Signed weighted gene co-expression network analysis of transcriptional regulation in murine embryonic stem cells

Constructing Signed Co-expression Networks

We first define a gene co-expression similarity measure which is used to define the network. We denote the gene co-expression similarity measure of a pair of genes i and j by s_ij. Many co-expression studies use the absolute value of the Pearson correlation as an unsigned co-expression similarity measure,

(1)

where gene expression profiles x_i and x_j consist of the expression of genes i and j across multiple microarray samples. However, using the absolute value of the correlation may obfuscate biologically relevant information, since no distinction is made between gene repression and activation. In contrast, in signed networks the similarity between genes reflects the sign of the correlation of their expression profiles. To define a signed co-expression measure between gene expression profiles x_i and x_j, we use a simple transformation of the correlation:

(2)

As the unsigned measure , the signed similarity takes on a value between 0 and 1. Note that the unsigned similarity between two oppositely expressed genes (cor(x_i, x_j) = -1) equals 1 while it equals 0 for the signed similarity. Similarly, while the unsigned co-expression measure of two genes with zero correlation remains zero, the signed similarity equals 0.5.

Next, an adjacency matrix (network), A = [a_ij], is used to quantify how strongly genes are connected to one another. A is defined by thresholding the co-expression similarity matrix S = [s_ij]. 'Hard' thresholding (dichotomizing) the similarity measure S results in an unweighted gene co-expression network. Specifically an unweighted network adjacency is defined to be 1 if s_ij > τ and 0 otherwise, i.e. two genes are considered connected if their similarity measure is above a given threshold τ, and are considered separated otherwise.

Because hard thresholding encodes gene connections in a binary fashion, it can be sensitive to the choice of the threshold and result in the loss of co-expression information [19]. The continuous nature of the co-expression information can be preserved by employing soft thresholding, which results in a weighted network. Specifically, we use a continuous measure to assess their connection strength:

(3)

where the power β is the thresholding parameter. As a default we use β = 6 and β = 12 for unsigned and signed networks, respectively. Alternatively, β and be chosen using the scale-free topology criterion [19]. Since log(a_ij) = β × log(s_ij), the weighted network adjacency is linearly related to the co-expression similarity on a logarithmic scale. Figure 1 shows the resulting adjacencies after applying the co-expression similarity measures and thresholding. Note that a high power β transforms high similarities into high adjacencies, while pushing low similarities towards 0. Since this soft-thresholding procedure leads to weighted adjacency matrix, the ensuing analysis is referred to as weighted gene co-expression network analysis or WGCNA [19,23,33,34].

A major step in our module centric analysis is to cluster genes into network modules using a network proximity measure. Roughly speaking, a pair of genes has a high proximity if it is closely interconnected. We will use the convention that the maximal proximity between two genes is 1 and the minimum proximity is 0. Specifically, we define the proximity as the topological overlap measure (TOM) [35-37] which can also be defined for weighted networks [19]. The TOM combines the adjacency of two genes and the connection strengths these two genes share with other "third party" genes (see equation 6 in the Methods section and Additional File 1). The TOM is a highly robust measure of network interconnectedness (proximity). This proximity is used as input of average linkage hierarchical clustering. Modules are defined as branches of the resulting cluster tree [38]. This module detection procedure has been used in many applications [23,25-30,32,39,40] and a comparison to alternative procedures is beyond the scope of this article.

We find it convenient to summarize the gene expression profiles of a given module with the module eigengene, which can be considered as the best summary of the standardized module expression data [33,41]. The module eigengene of a given module is defined as the first principal component of the standardized expression profiles (see equation 8 in the Methods section).

Quantifying Module Membership

To identify possible regulators within a given module, we looked for highly connected intramodular hub genes, i.e. genes that have strong connections within the module. In our effort to find these genes, we examined two types of connectivity measures, which can be applied relative to any module q. The first connectivity measure is intramodular connectivity defined as

(4)

where n^(q) is the number of genes in the q^th module. In the case of an unweighted network, simply counts the number of connections to gene i within the q^th module. Intramodular connectivity can be interpreted as a measure of module membership: the higher the intramodular connectivity, the more centrally located the gene is in the module and the more certain is its membership with regard to this module. In signed networks, these highly connected hub genes may up-regulate adjacent genes since they are positively correlated with them, while in unsigned networks they may activate or repress their neighboring genes.

The second connectivity measure is the module eigengene based connectivity, (also known as the signed module membership measure [33]), defined as

(5)

where E^(q) is the eigengene of the q^th module (see equations 9 and 10 in the Methods section) and x_i is the expression profile of the gene i. We denote modules by colors. For example, denotes the module membership measure of the i-th gene with regard to the blue module.

Module eigengene based connectivity has several advantages over intramodular connectivity: first, it is naturally scaled to take on values between -1 and 1; second, one can use a correlation test to calculate a corresponding p-value for a gene's module membership; third it can be used in signed networks to identify genes that are anti-correlated with a given module eigengene (i.e. they may repress genes in the module), and fourth, k_ME can be computed for any gene on the array (not just genes used in the network construction). In practice, we found that intramodular and module eigengene based connectivity are highly correlated (Additional File 2). A priori, the connectivity measures defined in equations 4 and 5 are quite different. But we show in the Methods section that a simple theoretical relationship between them can be derived in the context of a signed co-expression module. Due to its advantages, we used the module eigengene based connectivity as the measure of module membership in our applications.

Signed WGCNA Identifies Pluripotency Related Modules in Ivanova et al (2006) Data Set

We generated unsigned and signed co-expression networks to analyze over 17,000 genes measured across 70 expression arrays from data published in Ivanova et al (2006) [4]. This data set contains expression profiles of ES cells individually depleted for the transcription factors Oct4, Nanog, Sox2, Esrrb, and Tbx3 by RNA interference (RNAi). The data set also includes expression profiles for RNAi knock downs of Tcl1 a co-activator of AKT kinase, and an EST (Mm343880), along with expression profiles of control ES cells carrying an empty RNAi vector and of ES cells differentiated by retinoic acid (RA). Each of these treatments was sampled over approximately eight days. To compare the performances of unsigned and signed WGCNA in identifying gene groups that are important for the regulation of the pluripotent state, we defined gene modules in unsigned and signed networks and assessed module function and importance by determining gene ontology terms associated with each module and examining module membership of genes known to play a role in ES cells. In addition, we analyzed how genes of a given module are bound by chromatin regulators or pluripotency TFs by incorporating independent promoter binding information.

Figure 2a shows the dendrogram of the unsigned network for the Ivanova et al data set. Modules were found by cutting branches of the cluster tree (dendrogram), using the dynamic tree cut library in R [38]. Modules are indicated by the color bands below the dendrogram. Genes that do not clearly belong to a branch are colored grey. To compare modules in signed and unsigned networks, we show two color bands: the top color band shows the genes colored by module membership in the unsigned network (corresponding to the dendrogram), while the bottom color band shows genes colored by module membership in the signed network. Similarly, Figure 2b displays the dendrogram of the signed network with the top color band showing genes colored by module membership in the signed network and the bottom color band showing genes colored by module membership in the unsigned network. These figures show that while some large modules (turquoise, yellow, red, and blue) are preserved in both networks, the signed network has two distinct small modules (black and tan) hidden within the unsigned turquoise module. The black and tan modules from the signed network are scattered throughout the unsigned network's turquoise module and cannot be detected since there is no branch of the dendrogram corresponding directly to these modules (i.e. regardless of the tree cutting algorithm employed, these modules would not be found in the unsigned network, data not shown). Figure 2a also shows a heatmap of the expression profiles of genes in the turquoise module from the unsigned network. Genes in this module exhibit a positive (red) or negative (green) change in expression upon knock down of the master regulator Oct4, which is not surprising given that Oct4 RNAi has be shown to cause a distinct differentiation pattern from other TF RNAi's [4,42]. Heat maps are similarly shown in Figure 2b for the turquoise, black, and blue modules from the signed network. In contrast to the unsigned network, each module contains genes with similar expression profiles. Here the turquoise module is made of genes that are activated when Oct4 is knocked down, while the black module contains genes that are repressed when Oct4 is knocked down and are down-regulated during retinoic acid (RA) induced differentiation. The blue module contains genes that are activated upon differentiation. This analysis shows that signed WGCNA identifies modules with more specific expression patterns than unsigned WGCNA.

Functional Enrichment with Regard to Known ES Cell Related Genes

Next we used external data to further study the gene modules defined by the networks and reveal their functional roles. We used two different strategies for this evaluation: first we assigned transcription factors and other regulators with known roles in pluripotency, self-renewal or differentiation to modules [4,8] and second, we incorporated genome-wide binding data for transcription factors and other regulators implicated in ES cell regulation in order to determine if these modules contain genes that are directly controlled by ES cell related TFs or differentiation suppressors [5,6,16].

Many genes known to maintain the pluripotent state of ES cells are found in the black module in the signed network. We defined a measure of gene significance (GS) as the t-statistic from the paired Student's t-test of expression in control RNAi samples and ES cell samples with RNAi knock down of Oct4 (paired by day of treatment). Figure 2c shows GS plotted against its module eigengene based connectivity, k_ME, in the black and blue modules of the signed network with marker genes labeled. Since the signed module membership k_ME is defined as the correlation between a gene expression profile and the module eigengene, its values lie between -1 and 1 with values near 1 signifying strong module membership to the corresponding signed module. Figure 2c shows a strong linear relationship between k_ME and GS in the black module (correlation = 0.5, p-value = 6.5e-13). As expected, most of the genes whose RNAi knock down induced ES cell differentiation in Ivanova et al [4] belong to the black module (Oct4, Nanog, Sox2, Esrrb, and Dppa4, Fisher's exact test p-value = 3.2 × 10^-5). Oct4's high connectivity ( = 0.94) makes it a hub gene in the black module, consistent with its known role as a master regulator of the pluripotent state. Furthermore, many genes that are known to be highly expressed in ES cells are also in the black module (e.g. Klf4, Utf1, and Phc1). Klf4 is one of the four TFs that can reprogram differentiated cells into a pluripotency state [3]. Utf1 interacts with Oct4, affects chromatin regulation in ES cells, and has recently been shown to improve reprogramming efficiency [43-45]. Phc1 is a Polycomb Group (PcG) protein. PcG proteins repress genes that become active upon differentiation of ES cells by mediating histone H3 lysine 27 tri-methylation and histone H2a ubiquitination [6]. The blue module contains Gata6 and Gata4, which are both highly connected ( = 0.93 and 0.88, respectively). These TFs are markers of ES cell differentiation, particularly into endoderm. Below we provide further evidence that the black and blue modules are related to pluripotency and differentiation respectively.

Module Enrichment with Regard to Known ES Cell Regulators

We incorporated genome-wide binding data for TFs (Oct4, Sox2, Nanog, Stat3, Smad1, cMyc, nMyc, Zfx, E2f1) and other regulators (Suz12) implicated in the maintenance of pluripotency and self-renewal, which were obtained by chromatin immunoprecipitation (ChIP) and massive parallel sequencing (ChIP-seq) by Chen et al (2008) [16]. Oct4, Sox2, Nanog, Smad1, and Stat3 are referred to as the Oct4 group of TFs, as they have been shown to often co-bind genomic regions; cMyc, nMyc, E2f1, and Zfx are referred to as the cMyc group of TFs because they also co-bind genomic regions [16]. Together TFs in the Oct4 and cMyc group are thought to activate expression of genes involved in pluripotency and self-renewal. Suz12, is a subunit of the histone H3K27 methyltransferase PcG protein complex, which represses genes that are activated upon differentiation [6,16].

To determine if binding by the two TF groups and Suz12 occurs more often in certain modules we computed binding enrichment for each module. Enrichment is defined as the odds ratio, that is the probability of a gene being bound by a particular TF or TF complex for genes in a given module divided by the probability of being bound for genes not located in the module. Table 1 shows module enrichment of genes bound by TFs in the Oct4 group, the cMyc group, and Suz12 for modules in the unsigned and signed networks. A gene is called bound by the Oct4 or cMyc groups if it is bound by at least 4 of the 5 and 3 of the 4 TFs in each TF group, respectively (similar results are found when using 3 of 5 and 2 of 4, data not shown). In agreement with the notion that the black module found in the signed network contains genes that are implicated in pluripotency, this module is strongly enriched with genes bound by TFs in the Oct4 and cMyc groups and under-enriched for binding by Suz12. Specifically, the proportion of genes bound by the Oct4 group in the black module is almost twice the proportion of genes bound in the general population (1.94). Similarly genes in the black module are almost twice as likely to be bound by TFs in the cMyc group (1.87) and are almost three times less likely to be bound by Suz12 (0.344). These enrichments further support the idea that the black module is a pluripotency and self-renewal module. Other modules, like the blue, brown, and cyan, are enriched for Suz12 bound genes and under-enriched for genes bound by the two TF groups. The blue module is the most significantly enriched for Suz12 binding further supporting the idea that genes in this module are involved in ES cell differentiation. For modules preserved in both unsigned and signed networks, Table 1 shows that enrichment values are generally more significant in the signed network. Similar enrichments can be seen for Oct4 and Nanog binding from Loh et al (2006) and Polycomb Group (PcG) protein binding from Boyer et al (2006) (Additional File 3). The incorporation of binding data suggests that signed WGCNA better separates genes into modules based on function and regulation. Specifically, the black module can be considered a pluripotency/self-renewal module, while the blue, brown, and cyan modules can be considered differentiation modules in the signed network. It is important to reiterate that the pluripotency/self-renewal black module was only found using signed WGCNA.

Epigenetic Regulation and Module Membership

Recent studies suggest that chromatin structure and epigenetic modifications, like histone modification and DNA methylation, play a role in controlling gene expression during ES cell self-renewal and differentiation [46-51]. For example, gene repression by the PcG protein complex via histone H3 lysine 27 trimethylation (H3K27me3) is required for ES cell self-renewal and pluripotency [6,52]. To understand how epigenetic variables contribute to the regulation of ES cells we studied the relationship of the pluripotency and differentiation modules with ES cell H3K4 and H3K27 trimethylation, DNA methylation, and CpG promoter content from previously published data sets [50,51]. We related the epigenetic variables to module membership in the black (Figure 3, top row) and blue module (Figure 3, bottom row). Specifically, we determined what proportion of genes with a given epigenetic mark (H3K4me3 for example) are also in the top 1000 genes with the highest (or ). Below we show that our findings are highly robust with respect to the number of selected module genes (see also Additional File 4). H3K4me3 is associated with gene activation, whereas H3K27me3 is known to silence genes. Figure 3 shows that genes with H3K4 trimethylation in ES cells contain significantly more black module genes than genes without this marker (Kruskal Wallis test p-value = 6.7 × 10^-113) which may reflect the active role the pluripotency module plays in ES cells. Interestingly, genes with H3K4 trimethylation or bivalent methylation contain significantly (p = 1.3 × 10^-33) more blue module genes than other gene classifications. Promoters that are both H3K4 and H3K27 trimethilated in ES cells (referred to as bivalent promoters) are thought to poise key developmental genes for activation upon differentiation [50].

Mammalian gene promoters are known to fall into one of at least two major classes: 1) CpG-rich promoters are associated with both ubiquitously expressed 'housekeeping' genes, and genes with more complex expression patterns, particularly those expressed during embryonic development and 2) CpG-poor promoters are generally associated with highly tissue-specific genes. To understand the role of CpG content in our modules we analyzed three CpG content classifications from Mikkelsen et al(2007): high (denoted HCP), low (LCP), and intermediate (ICP). Figure 3 shows that HCP genes contain significantly more black module genes (p = 1.3 × 10^-51) and significantly more blue module genes (p = 2.4 × 10^-36) than ICP or LCP genes. The LCPs are known to have a very different trimethylation pattern than the HCPs. Few (6.5%) of LCPs have significant H3K4me3 in ES cells and virtually none have H3K27me3. HCPs and LCPs are subject to distinct modes of regulation. In ES cells, all HCPs seem to be targets of trithorax group activity, and may therefore drive transcription unless actively repressed by PcG proteins. In contrast, LCPs seem to be inactive by default, independent of repression by PcG proteins, and may instead be selectively activated by cell-type- or tissue-specific factors [50].

Figure 3 also shows promoter CpG methylation in relation to module membership. DNA methylation in mammalian cells plays multiple roles in cell physiology, including genome stability, repression of endogenous retroviral elements, genomic imprinting. Levels of DNA methylation are dynamically regulated during embroyogenesis but less is known about the role DNA methylation play in gene expression and maintenance of pluripotency in ES cells [51]. Figure 3 shows that methylated genes are significantly under-enriched for black module (p = 2.0 × 10^-14) and significantly under-enriched for blue module genes (p = 5.1 × 10^-11). In Additional File 5, we present the data used for cross-referencing module membership to epigenetic regulators.

Signed WGCNA Identifies a Pluripotency Module in data from Zhou et al (2007)

To further investigate WGCNA's ability to discover functionally important groups of genes, we turned to an independent data set from Zhou et al (2007) [8]. In this study, ES cells were removed from feeder cells and leukemia inhibitory factor (LIF) to induce differentiation. During the course of differentiation, cells were separated based on expression of an Oct4 green fluorescent protein (GFP) reporter gene. Multiple samples were taken from undifferentiated ES cells and cells sorted at days 2, 4, 8, and 15 for high and low Oct4 expression. As before, we first identified gene modules via signed and unsigned methods and then related module membership to external data. In the following we show that a pluripotency/self-renewal and a differentiation module can be found in this new data set. For consistency between data sets, we have colored these modules black and blue, respectively.

Functional Enrichment Analysis of the Pluripotency and Differentiation Modules

Given that there was significant overlap between the pluripotency modules and differentiation modules of the Ivanova and Zhou networks, we focused further analysis on the network constructed from the Zhou et al data set as this network was based solely on differentiation induced expression changes. We determined functional enrichment using the Database for Annotation Visualization and Integrated Discovery (DAVID) [53]. Table 4 shows the significantly enriched GO terms for genes with the 5% highest . In agreement with the finding that the blue module is related to differentiation, genes within this module are significantly enriched for a functional group containing organ development, system development, and cell differentiation (p-value = 0.002). Other highly enriched groups are involved in regulating protein localization (p-value = 9.7 × 10^-17) and membrane composition (p-value = 5.6 × 10^-8).

Table 4 also shows significant GO terms for genes with the 5% highest . Given that many pluripotency TFs are in the black module, it is not surprising that the functional classifications, DNA binding and transcriptional regulation, are significantly enriched (p-value = 5.4 × 10^-8). However, two functional classifications, DNA damage/repair and mitochondrial function, are more significantly enriched than the transcriptional regulation group (p-values = 2.0 × 10^-8 and 3.8 × 10^-8, respectively) suggesting that these pathways play important roles in maintaining pluripotency and self-renewal.

We also used Ingenuity Pathway Analysis, IPA, to compare functional enrichment in the pluripotency and differentiation modules (Ingenuity Systems, http://www.ingenuity.com). Additional File 7 shows that functional groups similar to those found using DAVID are enriched in the black and blue module respectively. Cell cycle and DNA replication, recombination, and repair are enriched in the black module compared to the blue module and skeletal, muscular, and cardiovascular system development are enriched in the blue module.

Comparison to a Standard Differential Expression Analysis

Here we compare some of our WGCNA results with those of a standard differential expression analysis. In Figure 2c and Figure 5 we showed that for some modules a strong relationship between module membership (k_ME) and differential expression (gene significance/fold change) can be observed. In [33], we provide a geometric description of modules for which such a relationship can be observed. While a close relationship may exist between k_ME and a Student t-test statistic, it does not imply that corresponding gene ranking procedures are equivalent.

Here we compare signed WGCNA to standard differential expression methods using three different approaches. First, we show that a gene ranking based on k_ME is more consistent (reproducible) than that based on the Student t-test in our data. Specifically, we computed two gene rankings for the Ivanova et al data set, one ranked by t-statistic and the other by connectivity to a module of interest. We similarly computed two such rankings for the Zhou et al data set and studied the overlap between the two data sets (Additional File 8). Of the 1000 genes most significantly down regulated upon differentiation in each data set 139 overlap (hyper-geometric p-value = 1.0 × 10^-20). However, when ranking genes by connectivity to each data set's pluripotency model there is an increase in overlap to 230 (p-value = 1.7 × 10^-75, Additional File 8). This increased consistency is also seen in genes up regulated upon differentiation where 77 genes overlap between the two data sets (p-value = 0.02) when ranking by t-statistic and 161 genes overlap when ranking by connectivity to the differentiation modules (p-value = 2.8 × 10^-31, Additional File 8).

A second approach for comparing gene rankings is to use the functional enrichment with regard to known gene ontologies. To compare the abilities of k_ME versus the conventional t-statistic in identifying functionally interesting groups of genes we consider functional enrichment of genes found by one ranking method but not the other (Figure 6). Of the 1000 genes most strongly connected to the Ivanova et al pluripotency module (black), 463 overlap with the 1000 genes most significantly down regulated upon Oct4 RNAi in the same data set. Figure 6 shows the functional enrichment from Ingenuity Pathway Analysis, IPA, of the 537 genes in each group that do not overlap. Genes found only by signed WGCNA are significantly enriched for functions important to ES cells like DNA Replication, Recombination, and Repair, Cell Cycle, Cancer, Protein Synthesis, and RNA Post-Transcriptional Modification, which have previously been found using network methods [54]. Generally, genes identified by signed WGCNA exhibit more significant enrichment of functional classifications than those found by standard differential analysis.

We similarly compare the 1000 most highly connected genes in the differentiation (blue) module and the 1000 genes that are most significantly up regulated upon Oct4 RNAi in the Ivanova network. Interestingly, only five genes overlap (Figure 6). By examining those genes that do not overlap we see that ranking by connectivity yields greater significant enrichment for many functional groups important in ES cell differentiation including Organ Development, Tissue Development, Cell Morphology etc.

Similar analysis of pluripotency genes in Zhou et al yields consistent results with DNA Replication, Recombination, and Repair being more enriched when ranked by connectivity (Additional File 9) while analysis of highly connected genes in the differentiation module shows that differential analysis moderately out performs ranking by connectivity. The differences in functional enrichment in the Zhou et al data set are subtle given that there is more overlap between the two rankings (Additional File 8). This large overlap is likely due to the simplicity of the expression array samples which are filtered into only two groups, those that exhibit Oct4 expression and those that do not. Meanwhile, signed WGCNA is especially useful in Ivanova et al where smaller overlap is caused by the complexity of the expression samples which are made of many different RNAi treatments.

A third approach for comparing gene rankings is to use the enrichment with regard to epigenetic and transcriptional regulators. In Additional File 4 we relate different gene rankings to enrichment significance with regard to the following variables (a) histone H3K4 alone versus all others, (b) bivalent H3K4 & H3K27 versus all others [50], (c) High CPG class versus all others (i.e. HCG versus ICG and LCG), (d) promoter CPG methylation status [51], (e) Oct 4 complex binding status, (f) cMyc complex binding status. We report results for 3 different gene rankings using the Ivanova data: the black and blue curve represent gene rankings according to and , respectively. The grey curve represents ranking according to a Student T-test of differential expression. Additional File 4 shows that black and blue module genes can have very different enrichment results that tend to be very different from those of a standard analysis. This analysis illustrates how module membership provides important complementary variables along the Student t-test for understanding differences between genes.

The increased functional enrichment and improved consistency between data sets suggest that signed WGCNA is a complementary method to standard differential analysis. In practice, we recommend to use both k_ME and the Student t-test to find highly differentially expressed intramodular hub genes.

Pluripotency Module Genes involved in Transcriptional Regulation and Chromatin Structure

To gain a better understanding of the regulatory network involved in maintaining ES cell pluripotency and self-renewal, we first looked at genes with high black and GO terms related to transcriptional regulation or chromatin structure. Figure 7 lists such genes and contains information about how these genes are bound by TFs in the Oct4 and cMyc groups, and Suz12 plus Klf4, Esrrb, and Tcfcp2l1. The sign of black allows us to distinguish genes that promote pluripotency from those that repress it. As expected, many of the genes with high positive module membership measure () are known to participate in ES cell regulation (Zic3, Mkrn1, Phc1, Esrrb, Jarid2, Nodal, Jarid1b, Tgif1, Utf1, Hells, and Rest) [20,55,56]. Importantly, four TFs capable of reprogramming differentiated cells into a ES cell like state are in this list (Sox2, Klf2, Nanog, and Pou5f1(Oct4)) [10,11], confirming that the black module captures the known core transcriptional regulatory network responsible for maintaining a cell's stemness. Genes with high positive values of black module membership () but have not been implicated in maintaining pluripotency and self-renewal should therefore be strong candidates for further functional study. Such genes include Msh6 (involved in DNA damage and repair [57]), Rbpj (involved in Notch signaling [58]), Zfp39 (spermatogenesis [59]), and Nrf1 (mitochondrial organization and biogenesis [60]). Similarly, many of the highly negatively connected genes in the black module are known to play a role in ES cell differentiation (Cited2, Gata4, Gata6, Tead4, Foxa2, and Sox7). Because and are highly negatively correlated (r = -0.99), genes with high positive have negative and vice versa. Highly negatively connected genes that are not known to be involved in ES cell differentiation are also candidates for functional investigation. Some genes like Maged1 are known to play a role in cell differentiation but have not been shown to be important in ES cell differentiation specifically [61]. Other genes like Lass2 have little known about their role in cell differentiation. Figure 7 also shows that genes positively connected to the black module tend to be bound by more ES cell related TFs compared to the negatively connected genes, supporting the idea that these TFs bind and activate pluripotency and self-renewal genes [13].

Pluripotency Module Genes not Involved in Transcriptional Regulation or Chromatin Structure

Genes that are not involved in transcriptional regulation or chromatin modification (as defined by GO analysis) but have high average are also of interest. Figure 8 lists such genes along with TF and Suz12 binding information, connectivity, and fold change. Once again the importance of some genes has been validated, while others should be candidates for further research. Genes like Dppa5, Dppa4, and Tcl1 are markers of pluripotency and Nup133, a nuclear pore complex subunit, has recently been shown to be necessary in the maintenance of pluripotency [62]. Nup133 highlights the usefulness of signed WGCNA. Using the t-statistic from standard differential analysis Nup133 is ranked 222^th most significantly down regulated upon differentiation while using connectivity its rank moves to 28^th. Other candidate genes include Sh3gl2, which binds lipids and proteins [63], Mrpl15, a mitochondrial ribosomal protein, and Ppif, involved in mitochondrial function and oxidative stress-induced cell death [64]. Of the negatively connected genes in Figure 8, Ctsl (cathepsin L) has recently been shown to cleave the histone H3 N-terminus during ES cell differentiation [65], while little is known about Ctsz, also a cathepsin, while Gnas and Ctgf are differentiation genes [66,67]. The high negative connectivity of Uqcrh, a mitochondrial inner membrane protein [68], along with the high positive connectivity of Mrpl15 and Ppif, confirms that mitochondrial regulation may be distinct in ES cells [69-71] and suggests that they may be important regulators of mitochondrial function in ES cells.

Pluripotency Module Genes that Lack Binding by Known Pluripotency TFs

Figure 9 shows genes with relatively high that lack binding by the TFs Nanog, Oct4, Sox2, and Klf4 [10,11] and have low binding (≤ 2) by other TFs that maintain pluripotency (Smad1, Stat3, cMyc, nMyc, Esrrb, Tcfcp2l1, Zfx, and E2f1). Transcriptional regulators that have high positive module membership in the black module but lack binding by the pluripotency TFs are of interest since their strong module membership cannot be explained via regulation by these TFs. Their high average suggests that they may be upstream regulators of Oct4, Sox2, Nanog, and other genes important to pluripotency.

To further investigate the role of these genes, we used motif scanning methods described in Zhou et al (2007) [8] to determine if the binding sites of these genes are contained in regions co-bound by TFs in the Oct4 group or cMyc group in ChIP-seq data from Chen et al (2008). We concentrated solely on Lrh1 (Nr5a2) and Elk1 since their motifs have known position specific weight matrices while the other genes lack known motifs. Table 5 shows the enrichment and significance of motifs scanned. Sox2 and Oct4 bind to the composite SoxOct motif besides their own, Stat3 binds the Stat1 motif, and cMyc binds the Ebox motif. Both the Oct4 and cMyc groups' expected motifs are enriched. For example, the SoxOct motif has over three fold enrichment in regions bound by TFs in the Oct4 group. Interestingly, the Lrh1 motif is more enriched than the Nanog motif in sequences bound by the Oct4 group, which contains Nanog binding by definition. This reinforces the hypothesis that Lrh1 co-binds regions bound by TFs in the Oct4 group [8]. Furthermore, Lrh1 sites are found in the promoter regions of Pou5f1 (Oct4), Klf4, Dppa5, and Suz12 with Pou5f1 having three separate sites. These motif sites and Lrh1's known importance in ES cells, suggest that it may be an upstream regulator of these pluripotency factors and as such is a candidate for experimental validation [72]. The Elk1 motif is also significantly enriched in sequences bound by the cMyc group, thus Ekl1 may co-regulate genes bound by TFs in this group.

A Geometric Interpretation of Signed WGCNA Modules

To understand how signed WGCNA is better able to separate genes into functional modules in the Ivanova data set, we plotted genes in the signed black or turquoise module relative to the unsigned turquoise module eigengene (Additional File 10). Note that genes located in the black and turquoise modules in the signed network are clearly separated into two clusters. Because a module eigengene is defined as the first principle component of its module, it describes the main direction in which the module's gene expressions vary. Note that the signed module eigengenes are oriented in the direction of their clusters. The direction of the unsigned turquoise module eigengene is more difficult to interpret. Because the turquoise module in the unsigned network contains two distinct signed modules (black and turquoise), its module eigengene describes the variance between these two sub-modules and the variance within the larger sub-module, the signed turquoise. As such, the unsigned turquoise module eigengene fails to quantify the true importance of highly connected genes in the signed black module. For example, Oct4's in the unsigned turquoise module is -0.74 while it is 0.94 in the signed black module. Thus, Oct4 is not identified as a hub gene in the unsigned network while it is clearly a hub gene in the signed network.

The Topological Overlap Matrix

The topological overlap, reflects the relative interconnectedness between genes i and j. It takes into account the relationship between the two genes and their shared connection pattern to other genes [19,35-37]. The topological overlap between two genes is defined as follows:

(6)

where a_ij is the above defined adjacency, l_ij = ∑_u≠i,j a_iua_uj, and k_i = ∑_u≠i a_iu.

The Effect of the Co-expression Similarity Measure on the Topological Overlap Measure

The choice of co-expression similarity measure (i.e. versus ) has strong implications for the resulting topological overlap measure. Note that cor(x₁, x_u) = -1 implies and . In case of a soft threshold β = 1, we find the following corresponding weighted adjacency measures: and . In the following, consider the simple network in Additional File 1 (part a) where adjacencies a_1u = a and a_u2 = a for u = 3, ..., n. Using equation (6), t₁₂ simplifies as follows:

where the latter approximation assumes that the number of genes n is large. For the above situation (with β = 1) this implies ≈ a^unsigned = 1 and ≈ a^signed = 0. Thus, the two genes have high interconnectedness in an unsigned network but zero interconnectedness in a signed network. Additional File 1 (part b) shows a simple network where genes 1 and 2 are oppositely correlated with their neighbors. Here the choice of gene co-expression measure results in a very different topological overlap measures, which in turn, leads to different modules.

The Module Eigengene and Module Membership

For the q-th module, we summarize its expression data, by a module eigengene, E^(q), found by singular value decomposition of the expression data [39]:

(7)

where X^(q) is the n^(q) × m matrix of standardized expression profiles of the n^(q) genes in the module across m samples, U^(q) is an n^(q) × m matrix with orthogonal columns, D^(q) is an m × m diagonal matrix of singular values, and V^(q) is an m × m orthogonal matrix of singular vectors. Then E^(q) is defined by:

(8)

where is the singular vector in V ^(q) corresponding to , the largest absolute singular value in D^(q). The module eigengene, E^(q), can be used to define the module eigengene based connectivity, k_ME, or fuzzy module membership [25,27,33,39] via

(9)

The Relationship between k_i and k_{ME, i}, when β = 1

To study the relationship between module eigengene based connectivity, k_{ME, i}, and intramodular connectivity, k_i, we consider a special case of a signed weighted network with β = 1 (i.e. ). Then the intramodular connectivity is given by

where n^(q) is the number of genes in the q^thmodule. It has been shown that network modules are approximately factorizable (i.e. cor(x_i,x_j) ≈ cor(x_i,E^(q)) × cor (x_j,E^(q))) [33,39]). This approximation implies

(10)

where E^(q) is the module eigengene of module q, , and . Equation (11) implies an approximate linear relationship between and if β = 1. Using real data, we illustrate this relationship in Additional File 2.

Motif Enrichment

Methods developed in Zhou et al [8] were used to scan sequences for motifs with pre-defined position specific weight matrices. Sequences were determined by extending bound ChIP-seq sites 150 bp up and downstream resulting in regions approximately 330 bp long. Any overlapping regions were then joined into larger meta-regions. A set of control sequences was scanned to determine a motif enrichment ratio. The control group was created by randomly sampling 5,000 probes from the Agilent Mouse Promoter Whole Genome ChIP-on-chip Microarray Set. These probes are distributed -5.5 kb upstream to +2.5 downstream of approximately 17,000 known gene transcription start sites from UCSC's version mm8 genome.

Overlapping control probes were merged into meta-regions as described above. Enrichment of a motif is defined as where M_T is the number of observed sites and λ is the number of expected sites, , where M_C is the number of sites in the control, N_C the length of all control sequences, and N_T the length of all bound sequences. Statistical significance is determined by the Poisson distribution with λ as the mean.