117

THE PREDICTION OF LOCAL MODULAR STRUCTURESIN A CO-EXPRESSION NETWORKBASED ON GENE EXPRESSION DATASETSYOSHIYUKI OGATA1
yogata@kazusa.or.jp

NOZOMU SAKURAI1
sakurai@kazusa.or.jp

HIDEYUKI SUZUKI1
hsuzuki@kazusa.or.jp

KOH AOKI1
kaoki@kazusa.or.jp

KAZUKI SAITO2,3
ksaito@faculty.chiba-u.jp

DAISUKE SHIBATA1
shibata@kazusa.or.jp

1

Department of Biotechnology Research, Kazusa DNA Research Institute, 2-6-7 KazusaKamatari, Kisarazu, Chiba 292-0818, Japan
2
Graduate School of Pharmaceutical Science, Chiba University, Chiba 263-8522, Japan
3
Plant Science Center, RIKEN, Yokohama, Kanagawa 230-0045, Japan
In scientific fields such as systems biology, evaluation of the relationship between network members(vertices) is approached using a network structure. In a co-expression network, comprising genes(vertices) and gene-to-gene links (edges) representing co-expression relationships, local modular
structures with tight intra-modular connections include genes that are co-expressed with each other.
For detecting such modules from among the whole network, an approach to evaluate network
topology between modules as well as intra-modular network topology is useful. To detect such
modules, we combined a novel inter-modular index with network density, the representative intramodular index, instead of a single use of network density. We designed an algorithm to optimize the
combinatory index for a module and applied it to Arabidopsis co-expression analysis. To verify the
relation between modules obtained using our algorithm and biological knowledge, we compared it tothe other tools for co-expression network analyses using the KEGG pathways, indicating that ouralgorithm detected network modules representing better associations with the pathways. It is also
applicable to a large dataset of gene expression profiles, which is difficult to calculate in a mass.
Keywords: network topology, local modular structure, network density, co-expression analysis

1.

Introduction

Complex systems are the focus of many studies in various fields, such as systems biology.
To visualize and analyze such complex systems, network analysis is considered to be a
powerful approach [1]. A network structure is composed of vertices that are linkable tomultiple other vertices on the basis of a threshold for vertex-to-vertex connections; e.g.,protein-to-protein in a protein interaction network [2-4], and gene-to-gene in a gene coexpression network [5-11], reviewed in [12-14]. Such a network includes ‘network
modules’, in which the vertices are tightly interconnected [15,16]. The detection of
network modules leads to the elucidation of novel vertex-to-vertex associations invarious fields of networks.
In a co-expression network, pairs of genes are connected on the basis of similarity intheir expression profiles, which are obtained from DNA microarray datasets, and areused for prediction their functional relatedness, such as the relationship of transcriptional

118 Y. Ogata et al.
regulation [6], close relationships in a metabolic pathway [10,17] and subunits of acommon protein complex [18]. To detect local network modules from among a coexpression network based on gene expression profiles, recent co-expression network
approaches have optimized a cutoff value for gene-to-gene links in the network. Gupta etal. [19] proposed that the ‘average clustering coefficient’ index can be used to optimize asingle cutoff value for their gene-to-gene correlation network. Margolin et al. [20]developed the ARACNE tool for detecting co-expressed genes, which exploits the ‘joint
probability distribution’ index for evaluating gene-to-gene correlations on the basis ofstatistical physics and information theory for reducing the number of false-positive coexpression links. The DP-Clus tool uses the ‘cluster property’ index, which is calculatedon the basis of network topology to improve the inference of protein-to-protein
interactions by removing false-positive interactions [21]. To detect communities in asocial network, in which a vertex represents a person and a pair of persons are connectedon the basis of human relationship, Bagrow and Bollt [22] proposed the ‘l-shell’algorithm, which can be executed without knowledge of the entire network. To use thesealgorithms, a cutoff value for the vertex-to-vertex association index must be selected.
To separate a local network module using an appropriate cutoff value, a combined
evaluation of intra-modular and inter-modular connections is useful. A high cutoff value
through the entire network may contribute to separating local network modules fromeach other and, conversely, may lose intra-modular connections. Although a low cutoff
value may contribute to intra-modular connections, it may prevent the separation of local
network modules. Network density is a representative index for intra-modular
connections. As indices for inter-modular connections, the betweenness [23] andcohesiveness [24] indices are used. It is, however, difficult to combine these indices with
network density due to the difference in object connections between them.
We devised a novel index for evaluating inter-modular connections and combining
with network density and developed an algorithm for detecting network modules using
the combination. For combining these indices, we calculated their harmonic mean. To
verify whether co-expression modules obtained from our algorithm are associated withbiological knowledge, we applied it to Arabidopsis co-expression analysis and compared
it to the publicly available algorithms ARACNE and DP-Clus for the similar purpose
using the KEGG pathways. Through comparative analysis, we demonstrate that ouralgorithm detected co-expression modules with higher network indices and better
assignment to the pathways than those obtained using the other tools.
2.

Method and Results

2.1. Definitions
We define an index (NB) for evaluating connections between network modules andcombining with network density (ND) as follows.

Local Module Prediction in a Co-Expression Network 119

 e(i)ND 

NB n  n  1(1)

 e(i)
 d (i)(2)

In Eq. (1), n represents the number of vertices included in the module, i ranges according
to 1  i  n , e(i) represents the number of vertex-to-vertex links (edges) between the ith
vertex in the module and the other module members over a threshold cutoff value forvertex-to-vertex association (TC; e.g. the Pearson correlation coefficient in theapplication to Arabidopsis genes in the present research), and d(i) represents the total
number of vertex-to-vertex links between the ith vertex and the all possible vertices
included in the whole network over a TC, irrespective of membership in the module(degree). Using these equations, the combined index (NC), the harmonic mean of ND and
NB can be calculated in the following equation.
NC 

 e(i)n  n  1   d (i )2(3)When NC of a network module is the maximal value of 1, the module is depicted as acomplete graph with no vertex-to-vertex link to vertices outside of the module. In thepresent paper, we hypothesize that vertices in a network module with a higher NC value
are more closely associated with each other.
We also define the combined index of a single vertex to a network module (VC) as
follows.
VC i  

2  e(i )n  n  1  d (i )(4)

In Eq. (4), e(i), d(i), and n are similar to those in Eq. (3).
2.2. Microarray datasets
For a comparative analysis between our algorithm, ARACNE [20], and DP-Clus [21]using the KEGG PATHWAY dataset, including not only metabolic pathways, but protein
complexes, we selected the AtGenExpress developmental dataset, composed of 237
DNA microarray data in various developmental stages [25], and 1752 Arabidopsis genes
included in the 136 metabolic pathways and protein complexes for Arabidopsis [26].
Using the dataset, Pearson correlation coefficients for all pairs of the genes were
calculated on the basis of similarity in their expression profiles. The size of this dataset
does not exceed the limitation in ARACNE and DP-Clus implementation.
For a comprehensive co-expression analysis in Arabidopsis genes, we obtained thegene-to-gene correlation dataset from the ATTED-II database [27], comprising 22 263
genes and 1388 DNA microarray data, which are available at [28].

120 Y. Ogata et al.
2.3. An algorithm to extract co-expression modules
Our algorithm starts with a single vertex, referred to as the seed vertex (SV), and thendetects a network module including the SV in the following steps (Fig. 1).

Figure 1: Schematic images of our algorithm. A hexagon and a circle represent a seed vertex (SV) and a vertex
except for the SV. A vacant and a filled circle represent a vertex with a high and low VC value, respectively. A
vertex-to-vertex link is depicted when a correlation coefficient for the relationship of the vertex pair are equal toor higher than a threshold cutoff value of the coefficient (TC). A link connected to a single vertex represents a
connection between a group member and a non-member. The detailed description of this scheme is in the An
algorithm to extract co-expression modules section.

Step 1. SV selection. An SV can be arbitrarily selected at the initiation of the algorithm.
We selected all individual genes as SVs.
Step 2. Highly correlated vertex (HV) selection. For the SV, we set a series of vertex
groups (HVs) composed of the SV (a hexagon in Fig. 1) and vertices (circles) with thehighest correlation coefficients to the SV. For example, the HV(i) group has i + 1 vertices
including the SV and the vertices that show the first to ith highest correlation coefficientsto the SV. The HV(i) group members are interconnected on the basis of the cutoff valueof TC(i), which is the ith highest (i.e. the lowest within this group) correlation coefficientto the SV. In this step, namely, the SV is connected to all vertices.
Step 3. False-positive vertex exclusion and NC calculation. Fisrt, the NC value of each
HV group is calculated. Next, the VC values of members of the group are calculated.

Local Module Prediction in a Co-Expression Network 121
Then, the vertex with the lowest VC value (one of filled circles in Fig. 1) is excluded as afalse-positive vertex from the group to create the new group. The NC calculation and theone-by-one exclusion are repeated until the group comprises three vertices. This step isperformed for all HV groups selected in Step 2.
Step 4. Kernel vertex (KV) selection. Of all temporal groups during Step 3 for each HVgroup, the group with the highest NC value is selected as the KV group.
Step 5. The best KV selection. From among the KV groups, the group with the highest
NC value is selected as the best KV group. When KV(i) group, originating from the HV(i)group, is selected as the best KV group, TC(i) is used as the cutoff value of vertex-tovertex correlation coefficient for the group.
Step 6. False-negative vertex detection. For the best KV group, vertices with high VC
values to the group but non-members of the group (i.e. false-negative vertices for thegroup) are extracted from outside of the group. The VC value of each non-member iscalculated at various TC (ranging from 0 to 1). If the highest value of a vertex is userselected cutoff value (e.g. 0.5) or higher, the vertex is incorporated into the group.
Finally, the group composed of the best KV group and the vertices extracted in this step
are selected as members of a ‘network module’ originating from the SV.
2.4. Testing
To verify the applicability of our algorithm to associating a network module detectedfrom our algorithm with a biological network, we performed a comparative analysis
using Arabidopsis genes to the ARACNE tool [20] and the DP-Clus tool [21], which canbe used to detect co-expression modules from gene expression datasets and areappropriate for a fair comparison to our algorithm, although they are designed for otherpurposes. In the analysis, the indices in their network topology and in their associations
to 136 pathways and protein complexes in the KEGG PATHWAY dataset werecompared. As a gene expression dataset for this comparative analysis, we selected the
AtGenExpress developmental dataset (see Microarray datasets), because ARACNE andDP-Clus can accept the sizes of the dataset, while our algorithm is feasible, irrespectiveof data size. The ARACNE and DP-Clus tools require setting threshold cutoff values forthe gene-to-gene association index. For fair comparison between our algorithm,
ARACNE, and DP-Clus in extracting co-expression modules, we used several cutoff
values for their execution; i.e. e-20, e-30, e-40, e-50 for ARACNE and a range of geneto-gene correlation coefficient of 0.45 to 0.90 at intervals of 0.05 for DP-Clus. To
perform the comparative analysis, we selected their adequate cutoff values with which
the average size of the co-expression modules from each approach is similar to that of the
KEGG PATHWAY dataset, resulting in values of e-40 for ARACNE and 0.75 for DPClus. For the co-expression modules obtained from the three network approaches, we122 Y. Ogata et al.
calculated the precision values of the modules to individual pathways in the following
equation and compared those between approaches.

(precision) = (module members assigned to a pathway) / (all of the module
members)(5)
The average precision values in a total of 1752 modules are 0.31 for our algorithm, 0.19
for ARACNE, and 0.18 for DP-Clus. Figure 2 shows the relationship of the precision
values to the ratio of genes included in co-expression modules with the precision value or
higher to 1752 genes. In Fig. 2, the ratios in any precision values are highest in using ouralgorithm, especially the number of genes in co-expression modules with 0.9 or higher
precision values (228 genes) are five-fold (45 genes) and ten-fold (22 genes) as much as
ARACNE and DP-Clus.

Figure 2: The number of genes included in co-expression modules well-assigned to the KEGG pathways. The
precision value represents the ratio of genes included in a specific KEGG pathway among genes included in aco-expression module. For the precision value of 0.9, our algorithm detects 228 genes; five-fold and ten-fold asmuch as those in ARACNE (45 genes) and DP-Clus (22 genes), respectively.

We performed comparative ROC curve analysis, a representative approach to evaluate
the ability to detect, in the assignment of co-expression modules detected by ouralgorithm and the other tools to a specific KEGG pathway. We selected the largest‘Ribosome’ pathway, to which 190 genes are assigned, for the analysis. The precision
values of all co-expression modules in the three approaches to the pathway were
calculated. On the basis of the assignment of SVs included in the modules to the pathway,
the false positive rates (FPRs) and the true positive rates (TPRs) were calculated to depict
ROC curves of the approaches (Fig. 3). In the ROC curves, our algorithm shows thebetter result than those of ARACNE and DP-Clus, especially in the region of low FPR.
These findings indicate that co-expression modules obtained using our algorithm areassigned to the KEGG pathways better than the other tools. The information of coexpression modules with high precision values (≥ 0.5) is listed in Table 1.

Local Module Prediction in a Co-Expression Network 123

Figure 3: ROC curves of the three approaches in the assignment of co-expression modules to ‘ribosome’
pathway. The FPR and TPR indices represent false positive rate and true positive rate, respectively. The FPR
index is calculated by the ratio of genes incorrectly assigned to the pathway by an approach among genes thatare not included in the pathway. The TPR index is calculated by the ratio of genes correctly assigned to thepathway by an approach among genes included in the pathway. The ROC curve of our algorithm shows thebetter result than those of the other two algorithms.

2.5. Implementation
To verify the applicability of our algorithm to a large dataset, we applied it to a large
expression dataset of Arabidopsis genes (see Microarray datasets). In this application,
many co-expression modules with high NC values were detected. Of 22 263 genes, 8446
genes (approximately 38 %) are included in co-expression modules with 0.5 or higher
NC values. The TC values of co-expression modules, in which the NC values are
optimized, vary in the both organisms as shown in Fig. 4; i.e. 0.59 ± 0.14 (mean ±standard deviation). It indicates that the TC values should be selected in individual coexpression modules in terms of tight intra-modular connections. The TC peak in Fig. 4(i.e. 0.59) are similar to a single TC value in the network that was selected in previous
reports. Saito et al. [13] used 0.6 as a single threshold cutoff for the coefficient. Aoki etal. [12] mentioned that, for TC ranging from 0.55 to 0.66, the density of the entire
network displays a minimal value, i.e. that the number of false-positive gene-to-genelinks is minimized within the range. To minimize such false-positive links, Gupta et al.
[19] optimized the average clustering coefficient value in a well-controlled dataset anddetermined a single TC value of 0.9 (the square of Pearson correlation coefficient). The
broadened distribution of TC values in optimized co-expression modules (Fig. 4)
suggests that TC values should be selected for each module when using the vast numberof microarray datasets, including various experimental designs.

124 Y. Ogata et al.

Table 1: Co-expression modules with high precision values to the KEGG pathways.
Genes in a
coexpression
module (A)
115
22
25
16

Genes in a
specific
KEGG
pathway
(B)
112
21
19
11

Precision(B / A)

KEGG pathway namePathway ID

One of SVs

Ribosome
ath03010
At2g17360
Proteasome
ath03050
At3g22630
Oxdative phosphorylation
ath00190
At1g01050
DNA replication
ath03030
At4g02060
Phenylpropanoid
15
8
0.53
ath00940
At5g42590
biosynthesis
10
7
0.70
Fatty acid biosynthesis
ath00061
At1g62640
107
0.70
Photosynthesis
ath00195
At4g12800
Alpha-linolenic acid540.80
ath00592
At1g17420
metabolism
Starch and sucrose
740.57
ath00500
At2g36390
metabolism
Valine, leucine and331.00
ath00290
At1g31180
isoleucine biosynthesis
33
1.00
Flavonoid biosynthesis
ath00941
At3g51240
Phosphatidylinositol
ath04070
At2g41210
3
3
1.00
signalling system
SNARE interactions in
ath04130
At5g58060
331.00
vesicular transport
Valine, leucine andath00280
At1g50110
43
0.75
isoleucine degradation
Phenylalanine, tyrosine and430.75
ath00400
At1g48860
tryptophan biosynthesis
43
0.75
Methane metabolism
ath00680
At1g22440
430.75
Mismatch repair
ath03430
At4g02460
53
0.60
Glycolysis / Gluconeogenesis
ath00010
At3g52930
Carbon fixation in
ath00710
At1g63290
530.60
photosynthetic organisms
Co-expression modules that show precision values of 0.5 or higher and comprise 3 or more genes are listed.

3.

0.97
0.95
0.76
0.69

Discussion

Our algorithm assembled network modules that are well-assigned to biologicalknowledge such as metabolic pathways. Especially, co-expression modules involved in‘ribosome’, ‘oxidative phosphorylation’, ‘proteasome’, and ‘phenylpropanoid
biosynthesis’ showed high (> 0.9) precision values of the module members to thepathways or complexes (Table 1), suggesting that such pathways and functional groups
are well co-expressed. Li [17] and Wei et al. [10] reported associations between
metabolic pathways and co-expression. However, many co-expression modules showlow precision values to the pathways because of the following possible causations. First,
all of metabolic pathways are not necessarily co-expressed, possibly due to regulatory
mechanisms other than transcriptional regulation such as post-transcriptional regulation,
as mentioned by Saito et al. [13]. Second, the paucity of information on the assignmentof enzyme genes to metabolic pathways may cause a low rate of assignment of enzyme

Local Module Prediction in a Co-Expression Network 125
genes included in co-expression modules. Third, co-expression relationships are notnecessarily equivalent to direct regulatory relationships. Co-expression network analysis
is just pre-screening for reconstructing regulatory networks. Furthermore information on
the assignment to biological pathways may lead to a reduction of co-expression modules
with low assignment to the pathways.
Co-expression modules with optimized NC values show broadened distribution of
TC values throughout all modules. Such distribution may arise from biases included in
combined datasets, e.g. a dataset including microarray data of different experimental
designs for different purposes. In the analyzed expression dataset of Arabidopsis genes,
which is composed of 1388 microarray data, includes 275 assay data (approximately
20 %) from leaves and 9 assay data (< 1 %) from pollen, indicating that expression
similarity in leaves between genes may have a strong influence on their gene-to-genecorrelation coefficients and vice versa in pollen. Under such influence, a single threshold
cutoff of the coefficient may by adequate for limited co-expression modules. Despite of
the heterogeneity of a gene expression dataset, Obayashi et al. [27] reported that theapplicability of such dataset in co-expression analysis.
Our algorithm can be applied to any sizes of datasets comprising vertex-to-vertex
correlation data. The algorithm starts with the association of a single vertex, such as
hierarchical clustering, and shows constant and processible computational loads throughthe implementation. In contrast, many publicly available tools for clustering approachsuch as ARACNE [20] and DP-Clus [21] start with the matrix operation of the wholedataset and thus depend upon the size of a dataset for their steady implementation.
Our algorithm is an approximate approach for optimizing an NC value for a network
module. In its application to Arabidopsis gene expression datasets, it occasionally causes
different memberships in co-expression modules involved in the common biologicalevent, indicating the insufficient maximization of NC values. The real maximization for agene cluster including an SV, from among 22 263 genes, requires verification of all
possible combinations of genes including the SV. The number of such combination,
however, reaches astronomical levels in the following equation.
N

 C
N

i

(6)

i 3

In Eq. (6), N represents the number of genome-wide genes. Using our algorithm, thenumber of such combinations is reduced into N2-order as follows.

N  2  N  1
2

(7)

Although our algorithm may provide some modules with insufficient maximization ofNC, it provided the NC values (0.76 on average) better than those in ARACNE (0.33)
and DP-Clus (0.48).

126 Y. Ogata et al.
Additionally, a gene may be included in multiple co-expression modules through ouralgorithm, indicating the possibility that a gene is associated with multiple biologicalprocesses.
4.

Conclusions

Our algorithm can be applied to vertex-to-vertex correlation datasets, based on which a
network including modular structure is constructed, such as gene expression datasets,
irrespective the size of the dataset.
Acknowledgments
This work was supported by the New Energy and Industrial Technology Development
Organization (NEDO) program, which is part of the “Development of Fundamental
Technologies for Controlling the Material Production Process of Plants” project.
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