Inferring interaction type in gene regulatory networks using coexpression data
 Pegah Khosravi†^{1, 2}Email author,
 Vahid H Gazestani†^{3},
 Leila Pirhaji^{4},
 Brian Law^{2, 5},
 Mehdi Sadeghi^{6, 1},
 Bahram Goliaei^{7} and
 Gary D Bader^{2}Email author
DOI: 10.1186/s1301501500544
© Khosravi et al. 2015
Received: 11 April 2014
Accepted: 16 June 2015
Published: 8 July 2015
Abstract
Background
Knowledge of interaction types in biological networks is important for understanding the functional organization of the cell. Currently informationbased approaches are widely used for inferring gene regulatory interactions from genomics data, such as gene expression profiles; however, these approaches do not provide evidence about the regulation type (positive or negative sign) of the interaction.
Results
This paper describes a novel algorithm, “Signing of Regulatory Networks” (SIREN), which can infer the regulatory type of interactions in a known gene regulatory network (GRN) given corresponding genomewide gene expression data. To assess our new approach, we applied it to three different benchmark gene regulatory networks, including Escherichia coli, prostate cancer, and an in silico constructed network. Our new method has approximately 68, 70, and 100 percent accuracy, respectively, for these networks. To showcase the utility of SIREN algorithm, we used it to predict previously unknown regulation types for 454 interactions related to the prostate cancer GRN.
Conclusions
SIREN is an efficient algorithm with low computational complexity; hence, it is applicable to large biological networks. It can serve as a complementary approach for a wide range of network reconstruction methods that do not provide information about the interaction type.
Keywords
Gene expression data Informationbased approach Interaction type Regulatory interactionBackground
With increasing amounts of biological data generated by modern highthroughput technologies, we are faced with a challenging problem: how to extract meaningful information from the data. A prominent direction for addressing this problem is using computational data mining approaches for the analysis of highthroughput biological data, such as gene expression data [1–4]. In particular, analysis methods have been developed to infer regulatory interactions from transcriptome data [5–14].These regulatory interactions link regulators, such as transcription factors and kinases, to their targets and may include the regulatory type of the interaction, which indicates whether there is an activating (positive) or inhibitory (negative) association between the interactor pair. Knowing the interaction type can be beneficial for a wide range of analyses including modulecentric analysis [15] and network simulation [16]. A growing number of approaches use coexpression measures, either correlationbased (generally linear) or information theorybased (can consider nonlinear relationships) [17], to infer GRNs.
Although information theorybased approaches have been widely applied to decipher GRNs [18–20], they are not currently used to determine the type of the regulation between two connected genes in a reconstructed GRN. Here, we present SIREN, a statistical framework that uses a new information theorybased measure to predict regulatory type. Our novel framework is capable of accurately predicting the type of regulation between two interacting genes. The fundamental assumption in our approach is that if two connected genes in the network have similar expression patterns, there is likely an activating (positive) association, among them. On the other hand, if their expression patterns are anticorrelated, the interacting genes likely have an inhibitory (negative) influence on each other. SIREN uses a mutual informationbased measure to predict the interaction type. Extending mutual information has been extensively used as a similarity measure for feature selection fields [21–24]. In our novel approach, a rescaling matrix was introduced to convert the MI function, which normally generates nonnegative scores, to a function that can have negative values. The resulting sign is used to predict the interaction type. While SIREN detects the regulation type, it cannot detect the direction of regulation. We evaluated SIREN by testing it on E. coli, prostate cancer, and in silico GRN benchmarks. In each case, SIREN reliably identified positive and negative regulatory types. Besides, comparison of SIREN with a baseline method based on Pearson coefficient correlation (PCC) revealed that it has a greater performance on biological GRNs. The R implementation of the algorithm is freely available at http://baderlab.org/PegahKhosravi/SIREN.
Methods
Information theory based metrics
If we define the function \(f(x,y) = \log (p(x,y)/p(x)p(y))\), then \(MI = \sum {p(x,y)f(x,y)}\) which is equivalent to the expected value of function f(x,y), i.e., \(MI = E(f(x,y))\). \(f(x,y)\) is defined to be the pointwise mutual information (PMI) [25]. PMI is a measure of how much the joint probability of a particular cooccurrence of events, p(x,y), differs from the expected joint probability, assuming the independence of x and y, p(x)*p(y) [25].
Discretization of continuous data
For a computationally feasible calculation of MI and PMI, discretizing and binning of the expression data is required. In conventional binning approaches, each data point is assigned to exactly one bin. This can be problematic for data points close to the margins between bins: small, noisy fluctuations may cause these data points to be improperly assigned to a neighboring bin. Additionally, the choice of bin size can strongly influence the MI values for datasets of moderate size [26, 27]. To address these problems, we used a Bspline approach for data binning [27–30] that allows each data point to be assigned to multiple bins. To accomplish this, the indicator function, which typically maps each data point to a specific bin, was improved to allow each datapoint to be assigned to several bins, with weights obtained from a Bspline function summing to one (schematically shown in Additional file 1: Figure S1). The Bspline function has a spline order that defines the shape of the function and influences the number of bins to which each data point is assigned.
MI versus correlation based methods
Both MI and correlation are measures of dependency between two random variables. MI calculates the amount of information that two genes provide about each other and is always a nonnegative value. Consequently, a higher mutual information value for two genes indicates that one gene is nonrandomly related with the other [19]. The Pearson correlation coefficient (PCC) is a measure that indicates the intensity and trend of the linear relationship between two variables [31]. Despite the fact that the PCC can characterize linear correlations with Gaussian noise, the mutual information measure is more powerful mainly because it is able to detect nonlinear dependencies that are invisible to PCC [27, 32]. Both PCC [33–37] and MI [6, 18, 27, 38–41] have been used to capture dependencies between random variables (e.g., genes).
Benchmarks GRNs and corresponding data
To assess our algorithm, we applied SIREN to three benchmark GRNs with associated gene expression data (Additional file 2).
In silico constructed GRN
Prostate cancer GRN
We extracted the prostate cancer GRN from the STRING functional interaction database [42], considering only regulatory interactions among genes with changed expression levels during cancer progression. To determine these genes, we focused on those which are up or downregulated significantly (fold change ≥2 and p value ≤0.05) in at least one state, considering the normal state as control. Thus the GRN is composed of genes which putatively play a role in prostate cancer and contains 1,436 interactions among 526 genes (53.8% of total genes with available transcriptome data).
Prostate cancer microarray data
We extracted gene expression data from the GEO database with accession number GDS2545. This dataset consists of 171 samples monitoring gene expression in four different cell states: normal prostate tissue free of any pathology (normal), normal prostate tissue adjacent to tumor (adjacent), primary prostate tumor tissue (tumor), and metastatic prostate cancer tissue (metastatic) [43].
E. coli high confidence GRN
We extracted the high confidence GRN of E. coli from the RegulonDB database [44]. This GRN contains 4,005 experimentally confirmed regulatory interactions among 1,696 genes. We created a subnetwork from RegulonDB GRN by considering genes with available data in our transcriptomics data set. The resulting subnetwork contains 2,687 interactions among 1,419 genes and was used for further analysis.
E. coli gene expression data
We extracted a microarray dataset consisting of 907 samples from the Many Microbe Microarray Database (M^{3D}) Web site [45].
SIREN algorithm
The fundamental assumption in our approach is that if two connected genes in the network have similar expression patterns, there is likely an activating (positive) association, between them. On the other hand, if their expression patterns are conversely related, the interacting genes likely have an inhibitory (negative) influence on each other.
Our method is useful for the analysis of reconstructed GRNs from any reconstruction method used to generate the input network. In addition to the GRN, SIREN also needs corresponding expression data. SIREN determines the regulation type for each pair of connected genes in the network by computing a similarity score between their expression profiles. As shown schematically in Additional file 1: Figure S2, SIREN determines the similarity between the expression profiles of two genes in four steps: (1) a Bspline discretization method is used to discretize expression data into ten bins, allowing overlap between bins to smooth the data as described in [27], (2) cooccurrence scores are calculated for each combination of bins of the genes (for example the first bin of first gene with the first bin of second gene, the first bin of first gene with the second bin of second gene, etc.) using the informationbased metrics, (3) the calculated cooccurrence scores are rescaled according to the given rescaling matrix, which enables SIREN to distinguish between cooccurrences resulting from an activating or an inhibitory effect between two genes; and (4) determines the SIREN score by calculating the expected value of the rescaled cooccurrence probability scores. Mathematically speaking, the expected value of function \(f(X,Y)\) is defined as: \(E(f(X,Y)) = \sum\nolimits_{x,y} {p(x,y)f(x,y)}\). Therefore, SIREN score for two genes of X and Y is: \(\sum\nolimits_{x,y} {p(x,y)(W(x,y)CoS(x,y))}\), where CoS(x,y) is the cooccurrence score, W(x,y) refers to the rescaling matrix and p(x,y) is the cooccurrence probability when X = x and Y = y.
To optimize results, we compared four distinct scoring functions and four rescaling matrices as well as tested a range of different cutoff scores for SIREN to determine a reliable threshold.
SIREN scoring functions
To optimize the algorithm, we investigated four possible interaction scoring functions:
S_{1}
Based on the definitions of MI and PMI, the first scoring function was defined as \(S_{1} = \sum {W(x,y)} p(x,y)\log (p(x,y)/p(x)p(y))\), where W(x,y) is the rescaling matrix. If \(g(x,y) = W(x,y)\log (p(x,y)/p(x)p(y))\), then \(S_{1} = \sum {p(x,y)g(x,y)}\) or \(S_{1} = E(g(x,y))\) according to the Law of the Unconscious Statistician (LOTUS) [46]. The defined S _{1} score is known as the weighted mutual information concept, as described in [47].
S_{2}
To give PMI a fixed upper bound, we normalize it to have a maximum value of 1 in the case of a perfect association. The advantage of normalized PMI over PMI is that the value of PMI is usually high for rare events. It is hoped that the normalized version will reduce the low frequency bias [25]. As stated above, \(PMI = \log (p(x,y)/p(x)p(y))\). In case of perfect association, we have \(p(x) = p(y) = p(x,y)\), consequently \(PMI = \log (p(x,y)/p(x)p(y)) = \log (p(x)/p(x)p(x)) =  \log (p(x))\).
S_{3}
The third scoring function is defined as \(S_{3} = \sum {p(x,y)[W(x,y)]}\). This score measures the expected value of the rescaling matrix W(x,y).
S_{4}
In a similar approach to the second scoring function, to give MI a fixed upper bound, we normalize MI to have a maximum value of 1 in the case of a perfect association [25]. As above, \(MI = \sum {p(x,y)\log (p(x,y)/p(x)p(y))}\). If \(p(x) = p(y) = p(x,y)\), then \(MI(x,y) = MI(x,x) = \sum {p(x,x)\log (p(x,x)/p(x)p(x))}\). Consequently, \(MI(x,x) = \sum {p(x,x)(  \log (p(x)))} =  \sum {p(x)\log (p(x))}\). Therefore \(Normalized\;MI(NMI) = {{\sum {p(x,y)} (\log (p(x,y)/p(x)p(y))} \mathord{\left/ {\vphantom {{\sum {p(x,y)} (\log (p(x,y)/p(x)p(y))} {  \sum {p(x,y)\log (p(x,y))} }}} \right. \kern0pt} {  \sum {p(x,y)\log (p(x,y))} }}\).
We defined Normalized Rescaled MI (NRMI) as \(S_{4} = {{\sum {p(x,y)\left( {W(x,y)\left( {\log (p(x,y)/p(x)p(y))} \right)} \right)} } \mathord{\left/ {\vphantom {{\sum {p(x,y)\left( {W(x,y)\left( {\log (p(x,y)/p(x)p(y))} \right)} \right)} } {  \sum {p(x,y)} \left( {W(x,y)\left( {\log (p(x,y)} \right)} \right)}}} \right. \kern0pt} {  \sum {p(x,y)} \left( {W(x,y)\left( {\log (p(x,y)} \right)} \right)}}\).
SIREN performance assessment
To measure SIREN performance, we defined two measures: true number (TNu) and false number (FNu). positive true positive (PTP) and negative true negative (NTN) is the number of interactions correctly signed positive and negative, respectively, whereas negative false positive (NFP) and positive false negative (PFN) is the number of interaction types that SIREN has assigned incorrectly positive and negative, respectively. TNu is the number of PTP plus the number of NTN, while FNu is the number of NFP plus the number of PFN. Accuracy, defined as TNu/(TNu + FNu), is the fraction of correctly signed interactions among all interactions signed by SIREN, while retrieval is the number of regulatory interactions signed by SIREN among all interactions (some of which are not signed). Performance of the algorithm is assessed using accuracyretrieval curves.
Results and discussion
Determining the interaction types in GRNs
MI is used extensively for reconstructing GRNs because it has a low computational complexity and is able to capture nonlinear dependencies among variables [48–50]. However, using MI has some disadvantages, including that it does not reveal the interaction type between two random variables. In the context of GRN reconstruction, this means that it does not identify whether a regulatory interaction is positive (activating) or negative (inhibiting) to overcome this challenge, we modified the conventional mutual information formula by adding a rescaling matrix. This rescaling matrix converts the MI function, which normally generates a nonnegative score to a function that can have negative values. The resulting sign is used to predict the interaction type.
Using expression data, our new method can determine the interaction type between two interacting genes in a GRN. SIREN discriminates activating from inhibitory associations based on the premise that the effect of two interacting genes on each other will be reflected in their expression patterns across multiple cellular conditions. Similar expression profiles indicate a positive interaction and dissimilar profiles indicates a negative interaction.
Selecting the optimum interaction scoring function
In silico GRN
Prostate cancer GRN
STRING is a functional interaction database that includes regulatory interactions. We extracted a functional interaction network based on genes that show alteration in their expression profile during prostate cancer progression, limited to regulatory interactions which we define as the prostate cancer GRN (see “Methods” for details). The extracted GRN was composed of 1,436 interactions among 526 genes (102 negative, 176 positive and 1,158 interactions with no sign). We then ran SIREN on this network and compared SIREN’s predicted interaction type to the known interaction type from STRING. As shown in Figure 3b, both S_{1} and S_{2} scoring functions had, again, comparable results on this GRN with about 70% accuracy.
E. coli GRN
We applied SIREN to an experimentally constructed GRN of E. coli containing 1,408 positive and 1,279 negative interactions. As shown in Figure 3c, both scoring functions S_{1} and S_{2} resulted in similar accuracies (56% maximum). In this test, the experimentally constructed GRN of E. coli and the gene expression data where from independent sources. The low performance of SIREN in this case is most probably because some regulatory circuits may not be reflected in the gene expression data. As an illustration, for two genes that have negative regulation on each other via negative feedback loop, we expect to observe that upregulation of the regulator leads to the downregulation of the regulated gene. However, if the expression level of regulator gene does not increase perceptibly, the expression patterns will not reflect the inhibitory effect. Thus, both genes will show similar expression patterns, and consequently, an activating interaction will be wrongly inferred. Considering this fact, our approach can reliably detect interaction types only for genes that show some level of alteration in the expression in the corresponding expression data set. Consistently, we found restricting the E. coli GRN to 10% most fluctuated genes (this subnetwork was composed of 31 positive and 13 negative interactions) resulted in the precision of 68.18% (22 PTP and 8 NTN) for SIREN algorithm.
Selecting the optimum rescaling matrix
For matrix 1, for the other negative bins, we repeatedly subtracted 0.166667 (one divided by six negative levels) as we moved towards the border between positive and negative. The positive cells were treated similarly. The positive cells in matrix 2 are the same as in matrix 1; however, to improve detection of negative interactions, and to compensate for the +10 discrepancy in overall weight arising from the matrix’s diagonal, we increased the weight on the negative cells such that the overall weight in the matrix was equal to zero. For matrix 3, we simply assigned −1 and +1 to each nonzero cell. The negative cells in matrix 4 are the same with matrix 1; for the positive cells, we used a multiplicative scaling factor rather than an additive one.
Selecting the best threshold
Deficiency in current knowledge about interaction types
Our SIREN predictions on the prostate cancer GRN included 454 interactions for which no regulatory type existed in STRING (Additional file 3). A literature search on a sample of these newly signed GRN supports the reliability of our results. For example, we predict a positive association between EGR1 and ATF3 genes, which is in line with previous studies that have shown these two genes associated with each other and ATF3 is a target of EGR1 that induces an upregulation of ATF3 [59].We also found a negative association between FASN and CAV1 which is consistent with previous reports that showed FASN interact with CAV1, a marker for metastasis state of prostate cancer, and inactivation of CAV1 mediates by FASN [52].
Conclusion
At present, there is no information theorybased framework to detect regulatory interaction types in gene regulatory networks. In this work, we tried to fill this gap by exploiting the notion that the effect two interacting genes have on each other can be observed in their expression patterns. This idea allowed us to develop an information theorybased solution, SIREN, to identify interaction types using gene expression data. SIREN increases the amount of information available for GRNs compared to standard GRN inference algorithms. SIREN runs reasonably fast; computing the 2,687 interactions among 1,419 genes took 6 min on an Intel Core i5 system with 4 GB of RAM; hence it is usable with large biological networks. Additionally, we have shown the method is applicable to prokaryotic and eukaryotic GRNs.
Notes
Abbreviations
 ARCs:

accuracyretrieval curves
 FNu:

false number
 GRNs:

gene regulatory networks
 LOTUS:

Law of the Unconscious Statistician
 M^{3D} :

many microbe microarray database
 MI:

mutual information
 NB:

number of bins
 NFP:

negative false positives
 NMI:

normalized MI
 NRMI:

normalized rescaled MI
 NTN:

negative true negatives
 PCC:

Pearson coefficient correlation
 PFN:

positive false negatives
 PMI:

pointwise mutual information
 PTP:

positive true positives
 SIREN:

signing of regulatory networks
 SO:

spline order
 TNu:

true number
Declarations
Authors’ contributions
PK, VHG and LP conceived of the study. PK, VHG, BL, MS, BG and GDB designed the experiments. PK and VHG devised the algorithm and PK performed analysis. PK, VHG, BL and GDB wrote the paper. All authors read and approved the final manuscript.
Acknowledgements
This work was supported by NRNB (U.S. National Institutes of Health, National Center for Research Resources Grant Number P41 GM103504). PK is supported by the School of Biological Sciences of Institute for Research in Fundamental Sciences (IPM).VHG is supported by CIHR Systems Biology Fellowship. The authors would like to warmly thank Dr. Scott Zuyderduyn in the Bader Lab for his critical comments.
Compliance with ethical guidelines
Competing interests The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Authors’ Affiliations
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