The proposed method was used to analyze a well-known data set in the microarray literature, colon cancer data, analyzed initially by Alon et al. 23. It consists of absolute measurements from Affymetrix oligonucleotide arrays, with 62 tissue samples of 2000 human gene expressions (40 tumours and 22 normal tissues).

This analysis started with building a gene forest, from which significant gene-gene relationships were extracted. To this end, a 5-fold cross validation resampling strategy was used to construct multiple replicates of training and test sets. In this procedure, colon cancer and normal samples were randomly divided into 5 non-overlapping parts of roughly equal size, denoted as D_i (i = 1, 2, ..., 5) for colon cancer and N_i (i = 1, 2, ..., 5) for normal samples, respectively. A combination of D_i and N_i constituted a test set and the rest of the data were used as the training set. Thus, all combinations produced 25 pairs of training and test sets, {L_d, T_d} (d = 1, 2, ..., 25). By repeating this procedure 20 times, we obtained 500 pairs of data. On each pair, a classification tree was constructed and tested using a computational statistic Matlab toolbox 24, where each gene was a node variable and in this way a gene forest with 500 trees was constructed. We used Gini's diversity index as the criterion for choosing a split. The tree growth was stopped if a further split at the current node did not improve the purity of its child nodes or when there were less than two samples. For the detail of construction of gene forest related to colon cancer, see the Methods section or the previous report 5.

From the newly built gene forest, we identified 780 gene pairs (involving 165 genes) appearing in the same trees. Per the definition and formula provided in the Methods section, the AFVs for these gene pairs ranged from 0.09 to 19.14, which was generally much smaller than the marginal relevance value that measured the contribution of a single gene feature 5. The distribution of the 780 gene pairs' AFV values is shown with blue circles in Figure 1. In order to determine their statistical significance, we performed 1000 permutations in which the sample labels were randomly shuffled. The estimated empirical null distribution of AFV obtained from estimating 8881 gene pairs in 1000 random trees gave the largest value of 4.43 and the threshold for significance level of 0.01 was estimated to be 0.53. The permuted distribution is shown with red circles in Figure 1. Apparently, both curves indicate that this metric follow an extreme value distribution and the curve for the real data shifted to the right of the null distribution. Thus, the gene pairs with AFV over the threshold were considered as having significant gene-gene interactions.

We found 200 significant (p ≤ 0.01, AFV threshold 0.53) colon cancer-specific gene-gene interactions among 74 genes, with the smallest p value <1.13 × 10^-4 (for details on all the gene pairs, see Additional file 1). All AFV values of the 200 significant gene pairs were used to create a graphical representation (Figure 2). The background of the heat map is red, and the AFV values are encoded by other colours, as indicated by the side bar. The heat map indicates that only a small proportion (7.40%, 200/2701) reached the significance level, but this number was much higher than expected (0.01 × 2701≈27), which was randomly selected under the null distribution. Intuitively, several genes (e.g. IL8, DES, RPL30, RBM9 and ENO1) had an unusually higher number of significant interacting genes (encoded by non-red colours), which suggests that they may play a central role in the disease process. By annotating the 74 genes to the Entrez Gene 25 and Unigene 26 databases at NCBI, we found 52 known genes that accounted for 109 gene interactions out of the identified 200 gene pairs. To simplify the further bioinformatics analysis, we only focused on the 52 genes whose function had been well characterized and documented in GO. By connecting two genes in each gene pair, we constructed an un-weighted gene network for colon cancer (Figure 3). One can easily identify that five genes (IL8, DES, RPL30, RBM9 and ENO1) had the highest connectivity scores. IL8, a chemotactic and inflammatory cytokine (a ligand), had 33 connections with the 52 known genes; next was DES, a type III intermediate filament found near the Z line in sarcomeres, which had 17 connections.

The functional implications of the constructed network remained to be elucidated. Thus, we used 'Functional Annotation' in DAVID Bioinformatics Resources to perform functional enrichment analysis based on Gene Ontology 27. We defined the 74 genes as the test set and the entire 2000 genes as the background. We set a minimal node size of five genes from the test set, and a nominal significance level of 0.05, given by the EASE Score method, a modified Fisher Exact test. We identified 13 significant GO terms, as shown in Table 1. In order to identify more specific functions, we eliminated the redundant but broad terms among the 13 GO terms. Finally, we obtained seven more specific GO terms (shown in bold type in Table 1). From the two dimensions 'Cellular Component' and 'Molecular Function', we found that the pathogenesis of colon cancer was consistently linked to ribosome (associated categories such as 'ribosome', 'ribonucleoprotein complex' and 'structural constituent of ribosome'). Based on the dimension 'Biological Process', we concluded that 'protein biosynthesis' largely accounted for colon cancer tumourigenesis. These conclusions are well supported by multiple lines of experimental evidence. One study has demonstrated that there is increased synthesis of ribosomes in colorectal tumours, and that this increase is an early event in colon neoplasia 28. In another recent study 29, it has been shown that perturbation of specific ribosomal proteins is likely to promote certain genetic diseases and tumuorigenesis.

We used a Poisson distribution to identify statistically significant hub nodes in the colon cancer specific network. Under the null hypothesis that the 52 genes were randomly connected, a gene with >10 connections in a random network was considered a rare event with probability of 0.0046. Thus, we set this threshold to claim a hub gene. By this criterion, we identified five hub genes: IL8 (33 connections; p ≈ 0), DES (17 connections; p = 2.71 × 10^-6), RBM9 (15 connections; p = 4.19 × 10^-5), RPL30 (14 connections; p = 1.50 × 10^-4) and ENO1 (14 connections; p = 1.50 × 10^-4). Even after adjusting for the number of genes tested, the five genes remained to be valid hub genes with highly significant connectivity. Their corresponding Bonferroni-corrected p values were ≈0, 1.41 × 10^-4, 2.18 × 10^-3, 7.79 × 10^-3, 7.79 × 10^-3, respectively. Three of the five hub genes (IL8, DES and ENO1) are proved cancer-related hub genes, while knowledge for the remaining two genes waits to be expanded. The detailed cross-talks with the three proved cancer-related hub genes are listed in Table 2.

The protein encoded by Interleukin 8 (IL8), is a member of the CXC chemokine family. This chemokine is one of the major mediators of the inflammatory response. IL8 can promote cell proliferation and migration through metalloproteinase-cleavage proHB-EGF in human colon carcinoma cells 30, and induction of IL8 preserves the angiogenic response in HIF-1alpha-deficient colon cancer cells 31. Desmin (DES) encodes a muscle-specific class III intermediate filament. Mutations in this gene are associated with desmin-related myopathy, a familial cardiac and skeletal myopathy (CSM), and with distal myopathies. It is also a negative marker for colon cancer discrimination 32. Enolase 1, more commonly known as alpha-enolase, is a glycolytic enzyme expressed in most tissues. It is a homodimer composed of 2 alpha subunits. Its gene, the ENO1, also encodes the Myc-binding protein-1, which downregulates the activity of c-myc protooncogene 33. However, there are few studies that can establish the hub roles of the remaining two genes (RPL30 and RBM9). The ribosomal protein L30 (RPL30) encodes a ribosomal protein that is a component of the 60S subunit. Disease specific humoral immune responses against TBP-1, p27(BBP), and RPL30 have been induced in patients with hepatocellular carcinoma (HCC), and the antibodies against these antigens may be also used as tumour markers 34. Gene RBM9 encodes an RNA binding protein that is thought to be a key regulator of alternative exon splicing in the nervous system and other cell types 35. The protein also interacts with the estrogen receptor 1 transcription factor and regulates estrogen receptor 1 transcriptional activity 35. However, there is a dearth of information that show its direct effects on the tumourigenesis in cancer.

To validate the newly identified five hub genes, we performed a pathway analysis using PathwayAssist software (Stratagene, La Jolla, CA, USA) 36. The knowledge-based gene network (Figure 4) was constructed by finding out all cellular processes directly linked to the hub genes. Based on this analysis, IL8, DES and ENO1 are proven central elements in this network, with 92, 24 and nine links, respectively. However, there are insufficient data to prove the hub roles of RPL30 (one link) and RBM9 (no link), as revealed by the above AFV-based networking, and these two genes may define new central elements in the gene network specific to colon cancer. Based on the cellular processes to which the hub genes were linked, the colon cancer-specific gene network captured the most important genetic interplays in several cellular processes such as differentiation, mitogenesis, proliferation, apoptosis, inflammation and immunity, which are known to be pivotal for tumourigenesis.

We also conducted a pathway analysis to identify all cellular processes (or proteins) that link the five hub genes by implementing "Find all shortest paths between selected entities" in PathwayAssist Software. Again, IL8, DES and ENO1 were the central elements (Figure 5). Interestingly, in this network, RPL30 and DES can be linked through GJA1 (connexin-43), the major protein of myocardial gap junctions, which are thought to have a crucial role in the synchronized contraction of the heart and in embryonic development. It was also interesting to note that the common cellular processes for the three hub genes IL8, DES and ENO1 greatly varied from cell proliferation and differentiation to maturity and death. This may have been due to the large number of cellular functions to which IL8 was linked (see also Figure 4).

In the colon cancer-specific gene network, 76 three-way interactions (triangles) among 60 genes were identified by an exhaustive searching algorithm for the network motifs. Based on 1000 random networks, only the triangle structure, which included all possible edges between the three nodes, was over-represented (p = 0.012) in this network at the significance level of 0.05 using MAVisto software 37. Hence, we focused on the triangle as the structural element in further analysis. In addition, we also searched for larger n-cliques, which were complete sub-graphs with n nodes. A maximum-size 5-clique was found that described the dense cross-talks between five genes: CD37, DES, MYH9, RBM9 and RPL30. However, this 5-clique could not be fully confirmed by our current knowledge of the five molecules, and further experimental validation is required.

Then, an enrichment analysis based on GO was performed. We defined the functional facets of the 60 genes using the DAVID resources 27, and the parameters were set as described above. We identified 11 GO functional categories, of which the terms, 'ribosome', 'ribonucleoprotein complex', 'structural constituent of ribosome' and 'protein biosynthesis', were the most specific functionalities, as shown in Table 3. These results were consistent with the enrichment analysis of two-way interactions, which suggested that the above categories largely captured the functional facets of the colon cancer specific gene network.

In our previous study 5, we identified 20 highly significant colon cancer relevant genes based on a marginal relevance index that measured their separate contribution to the gene forest for classification. Logically, the gene networks that included both the marginal and joint contributions of the colon cancer genes may better define the susceptibility risk for developing colon cancer. To verify this hypothesis, we compared the three gene sets: the 20 genes that extracted from our previous study, the 74 genes that extracted from gene-gene interactions and 60 genes that extracted from three-way interactions. We estimated the average accuracy of the three sets by leave-one-out validation using 5 classifiers: diagonal linear discriminate analysis (DLDA), 3 nearest neighbours (3NN), nearest centroid (NC), support vector machine (SVM) and Bayesian compound covariate (BCC), which were all implemented using the BRB-Arraytools software version 3.5.0 stable release 38. As a result, although the differences were not statistically significant, the gene network with gene-gene interactions, in most of the classifiers, had an equal or better power than the 20 marginally relevant genes in classifying tissue samples, or the gene set defined by three-way interactions, as conceptually this set was a subset of the data defined by two-way interactions (Figure 6). This result suggested that gene network may contain additional contributions from the gene-gene interactions and the three-way interactions.

A gene chip, a snapshot of the mRNA transcriptional activities of p genes in n tissue samples collected from either cancer or health patients, mathematically can be described by a n × p matrix, X = (x_ij), where x_ij represents the expression level for the jth gene (g_j) on the ith sample (X_i). The data for each sample consists of a vector of expression profile, X_i = (x_i1, x_i2, ..., x_ip) and a category label (y_i) describing the physiological (or pathological) condition that a subject has (e.g. diseased or healthy).

In our previous studies, we developed a systematic ensemble decision approach for hunting for disease genes using microarray expression profiling. The basic strategies were as follows. (i) To build all possible gene subsets by repeated learning and testing of multiple resampling-generated training and test datasets that were used for mapping the underlying molecular pathways that lead to complex disease. As a disease relevant gene subset was obtained by using a tree-based recursive partitioner, we named gene forest for the pool of such gene subsets; (ii) To identify all the disease relevant genes based on the behaviour and role of the molecular features in the gene forest. To this end, we defined a marginal relevance index that measured its contribution to the gene forest and derived a formula called ensemble vote, FV, which was the weighted frequency estimate of a putative disease gene that appeared in the trees of the forest. In the present study, we extended the ensemble-decision approach to identify disease-relevance gene-gene interactions and to build disease-specific gene networks.

First, a resampling technique was employed to build up pairs of training and test sets, {L_d, T_d} (d = 1, 2, ..., m), for learning and testing, respectively. Then, a binary decision tree was grown on L_d by a recursive partition algorithm. At each non-leaf node, a decision was made with regard to the choice of a feature gene and a threshold value (cut-off) such that the class impurity was reduced to a minimum when a branch was made by an induction rule. After the optimal bifurcation was made, the microarray samples were divided into two non-overlapping subsets (two child nodes). The same process was conducted successively until the stopping criteria for tree growth were satisfied. For each tree grown, it was tested on the holdout set T_d to evaluate its discriminating power for classification. This process was repeated on each pair of {L_d, T_d} (d = 1, 2, ..., m), which consequently resulted in a decision forest with m trees. In each tree, all the genes for bifurcation at non-leaf nodes composed a disease relevant gene subset (pathway), which denoted as G_d (d = 1, 2, ..., m). All G_d extracted from the m trees composed the gene forest. The aim of this step was to identify most, if not all genetic pathways that lead to complex disease.

Based on the gene forest established in the previous step, we extracted all gene pairs in the same gene subsets. In order to quantify the joint contribution of a gene pair, according to Definition 1 we designed a novel pair-wise relevancy metric, called Adjusted Frequency Value (AFV), which was formulated as follows:

A F V ( g i , g j ) = 100 × ∑ d ω d I ( g i , g j | G d ) ∑ d ω d , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@5905@

where I(g_i, g_j|G_d) was an indicator function and G_d was the gene subset that contains the gene pair:

I ( g i , g j | G d ) = { 1 , if both  g i  and  g j  appear in the  d th tree  ( G d ) 0 , otherwise . . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@8426@

A weight, ω_d, was a measure for the classification performance of G_d on a test set, such as the accuracy rate used in this study. In short, one gene pair's AFV value was weighted frequency of the two genes appear simultaneously in the same trees in the forest.

Because the asymptotic distribution of AFV could not be derived analytically, we resorted to a permutation approach to obtain its empirical null distribution. In the permutation approach, we randomly assigned a label (phenotype), y_i, to each microarray and then the same procedures for the field data were applied to the permutated data. Given the empirical AFVs and a user-specified significance level (e.g. α = 0.05 or 0.01), a critical value for AFV was determined by its (1-α)% percentile in the simulated null distribution. A gene-gene interaction was disease relevant if it's AFV≥AFVα0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemyqaeKaemOrayKaemOvay1enfgDOvwBHrxAJf2maeHbnfgDOvwBHrxAJf2maGabaiab=5NkYjabdgeabjabdAeagjabdAfawnaaDaaaleaacqaHXoqyaeaacqaIWaamaaaaaa@4030@, the threshold value at significance level α (one-tailed). According to Definition 2, if a gene network was built in such ways that every presented edge was a disease-relevant gene-gene interaction, it was a gene network specific to the disease, a sub-network enriched with pathogenic pathways that lead to the disease.

In order to characterize the functional facets of the constructed disease-relevant gene network, we performed functional enrichment analysis based on GO using 'Functional Annotation' in DAVID Bioinformatics Resources 27. All the 2000 genes analyzed in this study were used as the background. The probability of a GO term enriched with the gene-gene interactions was assessed by the EASE Score method, a modified Fisher Exact test. A smaller EASE Score was related to a higher likelihood of enrichment of a GO term with the gene-gene interactions. In this study, to avoid the possible loss of the true positive results, we did not perform multiple-test correction for the multiple GO terms evaluated. Therefore, the p-value quoted should be considered as a heuristic measure, useful for roughly rating the relative enrichment of each GO term. We removed all redundant terms if all the genes annotated to a term were also annotated to a child term. In this case, we retained the child term because its function was more specifically defined.

The topology and properties for most cellular networks were largely determined by a relatively small number of hub nodes (genes), which, in the context of a disease-relevant network, were key genes that lead to disease or maintaining health physiological condition. Connectivity (the number of links) was often used to measure importance of a hub node, which, in random network, follows a Poisson distribution 47. We used the following formula to determine whether a node could be categorized as a hub node. Suppose that p₁ was the probability of connecting any two nodes in a random network with n nodes, the probability of connectivity of equal or larger than t was as follows:

p ( x ≥ t ) = 1 − p ( x < t ) = 1 − ∑ k = 0 t − 1 λ k e − λ k ! , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemiCaa3aaeWaaeaacqWG4baEcqGHLjYScqWG0baDaiaawIcacaGLPaaacqGH9aqpcqaIXaqmcqGHsislcqWGWbaCdaqadaqaaiabdIha4jabgYda8iabdsha0bGaayjkaiaawMcaaiabg2da9iabigdaXiabgkHiTmaaqahajuaGbaWaaSaaaeaacqaH7oaBdaahaaqabeaacqWGRbWAaaGaemyzau2aaWbaaeqabaGaeyOeI0Iaeq4UdWgaaaqaaiabdUgaRjabcgcaHaaaaSqaaiabdUgaRjabg2da9iabicdaWaqaaiabdsha0jabgkHiTiabigdaXaqdcqGHris5aOGaeiilaWcaaa@544A@

where λ = n × p₁; p₁ was estimated using the number of links in the constructed disease-specific gene network divided by the number of all possible links. We claimed a hub gene if its p value was smaller than the nominal significance α.

To identify more specific pathways associated with the underlying pathogenic mechanisms of colon cancer, we used PathwayAssist software (Stratagene, La Jolla, CA, USA) to find all the cellular processes linked to the hub colon cancer genes using the option "Find all entities connected to selected entities (Expand Pathway)". Then, we identified all the cellular processes shared by the hub genes by implementing the option "Find all shortest paths between selected entities".

We further investigated high-order gene interactions. In this study, triangles (three-way interactions), which have all possible edges among the three vertices, were extracted from the network by an exhaustive searching algorithm and tested using MAVisto software 37. Then, in order to characterize the functions of these triangles, we annotated the gene pool of the triangles to GO, and assessed the enrichment of each GO term with these genes, using the DAVID resources 27, as described above. Again, for the reasons specified above, we did not perform multiple tests for multiple GO terms evaluated.

In order to better explain the novel network approach, we also made a graphic algorithm flow chart, as shown in Figure 7.