Model overview

BAGEA models gene expression as dependent on SNP genotypes in cis. In general, SNP effects on gene expression depend on both directed and undirected annotations (Fig 1A). BAGEA builds predictors of gene expression and ranks annotations by their impact on gene expression. For every gene j, BAGEA takes as input a genotype matrix X_j, an expression vector y_j, annotation matrices V^j, F^j and C_j. X_j has dimensions (n × m_j), where n is the number of individuals assayed, and m_j is the number of SNPs in cis of gene j’s TSS. The matrices V^j F^j and C_j are of dimensions (m_j × s), (m_j × q), and (m_j × t) respectively, where s, q and t are the number of annotations used. BAGEA models gene expression as a linear combination of SNP genotypes: (1) where ϵ_j is an i.i.d normal noise vector and b_j is a vector of SNP effects. The effect of SNP i on gene j b_ij is modeled as: (2) Detailed descriptions of the terms are as follows:

• encodes the directed annotaton for SNP i in the region of gene j. It is the ith row of V^j. In our applications of BAGEA, V^j is previously computed from sequence-based models, where column of V^j represents an epigenetic mark and each row V^j represents a SNP. Each entry in V^j expresses the predicted effect of a genotype change on the epigenetic mark in question.
• encodes the undirected annotations for SNP i in the region of gene j. It is the ith row of F^j. Each element in F^j expresses the presence or absence of the annotation at a SNP’s location. In our applications of BAGEA, F^j is derived from the relative positions of a SNP and gene j’s TSS, where each column represents a particular region around the TSS. For example, if a column F^j encodes a region of 20 kilobases (KB) upstream from the TSS, all entries for rows corresponding to SNPs within 20 KB upstream of that TSS will be set to 1 and entries for all other rows will be set to 0. By default, the first column of F^j is an intercept column consisting only of ones.
• ω and ν are vectors of lengths s and q respectively that are estimated by BAGEA. Specifically, ω and ν are the effects of annotations in F^j and V^j on the SNP effects b_j. By default, the effect of the interecept weight ν₁ is fixed at close to 1 via constraining priors.
• α_ij is a scalar estimated by BAGEA and models the variance of b_ij. Allowing different variances for the elements of b_j typically produces sparse estimates for b_j with many elements close to zero as integrating out the Gamma distributed prior α_ij yields a t-distribution for the different b_j [18]. Further, α_j, the vector of length m_j with α_ij its ith element, is modeled as dependent on the undirected annotation matrix C_j. C_j can potentially be identical to F^j but can model different undirected annotations as well (see Method Details).

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Fig 1. Illustration of BAGEA model components.

(A) The core components of the BAGEA model in the summary statistics formulation. Observed variables are in squares while estimated variables are circled. Given are z_j, the eQTL z-scores for gene j, as well as the LD matrix Σ_j, defining the correlation between summary statistics. Further, z-scores are influenced by the true eQTL effects b_j. These effects in turn depend on directed and undirected annotations, V^j and F^j respectively. While undirected annotations can cover regions of any size, directed annotation have the same size as the genomic variants themselves. The impact of annotations on b_j is estimated from the data via ω and ν. (B) An example of the modeling of different priors of elements of ω using meta-annotations via υ variable vectors. We assume that directed annotations are available for nine annotations, which were derived from tissues Liver, Blood and Brain via 3 assay types DHS, H3K27ac and H3K4me3. It is reasonable to assume that for a given eQTL study, particular tissues or cell types are more relevant than others. We model this by introducing a variable υ for each tissue (or cell type) that affects the prior distribution of only those elements of ω that are derived from this tissue, e.g. υ_Liver only affects elements of ω tied to experiments performed in liver. We model different priors for various for assay types analogously. Shown is the resulting network of influences of the variable υ_tissue, υ_assay on ω. (We used the actual group names as indices, while in the main text, elements of υ’s and ω are indexed by natural numbers).

https://doi.org/10.1371/journal.pcbi.1007770.g001

Typically, directed annotations are grouped by their cell type or assay type. BAGEA can use this grouping structure in order to select groups of annotations that are useful for predicting gene expression (Fig 1B). BAGEA selects annotation groups via a modeling strategy that yields sparsity on the annotation group level similar to the group lasso [19]. In BAGEA, this grouping strategy is implemented by partitioning annotations into multiple meta-annotations (such as different cell types, assay types etc.). When using this partitioning mechanism, BAGEA includes an extra random variable vector υ of the same length as the number of elements in the partition structure (e.g. the number of cell types, or the number of assay types) (See Methods as well as Fig 1B for an illustrative example). The kth element of υ, υ_k, controls the variance of the effect sizes for annotations that fall into partitioning group k. Specifically, υ_k is proportional to the inverse of the variance of the respective elements of ω. is therefore called the variance modifier of annotation partition element k (see Methods).

Importantly, the model can be reformulated in terms of the summary statistics and LD matrices . The reformulation enables the application of BAGEA to studies for which only summary statistics are available, by estimating Σ_j from external sources (see Methods).

Evaluation strategy for model fit

We developed an approach to evaluate the performance of BAGEA when fitting directed annotations to genotype and gene expression data. An important feature of BAGEA is that its results can be used to predict gene expression for a gene without using any expression data for that gene, but rather using genotypes and genome annotations whose weights are fitted from other genes. We can therefore validate BAGEA by training it on gene expression data for one set of genes, and then calculating the extent to which the trained model predicts gene expression for other genes.

We propose a so-called directed predictor , which predicts gene expression for gene j based on knowledge of directed annotations and genotype for gene j. Set as the expected mean shift of b_j due to the annotations. Using the same notation as in Eqs (1) and (7), we have (3) i.e. is the ith element of . the predictor is then computed by (4) The squared magnitude measures how much gene expression variance the model attempts to explain via the predictor . To evaluate the predictor’s accuracy and degree of overfitting, we use the directed mean squared error . The evaluation of the predictor is performed on a set of genes independent of the ones used to estimate ω and ν.

We can reformulate in terms of summary statistics , LD matrices , and : (5) if we assume that y^T y = n. In principle, the reformulation allows us to calculate a predictor’s directed mean squared error, even if only summary statistics are available, by approximating Σ_j from external sources.

Directed annotations derived from blood can partially explain cis-eQTLs in monocytes

We used BAGEA to determine the extent to which annotations can predict gene expression in CD14 positive monocytes. To this end, we aggregated data from two eQTL studies on expression genetics in CD14 positive monocytes [20, 21]. For directed annotations, we used predictions of genetic variant effects on epigenetic marks (12 different histone mark assays and DNase1 Hypersensitivity Site (DHS) calls with 4 different peak calling strategies) in various blood-derived cell types from the pre-trained ExPecto model. ExPecto is a deep learning framework that predicts epigenetic marks based on sequence context and performs in silico mutagenesis to evaluate the consequences of sequence variants [7]. ExPecto yielded 2002 directed annotations of which 253 were from blood related cell types. These are referred to as the Blood annotation subset in this paper. We partitioned these directed annotations by cell type and assay type, respectively, and modeled separate prior variance terms for each partition (Fig 1A).

To train BAGEA, we used gene expression data from human chromosomes 1 through 15. Only 2410 genes that had a SNP in cis showing a signficant association with a p-value lower than 10⁻¹⁰ were included. To test model fit, we predicted expression for 917 genes on chromosomes 16 through 22 with a top nominal cis-eQTL p-value below 10⁻¹⁰. Specifically, we used the model fit on the training set to derive the estimates and (see Eq 7). We then used these estimates to calculate the directed predictors for genes on the test set (see Eq 4). To assess the predictive power of , we calculated for every gene in the test set.

We observed that directed genome annotations can partially explain gene expression variance (Fig 2). The average MSE^dir across all genes was 99.5%, which was significantly smaller than 100% (as evaluated by bootstrap sampling genes; p-value smaller than 10⁻⁴). showed a dependence on predictor size S_j (where ), such that for the top quartile of genes when ranked by S_j, the directed component was estimated to predict 1% to 3% of expression variance (Fig 2A). For each gene, the variance explained is bounded by the additive genetic variance component in cis which is typically much lower than 100%. We estimated the variance of expression explained for each gene in cis in an unbiased way via Haseman-Elston (HE) regression [22]. This approach suggested that around 6.6% of the total genetic variance in cis was explained by the externally fitted directed component for genes in the top quartile w.r.t S_j (Fig 2B). Across all strong cis-eQTLs, we estimated that the directed component explained 2.5% of total additive genetic variance in cis. We further tested the impact of the distance modifier by constraining all elements of (except the intercept element) to zero, showing that the modeling the distance modifier leads to higher predictive power (S1 Fig).

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Fig 2. Gene expression variance can be partially explained by directed genome annotations.

The BAGEA model was fitted on genes in the training set (all genes on chromosomes 1 through 15) using monocyte eQTL data on genes with a top nominal p-value below 10⁻¹⁰, and with ExPecto-derived directed annotations. ExPecto includes 2002 annotations in total, of which one of two subsets were used: 253 annotations derived from histone and DHS assays in a blood related cell types (Blood), or, alternatively, 690 annotations derived from TF ChIP-Seq (TF). For each gene j in the test set (all genes on chromosomes 16 through 22 with a top nominal p-value below 10⁻¹⁰), we calculated the directed predictor of expression . As a measure of a predictor’s size, we use its squared magnitude . To evaluate the predictor’s performance, we calculated , the mean squared error (MSE) when predicting gene expression y_j from . To estimate what the smallest attainable would be, we estimated , the additive genetic variance in cis via Haseman-Elston regression per gene. (A) The relationship between the MSE of the predictor and its squared magnitude. We sorted results by predictor Size S_j and averaged within a sliding window containing 25% of genes and step size of 5% of data. Averaged Directed Predictor Size : The mean value of S_j per window on the horizontal axis; Averaged Directed MSE (): The averaged of genes falling into the window on the vertical axis. The 95% confidence interval for each window was derived by bootstrapping. Most variance is explained by genes in the top quartile when ranked by S_j. (B) The relationship between and for genes in the top quartile when ranked by S_j. Genetic Variance (): The estimated additive genetic variance in cis on the horizontal axis. Directed MSE () on the vertical axis. 95% confidence intervals for the mean of both the MSE^dir and are represented as the corners of the red diamond (i.e. the confidence interval for the average MSE^dir is given by the upper and lower corner, whereas the confidence interval for the average is given by the right and left corner respectively). A linear regression is plotted as the blue line, with 95% confidence interval shown in grey.

https://doi.org/10.1371/journal.pcbi.1007770.g002

These results show that BAGEA can be used to model how sequence changes affect gene expression. Note that this evaluation metric relies on global parameter (i.e. ω, ν) estimates only. This allows to form predictors for a gene’s expression even if the gene was not included in the training set, but has lower predictive power than approaches that use genewise local parameter estimates (i.e. b_j). These approaches can predict expression potentially in an out-of-sample fashion, but only for genes in the training set. To illustrate this, we fit BAGEA on a subset of the monocyte samples (134 samples from the Fairfax et al. study) and extracted the local parameter estimates [20]. We then used these estimates to predict gene expression in the other available monocyte samples [21] [20]. As expected, a substantial fraction of the genetic variance in cis could be predicted using the local parameter estimates (S2 Fig). Further, using BAEGA in an annotation uninformed manner lowered the variance explained.

Joint modeling of cis-eQTLs and directed annotations highlights biologically relevant epigenetic marks

We next evaluated if BAGEA can effectively be used to discover which annotations, or groups of annotations, are most predictive of gene expression. We grouped the directed annotations by cell type and assay type, and for each set of annotation groups, we modeled separate prior variance modifiers υ⁻¹ (Fig 1B). For each annotation group k we measured its contribution to gene expression as its estimated variance modifier (See Model Overview). For the monocyte data, BAGEA estimated the largest variance modifiers for annotations from DHS as well as H3K27ac and H3K4me3 assays (Fig 3A). This observation is consistent with results from a previous method, using undirected annotations, suggesting that SNPs with an effect on gene expression are enriched in open chromatin (DHS), activated enhancers and promoters (H3K27Ac, H3K4me3) [11]. Across cell type annotations, BAGEA estimated the largest variance modifiers for annotations from two blood cell types that were both CD14 positive (Fig 3B). This observation matches our expectations because the cells in the underlying expression data were derived from CD14 positive cells [20, 21]. Across all tested pairs of assays and cell types, BAGEA estimated the largest positive effect sizes for annotations from DHS, H3K27ac, H3K4me3 assays in CD14 positive cells (Fig 3C). Additionally, we saw a negative effect size for DHS assayed in CD3 positive cells, albeit with lower absolute effect size than CD14 positive cells. One explanation could be that DHS that occur in CD14 but not CD3 positive cells have larger predictive value than DHS that occur in both cell types.

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Fig 3. BAGEA, fitted on monocyte eQTL data, selects relevant epigenetic marks and increases directional effect sizes for SNPs close to a TSS.

Parameter estimates when applying BAGEA to monocyte eQTL data using as directed annotations histone and DHS ExPecto predictions derived from blood-related cell types (i.e. Blood from Fig 2). (A) For each chromatin assay type, BAGEA models an assay variance modifier that captures the extent to which that assay type is predictive of gene expression. Shown are the square roots for the assay types with the ten highest variance modifiers (from 17 assay types total). In the BAGEA model, DHS, H3K27Ac and H3K4me3 assays have the largest modifiers. (B) For each cell type, BAGEA models a cell type variance modifier , similar to the assay variance modifier in panel A. Shown are the square roots for the cell types with the ten highest variance modifiers (out of 61 cell types). In the BAGEA model, CD14 positive cells have the largest modifiers. (C) BAGEA reveals experiments underlying the directed annotations that were most predictive of gene expression. Assay Type x Cell Type: Each experiment is a particular assay type performed in a particular cell type. Effect Size (, for experiment i): The BAGEA-estimated effect on gene expression. Shown are the ten largest directed annotation effect sizes. In the BAGEA model, the experiments using DHS, H3k27Ac and H3Kme4 with CD14 positive cells have the largest effect sizes. We also see that most of the 253 annotations are estimated to have a close to zero effect. (D) Shown is the estimated distance modifier of the directed component, . We see a characteristic peak around the TSS, implying that the directed annotations are upweighted close to the TSS.

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It is well known that eQTLs occur more likely and increase in effect size closer to the TSS. This suggests that the effects of directed annotations might also be bigger for SNPs close to the TSS than for SNPs that are distal. BAGEA models SNP distance dependence of directed annotation effects by weighting the directed annotation effect term V^jω across SNPs, with a distance modifier F^jν (see Model Overview). We next tested whether BAGEA estimated directed annotation effect sizes to be dependent on a SNP’s distance to the TSS. We examined the value of a SNP’s estimated distance modifier against its position relative to the TSS. We observed a characteristic peak around the TSS (Fig 3D), suggesting that BAGEA can indeed produce a similar pattern of distance dependence for the effect sizes derived from directed annotations as for the eQTL effect sizes themselves.

We repeated this analysis with a different set of directed annotations, namely 690 ExPecto annotations derived from transcription factor (TF) ChIP-Seq in any cell type. We estimated the TF annotation subset to be similarly predictive of gene expression as the Blood annotation subset (Fig 2A). When looking at the estimates of ω, MYC assayed in the cell line NB4 had the largest effect size among all tested 690 annotations (S3 Fig). Additionally, SPI1, MAX CTCF had effects larger than 10% of the maximal effect size. For SPI1 and CTCF, effects from multiple cell lines reached this threshold with consistent effect size directions.

NB4 is a promyelocytic leukemia cell line that can be differentiated into neutrophils or monocytes [23]. NB4 is therefore expected to have similar expression genetics as CD14 positive monocytes, and, given that no TF ChIP-Seq experiment was performed in monocyte cell lines directly, the large ω values for NB4 data are consistent with our expectations. However, interpretations of cell type selection for the TF subset are complicated by the fact that the underlying TF ChIP-seq experiments did not sample each TF comprehensively across all cell lines which might lead to biases. When checking expression of the highlighted TFs in monocytes via Protein Atlas, we found all were TFs classified as expressed but not elevated in monocytes [24]. Effect size directions of CTCF and MAX were negative, which naively interpreted would suggest that their binding have a suppressive effect on gene expression. CTCF can act as repressor, activator and insulator [25]. Our data suggests that, globally and in the studied context, repressive effects outweigh. MAX and MYC are part of a family of TFs that form heterodimers [26]. The MYC/MAX dimer is usually regarded as an activator, which might seem to be at odds with our results as the MAX annotation had a negative effect size. However, another important family member MXD1 (also known as MAD) was not assayed. The MAD/MAX heterodimer is thought to act as a repressor. A positive effect size for MYC and a smaller negative effect size for MAX could just imply that MAX binding in absence of concomitant MYC binding has a negative effect on expression because it tracks with MAD/MAX heterodimer binding. Developmentally, during monocyte/macrophage maturation, a switch from high levels of MYC/MAX to MAD/MAX is observed [27]. A further highlighted transcription factor is SPI1 which is regarded as one of the most important transcription factor in monocyte and macrophage development [28].

Modeling directional components is robust to the use of summary statistics

In many cases it is not feasible to compute LD for the population from which the summary statistics were derived (i.e., the study population), and LD has to be derived from other sources (i.e., external genotypes) [29, 30]. The use of external genotypes allows publicly available summary statistics to be analyzed without access to restricted individual level genotype data [9]. However, LD computed on external genotypes can only approximate LD patterns of the study population. We therefore need to test the accuracy of methods when using external genotypes.

We evaluated if directed annotation effects were robust to the genetic source of LD information. We used 1000 Genomes data to compute LD [31]. We re-fit the BAGEA model to the monocyte data with the Blood annotation subset, using LD matrices derived from European 1000 Genomes data. We then compared ω estimates when using LD from 1000 Genomes to ω estimates when using LD from the monocyte data itself, for every annotation in the monocyte Blood data. We observed that the two approaches produced similar effect sizes with a linear regression R² of 97.5% and regression slope of 0.96 (S4A Fig). This suggests that directed annotation effect estimates are robust to the source of LD information. We then explored if the source of LD information affected our estimates of directed mean squared error (MSE^dir). To this end, we estimated MSE^dir on chromosomes 16 through 22 from summary statistics and external LD matrices derived from 1000 Genomes alone, and then compared these MSE^dir values to the original MSE^dir values computed with LD derived from monocyte data. We ensured that the same SNPs were included, by removing SNPs with low minor allele frequency (MAF) in either of the sets. We observed that the two sources of LD produced MSE^dir values that agree with each other, with a linear regression R² of 99.9% and regression slope of 1.002 (S4B Fig).

Exploring BAGEA’s ability to identify causal marks through simulation

To explore BAGEA’s ability to select the causal annotations among the set of annotations, we used simulation (see S1 Appendix). Naturally, this depends on the correlation structure of the directed annotations, as highly correlated annotations will make it difficult to isolate the causal one. We therefore used empirically observed directed annotations in our simulation experiment. We assumed a model were the truly causal annotations were sparse (with the non-zero effects varying from 3 to 12). We tested cases where the non-zero effects clustered in terms of meta-annotations (e.g. clustered in terms of cell types and assay type) (structured) and cases where non-zero effects did not cluster (unstructured). While fitting BAGEA we also used two parameter settings, either making use of the meta-annotations available for cell type and assay type, (group-lasso) or ignoring the meta-annotation information and letting each ω_i parameter be controlled by a separate υ_i parameter (lasso). While we saw high recovery of the causal annotations for lower number of causal variables, precision and recall tended to drop as more but individually smaller non-zero effects were added (S5 Fig). Drop-off was fastest when pairing unstructured data generation, with the group-lasso fitting procedure, presumably, because this parameter setting tried to enforce a structure that was not actually present. Conversely, structured data generation paired with group-lasso fitting procedure showed the highest performance of all settings. When evaluating the gene expression prediction power of the model fits, we saw that in all cases the results were close to optimal, suggesting that even in higher complexity settings when incorrect variables get picked, the chosen variables are highly correlated with the correct ones (S6 Fig).

Building predictive expression models from GTEx summary statistics

Having established that BAGEA performs well when using summary statistics, we next determined if BAGEA can identify relevant directed annotations for empirical data for which summary statistics are available but genotypes are not. Specifically, we fit BAGEA on summary statistics for eQTL studies of 13 tissues produced by the GTEx consortium with a sample size of at least 300 for each study [9]. We additionally supplemented this set with results for Lymphoblastoid cell lines (LCL) derived from a meta-analysis of GTEx and GEAUVADIS [17]. Because GTEx gathered eQTLs in complex tissues and sampled fewer individuals than were sampled in the monocyte studies, we expected lower power to produce robust parameter estimates. We therefore used different parameter values than in our monocyte analysis, including genes with top nominal cis-eQTL p-value lower than 10⁻⁷. We fitted models either using ExPecto derived annotation for all 1187 histone or DHS annotations derived from Roadmap consortium data or the derived annotations for non-histone ChIP-seq data from ENCODE [32, 33]. When using Roadmap annotations we used BAGEA in the group-lasso setting, whereas for ENCODE annotations we used the lasso setting, the rationale being, that the Roadmap consortium performed most assays for a given cell type, whereas ENCODE ChIP-seq was less complete, i.e. many TFs were assayed in only few cell lines, leading to potentially strong biases.

We again split genes into training and test set, fitting BAGEA on the training set and building directed expression predictors for all genes in the test set. We observed that the average MSE^dir per dataset was variable across GTEx datasets ranging from 100% to below 98.5% (Fig 4A). When looking at only the highest quartile of genes in terms of squared magnitude S_j, we saw the lowest average MSE^dir go to approximately 0.95 (Fig 4B). Furthermore the gains in average MSE^dir were in line with squared magnitude S_j values, suggesting that BAGEA does not substantially overfit. We saw that the ENCODE TF annotation set tended to outperform the histone and DNase1 Roadmap set and that in the Roadmap group lasso setting, BAGEA would not always select any annotations, potentially due to poor overlap between annotation and GTEx cell types and stringent regularization.

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Fig 4. Directed annotations partially explain gene expression variance in GTEx.

The BAGEA model was fit using various GTEx eQTL data (supplemented with GEAUVADIS eQTL data) and with ExPecto-derived directed annotations on genes in the trainig set (chr1,‥,chr15) with a top nominal p-value<10⁻⁷. ExPecto includes 2002 total annotations, of which either 1187 histone and DHS annotations from Roadmap (Roadmap) or 690 non-histone ChIP-Seq from ENCODE (TF) were used. For the Roadmap annotation set we enforced structure on the priors of ω by using the meta-annotations available for cell type and assay type, (group-lasso), while for the (TF annotation set, each ω_i parameter be controlled by its individual υ_i parameter (lasso). For each gene j in the test set (chr16,‥,chr22 and top nominal p-value< 10⁻⁷), we calculated an approximate version of S_j, the squared magnitude of the directed predictor , where the approximation uses external LD information. Further, we calculated an approximate version of , the mean squared error (MSE) when predicting gene expression y_j from . (A) Displayed is the average (approximated) across all genes for each GTEx experiment, and annotation subset. 95% Confidence intervals are computed by bootstrap sampling. (B) For each GTEx experiment and annotation subset, we sorted results by predictor size S_j and and averaged within the top quartile. Displayed is the relationship between the MSE of the predictor and its mean squared magnitude S_j. Averaged S_j, top quartile : The mean value of the directed predictor size S_j in the top quartile on the horizontal axis; Averaged Directed MSE (): The averaged of genes falling into the top quartile in terms of S_j on the vertical axis. The 95% confidence interval for each window was derived by bootstrap sampling. We see that the average squared magnitude S_j is of similar size as the gains in directed MSE suggesting that the BAGEA does not substantially overfit.

https://doi.org/10.1371/journal.pcbi.1007770.g004

We next compared the predictive power achieved by BAGEA on the GTEx data to results derived via ExPecto directly. To predict expression from genomic variants, the authors of ExPecto propose a strategy, where results from two statistical models are combined. The first model is a deep neural network that predicts the impact of genomic sequence variants on chromatin marks (the results of which are also used by BAGEA in this analysis). The second model is a l₂-boosting model that predicts gene expression from (spatially transformed) chromatin marks directly. as part of the ExPecto release, results of the second model were already publicly available for 13 relevant GTEx datasets [4]. Combining these results with the directed annotations and the z-scores from GTEx, allowed us to estimate the scalar product between the gene epression vector y_j and the corresponding directed predictor (see S1 Appendix). This allowed us to compare model quality in terms of the fraction of genes with a scalar product larger than zero. When comparing results from BAGEA (using TF annotation subset and lasso setting) and ExPecto (using all annotations) in terms of this metric, we saw that, while performance was comparable across all genes, BAEGA substantially outperformed ExPecto for genes in the highest quantiles in terms of effect size (S7 Fig).

We further compared BAGEA to Torus, a tool which allows to model SNP effect priors in terms of undirected annotations [14, 15]. We therefore made our annotations undirected by taking absolute values and adding the undirected annotations used in BAGEA to generate annotations for Torus (see S1 Appendix). We fit data from 13 GTEx experiment on the TF annotation subset filtering SNPs and genes as for BAGEA. Varying amounts of l₂ regularization were applied and an additional overfit strategy was added to have an upper bound on results achieved with an optimal regularization strategy (see S1 Appendix). As an evaluation strategy we recorded for each gene in the test set the SNP with the highest prior. As an evaluation metric, we used the fraction of genes for which that SNP had an absolute z-score close to the largest absolute z-score for that gene. We saw that, while BAGEA had a slightly higher value than even the overfit strategy, the increases were modest (S8 Fig). As Torus was run in ridge mode, few effects were very close to zero. BAGEA in its default parameter resembles lasso, in that it only selects a limited number of effect sizes substantially different from zero. To compare the variables selected, we split the distribution torus effect sizes estimates into 3 groups based on whether BAGEA estimated them as (close to) zero, negative or positive. We saw that the distribution of torus estimated effect sizes was substantially shifted for the non-zero BAGEA effect groups compared to the zero BAGEA effect group (S9 Fig).

GTEx expression models make use of biologically relevant annotations

To mitigate the impact of limited power during variable selection, we additionally fit models without splitting chromosomes into test and validation sets. Exploring the results from the Roadmap annotation subset first, we saw that the distribution of effect sizes of the directional annotations revealed a bias towards positive values (S10 Fig). Focusing on the largest positive effect sizes(top ten or ), we saw many biologically consistent pairings between the tissue assayed by GTEx via eQTL and the tissue assayed by Roadmap for epigenetic marks (Fig 5A). While some of the pairings are obvious from the annotation names themselves (such as correct pairings for lymphoblastoid cells, lung and adipose tissues) others were suprising yet on closer inspection, turned out to be consistent with the biological literature. For instance bone marrow derived mesenchymal stem cells (BMD MSC) are paired with fibroblast. A recent study found no functional differences between the two cell types leading the authors to support a longstanding opinion in the field that these two cell types should be classified as the same [34, 35]. The pairing between Esophagus Mucosa and keratinocytes can be explained by the fact that the Esophagus Mucosa is mainly composed of squamous cells, i.e. keratinocytes [36, 37]. The pairing between tibial artery and BMD MSC can be explained by the fact that fibroblasts are the main component of vascular adventitia [38]. Our model also paired tibial nerve and muscle, which seems physiologically the least biologically consistent among the ten pairings. When looking at the largest negative values, we saw some of the same tissue pairings repeated, with only one pairing with effect size smaller than -0.06 for the pairing between fibroblasts and BMD MSC) (S11 Fig). When looking at the variance modifier estimates for the different assay types, we saw that DHS and H3K27ac epigenetic marks were ranked consistently highly (Fig 5B). Interestingly, among various annotations derived from the same DHS experiments, some performed consistently better than others: DNase1 peak call annotations outperformed DNase1 hotspots calls. These two annotation types make use of the same underlying DNAse-seq data, but use different data processing pipelines [39]. Hotspots calls have variable length and are typically wider than peak calls which have a fixed length of 150 bp. BAGEA makes a clear choice which should be preferred to model sequence impacts on gene expression. Turning our attention to the results for the non-histone ChIP-Seq ENCODE annotation subset, we saw for most GTEx experiments, that the largest effect size was associated with a Pol2 assay (S12 Fig). As all annotations are fit together, it is possible, that large Pol2 effects could obscure transcription factor action as Pol2 binding could be a result of the binding of other factors. We therefore removed all Pol2 assay annotations, and refit the model again. We then counted the number GTEx experiment for which a given TF showed either a substantial positive or negative effect (S13A Fig). When looking for TFs That showed negative effect sizes in at least half of the assayed experiments, we found EZH2, SIN3A and ZEB1, which all have substantial literature support for being transcriptional repressors [40–42]. TFs that showed substantial positive effects in at least half of the experiments, we saw PHF8 and ELF1. PHF8 is known to demethylate H3K9me2, a mark strongly associated with transcriptional repression [43]. The fact that this assay shows significant positive effects in most analysed GTEx experiments, might highlight an underappreciated importance of this mode of transcriptional control. ELF1 has been cited in the literature as having both activator and repressor properties [44, 45]. Our results suggest that activator properties outweigh. To systematically evaluate whether our results are in line with the literature, we compared them to the most recent human version of TRRUST, literature database of regulatory interactions between TFs and their targets [46]. Interaction in TRRUST are annotated as repressive or activating in nature. We derived a TFs repressor activity score based on the fraction of annotated interactions defined as repressive. We derived a second TF repressor actvity score as the number of positive effects minus the number of negative effects (S13B Fig). We saw a strong correlation between these scores (p-value below 0.0001, R² = 0.43). When removing the 3 bona-fide repressors, results remained significant at the 0.05, level albeit less so (one-tail p-value below 0.015, R² = 0.15). Additionally, we saw that while distance modifier did vary between fits, the characteristic peak around the TSS was replicated in all cases (S14 Fig).

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Fig 5. Model fit for GTEx summary statistics selects directional annotations mainly from biologically consistent cell types.

Shown here are various parameter estimates from fitting 13 different GTEx eQTL summary statistics data (supplemented with GEAUVADIS eQTL data) using histone and DHS ExPecto predictions derived from Roadmap (1187 annotations). (A) BAGEA reveals the experiments underlying the directed annotations that are most predictive of gene expression. GTEx x Roadmap(Rm): Each GTEx eQTL dataset highlights particular Roadmap annotations. Shown here are the 10 largest positive effect sizes across all eQTL and annotation pairings. Effect Size: The estimate of for experiment i. (B) For each chromatin assay type, BAGEA models an assay variance modifier that expresses the extent to which that assay type is predictive of gene expression. Shown here is the distribution of the square roots of the assay variance modifier for any given assay type across all 13 GTEx eQTL datasets. Results are sorted by the maximal value achieved for each assay type and only the 10 highest scoring assay types are shown. We see that DNase.all.peaks H3K27ac annotations dominate. The DNase.fdr0.01.peaks was prioritized in Lung tissue, which had the lowest value for DNase.all.peaks among all experiments. The highest value in DNase.all.peaks was achieved in Fibroblast, an experiment that also showed low average MSE^dir.

https://doi.org/10.1371/journal.pcbi.1007770.g005

Model details

We assume individual level genotype and expression data for n individuals. For gene j, we model its n × 1 expression vector y_j as (6) where X_j is the n × m_j genotype matrix for the m_j SNPs surrounding gene j’s TSS. b_j is the m_j × 1 vector of SNP effect sizes and ϵ_j the expression noise unexplained by the genotype. The noise term precision λ_j is modeled in a hierarchical fashion: with hyperparameters λ₁, ρ₁ and ρ₂ (while this notation is overloaded, we expect it is clear from context which parameter is meant). We model the vector of effect sizes b_j as a multivariate normal, whose mean and covariance is affected by annotation matrices. For gene j we assume annotation matrix F^j and a directed continuous annotation matrix V^j, with dimensions m_j × q and m_j × s respectively, with the current implementation of BAGEA expecting F^j to be 0 − 1 coded due to performance reasons. Then the ith element of b_j is modeled as (7) with and being the ith row of V^j and F^j respectively. α_ij being an element of a vector of independently drawn gamma distributed random variables (the independence is conditional on its parental hyperparameters, the modeling of which is described further down). ω and ν are s and q dimensional multivariate normal distributed random variables respectively. ω denotes the vector of activities of directed annotations, whereas ν allows the overall weight that the directed annotations contribute to the effect size vary based on undirected annotations. This allows, for instance, the impact of the directed annotations to vary dependent on the distance to the TSS. ω is modeled in a hierarchical fashion where δ is again modeled as a random variable. The choice of model for δ enables the implementation of a grouping structure on the directional annotations (in our application, these groupings are the assay used to derive the annotation and the cell type in which the assay was performed). We allow the model to fit differences in prior variances based on group membership. Thereby, entire groups of directional annotation effects are shrunk to zero (akin to the group lasso [19]). Let d^l be a positive integer vector of length s taking h_l different values, i.e d^l partitions the vector of directed annotations into h_l groups (l = 1, ‥, w runs over the meta-annotations, e.g. if the modeled meta-annotations are cell type and assay type, l can either take the value one or two). Let υ^l be a random vector of length h_l (i.e. these are the group specific weights). Then, with hyperparameter χ_1l. χ_2l is modeled as with hyperparameters ζ₁ and ζ₂.

ν is modeled as where p and c are hyperparameter vectors of length q.

The vector of precisions of the effect size vector α_j is modeled as where γ₁ is a hyperparameter. Note that letting the precision for each SNP vary leads to sparse estimates for b_j; this is akin to automatic relevance determination (ARD) regression [18]. κ_j is a genewise parameter modeled in a hierarchical fashion where τ₁, ξ₁ and ξ₂ are hyperparameters. To model γ_ij, we again make use of annotation matrices. For gene j, assume undirected 0 − 1 coded annotation matrix C^j of dimension m × t (BAGEA currently only 0 − 1 coded matrices for C^j coded due to performance reasons). Then the SNP-wise precision modifier γ_ij is modeled as where if annotation k is active at index i in gene region j. Further, where ϕ₁ and ϕ₂ are hyperparameters.

Summary statistics adaptation

Instead of using individual level genotype and expression data, we can reformulate the model for the use of summary statistics. Multiplying Eq 6 with gives A natural model to use with summary statistics is therefore, where z_j is the vector of summary statistics, Σ_j is the LD matrix and . Σ_j can be approximated from external sources such as 1KG [31]. Alternatively, we can use an approximate and regularized version of the empirical LD matrix (see below).

Model fitting

The model was fit using a variational bayes approach [48]. As the model is in the conjugate exponential family, we can use the variational message passing strategy [49]. For detailed updating steps see S1 Appendix. Naive updates can be prohibitively expensive due to the requirement to invert many large matrices of the form (cX^T X + D_α), where c is a constant and D_α is a diagonal matrix. To speed up computation, we can approximate the LD matrix cX^T X with a low rank approximation , where A_t is a t × m matrix with t < m. This allows us to speed up a time critical matrix inversion step. If X is already low rank, it is computationally advantageous to use an A_t s.t. . If deviates from cX^TX, we need to use the summary statistics formulation to avoid convergence issues. For more detail, see S1 Appendix.

Deriving annotations

For common SNPs (minor allele frequency (MAF) above 2.5% in the 1000 Genomes European population [31]), we ran the ExPecto model to predict the effect of the variant on epigenetic marks [7]. For each SNP we predicted the epigenetic effects within the 200 bp region encompassing it. For most SNPs the effects are very close to zero, allowing us to sparsify the results. Absolute effects smaller than 0.008 were set to zero and all other effects were shrunk towards zero by 0.008 via x_new = x − 0.008 ⋅ sgn(x). Next, results for both strands were averaged and the shrinking procedure repeated with a threshold of 0.008. This yielded a matrix with 98.4% of entries zero. The directed annotations were then scaled to have all the same 2-norm. The magnitude of the 2-norm was set to the average of the unscaled 2-norms. These were the directed annotations used in BAGEA.

For undirected annotations, we used upstream and downstream distances to the TSS. Distance to TSS annotations as well as SNP positional annotations were downloaded from the UCSC genome annotation database with SNP and gene annotations taken from the refGene and snp147Common tables respectively (see link below) [50].

cis-eQTL datasets

For monocyte eQTL data, we used two preprocessed monocyte datasets with a combined sample size of 1176 (418 from Fairfax et al. and 758 from Rotival et al. respectively) [20, 21]. Expression matrices were quantile normalized and 10 PEER factors as well as 5 genotype PCs removed [51]. Genotype data was quality control filtered (4% SNP level missingness; 5% individual level missingness; Hardy-Weinberg p-value above 10⁻¹³ relatedness below 0.1875) and imputed using the human genome reference panel [52].

We further downloaded eQTL summary statistics for various tissues produced by the GTEx project if the number of samples was above 300 individuals [9]. Additionally, for LCL, we meta-analyzed eQTL summary statistics released for 117 samples by GTEx with summary statistics derived from 358 European PEER-controlled samples collected as part of the GEUVADIS study [17].

Running BAGEA

For the monocyte eQTL analysis, BAGEA was run with default hyperparameter settings (see S1 Appendix). Genotypes within a window of 150KB around a gene’s TSS were used to construct a genewise LD matrix. Each genewise LD matrix was approximated via singular value decompostion with a low rank symmetric matrix of equal top eigenvalues and eigenvectors, such that the trace of the approximation matrix was at least 99% of the trace of the original LD matrix. Then, a scaled identity matrix was added such that the trace of the resulting matrix was equal to the trace of the original LD matrix. As undirected annotations, distance windows around the TSS (50KB, 20KB, 10KB, 5KB, 2KB, 1KB, 0.5KB, 0.25KB) split into upstream and downstream windows were used. To analyse summary statistics with BAGEA, LD was approximated via 1KG European samples. Variants where the reference allele in 1KG did not agree with the reference allele in the UCSC SNP annotation table, were removed. Reference 1KG LD matrices were calculated and replaced with low rank approximations with 95% of the matrix trace kept, anlagously to the above procedure. For all GTEx summary statistics analysis, default hyperparameter settings where used except for c which was set to 0.3 instead of 0 to yield consistently positive signs for ν estimates. BAGEA was run for 300 iterations in each analysis.

URLs

• Code to run BAGEA can be found at https://github.com/dlampart/bagea
• Auxiliary preprocessed data automatically installed by BAGEA is downloaded from https://s3-us-west-1.amazonaws.com/bagea-data/bagea_data_freeze/

Links to publicly available external data sources are as follows:

• UCSC: http://hgdownload.cse.ucsc.edu/goldenpath/hg19/database/
• ExPecto: https://github.com/FunctionLab/ExPecto/
• 1KG: ftp://ftp-trace.ncbi.nih.gov/1000genomes/ftp/release/20130502/
• GTEX: https://gtexportal.org/home/datasets
• GEUVADIS: ftp://ftp.ebi.ac.uk/pub/databases/microarray/data/experiment/GEUV/E-GEUV-3/analysis_results/
• TRRUST-db: https://www.grnpedia.org/trrust/
• Protein atlas: https://www.proteinatlas.org
• First monocyte study: https://www.ebi.ac.uk/ega/studies/EGAS00001000411
• Second monocyte study: https://www.ebi.ac.uk/ega/studies/EGAS00000000109