Table 1 lists the mean value distributions used in a simulation of data where the standard deviation was set to be constant at 0.4. Figure 1 displays the mean values of the six probability distributions for generating the data, where Figure 1(A) and 1(B) are monotonically up and down, respectively. Figure 1(C) and 1(D) start and end at zero but peak at the third and second data points respectively. Figure 1(E) and 1(F) are flat at 3 and 0, respectively. The mean values equal to zero in Figure 1(F) reflect a flat response analogous to real gene expression data in which a number of genes may be not responsive to a given series of treatments. Normal deviates were drawn at random to generate 15 profiles for each of the six distributions. Figure 2 is a principal component analysis (PCA) of the simulated data, where the six distinct clusters of the profiles are distributed in 3-dimensional space. When the simulated data was analyzed by EPIG and CLICK for comparison, EPIG extracted 5 patterns from the data (see Figure 3A) corresponding to profile probability distributions depicted in Figures 1(A) to 1(E) and assigned all of the profiles to their proper patterns (15 profiles each) with 100% accuracy, except for the pattern from the distribution of the data shown in Figure 1(F) and its corresponding and uncategorized profiles (the ones generated to best represent a nonresponsive gene expression pattern). On the other hand, CLICK with the default homogeneity setting, only returned 3 clusters with the patterns of the centroids as shown in Figure 3B from the data omitting the clusters from the distribution of the data shown in Figure 1(E) which had all inter-group mean values at 3. CLICK merged the profiles from the patterns of the distributions of the data of Figure 1(C) and 1(D) together, despite the peaks at the distinct data points in the patterns. In addition, CLICK assigned 32 profiles (two of them from the distribution E in Table 1 or Figure 1(E)) to its Cluster 1 and 16 profiles (one of them also from the distribution E) to its Cluster 2. We also varied the homogeneity setting. With homogeneity settings at 0.83 or 0.84, CLICK generated the highest overall average homogeneity within the patterns and produced essentially the same three clusters as were produced using the default setting (data not shown).

A data set consisting of gene expression profiles from a C. elegans dauer recovery and L1 starvation time course study was obtained to compare EPIG's and CLICK's ability to extract patterns/clusters the data for identification of co-expressed genes from a publicly available microarray data set. The experimental design and other details of the study can be obtained from Wang et al 17. Applied to the data which included all the genes, EPIG extracted 18 patterns of gene expression and identified 1597 co-expressed genes (see extracted patterns in Figure S1 and heat map of the 1597 genes in Figure S2 in Additional file 1). The numbers of genes categorized to each of the patterns varied from 15 to 263 (see Table S1 in Additional file 1). On the other hand, CLICK generated only 6 clusters of genes from the data (1644 selected genes, based on SNR (>3) and magnitude of expression (>0.5), including all 1597 genes identified by EPIG) as shown by the patterns of the centroids in Figure S3 in Additional file 1. If we applied CLICK to the whole gene data set as what was done for EPIG, CLICK produced over 60 clusters (data not shown).

In dauer-specific processes, there were four groups of genes corresponding to the dauer recovery: transient, early, climbing and late 17. Table 2 lists the dauer recovery-specific gene expression patterns from both EPIG and CLICK corresponding to the four groups of co-expressed genes along with the dauer enriched state. There were no clusters generated by CLICK which contained genes with expression patterns related to either early or climbing states. However, there were four patterns which corresponded to the transient, late and dauer enriched states. On the other hand, the patterns of the genes extracted by EPIG corresponded to each of these dauer recovery response groups. For example, there were four patterns (Patterns 2, 3, 10 and 11 in Figure S1 in Additional file 1) in which expression levels of all of the genes decreased from early to late corresponding to the dauer transient state. However, these four patterns differed slightly from one another. For instance, Patterns 2 and 10 have no change at the start, then the expression of the genes gradually decreases to be substantially down-regulated at the end. However, their corresponding L1 starvation expression levels were either down-regulated (in the case of Pattern 2) or not changed (in the case of Pattern 10) across all the time points. Conversely, Patterns 3 and 11 reflect no change at all time points in L1 starvation, but the response of the genes to dauer recovery were either up-regulated at the start, then gradually decreased to no change at end (in the case of Pattern 3) or had no change at the start, were up-regulated early, then gradually decreased to no change at end (in the case of Pattern 11). Similarly, one may relate EPIG's Patterns 6, 7 and 9 to the dauer recovery early state and Pattern 14 to the climbing state. There were no clusters generated from CLICK that contained patterns of the centroids corresponding to these two dauer-specific states (see Figure S3 in Additional file 1). The patterns from EPIG and CLICK showing similar responses between dauer recovery and L1 starvation were not listed in Table 2.

We next applied EPIG to a microarray data set that combined gene expression profiles of both ionizing radiation (IR)- and ultraviolet (UV)-treated human fibroblast cells with two goals in mind: 1) to find similar and dissimilar responses between treatments and 2) to reveal differences in gene regulation upon DNA damage caused by IR or UV. In each of the two treatments, the data consisted of four biological states, i.e. sham-treated, 2 h-, 6 h-, and 24 h post UV- or IR-treatment. A gene expression profile consists of eight inter-groups, corresponding to four states from the two treatments. Each of the intra-groups contains six data points from three biological replicates and two technical replicates (dye-swap pairs) for a given treatment at a given time point. As such, each gene expression profile consisted of 48 data points. EPIG analysis using the whole data as its input resulted in total of 18 patterns as shown in Figure 4 with a total of 2661 co-expressed genes being identified. Each of the co-expressed genes was categorized to a particular pattern. Figure 5 is a heat map of the 2661 genes that are arranged in the order of pattern number from top to bottom. Table 3 lists the number of genes in each of the patterns and denotes their over-represented Gene Ontology biological processes19.

Each pattern shown in Figure 4 includes both UV (the first half of the profile from left to right) and IR (the second half) treatments. For each treatment there are three individual cell lines (F1-hTERT, F3-hTERT and F10-hTERT). Each one consists of four time-series points, i.e. sham-treated controls (red), 2 h- (green), 6 h- (blue), and 24 h- (magenta) post UV or IR treatments. Patterns 1 through 8 show UV-specific expression (either up- or down-regulated) with little or no changes in gene expression in IR-treated cells. UV-specific up-regulation of gene expression (Pattern 1 to 4) happened only at early time points, 2 and/or 6 h after UV irradiation, and fully recovered to baseline levels at 24 h. As shown in Table 3, the 324 genes in Patterns 1–4 were related to transcriptional regulation, RNA processing and nucleic acid binding. UV-specific down-regulation of gene expression (Patterns 5 to 8) also occurred at early time points (2 h and/or 6 h post UV). Biological processes related to regulation of transcription were also over-represented by these genes. In addition, over 600 genes in Pattern 8, which were substantially repressed at 6 h post UV, contained about 30 biological processes over-represented including purine nucleotide binding, protein modification, ubiquitin cycle, kinase activity, and cell growth.

Genes in Patterns 9 to 15 responded to both UV- and IR-treatment but with different time dependencies. In Pattern 9 expression was up-regulated at 2 and 6 h post UV-treatment and then recovered to near baseline levels at 24 h. Genes in this pattern were minimally up-regulated at 2 h post IR but showed substantial down-regulation at 6 h and even greater down-regulation at 24 h. Many genes that are maximally expressed in S phase were in this pattern, including CDC6, FEN1, MSH6, ORC6L, PCNA, POLG, RBM14, CCNE2 and TOP3A. The changes in expression of these genes were coincidental with changes in the S phase compartment of the cell cycle. S phase cells were increased over control at 2 and 6 h post-UV (data not shown), but were moderately reduced relative to control at 6 h and markedly reduced at 24 h post-IR 20.

Patterns 10 and 11 contained 42 genes including the p53-target genes GDF15, BTG2, PLK2, BAK1, PLK3 and CDKN1A (Pattern 10) and TP53INP1, SESN1, DDB2 and FDXR (Pattern 11). All of these genes were up-regulated at 2 h after UV treatment, they peaked at 6 h and then returned to baseline at 24 h. However, they responded differently to IR. In Pattern 10, the genes peaked at 2 h post IR then decreased to a stable level at 6 h and 24 h. The genes in Pattern 11 displayed continuous increases in expression until 24 h after IR.

Patterns 13 to 15 contained 997 genes in total that were mainly regulated at the late time point after treatment (24 h). More than 500 genes in Pattern 14 were down-regulated at 24 h post IR or UV treatment. This pattern included many cell cycle-regulated genes functioning in the G1/S and G2/M transitions, such as CDK2, CDC2 and MCM2. These genes were strongly down-regulated in IR-treated cells, but had substantially less change in UV-treated cells. Both Patterns 13 and 15 show late up-regulated responses, but genes in former were up-regulated after treatment with both UV and IR while those in the latter only responded to IR. The top biological process in Pattern 13 was complement activation with lysosome and lytic vacuole being the main cellular components that the products of the genes in the patterns might be associated with or located in. Finally, many genes exhibited co-expression in Patterns 16 (115 genes), 17 (61 genes) and 18 (56 genes) but UV and IR treatments induced opposite changes in their gene expression.

As a comparison, we also applied CLICK to the 2726 genes identified as differentially expressed using the SNR (> 3) and magnitude of expression (> 0.5) criteria. This set of genes included all 2661 co-expressed genes identified by EPIG. In this case, CLICK clustered the gene expression data into 11 groups (in Figure S4 in Additional file 1). Similar patterns from the centroids of the clusters were revealed in comparison to the ones extracted by EPIG. However, CLICK was unable to reveal some patterns extracted by EPIG which we consider to have biological importance related to DNA repair in responding to UV and IR treatments. Among them are EPIG's Patterns 9, 10 and 11 (see Figure 4). As presented above, many of the genes in Pattern 9 were S phase-related and the majority of genes in Patterns 10 and 11 were known to be related to p53-dependent cell cycle control.

It is vital that the extracted patterns and their associated genes can be inspected for their biological meaningfulness as presented above. On the other hand, it is also essential that that the extracted patterns or clusters of genes formed be validated objectively using some evaluation indicies. Although there were a number of methods developed to validate clusters, each of them is suitable to some specific applications. For example, one may use the General Silhouette (GS) to measure the stability of the clustering structure 6. The higher GS score indicates better formed clusters. But the GS measure seems to work best for cases where the number of clusters are small 2122. Given the above simulated data, we obtained GS scores of 0.77 and 0.71 for the pattern from EPIG and the clusters from CLICK, respectively. However, when we applied the GS measure to either of the above dauer recovery and L1 starvation or UV and IR DNA damage data sets, the GS scores was deemed inappropriate to judge cluster validity due to the higher numbers of extracted clusters. Other methods, for example, the Gap statistic 22, validate clusters depending on the number of groups in the data as determined a priori. However, both EPIG and CLICK are unsupervised approaches. Therefore, to objectively and appropriately compare the two methods, we calculated the overall average homogeneity 7 within clusters (i.e. the pattern-categorized gene expression profiles in the case of EPIG) and the averaged correlations between clusters/extracted patterns. The overall average homogeneity measures the amount of cohesion within a cluster/pattern whereas the averaged correlation measures the amount of separation between clusters/patterns 23. The results are listed in Table 4. As can be seen, in each of the cases of the three data sets, the overall average homogeneities from EPIG are consistently higher than those from CLICK and CAST (with affinity threshold set to 0.7, 0.8, or 0.85) suggesting that EPIG has better compactness of the profiles in the patterns than the clusters of the genes generated by the other clustering algorithms. The averaged correlations for all the methods were low (<= +/- 0.35), indicating that the expression of the genes in the patterns were quite dissimilar each other. However, it is clear from this result that CLICK performed better than EPIG and CAST with respect to the between cluster correlations (i.e., CLICK had a lower average correlation between clusters/extracted patterns than EPIG and CAST did).

Another observation is that the numbers of clusters generated by CLICK were always lower than the number of patterns extracted by EPIG and the number of clusters generated by CAST was about 8 (on average and not including singletons) when different affinity threshold values were used for analysis. However, if we use the whole data set as the input data for CLICK, as what was done for the analysis of the data with EPIG, CLICK produces over 60 clusters in both dauer recovery/L1 starvation and UV/IR DNA damage data sets. A Figure of merit (FOM) 24 analysis of the either data set showed that for cases when the number of clusters were larger than 20, the adjusted FOM values slowly decreased. This suggests that the number of clusters of genes in the data set could be between 5 and 20, well below the 60 clusters produced by CLICK when using the entire gene expression data set (Figure S5 in Additional file 1).

A data set comprised of numeric data with 90 profiles and 24 objects (four inter- groups i (i = 1,...,4) and six intra-groups j (j = 1,...,6)) was simulated from six distinct probability distributions. Normal deviates were drawn at random to generate 15 profiles for each of the six distributions. Table 1 lists the mean value distributions used in the simulation of the data where the standard deviation was set to be constant at 0.4.

Microarray gene expression data was acquired from three telomerized normal human fibroblast lines (logarithmically growing), F1-hTERT, F3-hTERT and F10-hTERT2829 that were sham-treated or treated with 1.5 Gy ionizing radiation (IR) or 6 J/m² ultraviolet (UV [from a 254 nm radiation source]) and harvested at 2, 6 or 24 h after the treatment. Briefly, total RNA isolated from cells harvested at each time point was subjected to microarray analysis by using Agilent 22,000 element human 1A arrays. The labelled cRNA from sham- or radiation-treated samples was hybridized to the microarray along with a labelled global reference cRNA. Hybridization of sample RNA against reference RNA was performed with a dye swap (cytofluor reversal).

The extracted intensity data were pre-processed by array-based Systematic Variation Normalization (SVN) 30, profile-based dye-swap correction to remove dye labelling affects, and to align biological reference states. The latter was done in such a way that the sham-treated samples were used as a reference state and the averaged sham-treated state of the dye-swapped pair was aligned to zero as a baseline. All other treated states were then adjusted by the same amount accordingly. It is plausible that each of the three human cell lines used for analysis may not have the same sham state. The alignment of the sham states aims at eliminating the biological variability and focuses the analysis on the relative changes upon IR or UV treatments. In this case, the 2-dimensional matrix of compiled gene expression data consisted of 16,757 rows of unique genes (grouped by UniGene) and 48 columns consisting of, for each of the three cell lines, dye-swapped replicates of IR-sham-treatment and 2 h, 6 h and 24 h after IR, UV-sham-treatment and 2 h, 6 h and 24 h after UV.

A compiled microarray gene expression data set (in this study, as conventionally presented, the log₂ pixel intensity ratio values) consists of a 2-dimensional matrix, in which each row represents a gene expression profile and each column represents an array. Upon sample perturbation or variation in biological factors, such as agent, dose, time or tissue, a gene expression profile can be made up of inter-group and intra-group samples. The arrays in an intra-group sample have a factor in common, e.g. biological replicates. The arrays in inter-group samples possess different factors, e.g., sham-treatment and time points post-UV or IR treatment. We denote each datum of log₂ ratio as g_ij in a gene expression profile, where i refers to a inter-group index from 1 to m, j is the intra-group index from 1 to n_i, m is the number of inter-groups and n_i is the number of arrays in i^th inter-group. To evaluate such a profile, we calculate each intra-group average g¯i MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGafm4zaCMbaebadaWgaaWcbaGaemyAaKgabeaaaaa@2EC9@ and sample variance si2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaem4Cam3aa0baaSqaaiabdMgaPbqaaiabikdaYaaaaaa@2FBC@. We define a gene expression profile's signal as

S = { max ⁡ { g ¯ i } , i f min ⁡ { g ¯ i } > 0 − min ⁡ { g ¯ i } , e l s e i f max ⁡ { g ¯ i } < 0 max ⁡ { g ¯ i } − min ⁡ { g ¯ i } o t h e r w i s e . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@8741@

where 1 ≤ i ≤ m. We define a profile's noise estimate as the square-root of the pooled variance, i.e.

N = ∑ i m [ ( n i − 1 ) ⋅ s i 2 ] ∑ i m ( n i − 1 ) ∑ i m 1 n i MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaemOta4Kaeyypa0ZaaOaaaeaajuaGdaWcaaqaamaaqahabaWaamWaaeaacqGGOaakcqWGUbGBdaWgaaqaaiabdMgaPbqabaGaeyOeI0IaeGymaeJaeiykaKIaeyyXICTaem4Cam3aa0baaeaacqWGPbqAaeaacqaIYaGmaaaacaGLBbGaayzxaaaabaGaemyAaKgabaGaemyBa0gacqGHris5aaqaamaaqahabaGaeiikaGIaemOBa42aaSbaaeaacqWGPbqAaeqaaiabgkHiTiabigdaXiabcMcaPaqaaiabdMgaPbqaaiabd2gaTbGaeyyeIuoaaaGcdaaeWbqaamaalaaabaGaeGymaedabaGaemOBa42aaSbaaSqaaiabdMgaPbqabaaaaaqaaiabdMgaPbqaaiabd2gaTbqdcqGHris5aaWcbeaaaaa@56BA@

where the sample variance

s i 2 = ∑ j n i ( g i j − g ¯ i ) 2 n i − 1 . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaem4Cam3aa0baaSqaaiabdMgaPbqaaiabikdaYaaakiabg2da9KqbaoaalaaabaWaaabCaeaacqGGOaakcqWGNbWzdaWgaaqaaiabdMgaPjabdQgaQbqabaGaeyOeI0Iafm4zaCMbaebadaWgaaqaaiabdMgaPbqabaGaeiykaKYaaWbaaeqabaGaeGOmaidaaaqaaiabdQgaQbqaaiabd6gaUnaaBaaabaGaemyAaKgabeaaaiabggHiLdaabaGaemOBa42aaSbaaeaacqWGPbqAaeqaaiabgkHiTiabigdaXaaakiabc6caUaaa@489A@

From Equations 1 and 2, we define a profile's signal-to-noise ratio as

S N R = S N MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaem4uamLaemOta4KaemOuaiLaeyypa0tcfa4aaSGaaeaacqWGtbWuaeaacqWGobGtaaaaaa@339C@

As can be seen, when m = 1, Equation 3 is equivalent to a two sample t-test, since by default the log₂ pixel intensity ratio is the treated against its control. Equation 3 includes the case for m > 1, i.e. multiple inter-groups.

In extracting gene expression patterns, EPIG uses a filtering process where all profiles initially are considered as pattern candidates. The pseudo code for the algorithm can be found in the Appendix. Briefly, using all pair-wise correlations, any candidate profile, whose local cluster size is less than a predefined size M_t or its correlation with another profile is higher (> R_t) but has a lower local cluster size M, is removed from pattern construction consideration. Among the remaining profiles, EPIG then creates representative profiles for the corresponding local clusters and removes those profiles with a SNR in Equation 3 less than 3 or magnitude S in Equation 1 less than 0.5. After this filtering processing, the remaining profiles consist of the extracted patterns, which are used to be the representatives to each of the local clusters. Each of the patterns has the highest local cluster size in comparison with other highly similar profiles (e.g. correlation larger than 0.8) in the same local cluster.

Subsequently, EPIG categorizes each gene to the pattern, for which it has the highest correlation with the gene profile. A gene not assigned to any extracted patterns is considered an "orphan" if its highest correlation r-value is lower that a given threshold R_c. Typically R_c is set to a value which corresponds to a correlation p-value of 10^-4 to assure the significance of the co-expression. A Java-based software tool and the source code for EPIG are publicly available 31.

CLuster Identification via Connectivity Kernels (CLICK) analysis of the gene expression data was performed using version 2 of the EXpression ANalyzer and DisplayER (EXPANDER) analysis and visualization tool 7. The default settings for CLICK were used for all analyses.

The Cluster Affinity Search Technique (CAST) for clustering data uses average similarity (affinity) between gene expression patterns and cluster cores (the current ones in the recursive portion of the algorithm) and then adds (and removes) elements from the current core one at a time 5. An affinity threshold is used to specify the cluster quality – it influences the number and the size of the clusters that are produced. The CAST implementation used for analysis of the data in this paper was based on a Java applet source code 32.