Learning Gene Networks under SNP Perturbations Using eQTL Datasets
[['Recent advances in the next-generation sequencing and other high-throughput technology has allowed researchers to collect various types of genome-scale datasets , providing unprecedented opportunities to discover detailed gene regulation processes in cells via analysis of the massive data .', 'The standard approaches for identifying the causal regulatory relationship among genes have been based on examining gene-expression data collected after an experimental perturbation of one or two genes such as in gene knockout studies [1] , [2] or over-expression studies [3] .', 'In a typical experimental design , genome-wide gene-expression levels are measured using microarrays under different experimental conditions such as strains with one or two genes knocked out [1] , [2] .', 'Then , the differential gene-expression patterns between control and experimental conditions are examined to obtain clues as to the organization of genes into functional modules and key regulators of those modules .', 'However , this standard approach for identifying the wiring of gene networks based on experimental perturbations of gene regulation system comes with many limitations .', 'Experimentally perturbing the activity of a gene can be very costly , time-consuming , and laborious and it is even more so for repeating such perturbation for every single gene in an organism to obtain a comprehensive picture of gene network wiring .', 'Furthermore , the experimental methods are usually limited to a perturbation of one or two genes at a time due to experimental infeasibility and combinatorial explosion in the number of experiments to perform .', 'Thus , they cannot be used to perturb more than two genes at the same time to obtain information on multifactorial gene interactions .', 'More importantly , it is often not possible to apply such experimental perturbation studies to humans for ethical reasons .', 'Finally , given the set of differentially expressed genes under each perturbation , it is difficult to distinguish between those genes that are directly regulated by the perturbed gene and those genes in the downstream of the pathway whose expressions are influenced as secondary/indirect effects .', 'Genetical genomics approach has been proposed as a less expensive but more powerful alternative to the approach with experimental perturbations [4] , [5] .', 'Genetical genomics treats genetic variation as naturally-occurring perturbation of gene regulatory networks and tries to learn gene networks by examining the effects of genetic variation on gene expression measurements within a large population of individuals .', 'The key advantage of genetical genomics approach is that unlike experimental perturbations that can be performed only on one or two genes at a time , there are more than millions of genetic variants across genomes in the case of a human population , enabling the effects of multifactorial perturbations to be observed directly in gene expression data .', 'Another advantage is that while experimental perturbation studies involve artificial perturbations in lab with often large perturbation effects , the perturbations of gene network by genetic variants occur in nature and usually induce more subtle changes in gene expressions .', 'Thus , understanding the consequences of network perturbations by genetic variants is likely to lead to more direct understanding of the gene networks that exist in nature .', 'In addition , genetical genomics approach can be easily applied to humans as well as other organisms , since genotype and gene-expression data are routinely collected for the purpose of expression quantitative trait locus ( eQTL ) mapping to understand the genetic architecture of complex phenotypes and diseases [6] , [7] .', 'However , the genetical genomics approach poses a significant computational challenge , because it is not obvious how to decode the effects of multifactorial genetic perturbations from genotype and gene expression data .', 'In an experimental approach with a perturbation of one or two genes , the genes that are differentially expressed under each perturbation experiment can be easily identified with a simple computation .', 'However , in genetical genomics approach , the gene expression variability is the result of aggregated effects of multifactorial perturbations by a large number of genetic variants and it is not obvious how to decouple the aggregated perturbation effects to identify the set of genes that are differentially expressed with respect to each perturbation by each individual genetic variants .', 'Furthermore , as many of the genetic variants do not have any functional consequences or perturbation effects , genetical genomics approach has an additional computational challenge of identifying eQTLs or the genetic variants that affect gene expressions , while identifying the gene network perturbed by these eQTLs at the same time .', 'Because of these computational challenges , given gene-expression and genotype data , researchers tended to limit their analysis to eQTL mapping , where eQTL mapping can be viewed as a special case of genetical genomics analysis that assumes an isolated effect of genetic perturbation on a single gene with no downstream effects in gene network .', 'While statistical methods such as graph-guided fused lasso ( GFlasso ) [8] have been developed to detect genetic effects on multiple correlated gene-expression traits , they focused only on eQTL mapping assuming a known gene network , instead of performing a more powerful genetical genomics analysis .', 'Those few existing computational methods for genetical genomics analysis have been limited in terms of computational efficiency and statistical power [9]–[12] .', 'For example , discovering eQTLs and reconstructing the gene network perturbed by those eQTLs were performed in two separate steps , leading to reduced statistical power [9] , or average gene-expression levels within each gene module were used as a trait to identify eQTLs rather than using the original gene-expression data , leading to the loss of information on individual gene activities [13] .', 'In this paper , we propose a statistical framework that directly addresses all of the above computational challenges of genetical genomics analysis within a single statistical analysis to achieve the maximum statistical power for identifying gene networks via single-nucleotide-polymorphism ( SNP ) perturbations .', 'Given SNP genotype and gene-expression data collected for a large number of individuals , our statistical method simultaneously identifies', '1 ) the gene network structure by decoding the effects of multifactorial perturbations of gene regulation system by a large number of SNPs ,', '2 ) eQTLs that perturb this gene network ,', '3 ) the genes whose expressions are directly perturbed by eQTLs and the genes whose expressions are indirectly perturbed as secondary downstream effects of the direct perturbations in the network , and', '4 ) detailed characterizations of the SNP perturbation effects on the gene network by decoupling the complex multifactorial SNP effects on the gene network with respect to each individual perturbation .', 'Our proposed statistical framework is based on probabilistic graphical models , and in particular , we introduce a new statistical model , called a sparse conditional Gaussian graphical model ( CGGM ) , that models a gene network under SNP perturbations as an undirected graphical model .', 'In our statistical model , the unknown gene network is represented as a graph over gene-expression traits and this graph is associated with an unknown probability distribution that models the strengths of gene-gene interactions in the gene network and the strengths of perturbation effects of SNPs ( Figure 1A ) .', 'Then , both the gene network structure perturbed by SNPs and probability distribution associated with the network structure are learned jointly from data .', 'We show that the learning problem is convex , leading to increased statistical power and a guarantee in the quality of the estimated model , and develop an efficient learning algorithm that scales to a large dataset .', 'Given the estimated graphical model , we describe inference methods to characterize the detailed mechanisms of how the effects of SNP perturbations propagate through the network ( Figures 1B–D ) .', 'From the computational point of view , addressing the challenges of genetical genomics analysis requires handling the computational challenges of both gene-network analysis given gene-expression data and eQTL mapping given gene-expression and genotype data , namely gene-network structure learning and SNP feature selection , at the same time .', 'We show that in fact our sparse CGGM subsumes as special cases both a sparse Gaussian graphical model [14]–[17] , which is popular as a model for gene network , and a sparse linear regression model [8] , [18] , [19] , which is widely used for eQTL mapping , thus providing a natural unifying representation for a gene network and eQTLs perturbing this network .', 'Moreover , by embedding the standard regression model for eQTL mapping within a probabilistic graphical model and leveraging the representational power of a graphical model , our approach allows to extract a significantly more detailed characterization of the functional roles of eQTLs than any of the existing methods for eQTL mapping .', 'In our experiments , we apply our statistical framework to HapMap-simulated and yeast eQTL datasets [20] , [21] .', 'Using HapMap-simulated data , we demonstrate our approach can recover the true underlying gene network under SNP perturbations , and at the same time , can recover true eQTLs with greater statistical power than other existing methods that have been developed for eQTL mapping .', 'In addition , we applied our method to yeast eQTL dataset collected for 112 segregants of two yeast parent strains , BY4716 and RM11-1a [21] .', 'Nearly all of the previous analyses of this dataset focused either on eQTL mapping or on gene network analysis ignoring the genetic information , with an exception of the genetical genomics analysis of this dataset performed by multivariate regression with covariance estimation ( MRCE ) [22] that has also been mistakenly called sparse CGGM [23] as we further discuss in the next section .', 'However , the analysis by MRCE [23] was performed for only a small subset of the full dataset because of its expensive computational cost .', 'In our experiment , MRCE took more than weeks of computation to analyze the full yeast eQTL dataset , whereas our method ran within a day .', 'We provide an in-depth analysis of SNP perturbations of a subnetwork over genes involved in stress response in yeast , and show that this subnetwork obtained from SNP perturbations by our approach is well supported by the network inferred from experimental perturbations in knock-out studies in the literature .'], ['A Gaussian graphical model defines a probability distribution over an undirected graph that models a gene network .', 'The nodes of the graph correspond to continuous-valued random variables for gene-expression traits and the edges represent probabilistic conditional dependence relationships between pairs of nodes [14]–[17] .', "Given microarray gene-expression measurements for genes and individuals , a Gaussian graphical model assumes that the gene-expression measurement for the th individual is an independently and identically distributed sample from a Gaussian distribution , where is a vector of 0's and is a covariance matrix .", 'Then , it is well-known that the inverse covariance represents a Gaussian graphical model , where the non-zero ( or zero ) value for in the th entry of represents the presence ( or absence ) of edges between the th and th gene-expression traits in the gene network .', 'While each non-zero element in implies conditional dependency between the th and th gene-expression traits given all the other gene-expression traits , computing the inverse of to obtain the covariance amounts to performing an inference in this graphical model to obtain marginal dependencies , or equivalently dependencies between the two nodes without consideration of any other nodes .', 'Graphical lasso [14] , [16] , [24] has been widely used to learn a sparse Gaussian graphical model , where only statistically significant gene-gene interactions have edges with non-zero entries in .', 'Graphical lasso minimizes the negative log-likelihood with sparsity-inducing penalty as follows: ( 1 ) where is the trace of matrix , is the sample covariance matrix , is the norm of , and is the regularization parameter that determines the amount of sparsity .', 'A large value of leads to a sparser estimate with a greater number of zero elements in .', 'The optimal value for can be determined using cross-validation .', 'The problem in Eq .', '( 1 ) is convex and can be solved efficiently [16] .', 'In eQTL mapping , the problem of identifying SNPs influencing gene-expression levels from eQTL data is often formulated as that of learning a multivariate linear regression model [8] , [18] , [19] .', 'Given genotype data for SNPs , where is a vector of length with each element taking values from for the number of minor alleles at the given locus , and the expression measurements for genes for samples , a linear regression model for the functional mapping from SNPs to gene-expression traits is given as: ( 2 ) where is the regression coefficient matrix representing the unknown association strengths , and is the matrix of noise terms whose rows are Gaussian-distributed with mean zeros and covariance .', 'Typically , a model without an intercept is considered , assuming that the genotype data are standardized to have mean and unit variance .', 'Since each gene-expression trait typically has only a small number of eQTLs , lasso [19] , [25] has been widely used to obtain a sparse estimate of .', 'Lasso minimizes the squared-error criterion with penalty as follows: ( 3 ) where is a regularization parameter that controls the amount of sparsity in .', 'Eq .', '( 3 ) is convex with a single globally optimal solution , and efficient algorithms are available for solving it [26] .', 'Lasso essentially performs separate regression analyses , treating the gene-expression traits as independent of each other .', 'In order to combine the statistical power across multiple correlated gene-expression traits , GFlasso [8] assumed a known gene network and extended the standard lasso by including an additional penalty , called graph-guided fusion penalty , that encourages multiple related genes in the gene network to be influenced pleiotropically by a common SNP .', "Given a gene network with a set of edges and edge weights 's for each edge between the th and th genes , the graph-guided fusion penalty takes the form of , where each term in the penalty encourages the amount of influence of the th SNP on the expression levels of the th and th genes to be similar if the two genes are connected with an edge in the network .", 'GFlasso was capable of identifying SNPs with pleiotropic effects , but it was restrictive in that the gene network should be known a priori .', 'Within the same statistical framework of linear regression method , MRCE [22] attempted to shift the focus from eQTL mapping towards genetical genomics analysis by identifying the gene network and eQTLs jointly .', 'Towards this goal , MRCE relaxed the assumption of uncorrelated noise ( i . e . , being a diagonal matrix ) in lasso and estimated the full noise covariance matrix .', 'Then , the inverse of the noise covariance corresponds to a gene network .', 'MRCE minimizes the negative log-likelihood of data with an penalty for both and : ( 4 ) where and are the regularization parameters .', 'We notice that unlike GFlasso , MRCE does not have any mechanisms to leverage the estimated gene network to model pleiotropic effects of SNPs on multiple correlated gene-expression traits in .', 'The optimization problem in Eq .', '( 4 ) is not convex , but bi-convex , since fixing either or and solving for the other is a convex optimization problem .', 'Thus , Rothman et al . [22] proposed to optimize for each of and alternately given the other over iterations .', 'However , they noted that this strategy often does not converge , and instead suggested to use an approximate method that prematurely terminates the iterative optimization procedure after two iterations .', 'As we discuss in the Results section , we found that even this approximate method was too slow to be applicable to a dataset of even moderate size .', 'The same statistical method for MRCE has been proposed independently in the literature under the name of sparse conditional Gaussian graphical models ( CGGMs ) [23] .', 'However , we emphasize that the statistical model that is learned in MRCE is not a graphical model , because as we further discuss in detail in the next sections , the parameters do not model conditional dependencies as in graphical models but only models marginal dependencies .', 'We believe that MRCE was mistakenly called a sparse CGGM due to the resemblance between the inverse noise covariance matrix in MRCE and the inverse covariance matrix in graphical lasso as well as the aspect of the standard regression model as a conditional model for given .', 'The sparse CGGM that we propose in this paper is set up as a proper probabilistic graphical model and as we show in the next sections , is significantly more powerful than MRCE in terms of representational power and computational efficiency .', 'Under the same name of sparse CGGMs , Li et al . [27] introduced a related but different statistical method that also models a gene network corresponding to in MRCE .', 'However , unlike our approach and MRCE , the estimation procedure for the sparse CGGM in [27] amounted to a two-stage process , where the gene-expression data are pre-processed to remove SNP effects in the first stage and then these pre-processed gene-expression data are used to learn a gene network in the second stage .', 'Thus , their graphical model was defined only on gene-expression traits and did not directly model the relationship between SNPs and gene expressions to identify eQTLs .', 'In contrast , our sparse CGGM is set up as a graphical model on both gene expressions and SNPs , performs a joint estimation of gene network and eQTLs , and infers various perturbation effects of SNPs on gene expressions via inference .', 'In this section , we introduce a statistical model for CGGM as a model for a gene network under SNP perturbations .', 'Then , in the next sections , we describe a learning algorithm for estimating a sparse model for CGGM from data and discuss inference schemes for the estimated sparse CGGM .', 'A sparse CGGM estimated from data captures the gene network structure and direct perturbations of the gene-expression levels by eQTLs via conditional dependency structure in the estimated graph structure .', 'By performing inference on this estimated sparse CGGM , we can obtain a detailed characterization of how the direct SNP perturbation effects propagate through the gene network to perturb the expression levels of other genes indirectly .', 'The key idea behind our proposed approach is to model a gene network under SNP perturbations as a Gaussian graphical model for a gene network conditional on SNPs .', 'We derive a CGGM as a conditional distribution from the Gaussian graphical model for a joint probability distribution for SNPs and gene-expression traits for the th individual .', 'Let us assume a Gaussian graphical model with covariance and inverse covariance , where zero mean is assumed after mean-centering each gene-expression trait and SNP .', 'Then , the conditional distribution of given can be obtained as .', 'We further re-write this conditional distribution , using the inverse covariance matrix and the partitioned inverse formula [28] to obtain a CGGM: ( 5 ) CGGM parameters and represent a gene network and SNP perturbation effects on this gene network , respectively .', 'A non-zero value for the th element of indicates that the th SNP is an eQTL for the th gene-expression trait .', 'This SNP perturbation captures the direct influence of the th SNP on the th gene expression , since the graphical model captures conditional dependencies .', 'While in experimental perturbation studies the expressions of only one or two genes can be directly perturbed ( e . g . , by knocking out the genes ) , our CGGM for genetical genomics study allows multiple gene-expression traits to be perturbed by multiple SNPs at the same time .', 'Then , this multifactorial genetic perturbations of gene-expression levels are decoded to learn a gene network by a learning algorithm that estimates and simultaneously .', 'In order to show the direct correspondence between a CGGM and a general undirected graphical model , we re-write Eq .', '( 5 ) by expanding the quadratic term in the Gaussian distribution in Eq .', '( 5 ) to obtain: ( 6 ) where is a constant , also known as a partition function in the literature of probabilistic graphical models [29] , which ensures that forms a proper probability distribution integrating to .', 'As Eq .', '( 6 ) is equivalent to the Gaussian-distribution form in Eq .', '( 5 ) , this constant can be obtained in a closed-form by directly comparing Eq .', '( 6 ) with Eq .', '( 5 ) .', 'If is positive definite , the integral in the partition function is finite and the probability distribution is well-defined .', 'The representation in Eq .', '( 6 ) explicitly shows that a CGGM is an undirected graphical model [29] defined over a graph with two sets of edges , namely the set of edges connecting each pair of gene-expression traits in gene network and another set of edges connecting each SNP to gene expressions that the SNP is influencing ( Figure 1A ) .', 'Then , following the definition of an undirected graphical model [29] , the numerator in Eq .', "( 6 ) is a weighted sum of features over the graph edges , where 's and 's are features and and define edge weights .", 'The gene network is modeled an undirected graph , but the directions from SNPs to gene-expression traits are implicit , since the model is a conditional probability model for gene-expression traits conditional on SNPs .', 'Since both gene-gene interactions and SNP perturbations of gene-expression traits are highly modular and localized , we are interested in learning a sparse model for CGGM .', 'In other words , only statistically significant gene-gene interactions should be represented as edges with non-zero entries in and only a small number of statistically significant direct SNP perturbations should be estimated as having non-zero effect sizes in .', 'In order to impose a sparsity constraint , we learn a sparse CGGM by minimizing the negative log-likelihood of data with an penalty as follows: ( 7 ) where is the negative log-likelihood of data based on Eq .', '( 6 ) or equivalently Eq .', '( 5 ) , and and are the regularization parameters that control the amount of sparsity .', 'It is not necessary to explicitly consider the positive-definite constraint for within the optimization problem in Eq .', '( 7 ) , because the partition function in the data log-likelihood contains term that acts as a log-barrier function for the positive-definite constraint [30] .', 'Within the penalty for , we do not penalize the diagonal elements of , since we found that this leads to a slightly better performance in our experiments , consistent with what has been reported for graphical lasso .', 'It is straightforward to prove that the problem in Eq .', '( 7 ) is convex ( Text S1 ) [30] .', 'Thus , the learning algorithm is guaranteed to find the globally optimal solution that achieves the maximum statistical power .', 'The main challenge for solving Eq .', '( 7 ) arises from the non-smoothness of the penalty function .', "We adopt a variant of accelerated proximal gradient algorithms , called a Nesterov's second method [31] , that has been developed as a general-purpose algorithm for handling a non-smooth component of the parameter estimation problem while improving the convergence ( and thus , computation time ) of the standard gradient descent algorithm [31]–[33] .", 'We provide details of the learning algorithm in Text S2 .', 'Lasso [25] and graphical lasso [14] , [16] , [24] can be viewed as special cases of the sparse CGGM estimation problem in Eq .', '( 7 ) .', 'When , the sparse CGGM learning problem in Eq .', '( 7 ) essentially reduces to applying graphical lasso to gene-expression data , ignoring genotype data , since the large encourages all or nearly all of the elements of to be set to zeros .', 'On the other hand , if , the sparse CGGM learning problem becomes equivalent to lasso that fits a regression model for each gene-expression trait separately , ignoring gene network , since the large tends to set all or almost all of the off-diagonal elements of to zeros .', 'The optimal values for and that strike the right balance between these two extreme cases can be found by cross-validation .', 'So far , we showed that by learning a sparse CGGM , it is possible to decode the underlying gene network and its direct multifactorial perturbations by SNPs from data .', 'Now , we show that given a sparse CGGM estimated from data , we can perform inference on this graphical model to characterize the mechanisms of SNP perturbations of gene network in detail .', 'Below , we discuss how inference schemes can be used on our estimated model to learn about indirect/secondary downstream effects of the direct SNP perturbations , a decomposition of the overall multifactorial SNP perturbation effects with respect to each individual direct perturbation , and a decomposition of observed covariance in gene expressions into genetic and non-genetic components .', 'We note that all of these inference schemes involve only few simple matrix operations and are highly efficient .'], ['In order to simulate eQTL datasets , we used the SNP genotype data from HapMap phase III release 2 [20] as SNP data and simulated gene-expression traits , given and known model parameters .', 'We used the SNP data for chromosome 21 of the 343 individuals of African origin , including ASW , LWK , MKK , and YRI population groups .', 'After removing SNPs with minor allele frequency and highly correlated SNPs with squared correlation coefficient , we obtained 4 , 901 SNPs .', 'In each simulated dataset , we randomly selected a region of 500 SNPs and simulated the values of 30 gene-expression traits for each individual , based on the CGGM in Eq .', '( 5 ) .', 'As almost all statistical methods for eQTL mapping assumes the standard linear regression model in Eq .', '( 2 ) , we performed experiments on gene-expression traits simulated from this model as well .', 'In order to set the model parameters , we first set the sparsity pattern and then assigned values to the non-zero elements of the parameters as follows .', 'While in our simulation studies we primarily focused on the relatively small datasets of 500 SNPs and 30 gene-expression traits as described above , in order to demonstrate the performance and scalability of our method , we also applied our method to larger-scale simulated datasets of 1 , 000 SNPs and 500 gene-expression traits .', 'Because MRCE , the main competing method to our approach , required substantially more computation time than our approach even on the smaller datasets , we were unable to compare the performance of MRCE on these larger simulated datasets .', 'Instead , we compared our method with other computationally efficient methods , including GFlasso and a base-line approach of applying graphical lasso [16] and lasso [25] sequentially to learn gene networks and eQTLs .', 'We use precision-recall curves and prediction errors as quantitative measures of the performance of different statistical methods .', 'Precision-recall curves summarize how accurately each method recovers the true eQTLs and gene network structure by plotting precisions and recalls on - and -axes .', 'In order to compute precisions and recalls , for each simulated dataset , we ranked all the elements of the estimated parameter matrices in a descending order according to their absolute values and compared this ranked list with the set of non-zero elements in the true parameters .', 'On the other hand , prediction errors evaluate the performance of different methods on how accurately each method can predict gene-expression levels , given SNPs and estimated model parameters .', 'Once the parameters are estimated using training data , prediction errors are obtained as , where is the prediction of gene expressions given by the model for the th individual in an independent test dataset and is the number of samples in the test set .', 'Given the 343 samples in the full dataset , we used 300 samples as a training dataset and the remaining 43 samples as a test dataset .', 'In order to determine the optimal regularization parameters in sparse CGGM , MRCE , and GFlasso during the training phase , we created a grid for different choices of regularization parameters , performed a five-fold cross-validation for each point on the grid , and selected the values that give the smallest cross-validation error as the optimal regularization parameters .', 'In order to illustrate the behavior of a sparse CGGM , we present the results from applying our method to a single dataset simulated from a CGGM parameterization and compare them with what we obtained from GFlasso and MRCE ( Figure 2 ) .', 'The non-zero elements of the true model parameters were drawn from for and for .', 'Since GFlasso requires the gene network to be known , we used the correlation coefficient matrix of gene-expression trait data thresholded at as a gene network in GFlasso estimation .', 'The true parameters for and along with are shown in the left , middle , and right columns , respectively , in Figure 2A .', 'The estimated parameters for sparse CGGMs , MRCE , and GFlasso are shown in Figures 2B–D , respectively .', 'In the plots for and , the rows and columns correspond to gene-expression traits and SNPs , respectively , and the results are shown only for the first 150 SNPs .', 'In each panel , the white pixels correspond to the zero elements of the parameters and the darker pixels to non-zero elements .', 'We note that while sparse CGGM provides the estimates of both and , MRCE and GFlasso provide a single estimate of eQTL effect sizes in and do not distinguish between direct and indirect effects of eQTLs on gene-expression traits .', 'As shown in Figure 2B , our method successfully recovers the three gene modules in the true gene network as the block-diagonal structure in along with the sparse direct perturbation of this network by eQTLs in .', 'When we perform inference in the estimated sparse CGGM by computing to learn indirect perturbation of the network by eQTLs , the direct perturbations of eQTLs in propagate primarily within each gene module in , leading to vertical stripes in .', 'Although MRCE learns a gene network from data , unlike sparse CGGM , it does not have any mechanism to leverage this gene network to learn pleiotropic or indirect effects of eQTLs on gene modules and the estimated in Figure 2C shows isolated eQTLs for individual gene-expression traits rather than vertical stripes .', 'As can be seen in Figure 2D , the GFlasso estimate of shows vertical stripes for eQTLs common within each gene module .', 'However , it is immediately clear that GFlasso results have significantly more false positives for eQTLs than sparse CGGM and MRCE .', 'This demonstrates that genetical genomics approach has the potential to improve the accuracy for detecting eQTLs than the conventional approach that focuses solely on eQTL mapping .', 'We performed a quantitative comparison of the performance of sparse CGGM , MRCE , and GFlasso , by obtaining precision-recall curves and prediction errors averaged over 50 simulated datasets .', 'Since MRCE and GFlasso are based on the standard linear regression model and sparse CGGM is based on a graphical model , we evaluate the different methods on datasets simulated from both models .', 'Since MRCE and GFlasso use the standard linear regression model , we also compared the performance of the different methods , using datasets simulated from the model in Eq .', '( 2 ) with known parameters for and ( Figure 6 ) .', 'We present the precision-recall curves for and averaged over 50 simulated datasets 
['Introduction', 'Materials and Methods', 'Results', 'Discussion']
['The standard approach for identifying gene networks is based on experimental perturbations of gene regulatory systems such as gene knock-out experiments , followed by a genome-wide profiling of differential gene expressions .', 'However , this approach is significantly limited in that it is not possible to perturb more than one or two genes simultaneously to discover complex gene interactions or to distinguish between direct and indirect downstream regulations of the differentially-expressed genes .', 'As an alternative , genetical genomics study has been proposed to treat naturally-occurring genetic variants as potential perturbants of gene regulatory system and to recover gene networks via analysis of population gene-expression and genotype data .', 'Despite many advantages of genetical genomics data analysis , the computational challenge that the effects of multifactorial genetic perturbations should be decoded simultaneously from data has prevented a widespread application of genetical genomics analysis .', 'In this article , we propose a statistical framework for learning gene networks that overcomes the limitations of experimental perturbation methods and addresses the challenges of genetical genomics analysis .', 'We introduce a new statistical model , called a sparse conditional Gaussian graphical model , and describe an efficient learning algorithm that simultaneously decodes the perturbations of gene regulatory system by a large number of SNPs to identify a gene network along with expression quantitative trait loci ( eQTLs ) that perturb this network .', 'While our statistical model captures direct genetic perturbations of gene network , by performing inference on the probabilistic graphical model , we obtain detailed characterizations of how the direct SNP perturbation effects propagate through the gene network to perturb other genes indirectly .', 'We demonstrate our statistical method using HapMap-simulated and yeast eQTL datasets .', 'In particular , the yeast gene network identified computationally by our method under SNP perturbations is well supported by the results from experimental perturbation studies related to DNA replication stress response .']
['A complete understanding of how gene regulatory networks are wired in a biological system is important in many areas of biology and medicine .', 'The most popular method for investigating a gene network has been based on experimental perturbation studies , where the expression of a gene is experimentally manipulated to observe how this perturbation affects the expressions of other genes .', 'Such experimental methods are costly , laborious , and do not scale to a perturbation of more than two genes at a time .', 'As an alternative , genetical genomics approach uses genetic variants as naturally-occurring perturbations of gene regulatory system and learns gene networks by decoding the perturbation effects by genetic variants , given population gene-expression and genotype data .', 'However , since there exist millions of genetic variants in genomes that simultaneously perturb a gene network , it is not obvious how to decode the effects of such multifactorial perturbations from data .', 'Our statistical approach overcomes this computational challenge and recovers gene networks under SNP perturbations using probabilistic graphical models .', 'As population gene-expression and genotype datasets are routinely collected to study genetic architectures of complex diseases and phenotypes , our approach can directly leverage these existing datasets to provide a more effective way of identifying gene networks .']
['systems biology', 'regulatory networks', 'population genetics', 'biology', 'computational biology']