Low Additive Genetic Variation in a Trait Under Selection in Domesticated Rice

Quantitative traits are important targets of both natural and artificial selection. The genetic architecture of these traits and its change during the adaptive process is thus of fundamental interest. The fate of the additive effects of variants underlying a trait receives particular attention because they constitute the genetic variation component that is transferred from parents to offspring and thus governs the response to selection. While estimation of this component of phenotypic variation is challenging, the increasing availability of dense molecular markers puts it within reach. Inbred plant species offer an additional advantage because phenotypes of genetically identical individuals can be measured in replicate. This makes it possible to estimate marker effects separately from the contribution of the genetic background not captured by genotyped loci. We focused on root growth in domesticated rice, Oryza sativa, under normal and aluminum (Al) stress conditions, a trait under recent selection because it correlates with survival under drought. A dense single nucleotide polymorphism (SNP) map is available for all accessions studied. Taking advantage of this map and a set of Bayesian models, we assessed additive marker effects. While total genetic variation accounted for a large proportion of phenotypic variance, marker effects contributed little information, particularly in the Al-tolerant tropical japonica population of rice. We were unable to identify any loci associated with root growth in this population. Models estimating the aggregate effects of all measured genotypes likewise produced low estimates of marker heritability and were unable to predict total genetic values accurately. Our results support the long-standing conjecture that additive genetic variation is depleted in traits under selection. We further provide evidence that this depletion is due to the prevalence of low-frequency alleles that underlie the trait.

ABSTRACT Quantitative traits are important targets of both natural and artificial selection. The genetic architecture of these traits and its change during the adaptive process is thus of fundamental interest. The fate of the additive effects of variants underlying a trait receives particular attention because they constitute the genetic variation component that is transferred from parents to offspring and thus governs the response to selection. While estimation of this component of phenotypic variation is challenging, the increasing availability of dense molecular markers puts it within reach. Inbred plant species offer an additional advantage because phenotypes of genetically identical individuals can be measured in replicate. This makes it possible to estimate marker effects separately from the contribution of the genetic background not captured by genotyped loci. We focused on root growth in domesticated rice, Oryza sativa, under normal and aluminum (Al) stress conditions, a trait under recent selection because it correlates with survival under drought. A dense single nucleotide polymorphism (SNP) map is available for all accessions studied. Taking advantage of this map and a set of Bayesian models, we assessed additive marker effects. While total genetic variation accounted for a large proportion of phenotypic variance, marker effects contributed little information, particularly in the Al-tolerant tropical japonica population of rice. We were unable to identify any loci associated with root growth in this population. Models estimating the aggregate effects of all measured genotypes likewise produced low estimates of marker heritability and were unable to predict total genetic values accurately. Our results support the long-standing conjecture that additive genetic variation is depleted in traits under selection. We further provide evidence that this depletion is due to the prevalence of lowfrequency alleles that underlie the trait.

heritability GWAS abiotic stress stress tolerance
Unraveling the mechanisms of phenotypic evolution under selection is a fundamental task in biology. Among-individual variation in most traits of interest results from a collective action of multiple genes in conjunction with environmental effects (Lynch and Walsh 1998). To a first approximation, only the component of the genetic variation that is due to the additive contribution of DNA sequence polymorphisms is passed on to the following generations. The fate of this additive genetic variance is thus of particular interest since it is an important factor determining both the speed and direction of adaptation. Fisher (1930) predicted that selection should deplete this component of phenotypic variance as beneficial alleles become fixed. This intuition has been extensively explored in theoretical studies. Most analyses focus on two scenarios: stabilizing selection, where the population is close to optimum fitness and extreme phenotypes are deleterious, and directional selection, where the optimal phenotype is shifted and the population is allowed to evolve toward the new state. While the studies involve different sets of assumptions and employ either analytical or simulation approaches, the general conclusion is that the predictions are very sensitive to model parameters. In some cases (Lande 1975;Turelli 1984) mutation-selection balance is expected to maintain a substantial amount of additive genetic variation. In other scenarios, the variance is either reduced or preserved due to segregation of a few large-effect alleles (Turelli 1984;Barton 1986;Johnson and Barton 2005;de Valdar and Barton 2014;Jain and Stephan 2017;Stetter et al. 2018). The assumed shape of the mutation effect size distribution appears to play a major role in determining whether substantial additive genetic variation is maintained in the face of selection.
Empirical observations can help us distinguish which theoretical models, and thus genetic mechanisms, are most consistent with data. The difficulty is that we cannot directly observe the components of phenotypic variance, but must rely on statistical models to estimate them. Assuming reliable estimates, the common approach is to compare additive genetic variances in fitness-related (often called life-history) traits to a control group (e.g., seemingly unimportant morphological characteristics). Since measurement scales differ among traits, some scale factor must be used to make comparisons meaningful. A popular scaling factor is the total variance. Its use is well-motivated by theory as the ratio of additive genetic to total variance yields estimates of narrow-sense heritability, a parameter important in predicting outcomes of single-generation selection on phenotypes (Lynch and Walsh 1998). Most empirical studies (Mousseau and Roff 1987;Gustafsson 1986;Kruuk et al. 2000;Johnson and Barton 2005) find that narrow-sense heritability of traits correlated with fitness is indeed low. However, two traits can have equal levels of additive genetic variance but different heritabilities due to differing amounts of environmental variance (Price and Schluter 1991). Fisher's prediction applies to the total amounts of additive variation, so arguably estimates of heritability are misleading. There is in fact some evidence (Price and Schluter 1991;Houle 1992;Johnson and Barton 2005) that scaling genetic variances by means reveals that life-history traits, if anything, possess a larger store of additive variation than other traits.
While proper scaling of variance estimates for among-trait comparisons is important, precise estimates of variance magnitudes are required for reliable inference. Observed (natural population pedigrees) or experimental (diallel) crosses have been necessary in the past to partition phenotypic variance into the various genetic and environmental components (Sprague and Tatum 1942;Lynch and Walsh 1998;Greenberg et al. 2010). Wide availability of dense DNA sequence markers makes it possible to dispense with crossing and instead fit models that directly incorporate nucleotide polymorphism effects (Meuwissen et al. 2001;Van Raden 2008). Plant species, as well as some model animals, offer the additional possibility of assaying genetically identical individuals in replicate. This makes it possible to separate the overall genetic contribution into that explained by markers and the rest (we refer to this quasi-residual as the "background" effect). Interpretation of this partitioning is not always clear, however, and depends on the model used for the molecular markers.
Cultivated rice Oryza sativa is a selfing species with extensive genomic resources, making it a useful model system to test quantitative genetic theory. It is a major food crop that is mostly grown under irrigated or lowland rain-fed conditions, where standing water and anaerobic soil conditions are at least intermittently present (Khush 1997;Bernier et al. 2008). However, a minority of area under cultivation is occupied by rice grown under upland (aerobic) rain-fed conditions, characterized by absence of standing water and frequent droughts Khush 1997;Bernier et al. 2008). Since initial rice domestication is associated with paddy rice (Oka 1988;Fuller et al. 2010), this arid growth regime has likely imposed a recent and strong selection for survival in the face of water scarcity (De Datta 1975). An additional difficulty for upland rice farmers is the prevalence of acidic soils (von Uexküll and Mutert 1995) that lead to metal toxicity (e.g., aluminum) and nutrient deficiency (e.g., phosphorus). These problems, in turn, exacerbate drought stress in plants. Thus, rice accessions adapted to upland conditions have been under additional selection pressure for tolerance to aluminum toxicity and nutrient deficiency.
Root architecture is an important determinant of drought tolerance in rice. Accessions successfully grown under rain-fed (rather than irrigated) conditions tend to have relatively long and thick roots Champoux et al. 1995;Bernier et al. 2008). There is evidence of a genetic correlation between root length, survival, and grain yield under arid conditions (Champoux et al. 1995;Bernier et al. 2009;Cairns et al. 2009;Lyu et al. 2014). Aluminum tolerance is also assessed by comparing root length of treated and control plants (von Uexküll and Mutert 1995;Famoso et al. 2010). While rice is relatively aluminum tolerant overall, japonica, particularly tropical japonica, cultivars are particularly resistant (Famoso et al. 2010(Famoso et al. , 2011. Accessions from this subpopulation are often grown on acidic, upland, rain-fed soils in Africa, Latin America, and South-East Asia (Khush 1997;Lyu et al. 2014). Root length, including under aluminum treatment, is thus a good candidate for a character undergoing strong directional selection. Examining its genetic architecture thus may shed light on the evolution of quantitative traits. In addition, any information gained can be used to explore the adaptive potential of rice root length in water-limited environments.
Using a world-wide panel of O. sativa accessions and a dense set of single-nucleotide polymorphisms (SNP), Famoso et al. (2011) performed a genome-wide association study (GWAS) to look for associations between genotyped loci and relative root growth under aluminum treatment. While they found associations in the aus and indica subpopulations, they did not detect any signal in tropical japonica despite significant overall variation in root growth. To determine if this result reflects a depletion of additive genetic variance under selection, we set out to build on this work. We first recapitulated the previous analyses using a new, ten-fold denser, genotyping platform (McCouch et al. 2016). We also introduced a more robust approach to modeling replicated data. Despite these improvements, we still failed to find any GWA signal in tropical japonica. Adding more accessions, increasing the duration of aluminum treatment, and measurement of additional traits did not produce a higher power to detect loci associated with aluminum tolerance. Furthermore, while total genetic variation accounted for a large portion of overall phenotypic variance, additive genetic variation captured by SNP genotypes was consistently low. We provide evidence that this is due to a low frequency of alleles conferring susceptibility or tolerance. These results are consistent with a set of theoretical expectations and allow us to deduce a possible genetic architecture of root growth in tropical japonica varieties adapted to upland cultivation. Our inferences provide testable predictions for future experiments and point to breeding strategies that are more likely to improve resistance to drought and cultivation on acid soils in this population.

Root measurements and accession sets
We expanded the tropical japonica accession set (n = 92) analyzed by Famoso et al. (2011) by adding 77 new accessions. These were treated and measured using the same protocol as before (Famoso et al. 2010(Famoso et al. , 2011. We made an average of six replicate measurements per accession under each treatment. We also chose a new set of 119 tropical japonica accessions (including some overlap with the original 169). The Al treatment in the latter experiment was conducted as before, but the duration was extended from three to six days, with an average of eight replicates per accession under each treatment. All plants were raised in a growth chamber (see Famoso et al. (2010) Supplemental File 1 for details). The root systems of seedlings were imaged and total and longest root length measured using RootReader 2D, a digital imaging and quantification system (Clark et al. 2013).

Genotype data
The accessions we chose have been previously genotyped using the HDRA array platform (McCouch et al. 2016). We filtered the original 700,000 SNPs for each accession set, including only variants with minor allele count greater than two and with fewer than 30% of the genotype calls missing. The minor allele count of two represents a genotype of a single diploid individual, but because our accessions are highly inbred (domesticated rice is a self-pollinating species) our threshold effectively eliminates singleton SNPs. We were left with 432,607 variants in the aus population, 434,753 SNPs in the tropical japonica set used by Famoso et al. (2011), 467,536 for the augmented tropical japonica set, and 461,298 for the new set of 119 accessions. Given that the rice genome is about 380 megabases in length (Kawahara et al. 2013), the average between-SNP distance in our data are roughly 0.8 kilobases. This is well within the linkage disequilibrium decay window for all rice populations (McCouch et al. 2016). Since average gene size is 3 kb http://rice.plantbiology.msu.edu/, each gene contains about three SNPs. Thus moderate to high-frequency causal variants should be tagged by at least several SNPs in our data set with high probability.

Modeling of replicated experiments
All experiments involved measuring root traits on replicates of genetically identical individuals (homozygous accessions). The plants were grown either in control or aluminum-containing media. This experimental structure informed our modeling approach. We sought to concurrently estimate contributions of genotyped SNPs, total genotypic effects, and random environmental fluctuations to overall phenotypic variance. In addition, in the analyses of the original Famoso et al. (2011) experiments and our augmentation of their tropical japonica data set, we needed to take into account the year each plant was measured, as there were a priori obvious year effects (Supplemental Figure 2).
We fitted multi-trait Bayesian hierarchical models (Gelman et al. 2004;Greenberg et al. 2010Greenberg et al. , 2011 to estimate parameters of interest. We treated each data set (the original Famoso et al. (2011) data in aus and tropical japonica populations, the augmented tropical japonica, and the new single-year measurements) separately. The single-year experiment measured two root traits (total and longest root length). We included them together in a multi-trait model, but analyzed the Al treated and control sets separately. The other experiments measured only total root length, but we modeled the treated and control measurements as separate traits. However, because the treatments were applied to different individual plants, the error covariance between the two pseudo-traits was set to exactly zero. Genetic covariances were estimated from data in all models, since the same accessions were subjected to both treatment and control conditions. The portions of our hierarchical models that estimate location parameters (accession means, aggregate marker effects, and year covariates) take the following form: Location parameters of the form y iÁ are row-vectors of the corresponding parameter matrices that have data points as rows and traits as columns (for example, as mentioned above, y iÁ has two values for the Famoso et al. (2011) data: the treated and untreated total root length). The year effect is included via the x is the row of the year contrast matrix, columns corresponding to years with 2016 set as the base; B year is the matrix of year regression coefficients). The correction is performed at the individual measurement level for the Famoso et al. (2011) data set. There is some risk of confounding between year and accession effects in the augmented data (see Supplemental File 1). Therefore, year effects were modeled at the level of accessions for that set. The third experiment was performed in one batch and therefore no year effect correction was required. The year effect coefficients (as well as the overall intercept m) were modeled with high-variance (Is 2 0 ; s 2 0 ¼ 10 6 ) Gaussian priors. The errors were modeled using a multivariate Student-t distribution with three degrees of freedom to dampen the effects of outliers (Greenberg et al. 2010(Greenberg et al. , 2011 for the last experiment. Our set-up did not allow for Student-t models for block-diagonal error matrices. Therefore, the first two data sets were analyzed with Gaussian models for errors. The marker effects enter through the eigenvectors U of the relationship matrix, weighted by square roots of their eigenvalues. We estimated the relationship matrix from all SNPs in each data set using the Van Raden method (Van Raden 2008). We are using all the eigenvectors that correspond to non-zero eigenvalues. If this regression is performed using a Gaussian prior on the coefficients g jÁ , it is the Bayesian analog of the (matrix-variate) mixed effect model (Kang et al. 2010;Hoffman 2013). In this case, Σ a is the additive marker covariance matrix. However, the first few principal components of the relationship matrix often correspond to largescale population structure and may have relatively large regression coefficients (if the phenotypes are also stratified by subpopulation). A Gaussian prior may over-shrink these coefficients. Therefore, we used multivariate Student-t prior distribution on G. This is a multivariate version of the BayesA model (Meuwissen et al. 2001). The results presented in the main text were obtained by setting the degrees of freedom to three (the minimal number that still results in defined moments of the Student-t). We repeated all analyses with the degrees of freedom set to 1000 (making the prior close to Gaussian) with the same results. This is likely because we work within populations, in the absence of deep population structure.
The covariance matrices Σ x were modeled using Wishart distributions with weakly-informative Wishart priors with two degrees of freedom for Σ e and Σ s and four degrees of freedom for Σ a (Gelman et al. 2004;Greenberg et al. 2011). This implies a uniform prior on narrow-sense (marker) heritability.
Detailed model descriptions are listed in the documentation provided in the Supplemental File 1. This supplemental file also includes the raw phenotypic data, scripts we used to manage the raw data, Markov chain convergence diagnostics, and post-processing and plotting of the results. The analyses were implemented in C++. The source code is included in the Supplemental File 1. The underlying library is also available from https://github.com/tonymugen/ MuGen.

Heritability estimates
The hierarchical model allows us to estimate heritabilities. Narrowsense heritability (Lynch and Walsh 1998) is the fraction of total phenotypic variance that is explained by additive genetic effects. If we were to genotype all causal variants and accurately estimate their effects, the fraction of total variance explained by the sum of these polymorphisms' contributions would be the narrow sense heritability. In practice, we only assay tag SNPs that may be in linkage disequilibrium with a fraction of causal variants. In addition, since linkage is likely imperfect, most linked genotyped SNPs do not capture the whole contribution of causal loci. We calculate marker heritability h 2 m;p of the p-th trait from the total marker effects estimated in our models: where N acc is the number of accessions and the rest of the notation is as described above. We used the variance of GEBV estimates ðUGÞ t Áp ðUGÞ Áp =ðN acc 2 1Þ rather than values from the marker covariance matrix Σ a p;p because the latter reflects the additive covariance only when the prior on the principal component regression (see above) is Gaussian. We use a Student-t prior, thus Σ a p;p may underestimate the additive genetic variance captured by SNPs.
The Σ s p;p values estimate the portion of total genetic variance not explained by genotyped markers. They include the non-additive effects and additive effects not tagged by SNPs. We call these estimates "background effects" (e.g., Figure 1). The sum of the GEBV and background variance, divided by total phenotypic variance, is broad-sense heritability (Lynch and Walsh 1998).
Our models were fitted using a Markov chain Monte Carlo approach, resulting in posterior samples of parameters. We used calculated heritabilities for each sample, yielding numerical approximations of posterior distributions of these statistics. Summaries of these distributions are plotted in the relevant figures.

GWA and genome prediction
Because full MCMC Bayesian model fitting is computationally demanding, we did not attempt to fit individual SNP effects within that framework. In addition, treated to control root length ratio (a measure of Al tolerance, Famoso et al. 2010Famoso et al. , 2011 was not modeled directly and could not be used as a response in our hierarchical models. We thus took a two-step approach, using the posterior modes of accession means as responses with mixed-model correction for population structure (Kang et al. 2010) using our own R package (Greenberg 2019), available from https://github.com/tonymugen/ GWAlikeMeth. This is a maximum-likelihood method, first fitting the mixed model where y is the response vector of posterior accession modes, 1 N is a column vector of ones of order, m is the overall mean, u is the column vector of random genetic values, s 2 u is the among-u variance, K ¼ XX t =N acc is the among-genotype covariance matrix estimated from the N acc · N snp genotype matrix X, e is the column vector of random errors, and s 2 e is the error variance. Residuals of this model are then used in individual SNP regressions. Each trait (including the logarithm of the treated to control root length ratio) is treated independently, although computations that are shared among traits are performed only once. We report 2log 10 p values as SNP effect scores.
To verify results obtained using the hierarchical Bayesian model outlined in the previous section, we also took the two-step approach to perform genome predictions. We again used modes of accession mean distributions, as described above for GWA. We started with a linear kernel K calculated as above. This is the genome best linear unbiased predictor (GBLUP) model of additive effects. To test whether modeling among-locus epistatic interactions improves prediction, we repeated the analyses with a Gaussian kernel for K, setting its elements to Kðx j ; x j Þ ¼ expð2hd 2 ij Þ. Here, d ij is the Euclidian distance between genotypes i and j, h is the bandwidth parameter that controls the speed of decay of K elements with genetic distance. We used the R package Bayesian generalized linear regression (BGLR, Pérez and de los Campos 2014) to fit the genome prediction models. The package implements a Gibbs sampler. We ran 55,000 iterations including 5,000 burn-in steps and fivefold thinning.
We assessed prediction accuracy using 50 training-validation random partitions. Pearson's correlations (r) between the predicted and the observed values were employed as accuracy measurements. To calculate summary statistics, we transformed raw correlation values using Fisher's z transformation (z ¼ 1 2 ½lnð1 þ rÞ 2 lnð1 2 rÞ). The mean and interval values presented in Table 1 were back-transformed to the original scale.

Introgression mapping
We used RFMix (Maples et al. 2013) to identify introgressions from aus and indica in the six admixed tropical japonica accessions. We used 10 training accessions from each of the three populations of interest (see the Supplemental File 2 for the full list). We ran Beagle v4.1 (Browning and Browning 2016) with default parameters to phase genotypes, after filtering loci with minor allele frequencies below 0.08 and more than 30% missing genotypes. Our final data set included 304,488 SNPs. We then ran RFMix with the option "TrioPhased" on the phased genotypes from 36 lines: three sets of 10 training pure accessions plus the six admixed accessions. The results were plotted using an R script included in the Supplemental File 1.

Data availability
Supplemental File 1 contains detailed models, raw phenotypic data, scripts to manage the raw data, Markov chain convergence diagnostics, post-processing and plotting of the results, and source code. The raw phenotype data include those generated by Famoso et al. (2011). Genotype data are from McCouch et al. (2016) and available at https://ricediversity.org/data/index.cfm. Supplemental File 2 lists the training accessions we used for admixture mapping. Supplemental material available at figshare: https://doi.org/10.25387/g3.10321538.

RESULTS
The initial study (Famoso et al. 2011) only examined relative root growth between Al-treated and control genotypes. To measure Al tolerance, separate individuals from the same accession are subjected to aluminum treatment and compared to plants grown under control conditions. The relative root growth is then estimated by calculating the ratio of the means among the treated and control plants within each genotype. Partitioning the phenotypic variance of this compound trait into genetic and environmental components using this experimental design is thus impossible. Therefore, we focused on estimating genetic parameters for root length itself under the two experimental regimes. This approach also fits better with results from previous studies that suggest a role for root length in water stress tolerance. The data are more sparse and ambiguous for relative root growth.
Since Famoso et al. (2011) did not report heritabilities for root length, we first re-analyzed their data, but using a denser SNP genotype map (HDRA, McCouch et al. 2016). All accessions in the original study were genotyped using this newer platform. Of the 700,000 variants available, we retained about 450,000 SNPs after filtering out loci with singletons or more than 30% of the data missing in our accession sets (see Methods for details). We built a Bayesian hierarchical model (Gelman et al. 2004, see Methods for details) that accounts for replicated measurements of accessions and the fact that experiments were performed across several years (Supplemental Figure  2). We considered root length under control and Al treatment conditions as separate traits and used a multivariate hierarchical model (Greenberg et al. 2011) that included a SNP-derived relationship matrix to account for marker effects (see Methods for model details).
Replication of identical genotypes, possible because rice is a selfing species with almost completely homozygous accessions (McCouch et al. 2016), allowed us to separate additive marker-based variance from the total genetic variation and residual environmental effects. We call the fraction of total phenotypic variance explained by the sum of the additive effects of all SNPs "marker heritability." If we estimate accurately all causal variant effects, this is narrow-sense heritability. However, most genotyped loci are likely only in linkage disequilibrium with causal variants and our estimates of even cumulative SNP effects are imperfect. Therefore, the residual genetic variance (total genetic variance, estimated from the hierarchical model as the accession effect, minus the marker variance) absorbs both additive effects of loci not tagged by our SNP panel and non-additive components. We call this portion of genetic variance the "background effect." We find that the heritability contributed by all markers considered together accounts for less than half of overall genetic variance in the tropical japonica population ( Figure 1A). This does not seem to be due to a lack of power resulting from a small sample size, since marker heritability is higher in aus under stress ( Figure 1A) despite a smaller number of accessions in that population (55 in aus vs. 92 in tropical japonica). To see if any individual SNPs are associated with either trait, we used the posterior modes of accession effects (as well as log 10 of the control to treated ratio, 2lgRRG) as predictors in a standard single-trait mixed-model GWA (Kang et al. 2010, see Methods for details). We were able to reproduce the GWA peak found by Famoso et al. (2011) for untransformed RRG in the aus subpopulation (Supplemental Figure 1A), but did not detect any new associations. We did not unearth any associations in tropical japonica (Supplemental Figure 1B), and no associations of SNPs with root length traits under Al treated or control conditions in either population.
Perhaps, as some theoretical results suggest, recent selection on root traits keeps adaptive alleles contributing to trait variation close to fixation. The resulting low minor allele frequency would diminish opportunities for linkage disequilibrium with our genotyped SNPs. In this case, increasing sample size should increase minor allele counts, boosting power to detect marker heritability. We therefore added more accessions to our tropical japonica sample, bringing the total to 169. These lines were already genotyped using HDRA (McCouch et al. 2016). We administered the same treatment protocol as Famoso et al. (2011) and analyzed the data as before. The measurements of each accession were highly reproducible, with total genetic effects explaining almost all phenotypic variance. However, the proportion of this genetic variation accounted for by the sum of marker effects was much smaller than in the initial limited sample ( Figure 1B). Neither did we find any significant individual SNP associations (Supplemental Figure 5A).
The estimates on a larger panel of tropical japonica accessions further supports the idea that the sum of all genotyped marker effects contribute little to phenotypic variation in root traits in this population. However, we wanted further confirmation that these estimates are not the result of experimental or modeling artifacts. Therefore, we selected a new panel of 119 tropical japonica accessions that represented the full diversity of the representatives of this population genotyped using HDRA (Supplemental Figure 4). To increase confidence in our estimates of accession means (and thus the fraction of phenotypic variance explained by all genetic factors), we increased the number of replicates per line (eight on average). We performed all experiments at the same time, eliminating the year effect as a potential confounding variable. We measured two root traits: total and longest root length (see Methods for details) and included them together in a hierarchical Bayesian model (see Methods for details). However, we fit the model to the Al-treated and control data sets separately.
Aluminum treatment of individuals from the aus population increases the fraction of genetic variation explained by markers ( Figure 1A). This is not evident in tropical japonica. Furthermore, we see a high total genetic correlation (mode: 0.886; 95% highest posterior density (HPD) interval: [0.803, 0.945]) between control and treated root lengths in our initial experiment. Therefore, we increased the duration of aluminum treatment to put extra stress on the plants. The longer treatment did reduce the genetic relationships between treatment and control, although the correlations remained fairly high: 0.770 (0.733, 0.811) for total root length (the trait measured in the previous set of experiments) and 0.816 (0.777, 0.844) for longest root length, closer to the correlation observed in aus (0.597 [0.484, 0.704]).
Despite all these changes in experimental protocol and analysis, we see the same low marker heritability despite the high fraction of the phenotypic variance explained by total genetic effects ( Figure 1C). No single locus stood out when we performed GWAS on these new accession means, either (Supplemental Figure 5B).
If our observation of extremely low marker heritability of root length in tropical japonica is correct, we expect that the genotyped SNPs would be poor predictors of phenotypic values. To test this, we implemented a two-step genome prediction scheme, using the posterior modes of accession means generated in the last experiment and used a separate implementation of a penalized regression on genotypes (Meuwissen et al. 2001;Pérez and de los Campos 2014) to model additive SNP effects (see Methods for details). Given the high total heritability, we are fairly confident in the robustness of accession mean estimates. We separated the data into two sets. The first set was used to train a model that used all markers to estimate model parameters. Given these training set estimates, we attempted to predict phenotypes of the remaining accessions. We repeated this process 50 times and generated empirical distributions of prediction accuracy. Consistent with the low additive marker heritability estimates presented above, genome-enabled prediction accuracy was low, with most distributions containing zero (Table 1). We also tried alternative models with the same results (Supplemental Table S1).
All lines of evidence presented so far point to the conclusion that additive marker effects do not explain much of the total genetic variance of the root growth traits measured in tropical japonica. Given the high broad-sense heritability, we wondered if non-additive SNP effects can explain our results. Given the very high homozygosity of individual lines (McCouch et al. 2016), dominance is highly unlikely to play a measurable role. We are therefore left with epistatic interactions among loci as the remaining possibility. However, the n■  (-0.143, 0.218) parameter space of all interactions is prohibitively large, making it statistically and computationally unfeasible to estimate individual effects. However, models that rely on summaries of interactions are available. It has been shown that using the Hadamard (element-wise) product of the SNP-based relationship matrix (Van Raden 2008) with itself in a genomic model gives approximate estimates of the sum of all pairwise epistatic interactions (Jiang and Reif 2015). Similarly, use of a Gaussian kernel (see Methods) approximates all by all interactions (Jiang and Reif 2015). We used the quadratic in place of the additive kernel in our hierarchical multi-trait model and the Gaussian kernel for genome prediction. We did not observe any measurable improvement in explanatory ability ( Figure 1D) or prediction accuracy (Table 1). Since we cannot find any evidence that epistatic interactions account for much genetic variance, we are left with the possibility that low-frequency DNA sequence variants underlie our root traits. Because such alleles are not shared widely among accessions in our panel, and are not in high enough linkage disequilibrium with the SNPs we genotyped, we do not detect their effects in our models. If this conjecture is correct, we would expect many regions of the genome to make small contributions to the genetic variance. We see some evidence of this when we look at accessions that carry introgressions from the less Al tolerant indica and aus populations into the tropical japonica background (Figure 2). We found six such lines in the Famoso et al. (2011) data set. All of them have shorter roots than the average tropical japonica accession in the sample, particularly under aluminum stress (Figure 2A,B). When we looked at the genomic regions that come from indica or aus in these lines using RFMix (Maples et al. 2013) we found an idiosyncratic collection of introgressed loci ( Figure 2C). This is expected if each tropical japonica accession carries a different set of rare causal variants.

DISCUSSION
Informative tests of the effects of selection on the genetic architecture of quantitative traits require partitioning of the total genetic variation into components. This can be achieved in selfing plants, O. sativa among them, because replicated measurements can be performed on genetically identical individuals. Relying on genotyping rather than diallel crosses saves on phenotyping costs. It also allows us to directly model additive and various non-additive effects of variants in our genotyping panel as well as loci that were not assayed but are in LD with those we assayed.
We employed this strategy to study the genetic architecture of root length in the tropical japonica population of rice. A significant fraction of accessions from this group are adapted to upland rainfed conditions (Khush 1997;Lyu et al. 2014) where root length is under strong selection because it underlies drought and nutrient toxicity tolerance (De Datta 1975;Champoux et al. 1995;Bernier et al. 2009;Cairns et al. 2009;Lyu et al. 2014). We find high broad-sense heritability using two experimental set-ups and three sets of accessions. However, we find that aggregate contributions of all genotyped loci, assessed using relationship matrices in and Bayesian hierarchical models, are very small in tropical japonica. Neither do we find any individual loci contributing significantly to phenotypic variation. This is in contrast to the situation in the aus population, where we see moderate marker heritability, particularly under aluminum stress, and a significant effect of the Nrat1 locus despite a smaller sample of accessions. Alternative modeling approaches, including genome-enabled prediction, yielded the same results. Employing models that account for among-locus epistatic interactions, the only plausible source of non-additive genetic variation in this highly inbred panel, does not increase estimated marker heritability or phenotype prediction accuracy. The possibility that we failed to detect major-effect loci with moderate to high frequency alleles seems unlikely. Our genotype map is dense enough to tag the vast majority of the euchromatic genome of rice (McCouch et al. 2016, and the "Genotype Data" section of Methods). Thus, if a moderate number of causal loci underlie the root traits, the chances of identifying at least one are high (as exemplified by the successful detection of Nrat1 in aus). On the other hand, if there are only one or a few loci that drive trait variation and reside in a region not tagged by our SNP collection, we should see a two or more modes in the phenotypic distribution, particularly in the last experiment where overall heritability is high ( Figure 1C). However, we see no evidence of this (Supplemental Figure 3).
We are then left with the possibility that the alleles underlying these traits are at low frequency. Such variants, even if they located next to genotyped SNPs of moderate to high frequency, would not be in linkage disequilibrium with the tag polymorphisms due to the frequency mismatch. Furthermore, the distribution of minor allele frequencies of the SNPs genotyped by our array is skewed toward rare alleles compared to the neutral expectation (Supplemental Figure 6). Even if these variants are in complete LD with ungenotyped low-frequency causal polymorphisms, we have no power to estimate their effects and thus such tag SNPs would not account for any phenotypic variance. This is true even if the effect of allele substitution is large, since the power to detect the contribution of the locus declines with allele frequency.
Models of directional selection on quantitative traits predict genetic architectures dominated by loci with low-frequency alleles under some parameter combinations (Jain and Stephan 2017;Stetter et al. 2018). The models envision a shift in the optimal phenotype value, followed by a change in the adaptive direction and finally stabilizing selection at the new optimum. As populations approach the new equilibrium, individuals carrying many positive alleles start to emerge. Phenotypes of these individuals exceed the optimum, generating selection against positive alleles that have not fixed yet. These alleles can be close to fixation, however, with the alternative allele at low frequency. This scenario fits with our understanding of the role of root architecture in the relatively recent adaptation of rice to cultivation under the arid upland rain-fed conditions Oka 1988;Khush 1997;Fuller et al. 2010). While selection for drought tolerance strongly favors longer roots, increasing root biomass can negatively affect grain yield. Stabilizing selection may therefore keep alleles driving root length from fixation, thus maintaining genetic variation but precluding robust estimation of variant effects. Genomic analyses of shifts in cultivation practices of domesticated plants may thus be a promising avenue for empirical studies of genetic architectures of traits under selection.
If our conjecture that low-frequency alleles underlie root traits in tropical japonica is correct, each accession should have a unique set of causal variants. Some would have a positive effect, some would be negative. The balance would then determine the overall genotypic value of the individual line. This, in turn, implies that most regions of the genome can contribute to phenotypic variance. We indeed see some evidence of this: introgressions from indica and aus populations reduce total root length of tropical japonica accessions, particularly under aluminum stress (Figure 2), and the introgressions are evenly distributed throughout the genome. Conversely, Famoso et al. (2011) report that introgressions from tropical japonica into indica increase relative root length under aluminum stress, consistent with our hypothesis.
Although the low-frequency alleles may not individually contribute much to within-population variance, they should be detectable in biparental crosses. This is because allele frequencies of the causal variants present in these lines will be 50% and thus at high enough frequency to measure. Indeed, multiple quantitative trait locus mapping studies found loci contributing to root length under various conditions (Champoux et al. 1995;Bernier et al. 2009;Famoso et al. 2011;Arbelaez et al. 2017). However, subsequent attempts to introgress favorable alleles into a variety of genetic backgrounds have been largely unsuccessful (Bernier et al. 2008). This is expected under our model, since every pair of parents should have a different set of causal loci. Furthermore, the predicted presence of at least some negative-effect alleles even in the high-value lines (Stetter et al. 2018) should lead to transgressive segregation in the progeny. This is indeed often observed (Champoux et al. 1995;Arbelaez et al. 2017).
Our results shed some light on a fundamental question in quantitative genetics. They also should be taken into consideration when designing breeding strategies to improve productivity of rice grown in aerobic soils under water-limited conditions. The upland cultivation method has been historically low-yielding De Datta 1975;Bernier et al. 2008) and has resisted efforts at genetic improvement (Bernier et al. 2008). If our model is correct, progress is possible as long as the number of founding accessions is not too high. All causal alleles would be at moderate frequency and thus visible to selection. Total genetic, rather than breeding, values of the founders should be used to select them from the base population. GWAS, marker-assisted selection, and even multi-parental mapping populations (Yu et al. 2008) are not expected to be effective to determine the best founders. However, genome prediction can still be used if the focus is within breeding populations. Finally, periodic addition of new founders can introduce additional beneficial alleles even if the new lines themselves are not top performers in the environments of interest.
Our study highlights the importance of detailed genome-enabled investigations of polygenic traits under selection to shed light on fundamental evolutionary questions and guide practical artificial selection decisions. We hope that it will stimulate further analyses of such traits in agriculturally important plants, taking advantage of sophisticated experimental designs and increasing availability of deep whole-genome DNA sequence data.