Figure 1.
Illustration of the tensor representation for multiple networks and a recurrent heavy subgraph.
(A) Microarray datasets are modeled as (B) a collection of co-expression networks; (C) These co-expression networks can be “stacked” together into (D) a third-order tensor such that each slice represents the adjacency matrix of one network. The weights of edges in the co-expression networks and their corresponding tensor elements are indicated by the color scale to the right of the figure. In (D), after reordering the tensor using the gene and network membership vectors, it becomes clear that the subtensor in the top-left corner of the tensor (formed by genes in networks
) corresponds to a recurrent heavy subgraph.
Figure 2.
Illustration of an RHS family and its tower-like structure in and
.
(A) Ten networks of 10 genes {A,B,C,D,E,F,G,H,I,J}, where the edge weight is associated with the color scale shown in (C); (B) The optimal membership vectors and
obtained by performing MSCR. Their significant components are ranked as follows:
, and
; (C) The tensor of networks and genes arranged in decreasing order of the elements in
and
. Three RHSs are discovered: the first RHS recurs in networks {1,2,3,4,5,6,7} with member genes {A,B,C}; the second recurs in networks {1,2,3,4,5} with member genes {A,B,C,D,E}; and the third recurs in networks {1,2,3} with member genes {A,B,C,D,E,F,G}; (D) A more intuitive illustration of three three overlapping RHSs, which form a tower-like structure.
Figure 3.
Evaluation of the functional, transcriptional, and protein complex homogeneity of RHSs with different recurrences and heaviness.
Four types of databases are used: (A) Gene Ontology (GO) and (B) KEGG pathway databases for functional enrichment, (C) ENCODE database for transcriptional enrichment, and (D) CORUM database for protein complex enrichment. It can be seen that the percentage of potential functional, transcriptional, and protein complex modules increases with the heaviness and recurrence of the RHSs.
Figure 4.
Comparison between weighted and unweighted network analysis.
The weighted networks were transformed to unweighted networks by dichotomizing edges with an expression correlation cutoff of 0.6. The proposed tensor method was then applied to both weighted and unweighted networks. We compared rates of functional homogeneity detected in the top modules, ranked by (A) recurrences or (B) average heaviness in their datasets of occurrence. Weighted graph analysis consistently outperforms unweighted graph analysis.
Figure 5.
An 8-gene module is enriched in the binding of multiple regulatory factors.
These regulatory factors are Pol2 (-value = 1.73E-3), H3K36me3 (
-value = 5.54E-3), E2F4 (
-value = 1.65E-4), and cFos (
-value = 2.68E-2). The module is active in 8 datasets, and its member genes are involved in DNA replication,
-value = 2.15E-2).
Figure 6.
Two modules are enriched in protein complexes.
The module in (A) is enriched in the U2 snRNP 17S protein complex (-value = 9.9E-5) and the module in (B) is enriched in the F1F0 ATPase protein complex (
-value = 1.8E-6). The members of the protein complexes are colored in yellow. The width of an edge is proportional to the average correlation of its genes in the datasets where the module occurs.
Figure 7.
Examples of phenotype-specific modules associated with (A) Glioma and (B) muscle.
The width of an edge is proportional to the average correlation of its genes in the datasets where the module occurs.
Figure 8.
The protein complex cooperativity network.
Nodes represent protein complexes, and edges represent high () second-order correlation between pairs. The second-order correlation quantifies the cooperativity of activities of the two RHSs modules across different datasets. The darker the color of the edges, the stronger the second-order correlation.
Figure 9.
Reconstruction of transcriptional regulatory networks.
(A) Three types of possible transcription networks that could explain a second-order correlation between two transcriptional modules. Given two modules controlled by two transcription factors, TF1 and TF2, respectively, the coactivation of the two modules implies cooperativity between TF1 and TF2. This relationship may be caused by a type I network in which the activities of TF1 and TF2 are controlled by common transcription factor(s) TF3; or a type II network, in which the activity of TF2 is controlled by TF1 or vice versa; or a type III network, in which TF1 and TF2 interact at the protein level. (B) A regulatory network reconstructed on the basis of the derived transcription networks. Green circles denote transcription factors, yellow boxes are transcription modules defined by RHSs (detailed information on these RHSs provided in Text S1), blue ovals denote protein complexes represented by the RHSs, and blue boxes highlight the biological processes in which the modules are involved.