Plant materials and phenotypic data

We formed a partial diallel population from the F1 progeny of 12 inbred maize lines (S1 Table, S1 Fig). Field performance of the 66 F1 hybrids and 12 inbred parents were evaluated along with two current commercial check hybrids in Urbana, IL over three years (2009-2011) in a resolvable incomplete block design with three replicates. To avoid competition effects, inbreds and hybrids were grown in different blocks within the field. Plots consisted of four rows (5.3 m long with row spacing of 0.76 m at a plant density of 74,000 plants ha⁻¹), with all observations taken from the inside two rows to minimize effects of shading and maturity differences from adjacent plots. We measured plant height (PHT, in cm), height of primary ear (EHT, in cm), days to 50% pollen shed (DTP), days to 50% silking (DTS), anthesis-silking interval (ASI, in days), grain yield adjusted to 15.5% moisture (GY, in bu/A), and test weight (TW, weight of 1 bushel of grain in pounds).

We estimated Best Linear Unbiased Estimates (BLUEs) of the genetic effects in ASReml-R (VSN International) with the following linear mixed model: where Y_ijkl is the phenotypic value of the l^th genotype evaluated in the k^th block of the j^th replicate within the i^th year; μ, the overall mean; ς_i, the fixed effect of the i^th year; δ_ij, the random effect of the j^th replicate nested within the i^th year; β_kij, the random effect of the k^th block nested within the i^th year and j^th replicate; α_l, the fixed genetic effect of the l^th individual; ς_i · α_l, the random interaction effect of the l^th individual with the i^th year; and ε, the model residuals. We calculated the broad sense heritability (H²) of traits based on the analysis of all individuals (inbred parents, hybrid progeny, and checks) following the equation: where i = 3 (number of years) and j = 3 (number of replicates per year).

The BLUE values for each cross can be found in S1 Table; values across all hybrids were relatively normally distributed for all traits (Shapiro-Wilk normality tests P values >0.05, S1 Fig), though some traits were highly correlated (e.g. Spearman correlation r = 0.98 for DTS and DTP, S2 Fig).

We estimated mid-parent heterosis (MPH) as: where , and are the BLUE values of the hybrid and its two parents i and j. Note that for ASI, lower trait values are considered superior. General combining ability (GCA) was estimated following Falconer and Mackay [33], and the estimated values can be found in S2 Table.

Sequencing and genotyping

We extracted DNA from the 12 inbred lines following [34] and sheared the DNA on a Covaris (Woburn, Massachusetts) for library preparation. Libraries were prepared using an Illumina paired-end protocol with 180 bp fragments and sequenced using 100 bp paired-end reads on a HiSeq 2000. Raw sequencing data are available at NCBI SRA (PRJNA381642).

We trimmed raw sequence reads for adapter contamination with Scythe (https://github.com/vsbuffalo/scythe) and for quality and sequence length (≥20 nucleotides) with Sickle (https://github.com/najoshi/sickle). We mapped filtered reads to the maize B73 reference genome (AGPv2) with bwa-mem [35], keeping reads with mapping quality higher than 10 and with a best alignment score higher than the second best one for further analyses.

We called single nucleotide polymorphisms (SNPs) using the mpileup function from samtools [36]. To deal with known issues with paralogy in maize [15], SNPs were filtered to be heterozygous in fewer than 3 inbred lines, have a mean minor allele depth of at least 4, have a mean depth across all individuals less than 30 and have missing alleles in fewer than 6 inbred lines. Data on the total number of SNPs called and the rate of missing data per line are shown in S3 Table. We estimated the allelic error rate using three independent data sets: for all individuals using 41,292 overlapping SNPs from the maize SNP50k bead chip [14]; for all individuals using 180,313 overlapping SNPs identified through genotyping-by-sequencing (GBS) [37]; and for B73 and Mo17 using 10,426,715 SNP from the HapMap2 project [15]. Alignments and genotypes for each of the 12 inbreds are available at CyVerse (https://doi.org/10.7946/P2WS60). Because these parents are highly inbred, knowing their homozygous genotype also allows us to know the genotype of the F1 derived from any two of the parents.

To test whether alignment to the B73 reference introduces a bias in relatedness estimation, we computed kinship matrices using both our SNP data as well as genotyping-by-sequencing data (version AllZeaGBSv2.7 downloaded from (www.panzea.org)) obtained from alignments to a set of sequencing reads ascertained from a broad germplasm base [38]. The two matrices were nearly identical (Pearson’s correlation coefficient r = 0.995), suggesting the degree of relatedness among lines is not sensitive to using B73 as the reference genome.

Identifying putatively deleterious alleles

We used genomic evolutionary rate profiling (GERP) [39] estimated from a multi-species whole-genome alignment of 13 plant genomes [40] including Zea mays, Coelorachis tuberculosa, Vossia cuspidata, Sorghum bicolor, Oryza sativa, Setaria italica, Brachypodium distachyon, Hordeum vulgare, Musa acuminata, Populus trichocarpa, Vitis vinifera, Arabidopsis thaliana, and Panicum virgatum; the alignment and estimated GERP scores are available at CyVerse (https://doi.org/10.7946/P2WS60). We define “GERP-SNPs” as the subset of SNPs with GERP score >0, and at each SNP we assign the minor allele in the multi-species alignment as the likely deleterious allele. Finally, we predicted the functional consequences of GERP-SNPs based on genome annotation information obtained from SnpEff [41]. The multi-species alignment made use of the B73 AGPv3 assembly, and to ensure consistent coordinates, we ported our SNP coordinates from AGPv2 to AGPv3 using the Gramene assembly converter (http://ensembl.gramene.org/Zea_mays/Tools/AssemblyConverter?db=core).

To compare GERP scores (for all SNPs with GERP > 0) to recombination rate and allele frequencies, we obtained the NAM genetic map [42] from the Panzea website (http://www.panzea.org/) and allele frequencies from the > 1,200 maize lines sequenced as part of HapMap3.2 [43]. To compare the burden of deleterious alleles in modern inbred lines to landraces, we extracted genotypic data of 23 specially-inbred traditional landrace cultivars (see [15] for more details) from HapMap3.2. For each line, we calculated burden as the count of minor alleles present across all GERP-SNPs divided by the total number of non-missing sites. We separated sites into fixed (present in all individuals of a group) and segregating for landrace and modern maize samples separately.

Estimating effect sizes and dominance of GERP-SNPs

We estimated the additive and dominant effects of individual GERP-SNPs using a GBLUP model [44] implemented in GVCBLUP [45]: where Y_i is the BLUE value of the ith hybrid, μ is the average genotypic value, α_j is the allele substitution effect of the jth GERP-SNP, d_j is the dominant effect of the jth GERP-SNP, X_ij = {2p, 2p − 1, 2p − 2}, ε is the model residuals, and W_ij = {−2p², 2p(1 − p), −2(1 − p)²} are the genotype encodings for genotypes A₁ A₁, A₁ A₂, and A₂ A₂ in the ith hybrid for the jth GERP-SNP with p of the A₁ allele.

The additive and dominance SNP encoding ensures that the effects are independent for a given GERP-SNP. We extracted additive (a = α − 2p(1 − p)d) and dominant (d) effects from the GVCBLUP output file (see supplemantary file of Da et al., [44] for more details). We first estimated the total variance explained under models of complete additivity (d = 0) or complete dominance (α = 0). Then, to assess correlations between SNP effects and GERP scores, we calculated the degree of dominance (k = d/a) [46] for SNPs that each explained greater than the genome-wide mean per-SNP variance (total variance explained divided by total number of GERP-SNPs). Because this approach can lead to very large absolute values of k, we truncated GERP-SNPs with |k = d/a|>2 for all further analyses.

To compare the variance explained by our model to that explained by random SNPs, we used a 2-dimensional sampling approach to create 10 equal-sized datasets of randomly sampled SNPs (including SNPs with GERP score < = 0) matched for allele frequency (in bins of 10%) and recombination rate (in quartiles of cM/Mb). For each dataset we fit the above model separately and estimated SNP effects and phenotypic variance explained by each SNP.

To test the relationship between GERP score and dominance under a simple model of mutation-selection equilibrium, we estimated the selection coefficient s by assuming that yield is a measure of fitness. We assigned the yield-increasing allele at each GERP-SNP a random dominance value in the range of 0 ≥ k ≥ 1 and calculated its equilibrium allele frequency p under mutation-selection balance using for values of k > 0.98 and for k ≤ 0.98. We then simulated datasets using binomial sampling to choose SNPs in a sample of size n = 12 inbreds.

Haplotype analysis

We imputed missing data and identified regions of identity by descent (IBD) between the 12 inbred lines using the fastIBD method implemented in BEAGLE [47]. We then defined haplotype blocks as contiguous regions within which there were no IBD break points across all pairwise comparisons of the parental lines (S3 Fig). Haplotype blocks at least 1 Kb in size were kept for further analyses.

Because there is no recombination in an inbred parent, this allows us to project the diploid genotype of each F1 based on the haplotypes of the two parents. In the projected diploid genotype of each F1, haplotype blocks were weighted by the summed GERP scores of all GERP-SNPs (python script ‘gerpIBD.py’ available at https://github.com/yangjl/zmSNPtools); blocks with no SNPs with positive GERP scores were excluded from further analysis. For a particular SNP with a GERP score g, the homozygote for the conserved (major) allele was assigned a value of 0, the homozygote for the putatively deleterious allele a value of 2g, and the heterozygote a value of (1 + k) × g, where k is the dominance estimated from the GBLUP model above.

Genomic selection

We used the BayesC option from GenSel4 [32] for genomic selection model training with 41,000 iterations. We removed the first 1,000 iterations as burn-in. We used the model where Y_i is the BLUE value of the ith hybrid, r_j is the regression coefficient for the jth haplotype block, and I_ij is the sum of GERP scores under an additive, dominance or incomplete dominance model for the ith hybrid in the jth haplotype block.

We used a 5-fold cross-validation method to conduct prediction, dividing the diallel population randomly into training (80%) and validation sets (20%) 100 times. After model training, we obtained prediction accuracies by comparing the predicted breeding values with the observed BLUE values in the corresponding validation sets. For comparison, we permuted GERP scores using 50k SNP (≈ 100Mb or larger) windows which were circularly shuffled 10 times to estimate a null conservation score for each IBD block. We conducted permutations on all GERP-SNPs as well as on a restricted set of GERP-SNPs only in genic regions to control for GERP differences between genic (N = 221,960) and intergenic regions (N = 123,216). We conducted permutation cross-validation experiments using the same training and validation sets.

We estimated the posterior phenotypic variance explained using all of the data to derive correlations between breeding values estimated from the prediction model and observed BLUE values. Note that the correlation used here is different from the prediction accuracy (r) used for the cross-validation experiments, where the latter is defined as the correlation between real and estimated values; the two statistics will converge to the same value when there is no error in SNP/haplotype effect estimation [48].

Finally, to compare our genomic prediction model to a classical model of general combining ability, we used the following equations: where Y_ij is the BLUE value of the hybrid of the i^th and j^th inbreds, μ is the overall mean, GCA_i and GCA_j are the general combining abilities of the i^th and j^th inbreds, G_ij is the breeding value of the hybrid of the i^th and j^th inbreds as estimated by our genomic prediction model, and ε the model residuals.

Data and code accessibility

Sequencing data have been deposited in NCBI SRA (SRP103329) database, and code for all analyses are available in the public GitHub repository (https://github.com/yangjl/GERP-diallel).

Heterosis in a partial diallel population

We created a partial diallel population from 12 maize inbred lines which together represent much of the ancestry of present-day commercial U.S. corn hybrids (S1 Table) [29, 30]. We measured a number of agronomically relevant phenotypes in both parents and hybrids, including flowering time (days to 50% pollen shed, DTP; days to 50% silking, DTS; anthesis-silking interval, ASI), plant size (plant height, PHT; height of primary ear, EHT), test weight (TW; a measure of quality based on grain density), and grain yield (GY). In an agronomic setting GY—a measure of seed production per unit area—is the primary trait selected by breeders and thus analogous to fitness. Plant height and ear height, both common measures of plant health or viability, were significantly correlated to GY (S2 Fig).

For each genotype we derived best linear unbiased estimators (BLUEs) of its phenotype from mixed linear models (S1 Table) to control for spatial and environmental variation (see Methods). We estimated mid-parent heterosis (MPH, Fig 1a) for each trait as the percent difference between the hybrid compared to the mean value of its two parents (see Methods, S1 Table). Consistent with previous work [28], we find that grain yield (GY) showed the highest level of heterosis (MPH of 182% ± 60%). While flowering time (DTS and DTP) is an important adaptive phenotype globally [49], it showed relatively little heterosis in this study, likely due to the relatively narrow geographic range represented by the parental lines.

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Fig 1. Heterosis and deleterious variants.

(a) Boxplots (median and interquartile range) of percent mid-parent heterosis (MPH). (b) Proportion of deleterious alleles in landraces (LR, green) and elite maize (MZ, blue) lines. (c) The allele frequency of the minor alleles in the multi-species alignment in bins of 0.01 GERP score (including GERP < = 0 sites). (d) The mean GERP score for putatively deleterious sites (GERP >0). Each point represents a 1 Mb window. In (c) and (d) the solid blue and dashed black lines define the best-fit regression line and its 95% confidence interval.

https://doi.org/10.1371/journal.pgen.1007019.g001

Annotation of deleterious alleles

We resequenced the 12 inbred parents to an average depth of ≈ 10×, resulting in a filtered set of 13.8M SNPs. Compared to corresponding SNPs identified by previous studies (see Methods), we observed a mean genotypic concordance rate of 99.1%. In order to quantify the deleterious consequences of variants a priori, we made use of Genomic Evolutionary Rate Profiling (GERP) [39] scores of the maize genome [50]. GERP scores provide a quantitative measure of the evolutionary conservation of a site across a phylogeny that allows characterization of the long-term fitness consequences of both coding and noncoding positions in the genome [51]. Sites with more positive GERP scores are inferred to be under stronger purifying selection, and SNPs observed at such sites are thus inferred to be more deleterious. At each site with GERP scores > 0 (hereafter called GERP-SNPs), we designated the minor allele from the multispecies alignment as putatively deleterious. Of the 350k total segregating GERP-SNPs in our parental lines, 14% are detected in coding regions, equally split between synonymous (N = 64,439) and non-synonymous (N = 65,376) sites (S4 Table). Each line carries, on average, 139k potential deleterious SNPs, 19k of which are in coding regions (S5 Table). The reference genome B73 contains only ≈ 1/3 of the deleterious SNPs of the other parents, likely due to reference bias in identifying deleterious variants. The F1 hybrids of the diallel each contain an average of ≈ 56,000 homozygous deleterious SNPs, ranging from 47,219 (PH207 x PHG35) to 77,210 (PHG84 x PHZ51) (S6 Table).

To compare the burden of deleterious variants between our elite maize lines and traditionally cultivated landraces, we used genotypes from the maize HapMap3.2 [43] for our diallel parents and 23 specially-inbred landrace lines [15] (S5 Table). Compared to landraces, the parents of our diallel exhibited a greater burden of fixed (allele frequency of 1) deleterious variants but a much smaller burden of segregating SNPs, resulting in a slightly lower overall proportion of deleterious sites (mean of 1.3M deleterious alleles out of 6.5M total sites vs. 0.6/3.3M; Fig 1b).

Population genetic theory predicts that deleterious variants should be at low overall frequencies, and that such variants should be enriched in regions of the genome with extremely low recombination [52]. Using data from more than 1,200 lines in maize HapMap3.2 [43], we find that allele frequency of the minor alleles in the multi-species alignment shows a strong negative correlation with GERP score (Fig 1c). This negative correlation holds using allele frequency derived from our 12 parental lines (S4 Fig), though as expected is less significant given the smaller sample size. SNPs found in regions of the genome with low recombination also show higher overall GERP scores (Fig 1d), a trend particularly noticeable around centromeres (S5 Fig). These results match previous empirical findings in maize that deleterious alleles are rare [19] and most abundant in the lowest recombination regions [17, 40, 53], and support the use of GERP scores as a quantitative measure of the long-term fitness effects of an observed variant.

Phenotypic effects of deleterious SNPs

We first investigate the impacts of deleterious variants on phenotype using simple linear regressions. Across all hybrids, the number of homozygote GERP-SNPs was negatively correlated with grain yield, plant height, and ear-height per se (see S6 Table for complementation data and S7 Table for correlations with all traits).

We next applied a genomic best linear unbiased prediction (GBLUP) [44] modeling approach to estimate the effect sizes and variance explained by GERP-SNPs for each of the phenotypes per se across our diallel (see Methods). GERP-SNPs had larger average effects and explained more phenotypic variance than the same number of randomly sampled SNPs (including SNPs with GERP score < = 0) matched for allele frequency and recombination (Fig 2a). We found the cumulative proportion of dominance variance explained by GERP-SNPs was higher for traits showing high heterosis (Spearman correlation P value < 0.01, r = 0.9), from ≈ 0 for flowering time traits to as much as 24% for grain yield (S6 Fig). Distributions of per-SNP dominance (see Methods) across traits were consistent with the cumulative partitioning of variance components (Fig 2b) and matched well with expectations from previous studies showing a predominantly additive basis for flowering time [54] and plant height [55] but meaningful contributions of dominance to test weight and grain yield [28, 30]. Although our diallel population is relatively small, our estimated values explain as much (for traits with low dominance variance like flowering time) or more variance (for traits with substantial dominance variance like grain yield) than sets of data with randomly shuffled values of dominance (n = 10 randomizations of k per trait; S7 Fig).

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Fig 2. Variance explained and degree of dominance (k) of GERP-SNPs for traits per se.

(a) Total per-SNP variance explained for grain yield trait per se by GERP-SNPs (red lines) and randomly sampled SNPs (grey beanplots). (b) Density plots of the degree of dominance (k). Extreme values of k were truncated at 2 and -2. (c-e) Linear regressions of additive effects (c), dominance effects (d), and degree of dominance (e) of seven traits per se against SNP GERP scores. Solid and dashed lines represent significant and nonsignificant linear regressions, with grey bands representing 95% confidence intervals. Data are only shown for SNPs that explain more than the mean genome-wide per-SNP variance (see Methods for details).

https://doi.org/10.1371/journal.pgen.1007019.g002

We then evaluated the relationship between GERP score and SNP effect size, dominance, and contribution to phenotypic variance. We found weak or negligible correlations between effect size and GERP score for flowering time and grain quality, but a strong positive correlation for fitness-related traits (Fig 2c and 2d). The variance explained by individual SNPs, however, was largely independent of GERP score (S8 Fig), likely due to the observed negative correlation between allele frequency and GERP score (Fig 1c). Finally, we observed a positive relationship between GERP score and the degree of dominance (k) for grain yield (Fig 2e), such that the putatively deleterious allele at SNPs with higher GERP scores are also estimated to be more recessive for their phenotypic effects on grain yield (larger k for the major allele).

We investigated a number of possible caveats to the results presented in Fig 2. First, to control for the potential inflation of SNP effect sizes in regions of high linkage disequilibrium, we removed SNPs from regions of the genome in the lowest quartile of recombination. While some individual correlations changed significance, our overall results appear robust to the removal of low recombination regions (S9 Fig). Second, we tested the impact of reference bias caused by inclusion of the B73 genome in the multi-species alignment used to estimate GERP scores. To do so, we removed the 11 hybrids which include as one parent the reference genome line B73 and repeated the above analyses. Doing so dramatically reduces the size of our dataset, but we nonetheless find significant correlations between complementation and phenotype (S7 Table), that GERP-SNPs explain a greater proportion of overall variation than randomly sampled SNPs (S10a Fig), and that the relative pattern of dominance among traits remains the same (S10b Fig). While most of the correlations between effect size and GERP score lose significance (S10c and S10d Fig), likely due to the decreased sample size, the positive correlation between dominance and GERP score remains significant even in the absence of B73-derived hybrids (S10e Fig). Finally, because natural selection will maintain dominant deleterious alleles at lower frequencies than their recessive counterparts, we investigated whether the ascertainment bias against rare alleles present in our small sample would lead to the observed correlation between GERP and dominance. Simulations of SNPs with random dominance at mutation-selection balance (see Methods), however, failed to find any relationship between dominance and GERP score (S11 Fig), though we caution that the dramatic demographic shifts involved in the recent history of maize [4] make such a simulation approximate at best.

Genomic prediction by incoporating GERP information

To explicitly test the informativeness of alleles identified a priori as putatively deleterious, we implemented a genomic prediction model that evaluates complementation at the haplotype level by incorporating GERP scores of individual SNPs as weights (see Methods). We explored the explanatory power with several different models and found that a model which incorporates both GERP scores and dominance (k) estimated from our GBLUP model explained a greater amount of the posterior phenotypic variance for most traits per se (Fig 3a) and heterosis (MPH) (Fig 3b). A simple additive model showed superior explanatory power for flowering time, however, consistent with previous association mapping results that flowering time traits are predominantly controlled by a large number of additive effect loci [54].

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Fig 3. Genomic prediction models incorporating GERP.

(a-b) Total phenotypic variance explained for traits per se (a) and heterosis (MPH) (b) under models of additivity (red), dominance (green), and incomplete dominance (blue). (c-d) Beanplots represent prediction accuracy estimated from cross-validation experiments for traits per se (c) and heterosis (MPH) (d) under a model of incomplete dominance. Prediction accuracy using estimated values for each GERP-SNP under an incomplete dominance model is shown on the left (red) and permutated values on the right (grey). Horizontal bars indicate mean accuracy for each trait and the grey dashed lines indicate the overall mean accuracy. Stars above the beans indicate prediction accuracies significantly (FDR < 0.05) higher than permutations. Results for pure additive and dominance models are shown in S13 Fig.

https://doi.org/10.1371/journal.pgen.1007019.g003

To explicitly test the utility of incorporating GERP information in prediction models, we compared cross-validation prediction accuracies of the observed GERP scores to those from datasets in which GERP scores were circularly shuffled along the genome (see Methods). Models incorporating our observed GERP scores out-performed permutations (Fig 3c and 3d), even when considering only SNPs in genes (S12 Fig). Our model improved prediction accuracy of grain yield by more than 4.3%, and improvements were also seen for plant height (0.8%) and testing weight (3.3%). While our model showed no improvement in predicting heterosis for traits showing low levels of heterosis (Fig 1a), including GERP scores significantly improved prediction accuracy for heterosis of grain yield (by 1%). Finally, our approach also significantly improved model fit for phenotypes of all traits per se as well as heterosis for GY and PHT compared to traditional models of genomic selection that use general combining ability (see Methods, S2 Table) calculated directly from the pedigree of the hybrid population [56] (ANOVA FDR <0.01 and difference in AIC < 0, S8 Table).

Conclusion

In this study, we use genomic and phenotypic data from a partial diallel population of maize to show that an incomplete dominance model of deleterious mutation both fits predictions of population genetic theory and explains phenotypic variation for fitness-related phenotypes and hybrid vigor. We find genome-wide support for hypotheses predicting that more damaging variants are more recessive. Finally, we show that leveraging evolutionary annotation information in silico enables us to predict grain yield and other traits, including heterosis, with greater accuracy. Together, these results help reconcile alternative explanations for hybrid vigor and point to the utility of leveraging evolutionary history to facilitate breeding for crop improvement.