The development of high-throughput Microarray technologies has provided a range of opportunities to systematically characterize diverse types of biological networks. Biological networks can be broadly classified as protein interaction networks [1–3], metabolic networks [4–6] and gene co-expression networks [7]. These networks provide an effective way to summarize gene and protein correlations. In this paper, we focus on gene co-expression networks, which is an undirected graph where nodes represent gene and nodes are connected by an edge if the corresponding gene pairs are significantly co-expressed. Gene co-expression networks provide the association between individual genes in terms of their expression similarity and a network-level view of the similarity among a set of genes. In co-expression networks, two genes are connected by an undirected edge if their activities have significant association, as computed using gene expression measurements such as Pearson correlation, Spearman correlation, mutual information. Compared to gene regulatory networks, a gene co-expression network is built upon gene neighborhood relations, which give interesting geometric interpretations of the network. One of the most important applications of gene co-expression networks is to identify functional gene modules [8] or network modules, which are represented by the strongly connected regions of the co-expression network.

In the literature, a number of techniques have been proposed for gene co-expression network construction. When inferring co-expression networks from gene expression data, the algorithms take a gene expression dataset as primary input and then, by using a correlation-based proximity measure, constructs the corresponding co-expression networks. Frequently used correlation-based measures are Pearson correlation coefficient, Spearman correlation coefficient and Mutual information. Approaches such as [9, 10] used Pearson correlation coefficient to extract the association among genes in a co-expression network. The Spearman correlation coefficient is used as a gene expression similarity measure to construct co-expression network in [10]. [11], Steuer et al. [12] reports the use of Mutual Information to find similarly expressed gene pairs in such networks. While some studies attempted to apply algorithms directly to the adjacency matrices of networks to partition network nodes into groups [13, 14], other studies rely on special purpose algorithms for identifying subnetworks with certain properties [15].

Generally, in a co-expression network, the connections between genes are obtained from the absolute values of a co-expression measure. Several researchers have suggested to threshold this value of the co-expression measure to construct gene co-expression networks. There are two ways to pick a threshold: one way is picking a hard threshold (a number) based on the notion of statistical significance so that gene co-expression is encoded using binary information (connected=1, unconnected=0). The other way is called soft thresholding which weighs each connection by a number between 0 and 1. The drawbacks of hard thresholding include loss of information regarding the magnitude of gene connections and sensitivity to the choice of the threshold. Generally, hard thresholding results in unweighted networks while soft thresholding results in weighted networks.

To construct the gene co-expression network, we use the general framework proposed by [16]. A new effective gene similarity measure called NMRS is used to construct the distance matrix. We use a hard thresholding based signum function to construct the adjacency matrix from the distance matrix. A spanning tree based approach is used to detect network modules in the co-expression network. Extracted network modules are projected as functional categories of genes and these modules are validated using p value and Q value. Our approach is explained next.

Define a gene expression measurement

To determine whether two genes have similar expression patterns, an appropriate similarity measure must be chosen [17]. To measure the level of concordance between gene expression profiles, we develop a gene co-expression measure called NMRS. The NMRS of gene d₁=(a₁, a₂,…, a _n ) with respect to gene d₂=(b₁, b₂,…, b _n ) is defined by

where

NMRS as a metric

NMRS satisfies all the properties of a metric. We establish The non-negativity, symmetricity and triangular inequality properties for our measure in additional file 1.

Significance of NMRS

The most widely used proximity measures in gene expression data analysis are Euclidean distance, Pearson correlation coefficient, Spearman correlation coefficient, Mean squared residue etc. In co-expression network, the used proximity measure is expected to effectively detect the linear shifting patterns in the gene expression data. But none of the widely used proximity measures can satisfactorily serve this purpose. The Euclidean distance measures the distance between two data objects. But in this domain, the overall shapes of gene expression patterns (or profiles) are of greater interest than the individual magnitudes of each feature [18]. So Euclidean distance can not straight away detect shifting patterns, but bringing down all the genes to the same range of expression values can make this measure to detect shifting patterns. This normalization process involves an extra overhead. Along with shifting patterns Pearson correlation coefficient also detects scaling patterns and some other patterns which is normally not desired in a co-expression network and may lead to inclusion of genes which have considerable amount of difference between their expression levels. Spearman Rank Correlation Coefficient uses ranks to calculate correlation which can neither detect shifting patterns nor scaling patterns. Mean squared residue is good enough to detect shifting patterns, but the aggregate measure can not operate in a mutual mode, i.e. it can not find correlation between a pair of genes. A general comparison of these measures is presented in Table 1.

Proximity measure	Mode	Normalization required	Detects shifting pattern	Detects scaling pattern
Euclidian	Mutual	Yes	Yes	No
Pearson	Mutual	No	Yes	Yes
Spearman	Mutual	No	No	No
MSR	Aggregate	No	Yes	Yes
NMRS	Mutual	No	Yes	Yes

The table 1 presents the comparison of different proximity measure.

Let us consider a random gene pattern a as presented in Figure 1(a). Gene pattern b1 in Figure 1(b) has a shifting relationship with gene a. Gene pattern b8 in Figure 1(i) is a shifted as well as negatively correlated form of gene a. Figures 1(b)-1(h) present gene patterns b2, b3, b4, b5, b6 and b7 which are uniformly distributed intermediate patterns between genes b1 and b8. Figure 2 shows Pearson, Spearman and NMRS correlation of gene patterns b1-b8 with that of a. As usual the Spearman correlation was found to be concerned only about the rank information about the gene patterns. Interestingly, Pearson correlation was found to produce some undesired correlation values for the pairs a and b2, a and b3, a and b4, a and b4, a and b5, a and b6 and a and b7, which are neither shifting nor scaling patterns. The values of these patterns are given in Table 2. Our measure is found to effectively distinguish patterns across this uniform distribution from a shifted pattern (with a value 1) to a shifted and negatively correlated pattern (with value 0) of a given pattern as can be seen in Figure 2.

a	4	7	6	3	6	5	8	7	3
b1	10	13	12	9	12	11	14	13	9
b2	10.4286	12.5714	11.8571	9.7143	11.8571	11.1429	13.2857	12.5714	9.7143
b3	10.8571	12.1429	11.7143	10.4286	11.7143	11.2857	12.5714	12.1429	10.4286
b4	11.2857	11.7143	11.5714	11.1429	11.5714	11.4286	11.8571	11.7143	11.1429
b5	11.7143	11.2857	11.4286	11.8571	11.4286	11.5714	11.1429	11.2857	11.8571
b6	12.1429	10.8571	11.2857	12.5714	11.2857	11.7143	10.4286	10.8571	12.5714
b7	12.5714	10.4286	11.1429	13.2857	11.1429	11.8571	9.7143	10.4286	13.2857
b8	13	10	11	14	11	12	9	10	14

The table 2 presents the random gene patterns for analysis of different proximity measures.

Compute an adjacency matrix

An adjacency matrix is obtained using a signum function based hard thresholding approach which encodes edge information for each pair of nodes in the co-expression network. Two genes d _i and d _j are connected if Dist(d _i ,d _j ) >δ, a user defined threshold. Based on the connected pairs, an adjacency matrix is computed as

Detect network modules

To detect subsets of nodes (modules) that are tightly connected to each other is an important aim of co-expression network analysis. In this paper, we use spanning trees and a topological overlap similarity measure [19] to find the network modules, since this measure is found to result in biologically meaningful modules. A tree T is a spanning tree of a connected graph G if T is a subgraph of G and it contains all vertices of G. We use Prim’s algorithm [20] to find a spanning tree of a undirected graph. However, unlike traditional Prim’s algorithm we find a spanning tree with maximum weight. For unweighed networks (i.e. a _ij = 1 or = 0), the topological overlap matrix is defined by

(1)

where l _ij = ∑ _u a _iu a _ij , and k _i = ∑ _u a _iu is the node connectivity.

Module Miner takes NMRS threshold, δ, as a input and works on a microarray gene data and constructs the gene co-expression network and finally network modules are extracted from the network. Our approach uses an effective similarity measure NMRS to form a co-expression network using signum function. The co-expression network is further explored to mine the potential network modules using a spanning tree based method and a connectivity measure called Topological Overlap Matrix.

Definition 1 A CEN can be defined by an undirected, graph G={V,E} where each v∈V corresponds to a gene and each edge e∈E corresponds a pair of genes d _i , d _j ∈D such that Dist(d_i, d _j )≥δ.

Definition 2 Connected regions in a CEN are parts of the network where each pair of vertices is connected by a path. The i ^th connected region extracted from G can be defined as a graph where and such that for any vertex , there is at least one vertex which are connected by an edge .

Definition 3 Maximum spanning tree of a weighted graph is a spanning tree obtained from i ^th connected region, can be defined as , where the sum of TOM values associated with edges in is maximum compared to other spanning trees.

Definition 4 Network modules are highly connected regions of the co-expression network. The i ^th network module derived from j ^th connected region is defined as a set of vertices if

Algorithm: Module Miner

The pseudo code of Module Miner is presented in Algorithm 1. In the pseudo code, lines 1-4 extracts the connected regions from the gene expression data. Lines 5-25 process each of the connected regions to extract the network modules. A maximum spanning tree is constructed using Prim’s algorithm [20] from a connected region with weights defined by topological overlap matrix in lines 6-8. Lines 9-10 find and remove the weakest edge from the spanning tree. Removal of this edge from the spanning tree leads to two subtrees which are processed in lines 11-23 to form either a connected module or a new connected region.

In this paper, an effective gene expression similarity measure NMRS is introduced, which is used to construct the co-expression network through a signum function based hard thresholding scheme. Finally, network modules are extracted from the network using maximum spanning tree and topological overlap matrix. However, soft thresholding method can be used to construct the adjacency matrix to reduce information loss. Generalized Topological Overlap Measure [25] can be used instead of Topological Overlap Measure to get more accurate results. There is scope to design supervised models to derive gene regulatory network from the co-expression network.