A crucial step toward understanding cellular systems properties is to analyze the topology of biological networks and biochemical progress in cells. Many graphic features are purposed to measure the role of proteins and identify local modularity structures of high connectivity in a PPI network. Laplacian matrix is a matrix representation of a given network. Here, we proposed the modularity structure matrix (MS-matrix), which is the pseudoinverse of the Laplacian matrix for describing the kernels on a graph, to evaluate the modularity structure properties of a PPI network. According to our knowledge, the modularity structure property is the first property to identify both global important proteins and local modularity structures within a network. For a given PPI network of S. cerevisiae, our results demonstrate that the important proteins identified by the MS-matrix are related to the essential biological processes (i.e. essential genes) and highly consistence with the topology features (i.e. degree, closeness centrality, and betweenness centrality). Then, the relationship between proteins derived from the MS-matrix could reflect the similarity of Gene Ontology and could be useful for the module identification. Furthermore, biological characterization (e.g. Gene Onotology) of the modules derived from the MS-matrix is similar to the modules collected from the experiment database (e.g. MIPS). Our results demonstrate that the MS-matrix would provide the insight for investigating a PPI network through important proteins and local modularity structures.
5-1. Introduction
A crucial step toward understanding cellular systems properties is to analyze the topology
interaction (PPI) network as completely as possible, genome-scale interaction discovery approaches, such as high-throughput yeast two-hybrid screening 25,26 and coaffinity purification
27 , have been proposed. Because of the complexity of a PPI network, many graphic features (e.g. degree, closeness centrality, and betweenness centrality) are purposed to measure the role of proteins in a PPI network 115. In addition, several agglomerative algorithmic approaches
116,117
have been developed to identify local modularity structures of high connectivity with relatively low connectivity to the rest of network. These dense sub-graphs are treated as potential functional modules.
In the mathematical and computational field of graph theory, the Laplacian matrix (or Kirchhoff matrix) is a matrix representation of a graph. In addition, the pseudoinverse of the Laplacian matrix plays a key role, has a nice interpretation in terms of random walk on a graph, and defines the kernels on a graph 118. Its application on biological field, the Gaussian network model has succeeded in describing the local modularity structures (e.g. flexible/rigid regions and domains of proteins) and the important residues of a given protein 119,120. However, a PPI network, which has the functional local modularity structures (i.e. module and complex) and the important hubs, is similar to the behaviors of a protein.
To address these issues, we proposed the MS-matrix to evaluate the modularity structure property within a PPI network. According to our knowledge, the MS-matrix is the first property to identify both global important proteins and local modularity structures within a network. For a given PPI network of S. cerevisiae, our results demonstrate that the important proteins identified by the MS-matrix are related to the essential biological processes (i.e. essential genes). In addition, the important proteins derived from MS-matrix are highly consistence with the topology features (i.e. degree, closeness centrality, and betweenness centrality). Then, the relationship between proteins derived from the MS-matrix could reflect the similarity of Gene Ontology and could be useful for the module identification. Furthermore, biological
characterization (e.g. Gene Onotology) of the modules derived from the MS-matrix is similar to the modules collected from the experiment database (e.g. MIPS). Our results demonstrate that the MS-matrix would provide the insight for investigating a PPI network through important proteins and local modularity structures.
5-2. Methods
Modularity structure matrix
Figure 5-1. The overview of the evaluating the importance of each node in a simple network through the
"MS-matrix"
(A) A simple network with three local density regions (red, blue and green nodes). (B) Laplacian matrix of the simple network. (C) MS-matrix is derived from the pseudo-inverse of Laplacian matrix.
Here, we consider a PPI network as an undirected graph. The Laplacian matrix is a matrix representation of a graph. Here, we use a simple network (Fig. 5-1A) with 17 proteins to
matrix M (Fig. 5-1B) for the network. The Mij is given as
Mij = {
−1, if i ≠ j and protein i interacts with protein j 0, if i ≠ j and protein i not interact with protein j
k, if i = j, k is the degree of protein i (1)
For example, the degree of node 8 is 3 (interacting with node 6, 9, and 14); and the M8 8, M6 8, M8 9, and M8 14 are 4, -1, -1, and -1, respectively. Then, the MS-matrix (MS) (Fig. 5-1C) is the pseudoinverse of Laplacian matrix M. Here, we got the pseudoinverse of Laplacian matrix based on the Scientific Tools for Python (SciPY).
According to the local modularity structure (MSij), these 17 proteins in this matrix MS can be clustered into three local modularity structures matching with the original network (red, blue and green regions). Additionally, the three lowest diagonal values (nodes 6, 9 and 14) of MS-matrix (MSii) are the centrality nodes; conversely, two highest values (nodes 7 and 13) of MSii are the peripheral nodes. These results are highly consistent with the graphic features, such as degree, closeness and betweenness centrality (Table 5-1).
Table 5-1. The degree, clustering coefficient, closeness centrality, betweenness centrality, and dynamic property of each node in the simple network (Fig. 5-1)
ID Degree clustering
coefficient
closeness centrality
betweenness
centrality Qii
1 3 0.667 0.421 0.004 0.52
2 3 0.667 0.421 0.004 0.52
3 3 0.667 0.421 0.004 0.52
4 3 0.667 0.421 0.004 0.52
5 3 0.667 0.421 0.004 0.52
6 9 0.222 0.64 0.563 0.183
7 1 0 0.4 0 1.066
8 3 1 0.516 0 0.36
9 6 0.4 0.593 0.4 0.242
10 3 1 0.41 0 0.595
11 3 1 0.41 0 0.595
12 4 0.5 0.421 0.125 0.566
13 1 0 0.302 0 1.448
14 5 0.3 0.571 0.329 0.272
15 2 0 0.39 0.058 0.845
16 2 0 0.39 0.058 0.845
17 2 0 0.296 0.004 1.036
Centrality properties
Here, we introduce two measures of centrality determining the relative importance of a node within a network. The betweenness centrality Cb(i) measures the node centrality in a network by computing the number of the shortest paths from all nodes to all others that pass through the node i. Cb(i) is defined as follows:
Cb(i) = ∑s≠i≠t(σst(i) σ⁄ st) (2)
where s and t are nodes different from i, σst denotes the number of shortest paths from s to t, and σst (i) is the number of the shortest paths from s to t that i lies on. The betweenness value of the node i is normalized by dividing by the number of node pairs excluding i: (N-1)(N-2)/2, where N is the total number of nodes in the paths that i belongs to.
The closeness centrality Cc(i) of a node i is defined as the reciprocal of the average shortest path length and is computed as follows:
Cc(i) = 1 avg(L(i, m))⁄ (3)
where L(i,m) is the length of the shortest path between two nodes n and m. The closeness centrality of each node is a value between 0 and 1.
The modular similarity between protein pair
The non-diagonal value of MS-matrix (MSij) could provide the relationship between related modularity properties of protein i and j. For a given protein A, we could identify the overall MSAi of A and all proteins to evaluate overall modularity relationships. Therefore, we are able to identify the similarity between a protein pair (A and B) based on the overall MSAi
and MSBi. Here, the similarity is evaluated by the Pearson correlation coefficient (r) and
computed as follows:
r(A, B) = ∑nk=1(MSAk−MSA)(MSBk−MSB)
√∑nk=1(MSAk−MSA)2√∑nk=1(MSBk−MSB)2
(4)
where 𝑀𝑆𝐴 and 𝑀𝑆𝐵 are the averages of MSAk and MSBk, respectively.
For example, the 𝑟(5,6) between nodes 5 and 6 located in the same region (red part in Fig. 5-1A) is 0.88. On the contrary, the r between nodes 6 and 9 which are in the different region (red and blue) is -0.53.
The protein-protein interaction network of S. cerevisiae
The high-through put data usually have the non-reliable protein-protein interactions. To construct a high-quality protein interaction yeast, we collected protein-protein interaction data from the core subset (named DIPc) of the DIP database 9 which consists of 1,882 proteins and 4,104 protein-protein interactions (the version dated 10 October 2010). Here, the DIPc consists of only the most reliable interactions 121.
Data set of module of S. cerevisiae
To evaluate reliability of modules which are identified through the MS-matrix, we collected a positive set of yeast module derived MIPS 85. For 193 modules derived MIPS, we selected 160 modules which have more than a half of proteins in the network constructed by DIPc. According to the definitions of module from the previous studies 84,122,123, a module should have a higher connectivity. Here, the connectivity is defined by previous study 124 and calculated as follow:
connectivity =No.of PPI within a module
k×(k−1) (11)
where, k is the number of protein within a module. Finally, we defined a golden positive dataset which includes 69 MIPS modules, which connectivity is more than 0.6.
5-3. Results
The diagonal value of MS-matrix infers essential genes in PPI network of S. cerevisiae
Essential genes usually involve in the fundamental cellular processes which required for the survival of an organism 96,97,125. As a result, the proteins which are products of essential genes should play an important role in the protein-protein interaction network of an organism.
To further investigate the relationship between essential genes and important proteins detected by the diagonal values of MS-matrix (MSii), we constructed the yeast protein interaction network by using the high-quality protein-protein interaction data extracting from the core sub set in DIP database (named DIPc). Figure 5-2 displays the progressive ration of essential protein for MSii from 0 to corresponding value. There are approximately one-half of the proteins recorded as essential proteins while whose MSii values are less than 0.2; and the proportion of essential protein decreases with the increasing value of MSii. Furthermore, YBR160W (main cell cycle cyclin-dependent kinase 126) and YJR045C (Hsp70 family ATPase
127) are the proteins with lowest value of MSii, are recorded as essential genes, and play a key role in the important biological processes (e.g. cell cycle and protein folding). These two proteins have enriched interactions and locate on the center of the network. On the contrary, YGL001C and YLR100W, which are related to a non-essential process (ERGosterol biosynthesis), have highest value and only one interaction in the network. These results suggest that those proteins with lower MSii are located within the steadier regions among the network and more critical for the survival of an organism.
Figure 5-2. The relationship between importance of protein and essential proteins
The importance of protein is calculated by MS-matrix diagonal value (MSii). The interval of MSii denotes the progressive ratios of essential proteins; the lower MSii value, more essential proteins are among the network.
The characterization and quantification of network topology derived from the diagonal value of MS-matrix
For a given network, there are various types of measurement for determining the relative importance of a node (protein) within a network. For example, degree (degree centrality) is defined as the number of links incident upon a node. According to the degree distribution, P(k), a network could be identified as a scale-free network, which is the architecture of many cellular networks 94. Closeness centrality is defined as the inverse of the average shortest paths of a given node. The average shortest paths can be regarded as a measure of how fast it will take to spread information from a node to all other nodes sequentially 128. The betweenness represents the fraction of all of the shortest paths between all nodes in a network that pass through a given node 115.
30 35 40 45 50
0 <0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 >1.8
Es sen tial p roet ins (%)
MS-matrix diagonal value (MS
ii)
Our experimental result confirms that the MSii could represent the essential gene within the yeast PPI network. Next, we evaluated the relationship between MSii and relative importance (i.e. degree, closeness centrality, and betweenness centrality) of protein within a PPI network (Fig. 5-3). Although the Pearson's correlation coefficient (r) between degree and MSii is only -0.50, the Spearman correlation (s) is -0.85. This result implicates that the relative importance detected by the older of MSii is related to the older of relative importance detected by the degree. For example, the protein with the lowest MSii, YBR160W (main cell cycle cyclin-dependent kinase 126), is also the node with highest degree (58). Furthermore, the r between closeness centrality and MSii is -0.78. For example, according to the network described in Figure 5-1, the node 8 is relative important by closeness centrality (0.52; top 4) and could also be identified by using MSii. In addition, the MSii is slightly similar (r=-0.3 and s=-0.70) to the betweennes centrality.
Figure 5-3. Evaluation importance of protein by (A) Degree centrality (B) Closeness centrality (C) Between centrality
(A) The Spearman correlation between degree centrality and MSii is -0.85. (B) The Pearson correlation between closeness centrality and MSii is -0.78. (C) The Spearman correlation between betweenness centrality and MSii is -0.70.
The non-diagonal value of MS-matrix reflects the relationship between proteins in yeast PPI network
Rank of MS-matrix diagonal value (MSii)
A B C
network, we utilize the similarity of Gene Ontology36 and distance of a given protein pair (i and j) to evaluate the MSij. The similarity of Gene Ontology is detected by the relative specificity similarity (RSS), proposed by Wu et al. 122, to measure the biological process, molecular function, and cellular component similarities.
For a given protein A, we could identify the overall MSAi of A and all proteins to evaluate overall modularity structure relationships. Therefore, we are able to identify the similarity of overall modularity relationships between protein pair (A and B) based on the MSAi and MSBi. Here, the similarity between A and B is evaluated by the Pearson correlation coefficient and derived from the equation (4).
Figure 5-4. The distribution of gene ontology similarities (i.e. RSS of BP, CC, and MF) and the shortest path between protein pairs under different modular similarity
The RSS-BP and RSS-MF have the highest value while modular similarity is more than 0.9; moreover, the average distance is lower than 2. The RSS-CC are higher than 0.7 while modular similarities are higher than 0.4.
Figure 5-4 illustrates the distribution of gene ontology similarities and the shortest path between protein pairs. While the protein pairs have ≥0.1 modular similarity, the average of
0 1 2 3 4 5 6 7
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Ave. shortest path
RSS of gene ontology
Pearson's correlation of MS-matrix value
BP CC MF Distance
their distance has an obvious decrease (from 4.88 to 3.48) and all of the average RSS have an obvious increase. In addition, the average of protein pair’s distance would be less than 2.5 and share the higher biological process and cellular component annotation (RSS-BP > 0.7 and RSS-CC > 0.8 ), while these protein pairs have more than 0.4 modular similarity. The RSS-BP and RSS-MF have the highest value while modular similarity is more than 0.9; moreover, the average distance is lower than 2. This result implies that a protein pair with a highly modular similarity would share a significant similarity of Gene Ontology, especially BP and CC, and are neighboring proteins (e.g. an interaction protein pair) in the PPI network.
Identification of modules based on the non-diagonal value of MS-matrix
According to the definitions of module from the previous studies 84,122,123, the proteins of a module should locate on the same component, join a same biological process, carry out similar or related function, and have relatively autonomous of the whole network. We have introduced that the modular similarity of protein pair (A and B) derived from the Pearson correlation coefficient of MSAi and MSBi, could infer the similarity of Gene Ontology and the relationship between A and B within the PPI network. Therefore, we believe that the MS-matrix could be useful for identifying modules of a give PPI network. Here, we utilize the hierarchical clustering method to identify the modules and the distance between protein pair (A and B) is calculated by using the modular similarity (i.e. Pearson correlation coefficient of MSAi and MSBi). Then, we identified 126 modules including 724 proteins derived from the MS-matrix. To further investigate the reliability of modules, we compare our modules with the modules recorded in MIPS and analysis the Gene Ontology and connectivity of our modules.
For 193 modules derived MIPS, we selected 160 modules which have more than a half of proteins in the network constructed by DIPc. According to the definitions of module from the
previous studies 84,122,123, a module should have a higher connectivity. Finally, we defined a golden positive dataset which includes 69 MIPS modules, which connectivity is more than 0.6.
The overlap between a reference MIPS module R and a predicted module M can be quantified by Jaccard index 129. The Jaccard index is calculated as follow:
Jaccard index = |R∩M|
|R∪M| (12)
where, the |𝑅 ∩ 𝑀| is the number of protein which is the intersection of R and M; the
|𝑅 ∪ 𝑀| is the number of protein which is the union of R and M.
For each reference module, we find the prediction that has the highest Jaccard index. Total 47 modules are related to our modules (Jaccard index > 0). If a module with Jaccard index ≥ 0.5 is considered as a hit module, our method has 36 (52%) hits of golden positive dataset.
Next, because modules have relatively autonomous of the whole network, the connectivity of modules should be higher than the proteins which include the module and the proteins connecting to the module (named "extent 1 layer"). Table 5-2 shows the connectivity of our module, 160 MIPS module, and golden positive dataset. Because the 160 MIPS modules are only filtered by number of protein within PPI network, these 160 MIPS have a lower connectivity. In addition, both of our modules and the golden positive dataset have a higher average connectivity (i.e. 0.73 and 0.84, respectively). The average connectivity of all set would have an obvious decreasing from modules to the extent 1 layer. In addition, all modules derived from MS-matrix and golden positive dataset have a higher connectivity than the extent 1 layer (Table 5-2).
Table 5-2. Connectivity of module and proteins which include the module and the proteins connecting to the
module (named "extent 1 layer") Module
Set
No. of Module
Average connectivity
Average connectivity of extent 1 layer
No. of module which connectivity >
connectivity of extent 1 layer
Our 126 0.73 0.32 126
MIPS 160 0.49 0.18 150
Golden
positive 69 0.84 0.28 69
Furthermore, we annotated modules by utilizing the consensus GO terms within a given module. To annotate a module with Y proteins, we define a consensus ratio (CRM) of GO term i as CRM=Yi/Y, where Yi is the number of proteins with GO term i in a module. Next, the enrichment for each module in each GO term was determined by the p-value of the hypergeometric distribution and then this p-value was adjusted based on Bonferroni correction
130,131
. Here, a GO term is considered as a representative GO term of a module if CRM > 0.6 and adjusted p-value of GO term ≤ 0.05 130,131 based on statistically analysis. Figure 5-5 illustrates the distribution of the number of representative GO term within a given module derived from MS-matrix and MIPS. Then, we applied the two-tailed T-test to further investigate the difference between MS-matrix and MIPS. However, all of the P-values (0.18, 0.30, and 0.13) imply that the number of representative GO term within a given module do not have a significant different between MS-matrix and MIPS. In addition, we also investigate the representative GO terms which have the top 5 ratio in our modules or MIPS modules. The Jaccard index of BP, CC, and MF are 0.67, 0.67, and 1, respectively. This result implies that the biological characterization (i.e. No. of representative GO terms in a module and top 5 terms) of our module derived from the MS-matrix is similar to the MIPS modules which are identified by the experiments.
Figure 5-5. The distribution of the number of gene ontology annotations (i.e. (A)BP, (B)CC, and C(MF) within a
0
Based on the two-tailed T-test between MS-matrix and MIPS, all of gene ontology annotations (i.e. BP, CC and BF) do not have significant different (i.e. P-values are 0.18, 0.30, and 0.13 respectively).
Example of modules derived from the MS-matrix
According to 126 modules including 724 proteins derived from the MS-matrix, Figures 6A and 6B illustrate the 9 modules, which sizes are greater than 10, on the network and their density region on the MS-matrix. Two modules with the lowest average MSii values (0.1 and 0.16) are the 19S proteasome and U4/U6 x U5 tri-snRNP complex (purple and light blue regions in Fig. 5A). The proteasome is a protease that controls diverse processes in eukaryotic cells; and snRNPs are large RNA-protein molecular complexes upon which splicing of pre-mRNA occur. Both of two modules are play essential roles in a yeast PPI network. In addition, two largest modules (19 and 17 proteins) are the F1-F0 ATP synthase and peroxisomes. Then, we use two modules (i.e. anaphase-promoting complex/cyclosome (APC/C) and peroxisomes) as examples to further introduce the module identification derived from the MS-matrix.
Figure 5-6. The modules derived from the MS-matrix
(A) Yeast protein interaction network with 9 colored modules (e.g. F1-F0 ATP synthase (red), 19S proteasome (purple), anaphase-promoting complex/cyclosome (pink), and peroxisome (light green)). (B) The MS-matrix of
19S proteasome F1F0 ATP synthase Peroxisomes
Rab family GTPase
U4/U6 x U5 tri-snRNP complex CCR4-NOT complex
U4/U6 x U5 tri-snRNP complex CCR4-NOT complex