 Research
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 Published:
Inferring interaction type in gene regulatory networks using coexpression data
Algorithms for Molecular Biologyvolume 10, Article number: 23 (2015)
The Erratum to this article has been published in Algorithms for Molecular Biology 2015 10:25
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.
Background
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
Mutual information (MI) is a measure of the information dependency between two random variables, defined as:
where p(x,y) is the joint probability of x and y and p(x) and p(y) are marginal probabilities.
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.
For the discretization of continuous expression data in SIREN, we tested different number of bins and the spline orders. Consistent with previous studies [27], we found the number of bins does not affect the SIREN performance remarkably as long as it is within a reasonable range (Figure 1a) and using spline order greater than three does not improve the quality of prediction significantly (Figure 1b), but rather it increases the computational cost of the algorithm.
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
To assess the performance of SIREN on a completely known network, we constructed an in silico GRN based on a prostate cancer gene expression data. This GRN is a clique network composed of two groups of genes. Genes in each group have a similar expression pattern to each other and opposite pattern with members of the other group (Figure 2). Because there is a clear expression pattern for each gene, we know the putative interaction types in this network by visual inspection.
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 \(PMI = \log (p(x,y)/p(x)p(y)) = \log (p(x)/p(x)p(x)) =  \log (p(x))\).
Therefore, we can define Normalized PMI (NPMI) as
We define the second scoring function \(S_{2} = E(W(x,y) \times NPMI)\).
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
To optimize SIREN, we defined four different scoring functions as well as four different rescaling matrices. To evaluate the performance of our algorithm in each case, we considered three different GRNs: an in silico constructed GRN, a prostate cancer GRN, and an E. coli high confidence GRN. The results, averaged across the four possible rescaling matrices, are presented in Figure 3.
In silico GRN
To evaluate the performance of SIREN algorithm, we selected nine (CXCR4, IGHM, TFAP2A, IGHV431, CLDN4, LOC100126583, CYAT1, TRPV6, FOXA1) and six (IFI27, RARRES1, GPX3, AGT, ACTA1, TTN) genes that were consistently upor down regulated during cancer progression, respectively (Figure 2). We then generated a clique network consisting of all possible interactions between these genes (105 interactions). This network consists of 51 positive (up–up or down–down) and 54 negative (up–down) interactions (Figure 4). Application of SIREN on the in silico network, as shown in Figure 3a, indicated that the scoring functions S_{1} and S_{2} have comparable performance while outperforming S_{3} and S_{4}. Closer examination of the results showed that S_{3} is not sensitive enough to detect negative interactions because negative interactions are often derived from small magnitudes of the changes in expression levels. The logarithmic component of the S_{1} and S_{2} scoring functions magnifies these small differences in expression levels, but this is not present in S_{3}. S_{4} had the lowest accuracy. Hence, we selected S_{1} and S_{2} scoring functions for further evaluations.
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.
Scores obtained from the S_{1} scoring function have a wider range compared with the scores of S_{2} scoring functions in all three GRNs (Figure 5). Also, the S_{1} scoring function has a lower computational complexity. Considering these points, we selected S_{1} as the best scoring function for further investigations.
Selecting the optimum rescaling matrix
We have used a rescaling matrix to convert the MI function, which normally generates a nonnegative score, to a function that can produce negative values. The resulting sign is used to predict either an activating (similar expression profiles) or inhibitory effect (different expression profiles) between genes. Using the Bspline approach, we smoothly discretized the expression profile of each gene into 10 bins. For each interacting pair, SIREN creates a twodimensional grid with 100 cells. The distribution pattern of expression data in these 100 cells is used for predicting the interaction type. The interaction type can be inferred from this grid because the distribution pattern for genes with positively correlated expression patterns will be different from the distribution pattern of genes with a negative association. To discriminate the distribution patterns from each other, we have introduced the rescaling matrix (Figure 6). The design of four rescaling matrices evaluated for use in SIREN. For all the matrices, we initially assigned −1 to the two most negative bins and +1 to the most positive bins, on the diagonal of the matrix. We also arranged the matrices to each have equal number of positive and negative cells (42 positive, 42 negative, and 16 zero cells).
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.
To select the best performing matrix from these four matrices, we examined the performance of each of them on the three selected benchmark GRNs (Figure 7). Our results indicate that SIREN is robust to the selected rescaling matrix, especially for the experimentally derived GRNs. However, the M_{3} rescaling matrix (Matrix 3) (Figure 6) performs better with the in silico network (Figure 7). This may be caused by the higher ratio of negative interactions in the in silico network (51.43%). Considering results obtained from this step, we chose M_{3} as the optimum rescaling matrix.
Selecting the best threshold
To select the best threshold on the resulting SIREN score, we applied it to the E. coli, prostate and in silico benchmarks, using the S_{1} scoring function and the M_{3} rescaling matrix. We tested a range of different cutoff scores (20 different thresholds between 0 and 1) for SIREN to determine a reliable threshold for various networks. The results showed that when the cutoff threshold is greater than +0.158 or smaller than −0.158, SIREN does not detect any interaction type in random data (generated by 10^{6} times of shuffling gene expression data) (Figure 8), while many interactions are predicted to be signed in the benchmark GRNs.
Figure 9 demonstrates the application of SIREN (with optimized parameters) on two example interactions, one activating and one inhibitory. As mentioned earlier, SIREN detects the interaction type in four steps: (1) discretizing the expression profile of each gene into 10 bins; (2) calculating the cooccurrence probability for each combination of bins using the PMI metric; (3) defining an activating or inhibitory relationship for each combination of bins by aid of a rescaling matrix; and (4) calculating a final score by integrating the calculated values for each combination of bins. For example, with two genes with a known activating relationship from STRING (JUN and ATF3) [52, 53], bins defined as activating have nonzero values while bins defined as inhibitory have mostly zero values (Figure 9a). This situation is reversed for two genes with a known inhibitory relationship (ZEB1 and CDH1) [54] (Figure 9b).
Deficiency in current knowledge about interaction types
PCC has been widely used to decipher the interaction type based on transcriptome data [55, 56]. We compared SIREN with PCC by applying both to all three GRNs. This comparison revealed that their overall results are similar, suggesting that most regulatory associations in the considered GRNs have a linear or monotonic nature (Figure 10). However, the results of PCC and SIREN are inconsistent for some interactions. For example two genes with an activating relationship from STRING (EGR1 and FGF2) [57] are determined to have an inhibition relationship using PCC while SIREN inferred a positive association. On the other hand MICA and IL10 show an inhibition association [58] by STRING and SIREN; while they have positive association based on PCC. Ultimately, SIREN shows superior performance in all but the in silico network case (Figure 11), indicating that consideration of nonlinear relationships in the gene expression data is useful.
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.
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
References
 1.
Cornuéjols A, Miclet L (2002) Apprentissage artificiel: concepts et algorithms. Eyrolles
 2.
Webb A (2002) Statistical pattern recognition. Wiley, New York
 3.
Mitchell T (1997) Machine learning. McGraw Hill, New York
 4.
Jaynes ET (2003) Probability theory: the logic of science. Cambridge University Press, Cambridge
 5.
Alon U (2006) An introduction to systems biology. Chapman and Hall, London
 6.
Basso K, Margolin AA, Stolovitzky G, Klein U, DallaFavera R, Califano A (2005) Reverse engineering of regulatory networks in human B cells. Nat Genet 37:382–390
 7.
van de Vijver MJ, He YD, van’t Veer LJ, Dai H, Hart AA, Voskuil DW et al (2002) A geneexpression signature as a predictor of survival in breast cancer. N Engl J Med 347:1999–2009
 8.
Wang Y, Klijn JG, Zhang Y, Sieuwerts AM, Look MP, Yang F et al (2005) Geneexpression profiles to predict distant metastasis of lymphnodenegative primary breast cancer. Lancet 365:671–679
 9.
Gardner TS, Faith JJ (2010) Reverseengineering transcription control networks. Phys Life Rev 2:65–88
 10.
Schulz MH, Devanny WE, Gitter A, Zhong S, Ernst J, BarJoseph Z (2012) DREM 2.0: improved reconstruction of dynamic regulatory networks from timeseries expression data. BMC Syst Biol 6:104
 11.
Awad S, Chen J (2014) Inferring transcription factor collaborations in gene regulatory networks. BMC Syst Biol 8(Suppl 1):S1
 12.
Awad S, Panchy N, Ng SK, Chen J (2012) Inferring the regulatory interaction models of transcription factors in transcriptional regulatory networks. J Bioinform Comput Biol 10:1250012
 13.
BarJoseph Z, Gitter A, Simon I (2012) Studying and modelling dynamic biological processes using timeseries gene expression data. Nat Rev Genet 13:552–564
 14.
Yeang CH, Jaakkola T (2006) Modeling the combinatorial functions of multiple transcription factors. J Comput Biol 13:463–480
 15.
Mason MJ, Fan G, Plath K, Zhou Q, Horvath S (2009) Signed weighted gene coexpression network analysis of transcriptional regulation in murine embryonic stem cells. BMC Genom 10:327
 16.
Guziolowski C, Kittas A, Dittmann F, Grabe N (2012) Automatic generation of causal networks linking growth factor stimuli to functional cell state changes. FEBS J 279:3462–3474
 17.
Song L, Langfelder P, Horvath S (2012) Comparison of coexpression measures: mutual information, correlation, and model based indices. BMC Bioinformatics 13:328
 18.
Margolin AA, Nemenman I, Basso K, Wiggins C, Stolovitzky G, Dalla Favera R et al (2006) ARACNE: an algorithm for the reconstruction of gene regulatory networks in a mammalian cellular context. BMC Bioinformatics 7(Suppl 1):S7
 19.
Butte AJ, Kohane IS (2000) Mutual information relevance networks: functional genomic clustering using pairwise entropy measurements. Pac Symp Biocomput 2000:418–429
 20.
Faith JJ, Hayete B, Thaden JT, Mogno I, Wierzbowski J, Cottarel G et al (2007) Largescale mapping and validation of Escherichia coli transcriptional regulation from a compendium of expression profiles. PLoS Biol 5:e8
 21.
Rivaz H, Collins DL (2012) Selfsimilarity weighted mutual information: a new nonrigid image registration metric. Med Image Comput Comput Assist Interv MICCAI Int Conf Med Image Comput Comput Assist Interv 15:91–98
 22.
Rivaz H, Karimaghaloo Z, Collins DL (2014) Selfsimilarity weighted mutual information: a new nonrigid image registration metric. Med Image Anal 18:343–358
 23.
Park SB, Rhee FC, Monroe JI, Sohn JW (2010) Spatially weighted mutual information image registration for image guided radiation therapy. Med Phys 37:4590–4601
 24.
Schaffernicht E, Gross HM (2011) Weighted mutual information for feature selection. In: Proceedings 21 international conference on artificial neural networks (ICANN 2011); Espoo, Finland, LNCS 6792. Springer, pp 181–188
 25.
Bouma G (2009) Normalized pointwise mutual information in collocation extraction. In: Proceedings of the Biennial GSCL Conference 2009. Gunter Narr Verlag, Tübingen, pp 31–40
 26.
Moon YI, Rajagopalan B, Lall U (1995) Estimation of mutual information using kernel density estimators. Phys Rev E 52:2318–2321
 27.
Daub CO, Steuer R, Selbig J, Kloska S (2004) Estimating mutual information using Bspline functions–an improved similarity measure for analysing gene expression data. BMC Bioinformatics 5:118
 28.
Unser M, Aldroubi A, Eden M (1993) BSpline signalprocessing .2. Efficient design and applications. IEEE Trans Signal Process 41:834–848
 29.
Deboor C (1978) A practical guide to splines. SpringerVerlag, New York
 30.
Li H, Sun Y, Zhan M (2007) Analysis of gene coexpression by Bspline based CoD estimation. EURASIP J Bioinform Syst Biol 2007:49478
 31.
Bolboacă SD, Jäntschi L (2006) Pearson versus Spearman, Kendall’s Tau correlation analysis on structureactivity relationships of biologic active compounds. Leonardo J Sci 2006:179–200
 32.
Numata J, Ebenhoh O, Knapp EW (2008) Measuring correlations in metabolomic networks with mutual information. Genome Inform Int Conf Genome Inform 20:112–122
 33.
Eisen MB, Spellman PT, Brown PO, Botstein D (1998) Cluster analysis and display of genomewide expression patterns. Proc Natl Acad Sci USA 95:14863–14868
 34.
Zhou XH, Kao MCJ, Wong WH (2002) Transitive functional annotation by shortestpath analysis of gene expression data. P Natl Acad Sci USA 99:12783–12788
 35.
Stuart JM, Segal E, Koller D, Kim SK (2003) A genecoexpression network for global discovery of conserved genetic modules. Science 302:249–255
 36.
Zhang B, Horvath S (2005) A general framework for weighted gene coexpression network analysis. Stat Appl Genet Mol Biol 4:17
 37.
Langfelder P, Horvath S (2008) WGCNA: an R package for weighted correlation network analysis. BMC Bioinformatics 9:559
 38.
Butte AJ, Tamayo P, Slonim D, Golub TR, Kohane IS (2000) Discovering functional relationships between RNA expression and chemotherapeutic susceptibility using relevance networks. Proc Natl Acad Sci USA 97:12182–12186
 39.
Meyer PE, Lafitte F, Bontempi G (2008) minet: A R/bioconductor package for inferring large transcriptional networks using mutual information. BMC Bioinformatics 9:461
 40.
Priness I, Maimon O, BenGal I (2007) Evaluation of geneexpression clustering via mutual information distance measure. BMC Bioinformatics 8:111
 41.
Cadeiras M, von Bayern M, Sinha A, Shahzad K, Latif F, Lim WK et al (2011) Drawing networks of rejection—a systems biological approach to the identification of candidate genes in heart transplantation. J Cell Mol Med 15:949–956
 42.
Snel B, Lehmann G, Bork P, Huynen MA (2000) STRING: a webserver to retrieve and display the repeatedly occurring neighbourhood of a gene. Nucleic Acids Res 28:3442–3444
 43.
Chandran UR, Ma C, Dhir R, Bisceglia M, LyonsWeiler M, Liang W et al (2007) Gene expression profiles of prostate cancer reveal involvement of multiple molecular pathways in the metastatic process. BMC Cancer 7:64
 44.
GamaCastro S, JimenezJacinto V, PeraltaGil M, SantosZavaleta A, PenalozaSpinola MI, ContrerasMoreira B et al (2008) RegulonDB (version 6.0): gene regulation model of Escherichia coli K12 beyond transcription, active (experimental) annotated promoters and Textpresso navigation. Nucleic Acids Res 36:D120–D124
 45.
Faith JJ, Driscoll ME, Fusaro VA, Cosgrove EJ, Hayete B, Juhn FS et al (2008) Many Microbe Microarrays Database: uniformly normalized Affymetrix compendia with structured experimental metadata. Nucleic Acids Res 36:D866–D870
 46.
Tsuei DJ, Hsu HC, Lee PH, Jeng YM, Pu YS, Chen CN et al (2004) RBMY, a male germ cellspecific RNAbinding protein, activated in human liver cancers and transforms rodent fibroblasts. Oncogene 23:5815–5822
 47.
Guiasu S (1977) Information theory with applications. McGrawHill Inc., NewYork
 48.
Zhang X, Zhao XM, He K, Lu L, Cao Y, Liu J et al (2012) Inferring gene regulatory networks from gene expression data by PCalgorithm based on conditional mutual information. Bioinformatics 28:98–104
 49.
Liang KC, Wang X (2008) Gene regulatory network reconstruction using conditional mutual information. EURASIP J Bioinform Syst Biol 2008:253894
 50.
Kim DC, Wang X, Yang CR, Gao J (2010) Learning biological network using mutual information and conditional independence. BMC Bioinformatics 11(Suppl 3):S9
 51.
Shannon P, Markiel A, Ozier O, Baliga NS, Wang JT, Ramage D et al (2003) Cytoscape: a software environment for integrated models of biomolecular interaction networks. Genome Res 13:2498–2504
 52.
Perinchery G, Sasaki M, Angan A, Kumar V, Carroll P, Dahiya R (2000) Deletion of Ychromosome specific genes in human prostate cancer. J Urol 163:1339–1342
 53.
Dasari VK, Goharderakhshan RZ, Perinchery G, Li LC, Tanaka Y, Alonzo J et al (2001) Expression analysis of Y chromosome genes in human prostate cancer. J Urol 165:1335–1341
 54.
Kurasawa Y, Kozaki K, Pimkhaokham A, Muramatsu T, Ono H, Ishihara T et al (2012) Stabilization of phenotypic plasticity through mesenchymalspecific DNA hypermethylation in cancer cells. Oncogene 31:1963–1974
 55.
Lee HK, Hsu AK, Sajdak J, Qin J, Pavlidis P (2004) Coexpression analysis of human genes across many microarray data sets. Genome Res 14:1085–1094
 56.
Bhardwaj N, Lu H (2005) Correlation between gene expression profiles and proteinprotein interactions within and across genomes. Bioinformatics 21:2730–2738
 57.
Iyoda T, Zhang F, Sun L, Hao F, SchmitzPeiffer C, Xu X et al (2012) Lysophosphatidic acid induces early growth response1 (Egr1) protein expression via protein kinase Cdeltaregulated extracellular signalregulated kinase (ERK) and cJun Nterminal kinase (JNK) activation in vascular smooth muscle cells. J Biol Chem 287:22635–22642
 58.
Chattopadhyay K (2011) A comprehensive review on host genetic susceptibility to human papillomavirus infection and progression to cervical cancer. Indian J Human Genetics 17:132–144
 59.
Lau YF, Zhang J (2000) Expression analysis of thirty one Y chromosome genes in human prostate cancer. Mol Carcinog 27:308–321
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.
Author information
Additional information
Pegah Khosravi and Vahid H Gazestani contributed equally
An erratum to this article is available at http://dx.doi.org/10.1186/s1301501500553.
Additional files
Additional file 1:
The Bspline function and SIREN algorithm schematically. This file contains the additional figure S1 and S2. In figure S1, data are discretized to 4 bins (Lowest, Low, High, and Highest) and the Bspline order is equal to 3. As shown, bins have overlap with each other. The dashed line represents a sample data point. The place that dashed line intersect with each bin, determines the weight of that bin. Note that sum of weights is always equal to one. In figure S2 given a GRN and corresponding transcriptome dataset, SIREN starts with discretization of continuous transcriptome data. Next, SIREN constructs a cooccurrence profile for each interacting genes in the GRN. Based on the patterns of cooccurrence, SIREN determines the type of interaction (Inhibitory or activatory).
Additional file 2:
Signed GRNs. The list of benchmark GRNs contains E. coli high confidence, Prostate cancer, and in silico constructed.
Additional file 3:
Prostate cancer GRN. 454 SIREN predicted interactions, not present in the STRING database.
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Keywords
 Gene expression data
 Informationbased approach
 Interaction type
 Regulatory interaction