Albugo occidentalis) in spinach (Spinacia oleracea). We used WR disease severity ratings (WR-DSRs) in a diversity panel of 267 spinach accessions to define resistant- and susceptibility-associated groups within the distribution scores and then tested the single-nucleotide polymorphism (SNP) variants to interrogate the MA prevalence in the most susceptible (MS) vs. most resistant (MR) individuals using permutation-based allelic association tests. A total of 448 minor alleles associated with WR severity were identified in the comparison between the 25% MS and the 25% MR accessions, while the MA were generally similar between the two halves of the interquartile range. The minor alleles in the MS group were distributed across all six chromosomes and made up ~71% of the markers that were also strongly associated with WR in parallel performed genome-wide association study. These results indicate that susceptibility may be highly determined by the disproportionate overrepresentation of minor alleles, which could be used to select for resistant plants. Furthermore, by focusing on the distribution tails, allelic mapping could be used to identify plant markers associated with quantitative traits on the most informative segments of the phenotypic distribution.Minor alleles (MA) have been associated with disease incidence in human studies, enabling the identification of diagnostic risk factors for various diseases. However, allelic mapping has rarely been performed in plant systems. The goal of this study was to determine whether a difference in MA prevalence is a strong enough risk factor to indicate a likely significant difference in disease resistance against white rust (WR;  Likewise, an individual carrying one or two copies of a particular allele is more likely to present the associated trait, such as susceptibility to a disease. The identification of the groups or individuals at high risk of developing a particular disease can be a daunting task, given that many major alleles may also be risk factors2. Shifts in allele frequencies are also common4 and can be challenging to track in fast-paced breeding programs relying on genome-wide association (GWA) mapping strategies alone. Therefore, the aim of this study was to determine whether particular minor alleles (MA) could be found more often than by chance in relation to a simple distribution using white rust disease severity ratings (WR-DSRs), in a diverse collection of spinach (Spinacia oleracea) accessions as a test model.Genetic association studies can be used to determine whether a genetic variant is associated with a particular trait. If an association is present, a particular allele, genotype, or haplotype will be seen more often than is expected by chance in individuals or groups carrying the corresponding traitAlbugo occidentalis, an oomycete obligate pathogen that attacks the vegetative and flowering structures of the affected plants and causes yellow lesions on the upper leaf surface and white pustules on the abaxial side, resulting in severe yield losses5. Resistance to WR in spinach is reported to be polygenic6; however, the mapping of its genic features is far from conclusive. It is perhaps not surprising that the genetic and molecular basis of resistance to WR is not well understood in spinach, and new approaches are required to elucidate the genetic and molecular basis of this resistance. One possibility is to perform a GWA to target minor alleles implicated as risk factors in susceptibility to the disease7. Determining the distribution of such alleles in diseased plants might illuminate the underpinnings of minor alleles important for WR susceptibility in spinach.WR in spinach is caused by 9. Despite these challenges, association studies comparing the contribution of alleles in different disease outcomes are more common in humans and animals than in plants, with the number of published articles  exploring human allelic associations to diseases significantly dwarfing the numbers in plants to date10. It is common to partition human study subjects into retrospective and prospective cohorts, and case and control groups11. Retrospective cohort studies are usually conducted on data that already exist, and the disease exposures are defined before the existing outcome data are explored to determine whether exposure to a risk factor is associated with a statistically significant difference in the disease development rate12. These studies then follow the participants for a defined period to assess the proportion that develop the outcome/disease of interest10. After this, the genetic associations can be explored to determine whether single-locus alleles or genotype frequencies  differ between two groups of individuals, such as diseased subjects and healthy controls11. Studies implementing such experimental designs have led to successful identification of loci harbouring risk alleles for important complex diseases13. Just as GWA studies (GWASs) were adapted from human studies and became even more successful in plants13, we propose that it is possible to adapt a related methodology involving the partitioning of study populations for use in genomic studies of plants.Compared to human studies, it is relatively easy to control and monitor the population structures and family mating systems of cultivated crops, the no sedentary nature of humans, coupled with ethical issues, means that providing the same treatment or exposure to disease is not a trivial matter11. Our segment definition drew on the strength of using a random, nonbiased selection of individuals, since the underlying genetic interventions in the defined segments were naturally occurring. The intention was to determine whether the difference in allelic prevalence is a strong enough risk factor to predict WR disease outcomes. By comparing these results with those of a GWAS in which relatedness among individuals is accounted for, we highlight the potential use of this technique in predicting breeding markers associated with WR in spinach.Unrelated, random individuals are included in retrospective cohort (segmented) studies of human diseases; therefore, we used individuals from plant populations that were assumed to be unrelated and random. We determined our study segments based on the WR-DSRs that we phenotyped. Studies of segmented populations can assess a range of outcomes, allowing an exposure level to be rigorously assessed for its impact on developing diseaseTwo parallel analyses were conducted: (1) a permutation-based basic allelic association to determine the relationship between the minor allele distributions and the WR-DSR, and (2) a GWAS adjusting for population and kinship structure to compare with the markers identified in the basic allelic association mapping.14 was used in this study. Briefly, the 267 spinach accessions consisted of a diverse panel with accessions from 33 countries and part of a collection maintained and provided by the USDA-National Plant Germplasm System (NPGS) at Ames, Iowa, USA. A detailed description of each accession can be found at https://npgsweb.ars-grin.gov/gringlobal/search.aspx. The 267 accessions were grown at the Texas A&M AgriLife Research and Extension Center located at Weslaco, Texas, USA, at a latitude of 26° 9′ 30′′N and a longitude of 97° 57′43′′W. The plants were grown in a randomized complete block design with three replicates per accession. Each plot contained two rows of ~14 plants spaced at ~10-cm intervals, with 15 cm between the two rows. Each plot was 1.22 m from the next adjacent plot. Conventional agronomic practices for spinach were used, from land preparation to the last date of data collection. The crop was fertilized with a generalized N–P2O5–K2O rate of 135–84–90 kg/ha.A panel of 267 accessions described in our previous report15; therefore, the spinach field was irrigated as needed to improve the spatter spread of the spores and enhance humidity within the plant canopy. The WR symptoms were evaluated on a per-plot basis, as described by Dainello et al.16. Briefly, the WR-DSR scores were based on the surface area covered by WR lesions as a percentage of total surface area of a plot. The percent ratings were categorized in 1%, 5%, and then at intervals of 5% up to 50%.The natural field inoculation was accomplished by planting crops in previously infected field. The experimental plots were intercalated with rows of the susceptible control cultivar “Monstrueux de Viroflay” to uniformly spread the disease. WR dispersal and infection is favoured by cool and humid conditions with mild wind and water conditions14, and the genotyping framework followed the double digest restriction site-associated DNA (ddRAD-seq) genotyping-by-sequencing protocol described by Peterson et al.17. The Illumina short-read sequencing (HiSeq 2500) and demultiplexing using individual indexes were performed using the Texas A&M AgriLife Genomics and Bioinformatics services. The paired-end raw sequencing reads (150 bp) were subjected to filtering to obtain high-quality reads for the downstream analysis. The raw reads were filtered and trimmed to remove adapter contamination and low-quality  or ambiguous sequences with uncalled nucleotides . The filtering and trimming of the raw reads were performed using an in-house pipeline developed using Python18 (https://github.com/reneshbedre/RseqFilt). The high-quality cleaned sequence data were aligned to the draft spinach reference genome (v1)19 using the bowtie2 alignment tool20.Tissue handling, DNA isolation and library preparations, and genotyping and single-nucleotide polymorphism (SNP) calling were performed as previously described by Awika et al.22, based on the draft spinach reference genome (v1). In brief, various modules of the Stack pipeline  were used to identify and filter the genotypes21. The Ada cluster of the TAMU High Performance Research Computing (http://hprc.tamu.edu/) was used to perform the bioinformatics analysis. The SNP pipeline-end cleanup criteria also included the removal of SNPs not anchored onto the six published draft chromosomes19. A final 6166 SNPs in VCF v4.223 were available for the downstream analyses of which 6,111 were used in this study. The SNPs in the spinach accession data can be downloaded from the supplementary table presented by Awika et al.14.The 267 spinach accessions were genotyped using Stacks (v1.48)24 and a kinship matrix25 were used to adjust for population stratification and allele sharing26 before running the GWAS (see below). STRUCTURE uses identity-by-state similarity, assuming that two random alleles drawn from the same locus are the same. An admixture model was run, with the distance of an individual from itself set to 0. The protocols for using STRUCTURE, visualization in STRUCTURE HARVESTER27, the associated algorithms29, and the kinship matrix (K) were described previously by Awika et al.14, except that, in this study, 5000 burn-ins instead of 1000 burn-ins were used in STRUCTURE, and 17 rather than 15 replications were performed for each of the seven population assumed (K). This was to further improve the accuracy of convergence toward reliable estimate of allele frequency in each (hitherto arbitrary) subpopulation.STRUCTURE (v2.3.4)30 and the statistical principles of case–control genotypic and allelic associations33 were used to define the study samples. The study groups (segments) were retrospectively defined based on the distribution of the WR-DSR scores, providing a random, nonbiased inclusion of individuals in each segment. The best linear unbiased predictor estimates (BLUPs), least square means (LSM), and arithmetic means (ArithMean) of the WR-DSR scores were used to construct the distribution.A modified form of the retrospective cohort of human diseasesij = µ + Ui + Wij + e, where µ is the mean WR-DSR for the whole population, U is the random plot mean, W is the random deviation of the jth replicate from the ith plot mean, and e is the residual error accounting for plot-to-plot differences. To test the accuracy of the BLUPs for predicting the true phenotypic means, the correlations between the LSMs and BLUPs and between the ArithMean and BLUPs were determined. One advantage of the BLUP estimates is that they have a reduced mean square error within the linear plot estimators and thus deviated less from the mean of the realized plot phenotypic values35. The data were then split into a random test sample (25%) and a training set (75%) to test the model fitness in different sample sizes. The slopes, R2, Akaike’s information criteria (AIC), and Bayesian information criteria (BIC) were compared for the three equations .The BLUPs and LSMs were determined by running the restricted maximum likelihood (REML) model on the three replicate WR-DRS scores in JMP Statistical Software v14  using the following formula: Y37 was applied at α = 0.05 to assess normality, while skewness (S), a measure of asymmetry38, was determined as S(G1) = [√n(n − 1)/(n − 2)]·g138. The standard error of skewness (SES) was calculated as SES = √[6n(n − 1)/(n − 2)(n + 1)(n + 3)]38, where n is the sample size and g1 is the squared variance (σ4).To partition the data into segments consisting of individuals with high WR-DSRs  and individuals with low WR-DSRs , the whole sample  was also tested for the sample distribution statistics of normality, skewness, and kurtosis. For these measures, a “sample statistics” computation was used because the material in this study is part of a larger diversity population maintained by the USDA. In the strict context of this study, the term “population” is used to refer to the 267 accessions used. The D’Agostino-Pearson Omnibus test39. This established how reasonable the distributions of the intermediate values and the extreme values are for the two WR-DSR distribution tails40. Kurtosis (G2) was determined as G2 = (n − 1)/(n − 2)(n − 3).[(n + 1)g2 + 6]38, and the standard error of kurtosis (SEK) was calculated as SEK = 2(SES)√[(n2 − 1)/(n − 3)(n + 5)]41, where n is the sample size and g2 is the squared variance (σ4).To assess the deviations in the distribution variance, kurtosis was determined to measure whether the variance results from infrequent extreme deviations or frequent modestly sized deviations42, Q1–Q4, each containing ~66 plots (accessions) used in the allelic tests. To define the quartile intervals, the nearest-rank method was applied43: n = [(P/100)*N], where n is the ordinal rank, P is the percentile in question, and N is the total number of ranks in the ordered list. For the cross-validation of the segments, the entire population was divided into mutually exclusive five percentile segments (k = 10). Each k consisted of ~13 accessions with WR-DSR scores, and ~10 accessions were drawn from each of the k segments for the downstream allelic analysis. The phenotypic raw data and assigned clustering can be found in Supplemental Tables The data, showing satisfactory (nonsignificant) kurtosis and skewness, was partitioned into four quartiles or segmentsk, five-percentile segments) and ~130 individuals for the Q1 vs. Q4 and the Q3 vs. Q4 tests. BLUPs and genotypic data were used to perform the basic allelic permutation tests to provide empirical P values that also control the familywise error rate31. In computing the allelic test for each segment pair, the individual members of segments with a mean larger than the population mean were populated with a phenotype code of 1. The individual members of the corresponding segments with a mean smaller than the population mean were populated with the code 2. The plots not included in either pair were coded as 0 (missing)44.The frequencies of alleles in the two opposite segments in the WR-DSR distribution were compared, i.e., the segments containing individuals with WR-DSR scores lower or higher than the mean population. The pairs of segments consisted of ~20 individuals 45, with 10,000 iterations for all SNPs. Max(T) performs all specified permutations without dropping what might be nonsignificant alleles45. The benefit of this is that two sets of empirical significance values could then be calculated: the pointwise estimates of the significance of an individual SNP and a value that controls for the fact that thousands of other SNPs were also tested. This was achieved by comparing each observed test statistic against the maximum of all permuted statistics  for each segment pair.The tested hypothesis was that the allelic coefficients associated with WR = 0, while the alternative hypothesis was that these allelic coefficients ≠ 0 . The max(T) permutation method was applied in PLINK and gPLINK  empirical significance values49 that are valid in small samples such as the segments used here where the likelihood ratio and Pearson tests become less reliable50. Because the permutation schemes preserve the correlational structure between SNPs, these tests provided a less stringent correction for multiple testing than the Bonferroni correction (BC), which assumes all tests are independent. In fact, because the corrected P value was of interest, it was not necessary to demonstrate genome-wide significance levels beyond 0.05 or 0.0145. A P value ≤ 0.01 was used to declare a significant association for the allelic tests, and an odds ratio (OR) > 1 was used to declare risk.The permutation-based Max(T) test uses a hypergeometric distribution to compute probabilities and thus does not depend on any large-sample distribution assumptions51. Individual missingness was set at 0.1; i.e., an accession was included in the analysis only if at least 90% of the SNPs were present in it.The pre-analysis thresholds were minor allele frequency (MAF) = 0.01, and max SNP missingness = 0.1; i.e., to be incorporated into the analysis, a SNP must be present in at least 90% of the accessions , otherwise it was omitted. This stringency was increased from the 80% used in the GWA, which was performed in parallel to the allelic tests (described below) to penalize what was believed to be-possible spurious associations during the multiple test permutation correction52, a GWAS was conducted using all 267 accessions and 6111 SNP variants. The GWA approach was implemented in TASSEL v5.2.52 and corrected for the population structure and possible kinship (as described above). For the purpose of this report, the significant signals identified in the GWAS were used as a baseline to assess the efficiency of the allele-based method . A basic genotypic association full (GAF) model was also performed for the Q1 vs. Q4 pair. The difference between the GAF and the allelic association tests is that instead of coding individuals as 1 or 2, the GAF uses the actual BLUPs to compute the genotypic association tests, while using similar permutation parameters to those specified for the allelic tests. The associated graphic plots were generated using qqman v0.1.453 in R, ggplot2 v3.1.054 in R, DAAG v1.22.255 in R, and e1071 v1.7–1 (https://www.rdocumentation.org/packages/e1071) in R, as well as Excel and jvenn56.Since basic allele tests may require corroboration using other testsz/T) of matching markers in the other methods as a factor of the GWA signals (L) and, thus:, (z/T)c ∙ Lkc, where z is the number of markers identified using the allelic or GAF model that were also identified in the GWAS; T is the total markers identified using the allelic or GAF model, and k is the efficiency multiple factor for model c1 compared to model cn.The markers identified via the GWA model were considered as the reference for true associations and were used to determine the efficiency of the allelic mapping using the case–control models at with the Q1 vs. Q4 and Q2 vs. Q3 comparisons, and the full model GAF. To achieve this, the “true” signals in the GWA model were used to standardize the ratio (α = 0.01).In order to evaluate how well SNP markers identified by allelic mapping can discriminate between WR-resistant and susceptible individuals, we compared the MAF of the MR accessions within the fifth percentile of the WR-DSR distribution tail vs. the MAF in the corresponding fifth percentile WR-DSR distribution tail consisting of the MS accessions. A disproportionately high MAF for a marker would suggest that the marker is a susceptibility factor, while a disproportionately low MAF would suggest a resistance factor. To be considered disproportionately represented, we defined that the frequency of the allelic base at a SNP locus had to be less than half the frequency of the corresponding allele on the second strand, and significantly lower than the mean of MAFs in the second strand (at 57 unweighted pair group method with arithmetic mean58. Phylogeny was tested using 500 bootstrap replications59 for the combined 20 (10 MR and 10 MS) accessions. The substitution model based on the maximum composite likelihood method60 was implemented for the nucleotides at the significant polymorphic sites in each of the accession. We used the default settings of uniform rates and assumed homogenous pattern among lineages. Pairwise deletion was selected for sites with gaps or missing data.To test the efficacy of this method, we determine if the significant markers could discriminate between WR resistance or susceptibility among the MR (WR-DSR < 20) and 10 MS (WR-DSR > 40) accessions  and the narrow-sense heritability (h) were determined as previously described14. Briefly, H was computed on a line mean basis using the variance components generated in the REML model, with replicates nested in three blocks in a single location: H = Vl/(Vl + (Vl·Vr[Block])/3 + Ve/3)), where Vl is the variance within a line, r is the replicate, and e is the residual.Both the broad-sense heritability (h) was calculated based on the marker mean genetic and residual variances. Since we specified the population parameters previously determined (P3D) in the optimal compression model (in TASSEL), h was obtained from the mean genetic variance and the mean residual variance in the compression models: h = (σa2)/(σa2 + σe2), where σa2 is the mean additive variance of the marker allele and σe2 is the variance of the residual (error).Narrow-sense heritability (g1) and excess kurtosis (Zg2)41, respectively, at the 0.05 significance level. The distribution statistics for ArithMeans and BLUPs are shown in Figs. The phenotypic distribution had a slight negative (left) skew. The ArithMean was 36.25 , while the MS-25 (Q4) segment had a significantly higher MAF than the mean MAF in the MR-25 (Q1) group (P ≤ 0.001)  Fig. . These oA total of 448 significant alleles Table , Fig. 4 z/T) of matching markers in the other methods as a factor of the GWA signals (L). Based on comparisons of the standardized ratios, the allelic mapping in the Q1 vs. Q4 DSR tails was 7.7 times more efficient at discovering true signals than the allelic mapping using the interquartile DSR range Q2 vs. Q3, and 2.4 times more efficient than the full model GAF. These observations indicate that mapping alleles in the Q1 vs. Q4 segments better identified the loci associated with WR than mapping using the Q2 vs. Q3 segments or the GAF model in a population with a near-normal phenotypic distribution. The difference in the number of individuals and markers used, or in the method used to account for population structure and stringency in declaring test significance, might have contributed to the large difference in the markers identified using the GWA in comparison with the other models.Taking the 142 markers in the GWA model as the reference for true associations, we compared the efficiency of the allelic mapping using the case–control models with the Q1 vs. Q4 and Q2 vs. Q3 comparisons, and the full model GAF. To achieve this, we used the 142 “true” signals in the GWA model to standardize the ratio  of the 15 markers identified by allelic mapping had disproportionately low MAF in the resistant plants  group on one hand and the most susceptible (MS) on the other hand Fig. . The MR r2 ranged between 0.00 and 0.09. These observations underpin the conundrum associated with the “missing heritability” in the GWA models.We observed a discrepancy in the predicted and actual heritability observed. The phenotype-based mixed model predicted 97% of the variation observed in WR-DSR, whereas the variance components explained a total of 64% of the heritability. At the marker level, the additive effect model predicted 98% of the heritable variance, even though individually, the per-marker 4. Indeed, for the associated loci, the mean MAFs were significantly higher in the WR-DSR distribution tail containing the most susceptible accessions compared to the WR-DSR distribution tail containing the most resistant accessions  of risk alleles associated with WR were identified in the most susceptible (Q4) individuals than in the most resistant (Q1) individuals ons Fig. ; Table 314, suggesting that large chunks of the genome in this material is are still well conserved and may contain long swathes of with many SNPs intervening in WR disease.The distribution of markers associated with WR showed no chromosome-specific pattern across the association models Fig. . This obP value distribution but can have reduced sensitivity63. The use of the Bonferroni correction (BC) in assigning significance may be highly conservative, however, generating many false negatives64 and causing a lower correlation percentage between the significant markers in the GWA and those containing the associated minor alleles 66. It was not clear in our study whether the partitioning that concentrated the risk alleles to a 25% subset of the population may also have experienced a low mapping resolution, such that many valid signals in the GWA went undetected in the basic allelic tests.Compared to the GWA model, the allelic tests may have exposed the presence of alleles reflecting stratification in the samples that could not be captured by permutation alone18906_141  and 41441_41  showed T/C and A/T changes, respectively, in both the GWA and in the MS-5 group. On the other hand, the markers 18167_81  and 23256_116  showed A/G and G/A polymorphisms, respectively, in the GWA, but were, T/C and C/A polymorphisms, respectively, in the MS-5 , Myles et al.13 identified a number of association signals of for alleles in strong LD regions with containing non-genotyped but functional alleles. This discrepancy in allele identity in this study may also have been an artefact of our artificial in silico “selection pressure” resulting in allelic drift3 in favour of the WR-DSR distribution tails. The alleles that are similar in both the MS-5 and the GWA analyses indicate they are common disease variants present in the whole entire population.Comparing the nucleotide identities for each given locus, some markers showed the same allelic pair in the GWA analysis as they did in the other models; for instance, of the 15 markers common to both the GWA and in the MS-5, a total of nine SNPs maintained the same minor and major allele sets (see Table 68.The variations explained by the field phenotyping highly overestimated the heritability (98%) compared to the heritability (64%) determined from the variance components. Similarly, the in the GWA model, the marker based mean heritability was 98% compared with a paltry <10% variation for the most heritable marker. We did not determine marker-based heritability for the markers identified using the allelic association and the GAF models. In any case, these observations underpin the conundrum associated with the “missing heritability” in the GWA models. For the purpose of this study, we will not try to untangle this ambiguity. Better and more extensive coverage of the probable causes of such wide disagreements between the predicted and observed heritability can be found in other studies and reviews8. Such variants may, for example, accumulate over time, and eventually forming novel transcription start sites, transcription factor-binding sites, protein-binding sites, and histone-binding sites69, which may be functional in pathogen–host plant interactions. In fact, some of these studied functionalities were represented in a small sample of 15 SNPs identified in plants displaying the highest and lowest WR-DSR scores in this study  is anchored on the gene Spo00225, which contains a predominantly plant-based protein domain known as a pentatricopeptide repeat (PPR). PPR mediates many plant functions, including the regulation of gene expression, by binding RNA and negatively regulating abscisic acid signalling and many other plant development and stress signalling pathways72. The marker 23256_116  is hosted by the gene Spo14439, which encodes Histone H3, the acetylation of which mediates many fungal pathogen–host plant interactions .It has been suggested that many rare variants may be functionless, yet they can still accumulate in the genome and eventually become functional if combined with other nearby variantsudy Fig. ; Table 474. In addition, the marker spectrum of all the models used in this study  show a discrimination of the resistant group from the susceptible group of accessions Fig. . The strSupplementary Table ST1 Phenotypic data and accession informationSupplementary Table ST2 Clustering of phenotypic dataSupplementary Table ST3 Markers and associated significant alleles in the most resistant Q1 and in the most susceptible Q4 accessionsSupplementary Table ST4 Markers and associated significant allelic tests in the resistant Q2 and in the susceptible Q3 accessionsSupplementary Table ST5 Markers and associated significant allelic tests in the most resistant 5% (MR-5) and the most susceptible 5% (MS-5) accessionsSupplementary Table ST6 Markers associated with white rust in the basic genotypic full model association testSupplementary Table ST7 Significant signals (Bonferroni correction) and related statistics in the genome-wide association analysis


Baccharis salicifolia. Here we build on these findings and provide a more in-depth analysis of terpene chemistry on these same plants from an experiment consisting of a common garden with male (N = 19) and female (N = 20) genotypes sourced from a single population. Our goal in the present study was to investigate quantitative and qualitative differences in terpene profiles associated with plant sex and genotypic variation. For this, we quantified leaf mono- and sesquiterpene amount, richness, and diversity (quantitative profile), as well as the composition of compounds . We found no evidence of sexual dimorphism in monoterpene or sesquiterpene profiles. We did, however, find significant genotypic variation in amount, diversity, and composition of monoterpenes, but no effects on sesquiterpenes. These findings indicated that genotypic variation in terpene profiles largely surpassed variation due to sexual dimorphism for the studied population of this species.Terpenoids are secondary metabolites produced in most plant tissues and are often considered toxic or repellent to plant enemies. Previous work has typically reported on intra-specific variation in terpene profiles, but the effects of plant sex, an important axis of genetic variation, have been less studied for chemical defences in general, and terpenes in particular. In a prior study, we found strong genetic variation  in terpene amounts in leaves of the dioecious shrub  They are typically classified based on the number of carbon atoms of a molecule, namely: monoterpenes (C10), sesquiterpenes (C15), diterpenes (C20), triterpenes (C30), and tetraterpenes (C40). Due to their low molecular weight, mono- and sesquiterpenes are highly volatile components found in scents and fragrances emitted by aromatic plants1. Many of these volatile compounds are considered as toxic or repellent to herbivores and pathogens2. In addition, they can also play multiple other roles in plant-insect interactions. These include attraction of predatory arthropods and parasitoids5, attraction of insect pollinators and seed dispersers8, insect-insect interactions such as co-factors for bark beetle aggregation pheromones9, plant-to-plant communication as warning signals to neighbouring plants of herbivore presence11, and plant protection against abiotic stresses (e.g. drought or elevated temperatures13).Terpenoids encompass a group of secondary metabolites which are often produced in high amounts in most plant tissues20. As for other plant defensive traits involved in herbivore resistance, studies have assessed broad-sense genetic variation in terpene levels  and in some cases addressed specific genes or groups of genes that code for focal compounds (e.g. candidate genes23). Plant sex is an ecologically important form of genetic variation in dioecious plants24. Dioecy is frequently characterized by the presence of sexual dimorphism in various traits25, which includes defensive traits associated with resistance to herbivores26. Female plants are expected to invest more resources into reproduction than males, such that allocation trade-offs are expected to lead to decreased growth and in turn higher investment in defensive traits relative to males28. Despite mounting evidence of sexual dimorphism in traits associated with resistance to herbivory, including plant physical defences  and secondary chemistry (e.g. phenolic compounds30 or coumarins31), few studies have tested for effects of plant sex on terpenes33. In addition, although there are a number of studies measuring the effects of plant sex on quantitative variation in chemical defences, including terpenes33, fewer have tested for effects on compositional variation or assessed the effects of plant sex relative to other sources of genetic variation. As a result, the degree of quantitative or qualitative variation in chemical defences between sexes (i.e. effects of sex on population genetic structure in defences) and the contribution of plant sex to variation in defences relative to total genetic variation associated with defences or that from other sources of ecologically important genetic variation are unknown. Disentangling these different sources of variation and their degree of control over plant phenotypes is important to gain a mechanistic understanding of genetic variation underlying plant chemical defences.A number of studies have reported substantial variation both within and among populations in terpene quantitative profiles, mainly for shrubs and treesBaccharis salicifolia (Ruiz & Pav.) Pers. (Asteraceae) is a woody shrub for which sex is likely genetically determined34. Our previous work with this plant species showed genetic variation in several traits related to growth and reproduction , in the emission of plant volatile organic compounds35, and in arthropod abundance and composition36, as well as sexual dimorphism in several plant traits  and arthropod community composition33. Likewise, in a recent study we also found substantial genetic variation  in leaf terpene amount for this species33. Here we build on these recent findings and provide a more in-depth analysis of terpene variation for these same plants from an experiment consisting of a common garden with male (N = 19) and female (N = 20) genotypes sourced from a single population of B. salicifolia33. Specifically, we quantify leaf mono- and sesquiterpene amount, richness, and diversity (i.e. quantitative profile) as well as compound composition . By replicating multiple genotypes within each sex, we are able to compare the effects of plant sex vs. those due to additional genotypic variation, and in doing so provide a unique assessment of multiple sources of genetic variation not only in quantitative but also qualitative terpene expression.B. salicifolia leaf tissue, of which 25 were monoterpenes and 31 were sesquiterpenes. Of this total, 46 were positively identified  and (E)-β-ocimene   Fig. .Figure 2B. salicifolia leaf samples, but only the latter exhibited significant genotypic variation in quantitative and qualitative profiles and we did not find sexual dimorphism in either terpene group. There was significant genotypic variation in monoterpene amount, diversity and composition (but not richness), with a few noticeable compounds dominating the samples (e.g. limonene and (E)-β-ocimene). In addition, we also found no sexual dimorphism in monoterpene or sesquiterpene amount, richness, diversity and composition. It therefore appears from our comprehensive analyses that plant sex is not a relevant axis of genetic variation in terpene quantitative and qualitative profiles in B. salicifolia. This, however, does not preclude the presence of sexual dimorphism in other chemical  or physical (e.g. toughness) defensive traits of potential importance to plant-herbivore or other types of interactions in this species.Our findings indicated that sesquiterpenes were overall more abundant than monoterpenes in the analyzed 28. Our findings do not support this prediction and add to a growing number of studies reporting inconsistent patterns with either male plants being more highly defended or no difference between sexes (reviewed by Avila-Sakar & Romanow37). For example, Stark and Martz32 found no sexual dimorphism in terpene concentration in shoots of Juniperus communis. Similarly, our previous work based on the same B. salicifolia experimental plants used here indicated weak or non-detectable sexual dimorphism in plant traits associated with growth and reproduction as well as in arthropod community structure associated with this plant33. Although speculative, the observed lack of sexual variation in terpene chemistry for B. salicifolia could have implications for herbivore preference or performance. For example, in another previous study we found that male B. salicifolia plants had higher abundances of the generalist aphid Aphis gossypii, whereas plant sexes did not differ in abundance of the specialist aphid Uroleucon macolai24. If terpene chemistry matters for herbivore preference or performance, taken together, results from that study and our current work suggest that higher A. gossypii numbers on male plants respond to plant chemical  traits other than terpenes that potentially do show differences between plant sexes . In the case of U. macolai, this aphid could be responsive to these compounds such that a lack of sexual dimorphism in terpene chemistry would preclude concomitant variation in this aphid’s abundance, or, alternatively, this aphid could also be affected but overcomes plant sex differences in other defensive traits.Our results indicated no evidence of sexual dimorphism in terpene quantitative or qualitative measures. Theory predicts that female plants should invest more energy in reproduction and defence and less in growth relative to male plantsB. salicifolia genotypes. Previous work has similarly reported variation in the amount of leaf monoterpenes both between and within populations for a number of shrub and tree species38. Several of these studies have found that the amount of these compounds is associated with resistance to insect herbivory41, suggesting a defensive role in plant-herbivore interactions. On the other hand, studies reporting on intraspecific genetic variation in monoterpene diversity are much more limited. One exception is a recent study of ours where we also found that monoterpene diversity exhibited significant variation among populations of Artemisia californica distributed along a latitudinal gradient in California42. Genetic variation in terpene diversity could also be potentially important as a number of studies have shown that greater chemical diversity is associated with increased resistance against herbivores44, and may buffer populations against other sources of biotic stress (e.g. pathogens) or abiotic (e.g. temperature) stress. Although there are a number of studies reporting on phenotypic variation in secondary chemistry in the genus Baccharis , these have not involved explicit assessments of sources of genetic variation in chemical traits47. In this sense, our results provide information on genotypic variation in the quantitative terpene profile for this species. The fact that we detected significant variation in these quantitative traits within a single population warrants future work assessing variation across populations and its potential biotic or abiotic correlates, as well as experimental studies investigating the influence of monoterpene amount and diversity on herbivore resistance.Our results showed that monoterpene diversity and amount exhibited variation among B. salicifolia genotypes. Similarly, a recent work by our group showed significant variation in monoterpene composition across populations of A. californica19. In addition, Thompson et al.48 similarly reported significant variation in monoterpene composition across populations of Thymus vulgaris. It should be noted, however, that despite observing genotypic variation in monoterpene composition, we did not find evidence of distinct ‘chemotypes’ within the studied population as reported for other species, primarily of Mediterranean climate origin50. Genotypic variation in monoterpene composition in B. salicifolia was primarily associated with changes in the relative amounts of two major compounds (limonene and (E)-β-ocimene), which did not separate into distinct genotypic groups but rather exhibited a range of variation in relative abundances across genotypes. Previous studies have reported that these two compounds may act as repellents or toxins to herbivores in woody species53. In particular, we previously found that the emission of both limonene and E-β-ocimene drastically increased after aphid herbivory54 so unaccounted differences in herbivory on our experimental plants could have influenced (via induced responses) observed patterns of genotypic variation in constitutive terpene profiles.We also found significant variation in monoterpene composition among 56, but the magnitude of variation in these compounds appears to be lower compared to monoterpenes57. For example, Sampedro et al.14 reported lower genotypic variation for sesquiterpene amount than for monoterpenes in young trees of Pinus pinaster in north-western Spain. Similarly, previous work of ours indicated that monoterpene but not sesquiterpene richness and diversity varied significantly across populations of A. californica19. In addition, and consistent with quantitative profiles, we found no evidence of genotypic variation in the composition of sesquiterpenes in B. salicifolia. To our knowledge, only two previous studies have tested for intra-specific variation in sesquiterpene composition and, in contrast to our study, both reported significant variation among populations57. In particular, Pratt et al.19 found that sesquiterpene composition in A. californica significantly varied among populations distributed along the Californian coast, whereas Moniodis et al.57 found that sesquiterpene composition in leaves of Santalum spicatum trees significantly varied across populations distributed in arid regions of West Australia. As compared to our study of within-population variation, both of these studies assessed variation among-population, and this distinction may underlie the contrasting results.We found no detectable genotypic variation in any of the quantitative or qualitative measures of variation in sesquiterpenes. With respect to quantitative measures, a number of studies have shown significant intra-specific variation in sesquiterpene amountB. salicifolia and its underlying genetic sources. Additional work involving multiple populations of B. salicifolia, as well as measurements of terpenes in other plant tissues (e.g. flowers), are necessary to reach stronger conclusions about sex variation in terpene chemistry as well as assess the independent effects of different sources  of genotypic variation on terpene expression. For example, there may be genes associated with variation in terpene profiles that are linked to genes that determine sex25, thus appearing spuriously associated. Conducting controlled crosses with different populations to produce segregating progeny would allow for a test of sex by genotype interactions to assess the linkage between sex-related and unrelated genetic variation. In addition, further work involving population variation, e.g. along ecological gradients in herbivory or abiotic variables, would provide a useful next step for identifying relevant factors associated with genetic variation in defensive chemistry in this species. Additional work could involve experimental tests of the effects of such factors on terpene expression and its consequences for insect herbivores. As a whole, the present study points at the need of assessing the independent effects of different sources of genetically-based variation, including plant sex, concurrently shaping plant defensive traits to uncover the mechanistic basis of plant defensive phenotypes. Likewise, our findings also emphasize the importance of increasing the level of detail and comprehensiveness of analyses of chemical traits putatively associated with defences to fully describe complex chemical defensive phenotypes in plants as well as their role in herbivore resistance.Our findings provide an assessment of quantitative and qualitative variation in terpene profiles in Baccharis salicifolia is a perennial, dioecious shrub widely distributed from the desert southwest of the United States and northern Mexico to South America59. It is typically found in riparian areas and mesic microhabitats in high-density monospecific stands, where multiple genotypes co-occur at small spatial scales60. In coastal southern California, B. salicifolia grows and flowers predominantly during the annual winter rains, but may also flower sporadically during the spring and fall. Notably, this species emits large amounts of volatile mono- and sesquiterpenes in all tissues , and previous work suggests that these compounds confer protection against herbivores61 and abiotic stress (e.g. drought47).et al.33. In that study, an experimental common garden was used to characterize sexual dimorphism and genetic variation in B. salicifolia traits and plant-associated arthropod communities. Monoterpene amount was shown to vary nearly 10-fold among 39 plant genotypes, while sesquiterpene amount did not vary significantly, and there was no sexual dimorphism in either compound class. In this study, we provide a detailed analysis of these data on chemical amount, richness, and diversity (i.e. quantitative profile) as well as compound composition .This study provides a detailed analysis of terpene data from Nell B. salicifolia occurring in 80 ha of habitat found within the University of California San Joaquin Marsh Reserve  that was also used in previous work of ours with this species62. In February 2008, we collected cuttings from 20 male and 20 female plants (i.e. genotypes hereafter). To maximize variation among genotypes, we collected cuttings from wild-grown plants that were separated by approximately 900 m. Cuttings were dipped in a 20% solution of Dip ‘N Grow Root Inducing Concentrate , planted in perlite, and kept in a greenhouse for six weeks. We then planted all cuttings in 1 L pots of soil  where they continued to grow for two months. One male genotype did not propagate successfully and was therefore eliminated from the study.We used a source population of B. salicifolia adjacent to the Marsh Reserve. We planted 39 genotypes and replicated each genotype 8–13 times . We randomly distributed plants throughout the common garden in rows and columns with 1 m spacing between them  to 90 μL of each sample extract. We injected the samples (4 μL) onto a gas chromatograph  with a mass spectrometer (MS) detector that was fitted with a 30 m × 0.25 mm × 0.25 μ film thickness DB-5 fused silica column. The GC was operated in splitless mode with helium as the carrier gas (flow rate 1 mL min−1). The GC oven temperature program was: 1 min hold at 50 °C, 5 °C min−1 ramp to 180 °C, 20 °C min−1 ramp to 290 °C, and 1 min hold at 290 °C. The MS was operated in electron ionization mode at 70.0 eV and we collected data between 50–650 m/z. We identified mono- and sesquiterpenes using a NIST Mass Spectral Library and comparing their Kováts indices  analysed under the same chromatographic conditions, with those reported in the literature63. It is important to note that, although our Kováts indices matched well with those previously reported63, terpene compounds should be considered as ‘putative’ until confirmation with standards. For each plant, we estimated the amount of mono- and sesquiterpenes by using normalized peak areas per fresh weight. The normalized peak area per fresh weight of each compound was obtained by dividing their integrated peak area by the integrated peak area of the internal standard and then dividing this value by the leaf fresh weight. To assess the relative abundance of terpenes across plant genotypes and sexes, we also calculated mono- and sesquiterpene diversity for each plant using the Shannon–Weiner index: H’ = −Σ(Pi log[Pi]), where Pi is the relative amount of a given terpene divided by the total terpenes in each plant. Finally, we also recorded the total number of mono- and sesquiterpene compounds (i.e. richness).In November 2011, we collected two fully expanded (undamaged) sun-exposed leaves from half of the replicates (5–6) for each genotype  scale as descriptive statistics.We ran linear mixed models including plant sex and plant genotype nested within sex as fixed factors to test for sexual dimorphism and genotypic variation in richness, diversity and amount of monoterpenes and sesquiterpenes (i.e. quantitative profile). We also included block as a random factor. We ran all analyses with PROC MIXED in SAS 9.4 65 including plant genotype as a fixed factor, constrained by plant sex to control for any effects of sexual dimorphism. A PERMANOVA is analogous to an ANOVA, but partitions similarity matrices between treatments and uses permutation tests with pseudo F-ratios. The PERMANOVA was based on 10,000 permutations using the ‘vegan’ package66 in R software67. To visualize the results of this analysis, we used pairwise Bray-Curtis dissimilarities as input to a principal coordinates analysis. The result of this analysis was then visualized in two dimensions, where each point reflected the genotype centroid. We selected influential terpenes based upon R2 > 0.35 (P < 0.001) for associations with the first two ordination axes (using ‘envfit’ in vegan) and displayed using biplot arrows with length scaled to R2 values.We tested for the effect of plant genotype on mono- and sesquiterpene composition  separately using data on the relative amount of individual compounds for each type of terpene. We used a permutational multivariate analyses of variance (PERMANOVA)2 > 0.60 (P < 0.001) for associations with the first two ordination axes.We used the same procedures described previously using PERMANOVA to test for sexual dimorphism on terpene composition  using genotype least square means. We visualized sexual dimorphism (with ordination) in terpene composition with the two sex centroids as well as the mean values for each male and female genotype displayed on the ordination plot. We selected influential terpenes based upon RSupplementary Material


In this paper, we consider the use of the least absolute shrinkage and selection operator (LASSO)-type regression techniques to detect important genetic or epigenetic loci in genome-wide association studies (GWAS) and epigenome-wide association studies (EWAS). We demonstrate how these techniques can be adapted to provide quantifiable uncertainty using stability selection, including explicit control of the family-wise error rate. We also consider variants of the LASSO, such as the group LASSO, to study genetic and epigenetic interactions. We use these techniques to reproduce some existing results on the Genetics of Lipid Lowering Drugs and Diet Network (GOLDN) data set, which collects from 991 individuals blood triglyceride and differential methylation at 464,000 cytosine-phosphate-guanine (CpG) sites and 761,000 single-nucleotide polymorphisms (SNPs), and to identify new research directions. Epigenome-wide and genome-wide models based on the LASSO are considered, as well as an interaction model limited to chromosome 11. The analyses replicate findings concerning 2 CpGs in carnitine palmitoyltransferase 1A (CPT1A). Some suggestions are made regarding potentially interesting directions for the analysis of genetic and epigenetic interactions. Blood lipids are linked to increased risk of heart disease, heart attack and stroke. Researchers have explored the effects of genetic variation  and epigWe propose an alternative variable screening method that considers a more global approach. Adaptations of the least absolute shrinkage and selection operator (LASSO) selector, first proposed by Tibshirani , have prA popular tool to tackle statistical problems with a large number of covariates is the LASSO. First proposed by Tibshirani , its comy of responses, and a design matrix X, the LASSO estimator is the solution of Ω(β) is a sparsity-inducing prior; for example, Ω(β) = ‖β‖1 = ∑i ∣ βi∣ in the case of the LASSO. Here, λ > 0 is a tuning parameter that trades off between goodness-of-fit and sparsity of the solution.The LASSO and its variants attempt to solve a sparse linear regression problem by optimizing a penalized least-squares objective function. Given a vector l2 penalty in addition to the sparsity-inducing penalty to reduce the effect of correlation among predictors. Such a procedure, called elastic-net, was first introduced by Zou and Hastie . Note that some variables may appear in several groups, which is often called overlapped group LASSO [Even though the LASSO is particularly adapted for high-dimensional problems, finding interactions in such data sets is challenging. Some proposals  adopt aup LASSO –17. We cIn addition, we note that in the case of methylation by genotype interactions, there is a natural biological structure to the problem. By considering the location of the methylation and genetic mutation sites on the chromosome, we may restrict the space of our interactions by requiring them to be colocated on the chromosome.We first consider the analysis of the effect of methylation on blood lipids as in Irvin et al. . We regrWe use an elastic-net and select the penalization by cross-validation on the data set. We subsequently apply the conjugate pair stability selection to 100 random pairs, and obtain the observed selection probability. The procedure is done using a custom-written group LASSO solver implementing block-coordinate descent  in Pythop values and cannot be interpreted as such.We have reproduced in Fig. CPT1A appear as significant in our data, as shown in Table τ = 0.33. We also note that 2 of the CpG sites are contained within genes (SREBF1 and APAOA5) that are related to the regulation of lipid metabolism by the peroxisome proliferator activated receptor α (PPARα) pathway.We observe a result similar to that in Irvin et al. . In partWe now consider the analysis of the association of blood lipids and genotype. The log of the mean fasting TG level is regressed on a feature matrix created from the genotype, controlling for age, smoking, and the scores of the first four principal components of the genotype. The regression is done using the group LASSO with a slight elastic-net penalization selected by cross-validation. Each SNP is coded using indicators that are considered as a group for the purpose of the group LASSO. Conjugate pair stability selection was applied on 100 random pairs, and the observed selection probability of the groups were obtained. The cross-validated penalty yielded an average of 230 selected groups. We produce a similar Manhattan plot as previously . Although this is adequate to model local variations in the methylation mediating gene expression, one may also be interested in taking further steps toward exploring interactions among genetic pathways that are not local to a single chromosome.The constructed feature matrix has 28,285 groups of size 1, corresponding to the methylation loci, 36,796 groups of size 2, corresponding to the SNP loci, and 190,073 groups of size 5, corresponding to the interactions of nearby methylation and SNP loci.The log of the mean fasting TG level is regressed on the feature matrix described above using the group LASSO with the described structure additionally controlling for age, smoking, the scores of the first four principal components of the methylation, and the scores of the first four principal components of the genotype.Figure We show that sparse regression techniques can be applied to tackle the problems of selecting variables in high-dimensional data sets as often arise in genome-wide or epigenome-wide association studies, including selecting variables in models containing interaction terms. These techniques enable us to improve on the power of multiple F-tests in high dimension as they adopt a more global, instead of locus-by-locus, point of view. The effects seem to appear in a clustered fashion, and further analyses could attempt to leverage the existence of such structure.


Findings can be investigated using a web application (http://mrcieu.mrsoftware.org/PRS_atlas/), which we envisage will help uncover both known and novel mechanisms which contribute towards disease susceptibility. To demonstrate this, we have investigated the results from a phenome-wide evaluation of schizophrenia genetic liability. Amongst findings were inverse associations with measures of cognitive function which extensive follow-up analyses using Mendelian randomization (MR) provided evidence of a causal relationship. We have also investigated the effect of multiple risk factors on disease using mediation and multivariable MR frameworks. Our atlas provides a resource for future endeavours seeking to unravel the causal determinants of complex disease.The age of large-scale genome-wide association studies (GWAS) has provided us with an unprecedented opportunity to evaluate the genetic liability of complex disease using polygenic risk scores (PRS). In this study, we have analysed 162 PRS (p<5×10 An individual’s risk of developing many diseases, including heart disease and schizophrenia, is influenced by a complex combination of lifestyle factors and the genes they inherit at birth. The total number of genetic variants that an individual has that increases their risk of developing a particular disease can be measured as their ‘polygenic risk score’. These scores allow researchers to predict whether it is likely that someone will develop a disease during their lifetime.Polygenic risk scores can also be used to link different conditions or traits to each other. For example, if high blood pressure can be caused by obesity, then genetic variants linked to obesity will also influence blood pressure. As a result, individuals with a high polygenic risk score for obesity will, on average, have a higher blood pressure than those with a low score. Comparing associations between polygenic risk scores and traits can therefore suggest whether one trait causes another.Richardson et al. have developed an ‘atlas’ that uses data from the UK Biobank study – which contains genetic data from over 300,000 people – to investigate how shared characteristics and risk factors in individuals relate to their genetic likelihood of developing a disease. The data currently includes 162 different polygenic risk scores and 551 traits.Richardson et al. used the atlas to evaluate which traits are most strongly linked to the polygenic risk score for schizophrenia. Analyses of these traits suggested that individuals with a high genetic risk of developing schizophrenia tend to perform worse in IQ and short-term memory tests, and that they are less likely to successfully quit smoking. These characteristics have previously been observed in studies of individuals with schizophrenia.In the future, the atlas could be used to identify possible relationships between a wide range of individual traits and diseases. This could help to prioritise which relationships should be investigated further as part of studies to understand the causes and consequences of disease. In the long term, such studies should improve our ability to prevent and treat many different medical conditions. Developing our understanding of how modifiable social, behavioural and physiological factors influence risk of disease is of vital importance to improve effective medical treatment and preventative interventions . Genetic−08), which may lack statistical power if these variants do not explain a large proportion of trait variance. In contrast, PRS derived using a more lenient threshold (e.g. p<5×10−05) can help recover some of this missing heritability due to a larger number of SNPs being included. This may help improve detection rates for causal relationships, which can be particularly useful when evaluating associations between genetic liability for a given trait and hundreds of diverse health outcomes. Such endeavours are commonly referred to as phenome-wide association studies  method ) Figure. Results−05) using findings from large-scale GWAS and evaluated their association with 551 traits in up to 334,398 individuals enrolled in the UK Biobank study  based on r2< 0.001 using genotype data from European individuals (CEU) from phase 3 (version 5) of the 1000 Genomes project .Overall, we undertook 89,262 tests to investigate the association between 162 different PRS derived from GWAS  and 551  project . As oppoOf the 162 GWAS we identified, 11 reported that they included UK Biobank participants in their analysis. As this may lead to overfitting, the PRS for these 11 traits were not weighted to reduce this source of bias. To demonstrate this, we evaluated the association of the sleep duration PRS in the UK Biobank study, weighting SNPs based on a GWAS involving the interim release of this dataset  Supplem. However−05 threshold. However, analyses have been repeated using scores derived using the conventional GWAS threshold of p<5×10−08 for future studies that wish to evaluate these results. Complex traits from the UK Biobank study were selected based on p<0.05 from previously undertaken heritability analyses within this study  PRS found that individuals had increased odds of 3.64 to develop this disease (based on the ICD10 code ‘I25’). A recent study by Khera and colleagues  reportedCombining this PRS with scores for established causal risk factors for CHD suggested that they can help improve polygenic prediction (namely low density lipoprotein (LDL) cholesterol and myocardial infarction), although integrating any associated scores in a hypothesis-free manner may hinder prediction . This coAmongst other findings, we observed that participants had increased odds of 2.43 in terms of obtaining a University or College degree when comparing top and bottom deciles for the years of schooling PRS. Other noteworthy examples included a 3.48 fold increase in odds of taking atorvastatin as medication when comparing the extreme deciles for the LDL PRS. We also observed that participants in the highest decile for the ulcerative colitis PRS had increased odds of 5.36 in terms of developing this disease in comparison to those in the lowest decile (based on the ICD10 code ‘K51’).−08 as instrumental variables to reduce the likelihood of weak instrument bias in our analysis   method aIf the IVW method provides strong evidence of association but in the presence of heterogeneity, we suggest undertaking two additional MR analyses using the weighted mode  and weigAs a sensitivity analysis, repeat steps 1 and 2 but only using SNPs as instruments which are not filtered out by applying the MR directionality test . We also−50). The schizophrenia PRS was also strongly associated with various neurological traits, such as neuroticism , being ‘tense or highly strung’  and self-reported depression .The top association with the schizophrenia PRS suggests that individuals with high schizophrenia genetic liability have increased odds of seeing a psychiatrist at some point in their lives due to nerves, anxiety, tension of depression . We also observed evidence that these individuals performed worse than others in an assessment of cognitive function concerning memorising pairs of cards . Follow-up MR analyses provided evidence from multiple methods that schizophrenia genetic liability (i.e. our exposure) influences both of these outcomes  and, accordingly reduced odds of being a past smoker . We observed strong evidence of association that schizophrenia genetic liability influences these outcomes  provided weak evidence of a causal effect in both directions . This reAnother strength of our atlas is that findings can be evaluated by selecting an outcome of interest and evaluating which of the 162 PRS are most strong associated with it. Doing so may motivate future endeavours to investigate the effect of multiple risk factors on disease risk. As a demonstration of this, we have evaluated the associations between all PRS and self-reported gout in the UK Biobank study . In thisA receiver operating characteristic plot  illustra−08 for each PRS were used as instrumental variables in MR analyses. There was strong evidence that BMI (i.e. our exposure) had a causal effect on each other trait in turn , where effect estimates appeared to be consistent between different MR methods  and the null model . We ran 1000 simulations using each model to compare the PRS approach with the IVW method using a dataset comprising of 10,000 samples and 50 SNPs. Further details and all the code used to conduct these simulations can be found at https://github.com/explodecomputer/prs-vs-mr.We have used the MR-Base platform  to ident−05. A threshold of r2 <0.001 was selected to identify independent SNPs using genotype data from European individuals (CEU) from phase 3 (version 5) of the 1000 genomes project  of one. Our PRS construction pipeline was also applied using a more stringent threshold of p<5×10−08. Although we have not interpreted any of the results using these more stringent scores in this report, they are available within our atlas for future use.PRS were constructed using SNPs for each GWAS trait based on p<5×10 project . When a K-means clustering (K = 4) were excluded from analysis. After exclusions there were up to 334,398 individuals with both genotype and complex trait data who were eligible for analysis.We selected traits from the UK Biobank study  which haWe evaluated the association between each combination of PRS and complex trait in the UK Biobank study using linear regression (for continuous traits), logistic regression (for case/control traits), ordinal logistic regression  and multinomial logistic regression . All analyses were adjusted for age, sex, the first 10 genetic principle components (to adjust for population stratification) and genotyping chip used to measure genetic data in participants. Only female participants were included in the ‘Age at menarche’ and ‘Age at menopause’ PRS analyses.2 coefficients for continuous traits and McFadden pseudo R2 coefficients for other models by repeating analyses unadjusted for covariates. McFadden’s R2 is defined as:We also calculated R2McF= 1 – ln(Lm/ln(L0)R0L is the value of the likelihood function of the model with no predictors and mL is the likelihood of the model being estimated. We note that pseudo R2 coefficients should not be interpreted in a similar manner to those derived using linear regression .We used various two-sample MR methods to evaluate associations detected in the PRS analysis. This involved using the observed effects of the genetic variants used in the PRS on both the GWAS trait that the score was based on  as well as the UK Biobank trait . For all MR analyses we only selected SNPs with p<5×10ent bias . In terment bias , weighedent bias  and weigent bias  was used−05 threshold. We note however that it is only advisable to apply techniques in MR using this threshold as long as in-depth sensitivity analyses  are also undertaken to robustly evaluate findings.To investigate the direction of effect for associations identified in the PRS analysis we undertook bi-directional MR . This inWhen undertaking our example of mediation MR in this study, we also calculated the proportion mediated along the causal pathway from exposure to outcome using effect estimates derived using the IVW method, where:Proportion mediated = The direct effect here is the IVW effect estimate derived for the association between the exposure (i.e. BMI) and our outcome (i.e. gout). The indirect effect was calculated as the product of all IVW effect estimates derived for all relationships along the causal pathway of interest .All analyses were undertaken using R (version 3.5.1). The R package ‘shiny’ v1.1 was used to develop the web application and ‘highcharter’ v0.5 was used to generate interactive plots. Figures in this manuscript were generated using ‘ggplot2’ v2.2.1.http://mrcieu.mrsoftware.org/PRS_atlas/). Our dataset was derived from the UK Biobank study as part of projects 8786 and 15825. The same dataset can be created with an application to use data from the UK Biobank study .All summary statistics for the analyses undertaken in this study can be downloaded using our web application . It would be good to go through the paper and make sure the causal relationships are described sufficiently.- Some of the PRSs were not weighed because the GWAS included the "interim release" of UKBiobank. The authors argue that without the weighing, they avoid overfitting of the PRS. It would be good to indeed demonstrate that; i.e. test a weighed PRS on the second part of the UKBiobank that was not part of the GWAS and compare it to the unweighted PRS in the full UKBiobank.- In the last paragraph of the subsection “An atlas of polygenic risk score associations across the human phenome”, the results for CHD are being discussed. While the ORs are similar as reported before, the AUC-ROC for CAD is much lower than reported in the Khera et al. paper; i.e. 0.60 (here) vs. 0.85 . Given the high profile of the Khera et al. results it would be worth explaining what this difference might be due to - likely because Khera et al. included age and sex in their model.- The fact that multiple PRSs reduce the predictive ability is interesting and probably worth a paper on its own. Has this been tested for other outcomes or has this been documented elsewhere? It currently leaves the readers a little on their own without a clear answer to why this might be.- The authors conclude that schizophrenia may lead to reduced cognitive function. However, in their results they state that there is some evidence for the opposite effect. This may need clarification.- How did the authors estimate that 77% of the effect of BMI on gout was mediated ?With reference to the atlas:- I was surprised to see that a high PRS for adiponectin was associated with high adiposity. I assume that this unexpected positive correlation is because the PRS was calculated based on BMI-adjusted adiponectin. I believe there are other traits that we adjusted (for BMI). It should be made clear in the browser which traits, as it changes the interpretation.Punctuation throughout the text could be improved.Reviewer #2:Richardson and colleagues develop an atlas of associations of 162 polygenic scores and 551 traits. They explore this atlas and develop two stories around causal inference for schizophrenia and gout.Strengths:1) The atlas may be useful for investigators to explore;2) There is a web application that will facilitate exploration of the results.Major limitations:1) From their exploration of this resource, they highlight two stories. The significance of the biologic insights presented is incremental and limited. As such, should the work be presented as a software or web resource rather than a publication of a scientific observation?Minor limitations:1) Discussion can be trimmed and simplified.Reviewer #3:Richardson et al. present a comprehensive study on an atlas of polygenic risk score associations to highlight putative causal relationships across the human phenome. The authors present a well qualified study pinpointing to the limitations of such an approach for investigating truly causal relationships.Regarding this manuscript I only have minor comments.1) Why only 334,398 individuals in the UK Biobank study?2) Simulation study details can be provided in main text. It was hard to follow what exactly was done in the simulation study.3) The first paragraph of the subsection “Elucidating risk factors which may play a mediating role along the causal pathway to disease”, contains a speculative statement about how likely it is representative of other findings within the atlas, where the PRS for the disease of interest may not always necessarily be the best polygenic predictor of it. Can you provide another example? Reviewer #1:[…] The paper illustrates how the atlas can be used with a few examples. While generally well written, I do have the following suggestions for the paper:- Throughout the paper, while the authors do a good job reporting the statistics and various methods, the readers are often left on their own with the final interpretation. E.g. often, the authors will state that there is evidence of a causal relationship, but from the sentence, it is not clear what the exposure and what the outcome is (not everyone will want to check the supplementary tables). It would be good to go through the paper and make sure the causal relationships are described sufficiently.Many thanks for this suggestion. We have added detail throughout the paper to clarify which exposures and outcomes are being evaluated when undertaking MR analyses. This includes some of the following additions:‘Furthermore, in these analyses we model schizophrenia as our exposure within an MR framework with associated complex traits as outcomes (unless stated otherwise).’‘Along with using MR to investigate the effect of PRS traits on outcomes, we recommend investigating the converse direction of effect where possible  influences both of these outcomes (Supplementary file 1G).’‘There was strong evidence that BMI (i.e. our exposure) had a causal effect on each other trait in turn , where effect estimates appeared to be consistent between different MR methods (Supplementary file 1N).’- Some of the PRSs were not weighed because the GWAS included the "interim release" of UKBiobank. The authors argue that without the weighing, they avoid overfitting of the PRS. It would be good to indeed demonstrate that; i.e. test a weighed PRS on the second part of the UKBiobank that was not part of the GWAS and compare it to the unweighted PRS in the full UKBiobank.We thank the reviewer for this excellent suggestion. We firstly thought it was important to emphasise in the paper that unweighted PRS cannot completely avoid overfitting, but rather as an approach to reduce this source of bias. As such we have included an additional sentence to clarify this. Furthermore, we have also undertaken the analysis suggested by the reviewer, using the PRS for sleep duration as an example. These results can be found in Supplementary file 1C and demonstrate the problems with overfitting due to overlapping samples, as well as how using an unweighted score in these scenarios can help mitigate this limitation:‘Of the 162 GWAS we identified, 11 reported that they included UK Biobank participants in their analysis. […] In case they are still useful for follow-up analyses despite overlapping with UK Biobank, these scores have been clearly flagged in Supplementary file 1 by being allocated to the ‘unweighted’ subcategory.’- In the last paragraph of the subsection “An atlas of polygenic risk score associations across the human phenome”, the results for CHD are being discussed. While the ORs are similar as reported before, the AUC-ROC for CAD is much lower than reported in the Khera et al. paper; i.e. 0.60 (here) vs. 0.85 . Given the high profile of the Khera et al. results it would be worth explaining what this difference might be due to - likely because Khera et al. included age and sex in their model.We agree with the reviewer that this would be a valuable addition to the manuscript. We have added the following sentence:‘A recent study by Khera and colleagues reported a similar odds ratio for CHD in their analysis . However, we note that they identified a higher area under curve in their analysis (0.806), which is likely attributed to tuning parameters such as LD clumping, along with covariates adjusted for their analysis.’- The fact that multiple PRSs reduce the predictive ability is interesting and probably worth a paper on its own. Has this been tested for other outcomes or has this been documented elsewhere? It currently leaves the readers a little on their own without a clear answer to why this might be.Again we thank the reviewer for this point. We have acknowledged that this work deserves further evaluation by future studies, following on from some possible explanations for the reader to why this result was observed:‘This could potentially due to the increase in variance incorporated into prediction analyses from scores that do not directly influence CHD, or alternatively may indicate that they are spurious associations. Additional research is required to evaluate the contribution of multiple PRS as predictors of a single outcome. Doing so may help develop a greater understanding into which traits can help predict disease outcomes using PRS.’- The authors conclude that schizophrenia may lead to reduced cognitive function. However, in their results they state that there is some evidence for the opposite effect. This may need clarification.Thank you for this suggestion. On the Results section we mentioned that there is ‘weak evidence of a causal effect in the opposite direction’. However, by this we mean that the results suggest that there is unlikely to be an effect in the opposite direction. To add some clarity to this point, we have rephrased this sentence as:‘In contrast, we did not detect strong evidence of a causal effect in the opposite direction for these associations , in particular after applying MR directionality filtering and when evaluating results from the weighted median and mode methods (Supplementary file 1H).’- How did the authors estimate that 77% of the effect of BMI on gout was mediated?Many thanks for this comment. We have added the following to the Materials and methods section of the paper to help clarify how we calculated the proportion mediated:‘When undertaking our example of mediation MR in this study, we calculated the proportion mediated along the causal pathway from exposure to outcome using effect estimates derived using the IVW method, where:Proportion mediated = directeffect-indirecteffectdirecteffectThe direct effect here is the IVW effect estimate derived for the association between the exposure (i.e. BMI) and our outcome (i.e. gout). The indirect effect was calculated as the product of all IVW effect estimates derived for all relationships along the causal pathway of interest .’With reference to the atlas:- I was surprised to see that a high PRS for adiponectin was associated with high adiposity. I assume that this unexpected positive correlation is because the PRS was calculated based on BMI-adjusted adiponectin. I believe there are other traits that we adjusted (for BMI). It should be made clear in the browser which traits, as it changes the interpretation.We are thankful to the reviewer for this suggestion. They are correct that the PRS for adiponectin was based on a GWAS which adjusted for BMI. We have therefore added this information to Supplementary file 1 and the web application as recommended.Punctuation throughout the text could be improved.Many thanks for this point. One of my co-authors has kindly gone over the manuscript to help improve the punctuation.Reviewer #2:[…] Major limitations:1) From their exploration of this resource, they highlight two stories. The significance of the biologic insights presented is incremental and limited. As such, should the work be presented as a software or web resource rather than a publication of a scientific observation?Many thanks for this suggestion. As discussed with the editor, we have now submitted this work as tools and resources paper.Minor limitations:1) Discussion can be trimmed and simplified.We thank the reviewer for this point. We have edited sections of the Discussion and subsequently reduced its overall length.Reviewer #3:[…] Regarding this manuscript I only have minor comments.1) Why only 334,398 individuals in the UK Biobank study?Thank you for raising this point as we realise it is not clarified in the main text of the paper. As such we have added the following sentence to address this:‘Our final sample size for analysis consisted of 334,398 individuals. This was determined using a strict exclusion criterion to reduce false positive associations, removing individuals with withdrawn consent, evidence of genetic relatedness or who were not of ‘white European ancestry’ based on a K-means clustering (K=4).’2) Simulation study details can be provided in main text. It was hard to follow what exactly was done in the simulation study.Thanks very much for this suggestion. We have added some further details regarding the simulation study to the Materials and methods. Furthermore, we have added a link to the GitHub repository with detailed methods and all the code necessary to run the simulations:‘Our simulation study concerned two different models; the causal model  and the null model . […] Further details and all the code used to conduct these simulations can be found at https://github.com/explodecomputer/prs-vs-mr.’3) The first paragraph of the subsection “Elucidating risk factors which may play a mediating role along the causal pathway to disease”, contains a speculative statement about how likely it is representative of other findings within the atlas, where the PRS for the disease of interest may not always necessarily be the best polygenic predictor of it. Can you provide another example?We thank the reviewer for raising this point, and on reflection feel that this statement may have been a little strong and have therefore toned it down. We have also provided another example where the strongest association for a PRS is a risk factor instead of outcome itself:-249).”Β=0.080, SE=0.002, P=1.31x10“Although many of the PRS in our analysis may be the best polygenic predictors for their target disease/trait, there may be other examples similar to this where the strongest association for an outcome is not the corresponding PRS. For example, the strongest association for birth weight as an outcome in our atlas was with the height PRS (


Methods for analysis of GWAS summary statistics have encouraged data sharing and democratized the analysis of different diseases. Ideal validation for such methods is application to simulated data, where some ‘truth’ is known. As GWAS increase in size, so does the computational complexity of such evaluations; standard practice repeatedly simulates and analyses genotype data for all individuals in an example study.We have developed a novel method based on an alternative approach, directly simulating GWAS summary data, without individual data as an intermediate step. We mathematically derive the expected statistics for any set of causal variants and their effect sizes, conditional upon control haplotype frequencies (available from public reference datasets). Simulation of GWAS summary output can be conducted independently of sample size by simulating random variates about these expected values. Across a range of scenarios, our method, produces very similar output to that from simulating individual genotypes with a substantial gain in speed even for modest sample sizes. Fast simulation of GWAS summary statistics will enable more complete and rapid evaluation of summary statistic methods as well as opening new potential avenues of research in fine mapping and gene set enrichment analysis.http://github.com/chr1swallace/simGWAS.Our method is available under a GPL license as an R package from Bioinformatics online. P values) more readily than the full genotype data. A wealth of methods have been developed to operate directly on the summary statistics, from fine mapping of genetic causal variants [e.g. PAINTOR . This approach, in particular, can be used very flexibly for generating multiple (quantitative) phenotypes, a design also common to Biobank datasets . HoweverInstead, when simulating case–control data, we typically simulate or sample genotype data conditional on a supposed distribution of phenotypes. Simulation options in this case are more limited because the problem is mathematically harder. For single causal variant scenarios, resampling from a reference population conditional on allele frequencies at a target variant may be used. For more complicated causal models, involving multiple variants potentially in LD, GWAsimulator , TriadSiThe general approach of simulating both genotype and phenotype on an individual level cannot scale well for Biobank-scale or large meta-GWAS situations, because of the number of individuals required. It is also potentially wasteful—the individual level data are not required when the goal is to evaluate methods that work on summary statistics.Here, we present an alternative approach, which simulates summary statistics directly, without needing to ever generate genotype data. It scales as a function of the number of SNPs, but is constant with regards to the number of samples, thus making it ideally designed for simulation of summary statistics for large case–control studies.Z score from a Cochran Armitage score test under an additive model at each SNP in the region. [Cochran Armitage score tests have been used for GWAS because of their computational simplicity, requiring no iterative maximization procedure, and because they allow for additive, dominant or recessive coding, although additive coding is the most commonly used . Let n be the total number of SNPs. For any SNP X, write i. Then, for the commonly used Cochran-Armitage score test, the Z-Score at SNP X is computed as:XV, YV denote Let iY given γ0 is chosen such that N0 controls and N1 cases and thus perform the conditional simulation needed for case–control studies.Write Y, we obtain the expected value of XU, the covariance between XG and Y:By conditioning upon the values of XU is XV is the variance of X.G As XV is a variance, a natural model is an inverse gamma distribution, Y, we show that the parameters of this distribution areXV are given in the XV from its distribution or calculateThe variance of Z score, Putting this together, we can now calculate the expected Equation . Note thP value is not the P value associated with the expected Z score. Instead, we must simulated ‘observed‘ GWAS results which vary randomly about Z score at two SNPs is equal to the correlation between their genotypes  and the same settings with HAPGEN2 + SNPTEST2, using reference data from 1000 Genomes Phase 3 , both methods were very fast (<30 s) for the simplest scenario of 1000 cases and 1000 controls. We found that both methods required slightly, but negligibly, more time as the number of causal variants increased from one to six a. As expSimulating GWAS summary statistics in the context of case–control studies, for any required causal model and set of odds ratios, has several potential applications. Primarily, simulated GWAS results have become the accepted gold standard for validating newly developed statistical models for the analysis of GWAS data. Our intent is to enable the faster simulation of summary statistics compared to individual level data simulation, while at the same time using considerably less disk space. Although the method focuses on region-level simulation, it can be used to generate genomewide statistics if required, by breaking the each chromosome into approximately independent blocks, according to recombination hotspots or breakpoints derived from examining correlation between genotypes .not additive is an interesting research question. A future direction to extend our method could be to adapt it to simulate data under any genetic model by expressing disease risk as a function of genotypes at causal SNP haplotypes. While we have focused on retrospective case–control designs as an obvious gap in the GWAS simulation toolbox, our methodology could be relevant in the area of extreme-sampling designs, where power is maximized for a fixed cost by sampling individuals with extreme values of a quantitative trait, for example in a study of blood pressure . The second seems more appropriate, because it acknowledges that enrichment tests are performed in the context of genome-wide significant associations having been already found. However, the second is also much harder to simulate.In addition to supporting method development, simulation of GWAS statistics is also used in tests aggregating information across sets of SNPs, e.g. for pathway analysis. Pathway analysis can test either the A common technique for simulating under a competitive null is permutation testing; the underlying dataset is maintained, and labels are permuted to generate new datasets where traits are still associated, but there is no possible correlation to the feature of interest. However, doing this so as not to destroy the genomic structure within the region, can require inventive generation of null distributions, for example, by circularization and permutation of genomic features to allow empirical null distributions to be calculated under a competitive null (P values for individual genes by their ranks (To allow more simple simulation techniques to be used, pathway-based tests of the competitive null have been adapted to have the same expected null distribution as tests of the global null. This requires replacing There is therefore potential to further develop pathway or enrichment test methodology if the distribution of test statistics under a competitive null hypothesis could be derived. Our method would naturally allow simulation of GWAS summary data under a specific hypothesis about the location and magnitude of genetic effects, in order to generate empirical null distributions for tests of the competitive null, preserving genomic structure even when analysis is performed across multiple regions.Finally, our method could be used to evaluate output of fine-mapping applied to real data. Particularly in regions where the patterns of LD between putative associated SNPs are complex, it can be hard to dissect what the true causal variants are. Different fine mapping methods make different assumptions about the number and independence of causal variants, which can produce conflicting results (Our method enables faster simulation of GWAS case–control summary statistics compared to individual level data simulation, at the same time using considerably less disk space. This should facilitate computationally simpler evaluation of existing and new summary GWAS methods and has the potential to underpin new method development in other areas.bty898_Supplementary_InformationClick here for additional data file.bty898_Supplementary_TableClick here for additional data file.


Current genomics methods are designed to handle tens to thousands of samples but will need to scale to millions to match the pace of data and hypothesis generation in biomedical science. Here, we show that high efficiency at low cost can be achieved by leveraging general-purpose libraries for computing using graphics processing units (GPUs), such as PyTorch and TensorFlow. We demonstrate > 200-fold decreases in runtime and ~ 5–10-fold reductions in cost relative to CPUs. We anticipate that the accessibility of these libraries will lead to a widespread adoption of GPUs in computational genomics. Current methodologies for analyzing genomic data were designed for datasets with tens to thousands of samples, but due to the continuing decrease in sequencing costs and growth of large-scale genomic projects, datasets are reaching sizes of millions of samples or single cells. The need for increased computational resources, most notably runtime, to process these growing datasets will become prohibitive without improving the computational efficiency and scalability of methods. For example, methods in population genetics, such as genome-wide association studies (GWAS) or mapping of quantitative trait loci (QTL), involve billions of regressions between genotypes and phenotypes. Currently, the state-of-the-art infrastructures for performing these tasks are large-scale clusters of central processing units (CPUs), often with thousands of cores, resulting in significant costs   QTL mapping , 10 whi. For SigTo further demonstrate the scalability of the Bayesian NMF to millions of data points, we used SA-GPU to identify the cell types and their associated transcriptional programs from single-cell RNA sequencing of 1 million mouse brain cells . Our implementation of cis-QTL mapping with permutations to estimate the empirical false discovery rate was > 250 times faster than the current state-of-the-art implementation  took less than 10 min, a ~ 200× increase in speed compared to running on a CPU  to efficiently read genotypes stored in PLINK [The core of tensorQTL is a reimplementation of FastQTL  in Tensoin PLINK  format iin PLINK .Cis-QTL: nominal associations between all variant–phenotype pairs within a specified window (default ± 1 Mb) around the phenotype (transcription start site for genes), as implemented in FastQTL.Cis-QTL: beta-approximated empirical p values, based on permutations of each phenotype, as implemented in FastQTL.Cis-QTL: beta-approximated empirical p values for grouped phenotypes; for example, multiple splicing phenotypes for each gene, as implemented in FastQTL.cis-QTL, following the stepwise regression approach described in [Conditionally independent ribed in .Interaction QTLs: nominal associations for a linear model that includes a genotype × interaction term.Trans-QTL: nominal associations between all variant–phenotype pairs. To reduce output size, only associations below a given p value threshold (default 1e−5) are stored.Trans-QTL: beta-approximated empirical p values for inverse-normal-transformed phenotypes, in which case the genome-wide associations with permutations of each phenotype are identical. To avoid potentially confounding cis effects, the computation is performed for each chromosome, using variants on all other chromosomes.The following QTL mapping modalities are implemented:trans-QTL mapping performance on a machine with and without an attached GPU, and cis-QTL mapping relative to the CPU-based FastQTL [cis-window using the --include-phenotypes and --region options, respectively. The cis-mapping comparisons were performed using skeletal muscle data from the V6p release of GTEx [trans-QTLs across a wide range of sample sizes, we used randomly generated genotype, phenotype, and covariate matrices. All tensorQTL benchmarks were conducted on a virtual machine on Google Cloud Platform with 8 Intel Xeon CPU cores (2.30 GHz), 52 GB of memory, and an Nvidia Tesla P100 GPU. For CPU-based comparisons, computations were limited to a single core.To benchmark tensorQTL, we compared its  FastQTL   to fit into the GPU memory; however, since high-memory GPUs are readily available , we do not foresee this limiting its practical use. In case data sizes were to exceed this limit, the method is easily extensible to multiple GPUs using shared memory with built-in PyTorch methods.SA-GPU is a PyTorch reimplementation of SignatureAnalyzer , a methoSA-GPU can run a single Bayesian NMF or an array of decompositions in parallel, leveraging multiple GPUs. Users should specify a data likelihood function (Poisson or Gaussian) and either exponential or half-normal prior distributions on the elements of W and H, corresponding to L1 or L2 regularization, respectively.To benchmark the performance of SA-GPU, we compared SA-GPU with the previous implementation in R. We ran the R implementation using R 3.2.3 with the “Matrix” package for efficient matrix operations. All SA-GPU benchmarks were conducted on a virtual machine on Google Cloud Platform with 12 Intel Xeon CPU cores (2.30GHz), 20 GB of memory, and a Nvidia Tesla V100 GPU. For CPU-based comparisons, a single core was used.Additional file 1:Figure S1. Performance of matrix multiplication on a single CPU core (2.30GHz Intel Xeon) vs. a GPU (Nvidia Tesla P100), using NumPy (compiled with OpenBLAS) and PyTorch. Runtimes were measured for multiplication of two random  square matrices (in 32-bit floating point) with the indicated dimensions. For the ‘PyTorch GPU’ runtimes, only the matrix multiplication itself was timed. For the ‘PyTorch GPU w/ copy’ runtimes, the copy of the two input matrices from CPU to GPU memory was included in the timing. Runtimes are shown as the median and median absolute deviation of 15 iterations each. Figure S2. Performance scaling of SignatureAnalyzer-GPU. (a) SignatureAnalyzer-GPU runtime scales linearly as a function of the number of samples. (b) Cumulative runtime for 20 runs of SignatureAnalyzer-GPU on a virtual machine configured with one or two GPUs (Nvidia Tesla V100). (c) Average number of signatures detected with one or two GPUs, indicating that the results are equivalent. The PCAWG mutation counts matrix was used for all comparisons. Error bars: standard deviation. Figure S3. GPU performance scaling of tensorQTL. (a) GPU-to-CPU runtime ratio for tensorQTL, across the indicated phenotype and sample sizes, for 107 common variants. The ratio is non-constant due to data load and CPU-to-GPU memory input/output times (“i/o”) that are more limiting for large sample sizes . (b) CPU runtime of tensorQTL for the indicated range of sample and phenotype sizes (left panel). CPU runtimes scale linearly, demonstrated by the collapse of the compute time when measured as a function of number of operations , whereas GPU runtimes do not show this collapse for large sample sizes due to i/o limitations (right panel). (c) Nominal significant trans-eQTL p values from the V6p GTEx release replicated using tensorQTL.Additional file 2. Benchmarking code from Additional file Additional file 3. Review history.