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Novel Distal eQTL Analysis Demonstrates Effect of Population Genetic Architecture on Detecting and Interpreting Associations

Mapping expression quantitative trait loci (eQTL) has identified genetic variants associated with transcription rates and has provided insight into genotype–phenotype associations obtained from genome-wide association studies (GWAS). Traditional eQTL mapping methods present significant challenges for the multiple-testing burden, resulting in a limited ability to detect eQTL that reside distal to the affected gene. To overcome this, we developed a novel eQTL testing approach, “network-based, large-scale identification of distal eQTL” (NetLIFT), which performs eQTL testing based on the pairwise conditional dependencies between genes’ expression levels. When applied to existing data from yeast segregants, NetLIFT replicated most previously identified distal eQTL and identified 46% more genes with distal effects compared to local effects. In liver data from mouse lines derived through the Collaborative Cross project, NetLIFT detected 5744 genes with local eQTL while 3322 genes had distal eQTL. This analysis revealed founder-of-origin effects for a subset of local eQTL that may contribute to previously described phenotypic differences in metabolic traits. In human lymphoblastoid cell lines, NetLIFT was able to detect 1274 transcripts with distal eQTL that had not been reported in previous studies, while 2483 transcripts with local eQTL were identified. In all species, we found no enrichment for transcription factors facilitating eQTL associations; instead, we found that most trans-acting factors were annotated for metabolic function, suggesting that genetic variation may indirectly regulate multigene pathways by targeting key components of feedback processes within regulatory networks. Furthermore, the unique genetic history of each population appears to influence the detection of genes with local and distal eQTL.

G ENE expression is highly heritable, indicating a strong genetic component (Cheung et al. 2003;Schadt et al. 2003). Expression quantitative trait loci (eQTL) mapping strives to uncover the underlying genetic architecture of transcriptional regulation. An important concept in dissecting complex regulatory processes is to identify both local and distal variants that are associated with gene expression. Local eQTL are largely thought to regulate proximal genes by affecting the activity of regulatory elements that directly influence transcription rates, such as through alterations in genomic sequence that affect binding affinities of regulatory factors. In contrast, distal eQTL map to genomic locations far from the affected gene, possibly on different chromosomes, and likely act initially on the expression or function of some nearby, intermediate gene that then affects the associated target gene in trans. Notably, in genetically diverse populations such as humans, the reported effect sizes and significance levels for distal associations are weaker than for local eQTL (Brem et al. 2002;Doss et al. 2005;West et al. 2007). This is likely attributable to the greater noise inherent in indirect effects that occur within the context of a protein-protein interaction network.
Initial eQTL discovery analyses performed association tests for all pairs of genomic variants and genes (Alberts et al. 2011;Holloway et al. 2011;Mehta et al. 2012), leading to challenges in both sensitivity and interpretation. Although recent methods have greatly reduced the computational burden for this approach (Shabalin 2012), the reduced statistical power due to multiple-testing correction still presents significant problems, especially in detecting distal eQTL. Using this technique, the reported frequency of distal effects has varied from 2% to 75% of all detected eQTL (Yvert et al. 2003;Göring et al. 2007;Mehta et al. 2012), and it remains unclear whether this is attributable to differences in regulatory architecture or statistical power. Indeed, in several recent eQTL analyses using human data, distal eQTL mapping was either not performed or not reported (Pickrell et al. 2010;Lappalainen et al. 2013), likely due to the inability to detect any distal eQTL whatsoever. Additionally, inferring the direction of effect of distal associations that result from protein interactions is difficult when dealing with gene expression data that are often noisy and highly correlated.
To detect distal eQTL with greater power, some recently developed methods assume an underlying regulatory architecture in which the local regulation of an intermediate gene leads to widespread expression variation in a large set of target genes (Bottolo et al. 2011;Duarte and Zeng 2011;Kompass and Witte 2011;Rotival et al. 2011). Modules of target genes are defined by factor analysis or gene-gene correlation statistics, and association testing is performed between genotypes and summary statistics of each module. In this setting, strong associations are thought to represent master regulators that exert broad, but potentially weak, effects in the regulatory network. These approaches reduce the multiple-testing burden, as thousands of genes are replaced by a few dozen modules; however, several drawbacks remain. First, if the regulatory activity of a transacting factor (TAF) affects only a handful of target genes, the initial clustering approach may not identify the small gene module. Second, the intermediate genes regulating the expression of gene modules are often not identified. Finally, expression for individual genes belonging to a module does not always correlate with the eQTL associated with the module, raising doubts about the validity of the results (Kompass and Witte 2011).
Others have developed methods focused on addressing interpretability and directionality of associations, using randomization of genetic variables (Chen et al. 2007) and causal model selection tests (Neto et al. 2013) as a foundation for statistical inference. In these methods, conditional dependence between expression of genes and/or latent variables is used to probabilistically determine whether the association between the genetic variant and the target gene is causal. In this study, we present a novel eQTL detection method, "network-based, large-scale identification of distal eQTL" (NetLIFT), which, rather than performing causal model selection or randomization, uses pairwise partial correlations derived from gene expression data to restrict distal association testing, thereby reducing the multiple-testing burden and highlighting candidate regulatory genes. In this framework, statistically significant local associations are first identified, and then local eQTL variants are tested for distal associations only for genes whose expression values show evidence of direct effects. We show that NetLIFT identifies individual SNP-gene distal associations with greater power than traditional pairwise eQTL testing, scales well to large data sets, and provides interpretability regarding the mechanism of association by highlighting potential trans-acting factors. In simulation studies, NetLIFT better identified distal eQTL, especially those with small numbers of target genes, when compared with a traditional all-SNPs vs. allgenes approach, a module-based approach (independent components analysis, adapted from Rotival et al. 2011), and a method designed to identify causal associations using randomization of genotype data (Chen et al. 2007). Applying NetLIFT to a data set consisting of 112 yeast segregants (Brem and Kruglyak 2005), we recapitulated previously reported distal associations and putative regulators, while discovering several additional eQTL with plausible biological mechanisms of association. In mouse livers, we discovered founder-of-origin effects for a subset of local eQTL that drive differential expression of target genes in a subspeciesof-origin specific manner, suggesting a possible role for these loci in transcriptomic and phenotypic differences between strains. Using data from human lymphoblast cell lines (Pickrell et al. 2010), we identified .1000 distal associations not previously reported. We note that individuals from each of these three populations (yeast, mice, and humans) have unique genetic histories, and our analysis suggests that this influences the number and type of eQTL detected in each study.

Description of the NetLIFT model
The analysis workflow for the NetLIFT model is outlined in Figure 1 and was designed to parallel our understanding of the mechanism of trans-regulatory effects. That is, if SNP s i affects the transcription of gene g j in trans, we expect that s i first directly affects the transcription level of an intermediate gene g i and that the transcription rate of g i directly or indirectly affects the transcription rate of g j . There are three main steps to the NetLIFT algorithm.
Step 1: Identify local eQTL: Local association tests are performed for all variants that lie within an a priori-defined window of each gene ( Figure 1A). Allele counts are regressed on the gene's expression values, using a univariate, additive linear model. Since some genes contain many more variants than others, we control the false positive rate in local testing by retaining only associations that meet a Bonferronicorrected significance cutoff of 0.05. Significant associations represent variants that may have a direct effect on the transcription rate of nearby genes, likely by altering activity of cis-regulatory elements.
Step 2: Estimate pairwise partial correlations for all genes: Pairwise partial correlations are estimated for all gene pairs ( Figure 1B) to identify genes with expression level dependencies. The distribution of connections for gene networks has been shown to follow a power-law distribution (Jeong et al. 2000;Barabási and Oltvai 2004;Yook et al. 2004;Lorenz et al. 2011) with an overall small number of edges. Therefore, we estimate the partial correlation matrix G, using a method that enforces sparsity on the entries of G via L1 regularization and has been shown to accurately identify network hubs (Peng et al. 2009;Allen et al. 2012).
Briefly, this method performs joint sparse regression on all p variables (genes) simultaneously, by minimizing the penalized loss function where g i and g j are the expression vectors for genes i and j, r ij denotes the partial correlation between genes i and j, and s ii and s jj are the ith and jth diagonal entries of the inverse covariance matrix. The L1 penalty l controls the sparsity of the network and was optimized by minimizing the Bayesian information criterion outlined in Peng et al. (2009). For p genes, the resulting p 3 p matrix G consists of entries G i,j that represent the correlation between expression vectors g i and g j , conditioned on the expression of all other genes' expression: G can be interpreted as an undirected network, where each node represents a gene, and an edge is drawn between two nodes if and only if the corresponding entry in the matrix G is nonzero.
Step 3: Distal eQTL testing: Distal eQTL are called by integrating the results from these two steps ( Figure 1C). For each variant s i that shows significant association to a local gene g i , we test s i for association with distal genes g j that are nearby g i in the partial correlation network defined by G.
Since the edges of G account only for direct relationships between two genes, we exploit the network structure to search for second-degree (downstream) regulatory effects as well. Specifically, we require two conditions for s i to be tested for a distal effect on g j : 1. s i must be strongly associated with expression of the putative TAF, g i . 2. Genes g i and g j must be separated in the partial correlation network by no more than two edges; i.e., either G i,j 6 ¼ 0 or there exists a third gene g k such that G i,k 6 ¼ 0 and G k,j 6 ¼ 0. Additionally, we incorporate a threshold whereby two-degree genes are tested only if the association between s i and the intermediate gene g k meets a user-defined significance level (we selected P , 0.2 for this cutoff in all analyses presented here). Although longer-range interaction effects could be considered by testing genes at increased distances within the network, doing so would exponentially increase the number of tests performed at each distance cutoff. We sought to balance this trade-off by limiting the edge distance to two.
If a locally affected gene contains many significantly associated variants, only the variant with the strongest local association is tested with distal genes. Furthermore, we . . , s m ) and "p" genes (g 1 , g 2 , . . . , g p ) are assayed for the same "n" individuals (a 1 , a 2 , . . . , a n ). Markers and genes that map to the same locus are color coded. Local eQTL mapping is performed for markers and nearby genes using an a priori-defined genomic distance for local effects (A), yielding a local eQTL effect matrix (significant marker-gene associations depicted in green). A sparse partial correlation matrix is inferred from the expression data, representing a network of gene-gene interactions (B). Finally, significantly associated local eQTL markers are tested for distal eQTL effects on genes near the locally affected gene in the interaction network (C).
impose directionality in the ambiguous case where two directly connected genes both have local eQTL, by recording only the direction with the strongest distal association. We note that since G is a symmetric matrix representing an undirected network of correlated genes, we make no assumption regarding the direction of potential gene-gene effects and therefore no assumption about how variant-to-gene effects may propagate through the network. Instead, we use the network structure only to select which variant-gene pairs to test for associations. Although significant associations do not provide conclusive evidence of trans associations, we expect that many of the distal eQTL will be acting in trans, potentially through the putative TAF identified by our method.
We note that the correlation-based network structure used to guide the distal association tests will likely lead to correlations among test statistics. The Benjamini-Yekutieli (BY) false discovery rate (FDR) correction holds rigorously under general dependence of test statistics (Benjamini and Yekutieli 2001); however, this correction is generally considered to be overly conservative. Instead, we use the standard Benjamini-Hochberg FDR (Benjamini and Hochberg 1995), which in simulation studies was shown to perform comparably with the BY correction in the case of general dependency and in particular for two-sided t statistics (Romano et al. 2008).

Independent components analysis method
The independent components analysis (ICA) methodology was adopted from Rotival et al. (2011) and applied to the simulated data for comparison with NetLIFT. ICA identifies a predefined number of hidden variables ("independent components") by factoring the gene expression data matrix, X, into a product of two matrices: X SA. Each column of matrix S corresponds to an independent component or factor, and the ith element of a column is the "activation" level of the ith gene in that factor. These factors are meant to model some latent or underlying biological process. The kth row of matrix A reflects the amount of activation of the kth independent component across all individuals, and A ij is activation on the jth individual for component i. Rows of A serve as the response vector when testing SNPs in a linear model. We used the fastICA function implemented in the R programming language to factor the expression data. This algorithm minimizes the statistical dependencies between the columns of S, so that each column of S defines groups of coexpressed genes. Since the method requires an a priori-defined number of components to use in factorization, we set this parameter to 14, the number of modules in each simulated expression data set. To assign individual genes to components, we used the fdrtool function, which models a column's scores as a mixture of null and alternative distributions. Each entry of the column is assigned an FDR corresponding to the likelihood of belonging to the null. For each component (column of S), a corresponding component set was defined for genes with FDR , 0.05. Association tests were performed by regressing allele counts on rows of A, which represent the activation of each component across individuals. SNP-component associations with Benjamini-Hochberg-corrected FDR , 0.05 were considered significant. For each association between a true local eQTL and a component, we defined the number of true positives to be the number of component-set genes that were downstream of the locally affected driver gene. False positives were defined as any other gene assigned to that component set.

Trigger method
The Trigger method is described in Chen et al. (2007). This method aims to infer causality of a genetic variant on expression of a gene by treating genetic variants as randomized variables and leveraging the causality equivalence theorem to identify the direction of effect. Briefly, let s i be the genetic variant to be tested for association, and let g i be a nearby gene. Trigger first tests for association between s i and g i (graphically: s i / g i ), using a standard likelihood-ratio test. This gives Pr(s i / g i ). If the probability of a local association exceeds a defined threshold, the variant is then considered for distal association testing. A similar likelihood test is used for defining the probability of linkage between s i and g j , for all other genes g j , under the condition that Finally, we test whether s i and g j are independent, given the expression of g i : Pr(s i ' g j | g i | s i / g i and s i / g j ).
The causality equivalence theorem can be used to show that Pr À s i 'g j g i js i /g i and s i /g j Á ; so multiplying the probability estimates yields an estimate for direct effect of s i on g j . We use the R package "trigger" for implementation of this algorithm.

Data simulation procedure
A total of 10 gene expression data sets were simulated, each with 500 genes and 250 samples. For each set of 500 genes, a network gene structure consisting of 14 disconnected gene modules of varying numbers of genes was imposed. Sizes of gene modules in each data set were as follows: 100 (32), 50 (32), and 10 (310), leaving 100 genes that were independent of any module. Module topologies are depicted in Supporting Information, Figure S1. For each module, the hub gene's expression values for 250 samples were simulated first, by drawing from a standard normal distribution. Each successive downstream gene's expression was modeled as a linear combination of the upstream gene plus random error, using an effect size of 61 and a random error drawn from a standard normal distribution, represented as where g ds and g us represent expression of the downstream and upstream genes, respectively, and e N(0,1). Genes directly downstream of either the hub gene or a highly connected gene (defined as a gene with degree .20) were chosen to have effect sizes of 1, while all other effect sizes were assigned randomly as 21 or 1 with probabilities 0.3 and 0.7, respectively. Next, for each gene, the total number of SNPs for that gene was drawn from a gamma(4, 0.2) distribution and rounded to the next highest integer. Minor allele frequencies for each SNP were drawn from a uniform(0.05, 0.5) distribution; from these, diploid genotype frequencies encoded 0, 1, and 2 were derived under the assumption of Hardy-Weinberg equilibrium.
For each module, a single gene, not necessarily the hub gene, was chosen to have a local eQTL effect. Since the network topology is undirected, local eQTL effects on nonhub driver genes may lead to spurious distal associations in the analysis. To investigate the sensitivity and specificity of the method under these potentially confounding circumstances, we assigned local eQTL effects to hub genes in some modules and to genes downstream of the hub in others. Furthermore, 30% of the 100 independent genes were assigned at random to have local eQTL effects. If a gene was not chosen to have an eQTL, genotypes were assigned randomly to the 250 samples. For genes chosen to have an eQTL, the direction of effect was chosen to be positive or negative with probabilities 0.7 and 0.3, respectively. Genotype labels were assigned using a genetic algorithm that sought to maximize the effect size under the condition that the significance of association lie within a certain range (here, between 5e-05 and 1e-08). In cases where the eQTL was assigned to the hub gene, all genes in the module were considered as distal targets; however, to model cases where confounding associations may occur between the eQTL SNP and genes "upstream" of the locally affected gene, we also assigned eQTL effects to nonhub genes.
The retrospective allele assignment allowed the specification of desired eQTL effect sizes and significance levels without the need to explicitly consider the pairwise correlations between genes when performing the genotype simulation. This procedure was carried out for 10 simulated data sets. Each data set consisted of gene expression networks for the same module topologies, and each module's expression was characterized by an identical underlying genetic architecture. We defined true distal associations as those genes downstream of the locally associated gene in the expression topology. Working code and a representative simulated data set are available for download at http://fureylab.web.unc. edu/software/netlift/.

Yeast data
Gene expression and genotype data, described previously (Brem and Kruglyak 2005), were obtained from R. Brem (Buck Institute, Novato, CA). A total of 112 yeast segregants were mated from parent strains BY4716 and RM11-1a and grown in culture. Strains were genotyped at 2957 markers and expression measurements were assayed for 6216 ORFs. Genes with no available annotation information were removed, leaving a total of 5647 genes for analysis.

Mouse liver data
Gene expression data were previously assayed on the Affymetrix Mouse Gene 1.0 ST array and were obtained from GEO (accession no. GSE22297) (Aylor et al. 2011). Expression values were normalized using the "rma-sketch" option in the Affymetrix Power Tools package. Probes containing SNPs were masked in the normalization procedure. Probe sets that were expressed at a level .6 on a log2 normalized scale in at least 87.5% of mice were retained, leaving a total of 9377 probe sets for further analysis. Genotypes for 181,752 markers from the "A" test array for the Mouse Diversity Array were obtained from D. Aylor (North Carolina State University, Raleigh, NC).

Human lymphoblastoid cell line data
Gene expression data and HapMap phase 2 and 3 genotypes were obtained from http://eqtl.uchicago.edu. Normalization and processing were performed as described previously (Pickrell et al. 2010). Additionally, the top 25% of transcripts ranked by expression level were retained for further analysis, based on median expression level of the prequantile normalized data across all 69 individuals, leaving 9810 transcripts that were retained for analysis.

Simulation analysis
To assess the sensitivity and specificity of NetLIFT for identifying distal eQTL, we applied the method to 10 simulated data sets consisting of paired expression and genotype data (see Materials and Methods).
For comparison, we also tested three previously described eQTL detection methods: ICA, Trigger, and an all-vs.all pairwise testing approach (AvA) ( Figure S2). The ICA method is primarily suited to identify eQTL that drive the expression of large numbers of distal genes; however, we note that the number of desired components must be defined according to some empirical criteria, and no specific intermediate gene is pinpointed as the trans-acting factor responsible for large-scale variations. Therefore, this method does not identify local eQTL.
We first compared the network structures inferred by NetLIFT's partial correlation analysis to the true simulated regulatory architecture. We found that NetLIFT estimates the gene-gene partial correlation structure with high sensitivity, but note that as module connectivity increases, specificity decreases (Table S1, Figure S3). However, since the network structure is used primarily to determine which SNP-gene tests to perform, the main effect of false network edges is a slight increase in testing burden. As a result, we were willing to tolerate a reduction in network accuracy as long as the sensitivity remained high.
For detection of local eQTL effects, NetLIFT, Trigger, and AvA all identified true positives with 100% success (FDR , 0.05, Table S2). The local eQTL false positive rate for Net-LIFT was identical to that for AvA under this FDR; setting a stricter FDR cutoff of 0.001 resulted in only one false positive for both methods. Additionally, we observed a large number of false positive local eQTL for Trigger, likely due to a lenient default thresholding criterion in the local eQTL testing step. Since we are particularly interested in this method's ability to detect distal eQTL and since distal eQTL identification is conditional on local linkages for this method, we chose to retain the permissive threshold and focus primarily on results for distal associations.
Intramodule distal eQTL were predicted using each method simultaneously, considering all genes and SNPs from all simulated modules. For each module, the true set of distal effects was defined as all SNP-gene associations between the module eQTL and genes downstream of the locally affected gene. Thus, for modules where the eQTL acted on the hub gene, all combinations of the local eQTL SNP with nonhub genes were considered "true positives." For modules with eQTL acting on nonhub genes, the true positives were defined as the eQTL-gene pairs in which the associated genes were downstream of the locally affected, driver gene. False positives were defined as eQTL-gene associations where the associated gene was not downstream of the locally affected gene. Figure 2 details the performance of each of the four methods.
In this case, NetLIFT identified true distal associations at a higher rate for all module topologies (overall 77.9% detection rate), at the cost of a slightly elevated false positive rate. These false positives were mostly due to eQTL SNPs being linked distally to genes that were in the same module, but that were not downstream of the locally affected gene. Since our network estimation step cannot infer directionality of expression effects, these false associations reflect our inability to distinguish true functional associations from those that are due to confounding gene expression correlations present in the data. However, we note that the estimation of direct gene-gene effects and the subsequent testing procedure prevent many upstream genes from being tested against the eQTL SNP, reducing the overall burden of these false associations. Moreover, in a rankbased test performed on FDR values, true positives were found to have higher significance values than the false positives (P = 4.92e-96), again suggesting that the false positive count is strongly dependent on the FDR threshold chosen.
The AvA approach performed poorly, as most true associations were lost after correcting for multiple-hypothesis testing. ICA performed well in large module settings, but poorly for small modules, suggesting that this approach is underpowered for detecting small coregulated gene modules under the influence of a common variant. Trigger performed better than the AvA approach, although in general identified ,12% of true distal associations. NetLIFT was the only method to consistently identify distal effects in all network topologies.
We next evaluated NetLIFT's performance in detecting "hotspot" eQTL loci, where a hotspot is defined as a locus that is associated with more transcripts than are expected by chance. To derive a family-wise error rate (FWER) for each locus, we used the procedure described in Breitling et al. (2008), which permutes genotypes among samples but preserves the correlation structure present in the gene expression data. Performing association testing with the permuted genotype data sets yields a distribution of the expected maximum number of linkages under the null hypothesis of no eQTL associations. When restricting to a FWER of 0.05, NetLIFT identified the eQTL for all hubbased gene modules as hotspots in 10/10 simulated data sets, while the AvA approach identified these eQTL as hotspots only 20-60% of the time and with many fewer linkages (Table S3).
To investigate whether a larger simulated data set affected the sensitivity and/or specificity of our method, we generated and analyzed an additional simulated data set consisting of 2000 genes. We observed that the overall fraction of true and false positives remained similar in this analysis (data not shown). These simulation results indicate that in addition to scaling well to large data sets, NetLIFT may discover distal eQTL that are not readily identifiable with existing detection methods.

Analysis of 112 yeast segregants
We applied NetLIFT to previously analyzed paired genotype/gene expression data for 112 haploid yeast segregants (Brem and Kruglyak 2005). After filtering for genes with available annotation, 5647 genes and 2956 variants were retained for analysis. Variants within 10 kb of the gene's transcribed region were considered "local," and all other linkages were denoted as distal eQTL. At an FDR of 0.05, we identified a total of 1124 (19.9%) and 1642 (29.1%) genes with local and distal eQTL effects, respectively ( Figure  S4). Local and distal effects were observed to have a similar effect size and level of significance (Table S4). The large effect sizes for distal eQTL are in line with previously reported results and are likely attributable to the extreme diversity between the two strains of yeast.
A Gene Ontology (GO) analysis using all 143 genes identified as intermediate TAFs for at least 10 downstream targets revealed enrichments for a wide range of functions, with top hits reserved for metabolic function and transport (Table 1). This corroborates previous findings where putative regulators located near hotspots were not found to be enriched for transcription factors; instead, evidence suggests that many trans regulators exert widespread transcriptional effects by mediating levels of key metabolites or regulating post-translational processes (Yvert et al. 2003;Litvin et al. 2009). A comprehensive list of all putative regulators is provided in Table S5.
For most previously identified hotspots, NetLIFT correctly identified biologically validated regulators (Table 2). Several predicted novel regulators with .15 target genes were also found, many involved in metabolic and biosynthetic processes. In some cases, we provide regulatory evidence for novel drivers not identified previously for detected hotspots; furthermore, our results suggest that there may be numerous secondary drivers within previously identified hotspot regions, indicating that local association signals arising from two or more distinct loci may influence a similar set of distal target genes. One example is the hotspots on chromosome 2 where target genes are enriched for ribosome biogenesis and noncoding RNA (ncRNA) processing (Table  2). Previous results implicated AMN1 and MAK5 as transacting factors for subsets of the target genes; however, patterns of linkage to distinct regions within this locus suggest that additional regulators lie on chromosome 2 (Brem et al. 2002). In addition to AMN1, NetLIFT implicated at least seven new candidate regulators on chromosome 2-TBS1, ARA1, YSW1, TOS1, UMP1, NPL4, and YBR197C-that were strongly linked with local eQTL (P , 1.0e-05) and were associated with highly overlapping sets of distally associated genes ( Figure S5). Notably, we failed to identify MAK5, as this putative regulator was shown to contain a loss-of-function mutation that has no effect on transcription (Brem et al. 2002). By definition, distal effects arising from amino acid substitutions affecting protein function of the trans-acting factor will be undetectable using NetLIFT, as we specifically seek to identify distal effects that arise from local, cis-regulatory effects.
Given the strong enrichment for ribosome function among target genes linking to the chromosome 2 loci, we hypothesized that causal variants would significantly affect growth rates via widespread differential transcription originating from direct up/down local regulation of the candidate TAF. To investigate this, we used segregants' gene expression profiles to predict relative growth rate, using previously described methods (Airoldi et al. 2009). We then tested each of the candidate regulators' distal eQTL for association with the growth rate phenotype. After correction for multiple testing, we found that nearly all of the underlying variants attained significance at FDR , 0.05. We propose that differential expression of the putative regulators influences growth rate by perturbing common, growth-related pathways in trans.
We found numerous loci linking to small sets of target genes that are functionally related, as might be expected from the simulation results. TEC1, a transcription factor that targets filamentation genes, was found to have a significantly associated local variant that was distally linked to 16 genes enriched for pseudohyphal growth annotation (P = 1.03e-03). Additionally, for 5 of these 16 genes (31.2%), the YEASTRACT database shows direct evidence of TEC1 DNA binding and transcriptional regulation (Teixeira et al. 2014). Of the 25 genes that mapped to the lead variant (defined as the variant with strongest local effect on TEC1) in an all-vs.-all test, only 4 (16%) showed direct evidence of TEC1 binding and regulation, suggesting that NetLIFT is better able to identify biologically relevant associations.
We identify several putative regulators that are metabolic enzymes and whose target gene sets are enriched for metabolic and biosynthesis annotations. For example, a locus on chromosome 2 that acts as a local eQTL for LYS2 was distally associated with 167 target genes enriched for the GO term "lysine biosynthetic process via aminoadipic acid" (P = 1.27e-07). LYS2 catalyzes the reduction of a-aminoadipate to a-aminoadipate semialdehyde (aAASA), the fifth step in the lysine biosynthesis pathway. Downstream of this reaction, glutamate-forming saccharopine dehydrogenase, which consists of the structural determinant LYS9 and the regulatory product LYS14, converts aAASA to saccharopine. LYS9 loss of function increases intracellular levels of aAASA, which induces the regulatory activity of Lys14p and results in the upregulation of several genes in the pathway, including LYS1, LYS9, LYS2, LYS4, LYS20, and LYS21 (Becker et al. 1998). In a previous experiment, a mutant strain with loss of function for both LYS2 and LYS9 was shown to have decreased intracellular aAASA and lower levels of transcriptional activation of pathway genes, relative to the LYS9 single mutant (Ramos et al. 1988;Feller et al. 1999). We hypothesize that strains harboring the genomic variant associated with decreased transcription of LYS2 will have a similar reduction of intracellular aAASA concentration and thus a decreased potential for transcriptional activation of Lys14p. Of the previously mentioned lysine biosynthesis genes that are targeted by Lys14p, we find four linked distally to the putative eQTL (LYS1, LYS9, LYS20, and LYS21). We note that the direction of effect between the eQTL and the downstream genes reflects what we expect under the proposed mechanism ( Figure S6). Within the set of transcriptional targets are four additional genes whose promoters contain the Lys14p binding motif, TCCRNYGGA, one of which, LYS12, is involved in lysine biosynthesis and has a directional expression pattern matching the other Lys14p targets ( Figure S6).

Analysis of 156 partially inbred mouse lines
To test how well NetLIFT scales to larger data sets and for organisms with more complex mechanisms of gene regulation, we analyzed paired genotype and liver gene expression data from 156 partially inbred mice originating from 8 founder mice (A/J, C57BL/6J, 129S1/SvImJ, NOD/LtJ, NZO/HlLtJ, CAST/EiJ, PWK/PhJ, and WSB/EiJ), part of the Collaborative Cross (CC) project (Churchill et al. 2004;Collaborative Cross Consortium 2012) (Figure 3). Founder strains of the CC were chosen to provide a high level of genetic diversity and represent three subspecies of origin: Mus mus domesticus, M. m. castaneus, and M. m. musculus. Wild-derived WSB/EiJ and classical inbred strains A/J, C57BL/6J, 129S1/SvImJ, NOD/LtJ, and NZO/HlLtJ have a genetic background composed mostly of the M. m. domesticus subspecies, while the wild-derived CAST/EiJ and PWK/ PhJ founder strains are primarily representative of the M. m. castaneus and M. m. musculus subspecies, respectively (Churchill et al. 2004;Collaborative Cross Consortium 2012).
We filtered for probe sets expressed above background levels and retained 9377 genes for analysis. PCA analysis revealed no batch effects in the data ( Figure S7). Genotypes for the same mice were available for 171,761 markers. In a previous analysis, a total of 6182 eQTL were discovered  for 5733 genes at a 5% genome-wide threshold; 75% of eQTL were within 10 cM of the affected gene (Aylor et al. 2011).
For eQTL testing, we defined local effects as those where variants were within 1 Mb of the affected gene, based on the marker-to-gene distances for linkages reported previously for these data (Aylor et al. 2011). We detected a total of 5744 genes (61%) with a local eQTL and 3322 (35%) with at least one distal eQTL (FDR , 0.05). Of the genes with a distal eQTL, 1102 (12%) were linked to one SNP, 574 (6%) were linked to two SNPs, 400 (4%) were linked to three SNPs, and 1246 (13%) were linked to four or more SNPs.
We next investigated patterns of large-scale effects on the regulatory architecture that are attributable to founder and/ or subspecies of origin. For the 293 genes with a local eQTL that was linked to at least 5 genes on different chromosomes, genes inherited from a PWK genetic background showed more extreme expression variation than genes inherited from the other founder strains ( Figure S8). Mice from the CC have been shown to be phenotypically diverse for various immune-related phenotypes (Ferris et al. 2013;Phillippi et al. 2013), body weight (Philip et al. 2011), and behavior (Philip et al. 2011), with variance for some traits exceeding that observed in the founder strains (Philip et al. 2011). One plausible reason for this is that epistatic interactions between alleles inherited from distinct subspecies (castaneus, domesticus, and musculus) may severely misregulate gene expression and homeostasis. To investigate whether allele inheritance from different subspecies of origin led to more extreme expression for particular combinations of locally acting eQTL alleles and target genes, we mapped both eQTL SNPs and target genes to their subspecies of origin. Since alleles inherited from PWK mice appeared to be driving extreme expression variation in locally affected genes, we reduced the locally affected set of genes to a subset of 61 genes for which the M. m. musculusderived PWK allele explained at least half of the overall genetic effect on expression (Figure 4, top). We observed that for these SNPs, expression of distally linked genes showed differential variation based on the combinatorial genetic backgrounds of the locally associated variant and the target gene (Figure 4, bottom).
These transcriptomic differences may in turn affect phenotype. Body weight for wild-derived founder strains (CAST/EiJ, PWK/PhJ, and WSB/EiJ) used in the Collaborative Cross is lower than in classical laboratory strains (Aylor et al. 2011). A GO analysis performed for the 142 distal genes linking to the PWK-driven eQTL revealed annotation for various terms related to metabolism and lipid processes (Table 3). This enrichment suggests a possible role for the candidate trans-acting factors in regulating weight, via a broad but subtle effect on gene expression.

Analysis of 69 human individuals
RNA-seq data from lymphoblastoid cell lines and HapMap genotype data for 69 Nigerian individuals were recently  interrogated for eQTL (Pickrell et al. 2010). For NetLIFT analysis, expression data were corrected for GC content and batch and were normalized as described previously. We selected 9810 Ensembl transcripts in the top quartile based on median expression level for further analysis. Genotype data for the same individuals, consisting of 9.5 million SNPs, were obtained from HapMap phases 2 and 3, release 27. Using a local regulatory window of 200 kb, similar to the original analysis (Pickrell et al. 2010), we identified 2483 transcripts (25.3%) with a local eQTL effect (FDR , 0.10). Of the 929 transcripts previously identified as having local associations at the same FDR, we replicated 538. The remainder not found consisted of transcripts that we removed from the data set due to low median expression level, with the exception of 3 transcripts that were not identified in our analysis. In addition, we identified 1945 novel local associations, likely attributable to greater power resulting from testing only the most highly expressed quartile of transcripts.
NetLIFT identified 1274 transcripts (13.0%) with at least one distal eQTL (FDR , 0.10, Figure S9). None were reported in the previous analysis (Pickrell et al. 2010). A traditional all SNPs-vs.-all genes testing approach on this filtered set of genes and variants yielded only five significant distal associations at this FDR, indicating that our method is better powered for detecting these associations. A GO analysis for the 64 candidate regulators that were linked to at least 3 transcripts (FDR , 0.1) again suggested enrichment for metabolic and biosynthetic processes (Table 4).

Discussion
Genome-wide association studies (GWAS) have so far identified thousands of quantitative trait loci associated with hundreds of complex traits (Hindorff et al. 2009). However, the success of GWAS has been tempered by a lack of understanding of the mechanism of association for many variants. eQTL studies have shown excellent promise in highlighting potential biological mechanisms of SNPphenotype associations and prioritizing particular variants for follow-up studies (Mehta et al. 2012). Furthermore, the correlation between significance levels of SNP-phenotype associations and eQTL associations may help to identify tissue types that play a key role in disease etiology (Kang et al. 2012). Recently, gene-gene interaction evidence has been incorporated in the GWAS setting to identify epistatic effects on phenotype (Ma et al. 2013), suggesting that correlationbased testing may increase power to detect associated variants. We described here a novel method, NetLIFT, that addresses the problems of computational burden and power in traditional eQTL testing, by reducing the search space and using conditional dependencies between genes' expression to prioritize variant-gene testing. The reduced multiple- Figure 3 Distal eQTL associations in pre-Collaborative Cross mice. The x-axis gives the genomic coordinates of marker SNPs; the y-axis represents gene position. Each dot represents a significant marker-gene association at FDR , 0.05, for markers that were at least 1 Mb from the associated gene.
testing correction penalty under our algorithm allows detection of weaker eQTL effects that are missed by currently available methods. Furthermore, our results provide immediate interpretability of the mechanism of association, by highlighting potential regulatory genes that mediate discovered distal effects. We note that in the current implementation of our code, runtime and memory usage increase nonlinearly as the number of genes increases and the major bottleneck in runtime is the estimation of the partial correlation matrix. Therefore, when the number of genes exceeds 10,000, users may wish to filter gene expression data sets by most highly expressed or most variable genes.
Importantly, we showed through simulations that Net-LIFT can identify instances where distal eQTL affect only a small number of genes, not just the large hub genes found by other methods. Additionally, candidate regulators that are putatively affected in cis by the causal variant can be identified, highlighting potential mechanisms of association. We note that since our method seeks to identify distal effects that arise via alterations in the expression level of transacting factors located nearby the eQTL, we are unable to detect associations mediated by a loss-of-function coding variant in the trans-acting factor.
We demonstrated the ability of NetLIFT to identify distal eQTL in three very different data sets. In yeast segregants, we replicated numerous distal eQTL reported previously, as well as the biologically validated regulators for many of the associations. Additionally, we identified several novel biologically Figure 4 Expression variability for PWK-driven trans-acting factors and target genes, in pre-Collaborative Cross mice. (Top) Distribution of absolute expression deviation from median, for putative trans-acting factors with a PWK-driven local eQTL, grouped by founder strain genetic background at the eQTL locus. Only putative trans-acting factors that were linked to at least five target genes on a different chromosome were considered. (Bottom) Expression distribution for target genes of PWK-driven eQTL loci, stratified by subspecies-of-origin allele (castaneus/domesticus/musculus) at both the local and distal loci. Each boxplot represents the expression deviation for all target genes, for each possible combination of local/distal alleles. plausible distal associations. In inbred lines from genetically diverse founder mice, we detected an interesting pattern of eQTL effects driven by PWK-derived alleles, which may provide clues as molecular underpinnings of downstream phenotypes such as reduced mouse size in the wild-type derived PWK mice. Finally, in a set of 69 human individuals, NetLIFT was able to find .1200 gene transcripts with significant distal eQTL due to its increased power, whereas previously only 5 had been identified.
Intuitively, one might think that the best candidates for asserting regulatory influence on distal genes would be transcription factors that directly participate in controlling gene transcription rates. In accordance with previous results, however, we found no enrichment for transcription factor annotation among genes implicated by our method as trans-acting factors; instead, we find that many of these genes play a role in metabolic and biosynthesis pathways. This suggests that more commonly, the regulation of key genes in these pathways plays a role in feedforward or feedback processes that then affect transcription rates of downstream target genes within the same pathway. These indirect effects are more subtle than the direct effects associated with local eQTL, but they can have significant effects on phenotypes, such as growth rates (seen in yeast) and size (seen in mice).
Our results also highlight an often unaddressed topic in complex trait mapping, namely, that eQTL discovery and interpretability of mapping results are significantly influenced by the genetic and genomic diversity within the sample population. The two yeast strains from which the analyzed segregants were derived were extremely diverse, with an estimated sequence divergence of 0.5-1%. This, and overall genome complexity, likely contributed to many distal effects being found to be as strong as local effects, enabling their easier detection. Genetic incompatibilities between progenitors can result in atypical patterns of linkage disequilibrium, which present challenges in identifying causal vs. linked markers. In an inbred mouse model, we were able to identify numerous distal linkages where expression variation in the distally affected genes appears to be driven by differences in the genetic background at the local and distal loci. However, the resolution of the eQTL mapping is ultimately restricted by the randomization of the genome that is mediated by recombination events. On the other hand, human studies typically involve genetically diverse individuals, whose genomes are randomized to a greater extent. Thus a model organism may allow for accurate eQTL mapping at the expense of precision, whereas in human populations we expect to identify eQTL with precision, but reduced accuracy.

Literature Cited
Airoldi

Figure S1
Simulated gene module topologies. Each module's expression effects were simulated by first generating the hub gene's expression; each successive downstream gene's expression values were simulated using the upstream gene's expression as a baseline (dependencies indicated by arrows). For each module, a single local eQTL effect was simulated for a SNP assigned to either the hub gene (black), or to a gene downstream of the hub (red), but not both.

Figure S2
Illustration of eQTL detection methods. SNP is depicted as a red node; genes depicted in green. All vs All (A) performs a standard regression significance test for all pairs of SNPs and genes. Trigger (B) seeks to identify distal associations that are mediated by a locally associated variant-gene pair (local associations depicted with blue arrows). Genes downstream of the inferred direction of gene-gene effects (represented by green arrows) should be associated with the variant (true distal associations = solid black arrows), while genes upstream of gene effects will not show association (dashed black arrows). Independent Components Analysis (C) first factors expression data into Independent Components, then performs association tests between allele frequency and the activation levels of components across samples. NetLIFT (D) first performs local linkage tests for a SNP and nearby genes (blue arrows). For significant linkages (solid blue arrow), distal eQTL tests are performed for all genes in the network which are one-or two-edges removed from the locally affected gene (black arrows).

Figure S3
Partial correlation structure from network detection step, for representative 100 gene, 50 gene, and 10 gene modules. True positive correlations depicted with green edges, false negatives correlations in red, false positives in gray.

Figure S4
Local and distal eQTL linkages in yeast. X axis shows the genomic coordinates of marker variants; Y axis represents gene position. Each dot represents a significant marker-gene association at FDR < 0.05.

Figure S9
Local and distal eQTL linkages in human lymphoblastoid cell lines. X axis shows the genomic coordinates of SNPs; Y axis represents gene position. Each dot represents a significant marker-gene association at FDR < 0.1.

Table S1
Sensitivity and specificity of partial correlation detection, by module. Column 1 shows the mean and standard deviation for the fraction of true edges detected, for ten simulated data sets. Column 2 estimates the intra-module false-edge detection rate. For each module, the ratio of false positive edges detected to the total number of possible false edges is reported.

Module Topology
Mean (