS estimates were lower than our broad-sense heritability estimates, with the exception of branches at the senescence stage on HN. The apparent increase in h2GWAS for this trait between the 2-silique and senescence stage could be due to higher variability within genotypes at the 2-silique stage than at the senescence stage. The generally low h2GWAS values suggest that the markers do not capture the genetic component of the variance in this trait (a case of “missing” heritability). One explanation for this is that the markers used here are not in linkage disequilibrium with the causal loci. However, we also calculated h2GWAS using a panel of ~1.7M imputed SNPs, which did not improve the result, suggesting this “missing” heritability might be due to other complexities of the genetic architecture of this trait that cannot be captured by h2GWAS.To investigate whether the failure to identify any QTL for the remaining traits was due to a lack of markers in linkage disequilibrium with causal loci, we estimated the heritability of each trait using a SNP-based relatedness matrix , thus a−5, allowed inclusion of these two flowering QTL on both LN and HN , when compared with other studies (nowadays on the order of one thousand ). Furtheerecta (Ler) accession, which is apparent in the predicted allelic effects on height at this locus in the MAGIC population  do not show a clear large effect allele, but rather a range of lower size effects within 1 standard deviation of the overall mean. This is consistent with a more subtle effect of multiple alleles at the locus affecting shoot branching plasticity, which might not be captured with simpler bi-allelic genotype associations.Another hypothesis to explain the failure of GWAS to find significant associations is that multiple alleles at a single locus might be associated with the trait of interest ,51,60,61pulation . BesidesTheoretical analyses of the circumstances under which developmental plasticity is adaptive identify a number of important factors that influence the balance between the costs and benefits of plastic responses ,5–7,62. Optimal foraging strategy for N has been studied extensively in roots (reviewed in ). There Our results demonstrate that in natural accessions of Arabidopsis, and in a collection of MAGIC lines derived from 19 natural accessions, shoot branching plasticity correlates positively with flowering time and fruit set on high nitrate, but negatively with fruit set on low nitrate. In contrast, flowering time was relatively insensitive to N supply. Our results are comparable to those observed in a series of studies in Arabidopsis by Pigliucci and Schlichting, who found extensive variation in plasticity for shoot branching and other traits in response to nutrient levels ,23,67. IOne framework to interpret these data is to consider the relationship between phenotypic plasticity and the evolution of specialist and generalist lifestyles. Often, generalist species, which occupy a wide range of habitats, are associated with high plasticity , andDA locus ,51,60). DA locus ,80. AlthDA locus ,81).Although our approach has the advantage of including a wide range of the natural genetic diversity in Arabidopsis, providing a broad picture of shoot branching GxE in this species, it comes at the expense of a lower statistical power compared to traditional bi-parental populations. Studies using bi-parental RIL populations are often able to identify many QTL for growth and physiological traits under different environmental conditions including N supply –84. For Consistent with this idea, mapping in the accessions was less effective than in the MAGIC lines, where there are fewer parental haplotypes involved ,80,86. FBRC1 and BRC2 genes, which have been implicated in branching and its plasticity lie within these two regions, respectively [Despite the relatively few loci identified, the mapping results from the MAGIC lines reveal some interesting features. First, there are significant peaks for shoot branching that are not significant or even clearly detectable for shoot branching plasticity. For example, two QTL for branch number on low N, SB~FT.LN.3 and SB~FT.LN.1 , were deectively –89. ConvFT, FLC and FRI [FRI gene, which besides controlling flowering time also affected the number of inflorescence nodes and associated branches of plants carrying recessive non-functional alleles [FLC genotype, which is expected from the epistatic interaction between these two loci [Several significant peaks for flowering time were detected in regions of the genome known to include major flowering time regulators. In some but not all cases, these correlated with significant peaks for shoot branching traits, for example FT.HN.5 and SB.Pl.5 . The flo and FRI ,90. This alleles . This eftwo loci .FRI and FLC are typically studied in the context of vernalization requirement, so it is interesting that they may also contribute to flowering time in these early flowering lines. Although under our conditions flowering was not plastic, this trait is sensitive to seasonal and temperature changes [It is important to note that the lines we studied all flower rapidly, so the effects we see are primarily of relevance to a rapid-cycling lifestyle.  changes ,90,92. IOur findings might also be of agronomic relevance, where breeding for increased nitrogen use efficiency is of importance . There aOverall, our work identifies intriguing associations between branching, its plasticity and flowering time, which may have adaptive significance. Our analysis suggests the hypothesis that a rapid escape strategy combining early flowering with uncoupling of shoot N status from branching suppression allows seed yield to be maintained in N-poor environments. This may provide tools to understand better shoot N status sensing, which is currently enigmatic. However, the genetic complexity of the natural variation we have identified suggests that selected bi-parental mapping populations may be more powerful in determining the underlying genetic basis for these traits and their association than the multi-accession approaches we used here.Arabidopsis thaliana populations for our experiments: 297 natural accessions and 374 MAGIC (Multiparent Advanced Generation Inter-Cross) lines. The MAGIC lines are derived from 19 natural accessions that were randomly inter-crossed for 4 generations, followed by 6 generations of self-fertilisation to generate inbred lines, typically used for QTL mapping [www.arabidopsis.info accessed Dec 2018) and were selected from several collections [http://abrc.osu.edu accessed Dec 2018; own experimental data).We used two  mapping . The natlections . Only aclections ,48; ArabArabidopsis thaliana salts (ATS) solution [3- , which we have previously shown represent N-sufficient and N-deficient conditions for Col-0 [For each line, seeds were sown on wet filter paper and stratified for five days at 5°C in the dark, and then transferred to 5.5 cm diameter pots filled with low nitrate substrates consisting of 50% sand  and 50% Terragreen (Oil-Dri). The substrates were wetted with solution , containor Col-0 . After tFor our main natural variation experiments we measured flowering time, total branches and height for each plant. Flowering time was measured as the number of days from germination to the day at which the first flower buds were visible at the rosette centre. Total branches were counted as the number of secondary shoots (from the axils of rosette + cauline leaves) that were more than ~1cm in length. Height was measured as the length of main inflorescence stem.Total branches and height measurements were made when plants had formed two full siliques (2-silique stage). For 278 of the 297 accessions, we also obtained measurements at a later stage when plants had at least two senescing siliques (senescence stage). Traits are reported as averages for each line, with each line represented by 4–8 replicate plants (median n = 8).Branching plasticity was calculated for each line as the difference between the mean number of secondary shoots formed on HN vs LN. This measure was chosen taking into consideration the biology of shoot branching. It may seem attractive to normalise the number of branches to the number of nodes, and thus the total number of possible branches. Similarly, branching plasticity could be expressed as a proportional change in the number of branches. These measures were rejected based on the fact that branching occurs in a strict basipetal sequence and nitrate supply modulates the stopping point of that sequence ,104. SinTo allow plasticity comparisons across traits, we’ve also calculated a relative metric, the “relative distance plasticity index” adapted from . For eac3 or KCl for 2 hours. After this treatment, they were quickly dried and flash frozen in liquid nitrogen, and stored at -80°C until RNA extraction. Total RNA was extracted using the RNeasy Plant mini kit including DNase I treatment (Qiagen), following the manufacturer’s instructions. RNA was quantified using a NanoDrop 1000 and 1 μg was used to produce cDNA using Superscript II (Invitrogen), following the manufacturer’s instructions. qPCR reactions were prepared using LightCycler 480 SYBR Green I Master (Roche), with 5ng of cDNA in 20μl reactions, following the manufacturer’s instructions. Reactions were performed in a LightCycler 480 II (Roche) machine and Cp values were determined based on the “second-derivative maximum” method implemented in the manufacturer’s software.The expression of primary nitrate responsive genes was analyzed by RT-qPCR in four natural accessions , two MAGIC lines (MAGIC.11 and MAGIC.345) and the standard laboratory line Col-0. For each sample, 10mg of seeds were surface sterilized and stratified for 4 days at 5°C and then transferred to 25ml of liquid ATS in which nitrate was replaced by 0.5mM ammonium succinate. The seeds were left to germinate and grow on a shaker (100-120rpm) in a controlled environment room . After 10 days, the seedlings were treated with 5mM KNOp values of the technical replicates were averaged for each biological replicate and used in subsequent calculations. For each sample, the transcript levels of the primary nitrate response genes were normalised relative to the mean Cp value of two reference genes: APX3 (AT4G35000) and UBC9 (AT4G27960). Finally, we estimated the relative expression of those genes in the treatment (KNO3) relative to the control (KCl) conditions. These estimates were made using the ΔΔCp method, assuming equal primer efficiency [There were two biological replicates for each line, with three technical replicates each. The Cficiency . All pri3-. From thereon, the grafting experiment was performed as described in [Two pairs of lines were used for grafting: two MAGIC lines (MAGIC.11 and MAGIC.345) and two accessions (Shahdara and Rsch-4). For each of these pairs we made four pairwise grafting combinations: two autografts and two allografts (one in each direction). Plants were germinated on ATS medium containing 0.8% bacto-agar and either 9mM or 1.8mM of NOribed in . Flowerilmer function of the lme4 package [Anova function in the car R package [The lines used in these experiments represented the two ideotypes of focus: low plasticity and early flowering (MAGIC.11 and Shahdara) and high plasticity and later flowering (MAGIC.345 and Rsch-4). To assess whether each ideotype’s characteristic phenotype was mainly shoot or root driven, we tested the hypothesis of no effect of root and shoot ideotypes on each trait using a mixed model ANOVA. This model included fixed terms for nitrate treatment, shoot ideotype, root ideotype and experiment (this was included as a fixed rather than random term since there were only two levels for this factor). We further included interaction terms between nitrate and each of the root and shoot ideotypes to account for their contribution to the trait’s plasticity (ideotype-by-nitrate interaction). Finally, we included a random term for each graft’s ID  and a random slopes term for nitrate (to account for the specific plasticity of each graft combination). The mixed model was fitted with the statistical program R , using t package . The hyp package .We estimated broad-sense heritabilities based on replicate measurements of each genetic line. Strictly speaking, this is a measure of “clonal repeatability”, but in a randomized experiment like ours it should give a good estimate of the degree of genetic determination of the trait, i.e. its broad-sense heritability (p. 123 in ). BrieflYij is the trait value for the i-th individual from the j-th genotype (MAGIC line or accession ID); β0 is the intercept of the model, which in our specification is the mean on LN; β1 is the response when on HN; NITRATE is a dummy variable indicating whether the individual was grown on HN; u0j is the random term for varying intercepts of the j-th genotype (genotype-specific average on LN); u1j is the random term for varying slopes of the j-th genotype ; ϵij is the residual term with To partition the variance into genotype, environment and genotype-by-environment (interaction) components, we fitted a more complex random slopes linear mixed model . We incl2u0 is the variance of the trait on low nitrate; σ2u1 is the variance of the trait responses on HN (the GxE component); σu01 is the covariance between the two. The covariance parameter was used to calculate the correlation of the trait values between LN and HN. The estimates from the model are presented in ijY = β0 + β1NITRATE + uj0 + ijϵ. We assessed differences between the models using a likelihood ratio test  and difference in the Akaike Information Criterion .Where: σR [lmer function of the lme4 package [In all cases, linear mixed models were fitted with R , using t package  and variR/qtl2 package [https://github.com/tavareshugo/atMAGIC). In summary, for each of the 1254 available markers, the probability of ancestry of an individual’s genotype at that marker was inferred using the function qtl2::calc_genoprob, assuming a 1% genotyping probability error. The qtl2::scan1 function was then used to fit the QTL model to each marker for each trait analysed. For shoot branching, we also fitted a model that included flowering time as a covariate (to account for the correlation between these traits). We used a 5% genome-wide significance threshold obtained by permutation using the qtl2::scan1perm function. The founder accession’s effect at each candidate QTL was estimated using the qtl2::scan1blup function following [Association (QTL) mapping in the MAGIC lines was performed using the  package  and a cuollowing . These aR/qtl2 to a single genotype value corresponding to the founder allele with maximum probability at each marker for each individual . LOD scores were obtained for two model contrasts: the full model compared to a genetic model with no genotype-by-nitrate interaction (GxE) term; the genetic model compared to a null model (no genotype term). We obtained a 5% genome-wide threshold by permutation of the genotype data. This analysis was done using custom R scripts (see data availability section).For shoot branching, we also fitted a more complex “multi-trait” model, to assess the independence of GxE QTL from common effect QTL ,49,50. TFinally, we estimated the variance explained by each marker by comparing the genetic variance using the mixed models just described with a model excluding the QTL marker as a predictor variable, similarly to .http://github.com/Gregor-Mendel-Institute/atpolydb, last accessed Jul 2019), which were converted to plink format  using a custom python script. SNPs with minor allele frequency below 5% were discarded, leaving 192863 biallelic SNPs. A 5% genome-wide significance threshold was obtained by Bonferroni correction. The “--reml” function in GCTA 1.26.0 was used to estimate the proportion of phenotypic variance explained by the SNPs used in the GWAS, referred to as GWAS heritability, h2GWAS.Association (GWAS) mapping in the accessions was performed using the “--mlma-loco” function in GCTA 1.26.0 . This peh2GWAS using a panel of 1 763 004 imputed SNP genotypes provided by Ümit Seren in the group of Magnus Nordborg. This is the set of imputed SNP genotypes used in the web application “GWA-Portal” .We also estimated limix 0.7.12 Python package [For shoot branching, we also fitted a “multi-trait” model using the  package . Similar package ). Finall package . In all tidyverse [https://github.com/tavareshugo/publication_deJong2019_NplasticityData analysis and visualisation were carried out using the statistical software R version 3.4.1 . The metidyverse  was usedS1 Fig−6). The dashed line is the identity line (x = y).Data are shown separately for plants growing on high  or low  nitrate. Data are means from 278 accessions from n = 4–8 replicates per line in each nitrate treatment (median n = 8). Pearson’s correlation coefficient (r) is shown in each panel with the 95% confidence interval shown in brackets Click here for additional data file.S2 Fig−6).Correlation between seed weight and total branches (A) or total siliques (B) at senescence in a set of 4 MAGIC lines (cross symbols) and 7 accessions (dot symbols). Data are means of n = 12–15 replicates per line (error bars show 2x standard error of the means). (C) Correlation between number of branches and number of siliques at senescence stage in 278 accessions. Data are means of n = 4–8 replicates per line (median n = 8). Pearson’s correlation coefficient (r) is shown in each panel with the 95% confidence interval shown in brackets Click here for additional data file.S3 FigThe top panels show the heritability for all measured lines in each population: 374 MAGIC lines and 297 accessions scored at 2-silique stage; 278 accessions scored at senescence (sen). The bottom panels show estimates for the subset of early-flowering lines (mean flowering time <25 days on LN): 258 MAGIC lines and 266 accessions scored at 2-silique stage; 260 accessions at senescence. Error bars are the 95% confidence interval of the estimate obtained by a bootstrap procedure see . For eac(TIFF)Click here for additional data file.S4 FigErecta gene. The QTL on chromosome 5 co-localises with QTL for flowering time (−16).The plot shows the LOD score of the association test carried out for each marker along the genome see . The numing time , and is (TIFF)Click here for additional data file.S5 FigManhattan plots showing the association mapping results of a multi-trait model simultaneously modelling shoot branching variation on high (HN) and low (LN) nitrate. Two model contrasts were performed to test for a genetic effect common to both nitrate treatments  and an interaction GxE effect . The horizontal dashed lines show the 5% genome-wide significance threshold, obtained with bonferroni correction.(TIFF)Click here for additional data file.S6 FigMulti-SNP QTL mapping for (A) flowering time and (B) shoot branching in accessions. Manhattan plots showing the results of a set-based test that used the results from the test shown in (TIFF)Click here for additional data file.S7 Fig2GWAS [2GWAS is estimated from a linear mixed model that regresses the phenotype to a genetic relatedness matrix, arriving at an estimate of the variance explained by it. h2GWAS is thus the proportion of total phenotypic variance that can be attributed to variance in the relatedness between individuals. h2GWAS was estimated based on two sets of SNPs: 192 853 SNPs from [Heritability estimates based on the genetic relatedness between samples, inferred from all SNPs used in the GWAS for 240 accessions, h2GWAS . h2GWAS NPs from  (upper p(TIFF)Click here for additional data file.S1 TableThe location of each QTL peak is given in relation to the Arabidopsis TAIR10 reference genome. For each peak marker we give the LOD score and an estimate of the explained variance at that QTL . Where (PDF)Click here for additional data file.S2 TableLN mean” is the intercept of the model (β0); “Plasticity mean” is the mean response to HN across lines (β1). Random terms: “LN” is the variance of the trait on low nitrate (σ2u0); “GxE” is the variance of the trait responses to HN ; “ρ” is the correlation between the trait on LN and its response to HN; “ρ” is the correlation of the trait on LN and HN. Both correlations were calculated from the variance-covariance estimates. The full model was compared to a reduced model that excluded the GxE component: ijY = β0 + β1N + uj0 + ijϵ. We assessed differences between the models using a likelihood ratio test  and changes in the Akaike Information Criterion, “ΔAIC” . To allow easier comparison across traits, their values were centered on the mean and scaled to the standard deviation.The model specification is detailed in the methods. For each trait, model estimates are shown for the fixed and random parts of the model. Fixed terms: “(PDF)Click here for additional data file.S3 Table(PDF)Click here for additional data file.