Maximizing Modularity Density for Exploring Modular Organization of Protein Interaction Networks

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1 The Third International Syposiu on Optiization and Systes Biology (OSB 09) Zhangjiajie, China, Septeber 20 22, 2009 Copyright 2009 ORSC & APORC, pp Maxiizing Modularity Density for Exploring Modular Organization of Protein Interaction Networks Shihua Zhang 1, Xueei Ning 2 Chris Ding 3 1 Acadey of Matheatics and Systes Science, CAS, Beijing , China 2 College of Science, Beijing Forestry University, Beijing , China 3 Departent of Coputer Science and Engineering, University of Texas at Arlington Arlington, TX 76019, USA Abstract The purpose of this study is to introduce a new quantitative easure odularity density into the field of bioolecular networks and develop new algoriths for detecting functional odules in protein-protein interaction (PPI) networks. Specifically, we adopt the siulated annealing (SA) to axiize the odularity density and evaluate its efficiency on siulated networks. In order to address the coputational coplexity of SA procedure, we devise a spectral ethod for optiizing the index and apply it to a yeast PPI network. Our analysis of resulted odules suggests that ost of these odules have well biological significance in context of protein coplexes. Coparison with the MCL and the odularity based ethods shows the efficiency of our ethod. Keywords Modular organization; network clustering; odularity density; spectral ethod; protein interaction network 1 Introduction Modularity has been considered to be one of the ain organization principles of biological networks in the past decade years. Biological odules as a critical level of biological hierarchy and relatively independent units play special roles in biological systes [8]. How to uncover odular structures in various biological networks is a basic step for understanding cellular functions and organizational echaniss of biosystes. For exaple, by using the network partition, Zhao et al. (2006) investigated the functional and evolutionary odularity of huan etabolic network fro a topological perspective. One popular class of ethods for dissecting odular structure in the field of general coplex networks is based on optiizing a global quality function called odularity [2, 11] to partition the network into odules. And it has been coprehensively adopted to analyze biological networks [12, 4, 3, 5]. However, it has recently been shown that the resolution of the odularity based ethods is intrinsically liited. It fails to find sall This work is partially supported by the National Natural Science Foundation of China under grant No , No , Innovation Project of Chinese Acadey of Sciences, kjcs-yw-s7 and the Ministry of Science and Technology, China, under Grant No.2006CB zsh@ass.ac.cn

2 362 The 3rd International Syposiu on Optiization and Systes Biology counities in large networks instead, groups of sall counities turn out erged as larger ones [13]. Li et al. (2008) proposed a novel quality function called odularity density (D) which ais to conquer the resolution liit proble in odularity. They have tested it on any kinds of sall networks for illustration but not on large real networks. In this study, we ai to introduce the new quantitative easure odularity density into the odular analysis of bioolecular networks and develop new algoriths for detecting functional odules in protein-protein interaction (PPI) networks. We first adopt the siulated annealing (SA) technique to axiize the odularity density and evaluate its advantages on a suit of siulated networks where the odules are known. In order to conquer the coputational burden of SA procedure, we adopt a spectral k-eans ethod for optiizing the easure and apply it to a yeast PPI network. Our biological analysis of resulted odules suggests that ost of these odules carry distinguished biological significance. We also ake a coparison of our ethod with other two ethods including the popular MCL and odularity based ethods to verify its effectiveness. 2 Materials and Methods 2.1 Definition of odularity and odularity density The popular odularity Q is defined by Newan and Girvan (2004). Briefly, when the nodes of a network are divided into odules, one can copute it as follows: [ ( ) ] l 2 i Q = L di, 2L s=1 where is the nuber of odules, L is the total nuber of edges in the network, l i is the nuber of edges between nodes in odule i, and d i is the total nuber of degrees of the nodes in odule i. The highest Q value of all possible odule separations is called the network odularity. In the past studies, epirical and siulation studies showed that the network partition ethod of axiizing odularity Q (MQ) has good perforance. However, Fortunato and Barthéley (2007) recently pointed out the serious resolution liits of this ethod, and claied that the size of a detected odule depends on the size of the whole network. The ain reason is that the odularity Q does not capture the inforation of the nuber of nodes in a odule, and the choice of partition is highly sensitive to the total nuber of links in the network. In the following, we introduce the so-called odularity density D which was proposed as an alternative easure for describing the odular organization [14]. The characteristic of this easure is that it is related to the density of subgraphs. We first define the average odularity degree of subgraph G i (V i,e i ) as follows: ad(g i ) = aid(g i ) aod(g i ) = 2l i l i n i, where aid(g i ) is the average inner degree of the subgraph G i, which equals to twice the nuber of edges in subgraph G i divided by the nuber n i of nodes in this subgraph. aod(g i ) is the average outer degree of the subgraph G i, which equals to the nuber of edges with one node in the subgraph and the other node outside it divided by the nuber

3 Exploring Modular Organization of Protein Interaction Networks 363 n i of nodes in the subgraph. The intuitive idea is that ad(g i ) should be as large as possible for a valid odule. Then the odularity density D of a partition G 1,,G is defined as the su of all average odularity degree of G i for i = 1,,. In contrast to Q, D can be calculated as follows: D = ad(g i ) = 2l i l i n i. (1) This easure provides a way to deterine if a certain esoscopic description of the graph is accurate in ters of odules. The larger the value of D, the ore accurate a partition is. So the counity detection proble can be viewed as a proble of finding a partition of a network such that its odularity density D is axiized. The search for optial odularity density D is a NP-hard proble due to the fact that the space of possible partitions grows faster than any power of syste size. Moreover, the phenoenon of ultiple resolutions or/and hierarchy of odular structures have been observed in biological networks [16]. The odularity density D can be extended for this ore general case using a tuning paraeter λ as follows [14]: D λ = 2λ(2l i ) 2(1 λ)l i n i (2) where λ is a value ranging fro 0 to 1, and when λ = 0.5, the D 0.5 corresponds to odularity density D. By varying λ, we can detect detailed and hierarchical organization of biological systes. In other words, we can divide the network into large odules and sall odules using a sall λ and a large λ respectively. 2.2 Siulated annealing for axiizing D (MD) In principle, the goal of a odule detection is to find the optial partition with largest odularity Q or odularity density D. Several ethods have been proposed for optiizing Q. Most of the rely on heuristic procedures or approxiate strategies. Here, we eploy the siulated annealing (SA) technique to axiize Q and D to obtain the best deterination of the odules of a network for evaluating. Siulated annealing is a kind of stochastic search technique for optiization probles. It enables one to find low cost configurations without getting trapped in high cost local inia and has any applications in cobinatorial optiization probles. In the searching process, a global paraeter T representing teperature is introduced. When T is high, the syste can explore configurations of high cost while at low T the syste only explores low cost regions. Along with the decrease of T, low cost configurations can be reached step by step by overcoing sall cost barriers. When identifying odules, the objective is to axiize the quantitative indexes (i.e. Q or D), thus, the cost is C = Q or D. At each teperature, we perfor a nuber of rando updates and accept the with probability: { 1, ( if Cf C i p = exp C f C i T where C i (C f ) is the cost before(after) the update. ), if C f > C i (3)

4 364 The 3rd International Syposiu on Optiization and Systes Biology Specific ipleentation detail can be seen in [12]. Note that we add a decision cause to ensure that each potential odule be connected. The one that perfors best consists in isolating the odule fro the rest of the network, and perforing a nested SA, entirely independent of the global one. In using Q and D as fitness functions, the ethod is ore direct than those relying on heuristic procedures. Moreover, SA enables us to carry out an exhaustive search and to iniize the proble of finding sub-optial partitions. We should note that the SA ethod can t scale to very large networks, but it is an efficient evaluation ethod for its exhaustive characteristic. Several efficient ethods for optiizing Q have been proposed, but designing efficient algoriths for optiizing the new easure (D) is still an essential and challenging proble. 2.3 Spectral ethod for axiizing D (SpeMD) Given a network G = (V,E), and denote its vertex set as V, edge set as E and adjacent atrix as A. Given a -partition P, define a corresponding n assignent atrix X = [h 1,h 2,,h ] with h ic = 1 if v i V c, and h ic = 0 otherwise, for 1 c. Observe that since each vertex can only be in one cluster, X1 = 1 n. We can reforulate D in ters of the assignent atrix X as follows: D = = = = 2l i l i n i h T i Ah i (h T i Bh i h T i Ah i) h T i h i 2h T i Ah i h T i Bh i h T i h i h T i (2A B)h i h T i h i (4) where B is the degree atrix. Let h i = h i h i, H = [ h 1, h 2,, h ], note that h T i H T H = I, then, we can obtain D = h T i (2A B) h i = Tr H T (2A B) H. So the proble of axiizing D can then be expressed as: axd = Tr H T (2A B) H s.t. H T H = I h j = δ kl or (5) (6) Fro the standard result in linear algebra, the optial H of the above trace axiization has close relationship with the leading k eigenvectors of 2A B by relaxing H as an arbitrary orthonoral atrix. We can adopt the corresponding spectral algoriths and use the leading k eigenvectors of 2A B to optiize the odularity density D. To obtain the final network partition, we apply the k-eans clustering ethod to cluster eigenvectors. Iportantly, the sae principle can be derived for D λ.

5 Exploring Modular Organization of Protein Interaction Networks The procedure of the algorith Given an upper bound K of the nuber of odules and the adjacent atrix A = (a i j ) n n of a network. The procedure of the algorith is stated straightforward as follows: Spectral apping: 1. Copute the diagonal atrix B = (d ii ), where d ii = k a ik. 2. For the eigenvector atrix U K = [u 1,u 2,,u K ], corresponding to the K largest eigenvalues of 2A B. k-eans: for each value of k, 2 k K 1. For the atrix U k = [u 2,u 3,,u k ] fro the atrix U K. 2. Noralize the rows of U k to unit length using Euclidean distance nor: u i j. j u 2 i j 3. Treat the rows of U k as points in R k and cluster the into k clusters using k-eans or even other clustering ethods. Maxiizing odularity density D or D λ with given λ: Pick the k and the corresponding partition P k that axiizes D or D λ. We should note that this type of spectral clustering technique has been successfully applied to general clustering probles as well as graph clustering probles [10, 17]. Here, we explore the characteristic of odularity density D, and derive a new spectral clustering based ethod for axiize D (D λ ) (SpeMD). And the SpeMD procedure described here can be seen as a particular anner of eploying the standard k-eans algorith on the eleents of the leading k eigenvectors to extract k clusters siultaneously. Convergence and coputational coplexity of the SpeMD procedure are key probles when this ethod is applied to large coplex networks. Fortunately, several strategies can be eployed to iprove these probles. First, we can initialize the k-eans such that the starting centroids are chosen to be as orthogonal as possible [18]. This strategy does not change the tie coplexity, but can iprove the quality of convergence, thus at the sae tie reduce the need for restarting the rando initialization process. Second, several fast techniques for solving eigen syste have been developed and several ethods of k-eans acceleration can also be found in the literature. Based on this type of techniques, for large sparse networks with n, and k n, the SpeMD procedure will scale roughly linearly as a function of the nuber of nodes n [17]. Here we didn t consider these aeliorative techniques and only focus on the validity of the SpeMD ethod. 2.5 Perforance easures All perforance easures can be seen in the extended version. 3 Results In this section, we apply the present ethod to a suit of siulated networks and a yeast PPI network to test its efficiency. We first present detailed nuerical results to show the difference of network partition deterined by axiizing the odularity density D and odularity Q with siulated annealing (SA) technique. In general, axiizing D (MD) can give ore detailed and valid results, while axiizing Q (MQ) encounters serious resolution liit in siulated networks.

6 366 The 3rd International Syposiu on Optiization and Systes Biology Then we apply the new spectral ethod for axiizing the generalized D λ (SpeMD) to a yeast PPI network to identify functional odules which show significant biological relevance. Coparison with MQ and MCL, we show that the SpeMD can obtain copetitive perforance with the well-known MCL ethod and resolve uch finer odular structure than MQ ethod. To extract appropriate odules, the SpeMD and MCL both rely on one paraeter. Here, we perfor the SpeMD and MCL with adjusted paraeters to obtain the best geoetric accuracy and separation. For SpeMD, we tune λ fro 0.4 to 0.7 in step of 0.05, and for MCL, we saple inflation paraeter values fro 1.5 to 2 in steps of Siulated networks First we do the coprehensive tests on a group of siulated networks which take on significant odular characteristics. In the work of [14], D-based ethod has been showed to be able to obtain copetitive perforance with Q-based ethod. However, the size of artificial networks generated by using Newan s popular procedure as well as its variant are too sall to show the serious resolution liit proble of Q. Therefore, we devise a new type of artificial networks. The network is coposed of 2 cliques ( n 1 -clique and n 2 -clique), and external edges are placed randoly with a fixed expectation values so as to keep the average edge connections k out of each node to nodes of other cliques. So each network has (n 1 + n 2 ) nodes and about (n 1 (n 1 1)/2 + n 2 (n 2 1)/2) + (n 1 + n 2 )k out /2 edges. In the following test, we let n 1 = 10 and n 2 = 15. Note that we can also relax cliques as dense odules for testing, but here we just show the clique case for convenience. Precision MQ MD Nuber of odules MQ MD Precision kout kout Nuber of odules kout kout Figure 1: Left figure: Coparative test of MD and MQ on siulated networks with known counity structures. It is a plot of the fraction of nodes correctly classified with respect to k out. Each point is an average over 50 realizations of the networks. Right figure: Nuber of odules detected by MD and MQ with the real nuber of cliques (NC), averaged over 50 network realizations. The coputational results for this experient are suarized in Fig. 1, where NC is the nuber of cliques, i.e., NC = 2. The left plot of Fig. 1 shows the fraction of nodes that are correctly classified into the counities (Precision) with respect to k out by MD and MQ respectively. We can see that MD ethod based on D-value perfors

7 Exploring Modular Organization of Protein Interaction Networks 367 uch better than MQ ethod under all the different NC. For instance, for 50 rando networks with NC = 60 and k out = 5, on an average 99.97% nodes are classified correctly by MD, while only about 72.23% nodes by the MQ. When k out = 8 which indicates the corresponding networks are difficult to be partitioned, MD still has very high accuracy (>86%). The ost interesting observation is that perforance of MD is alost the sae, while that of MQ is greatly decreasing with the increase of NC (also the size of networks). For exaple, for 50 rando networks with k out = 6, always on an average >99.9% nodes are classified correctly by MD on four different sizes of networks with NC = 20, 40, 60, 80, while about 92.40%, 78.18%, 69.75%, 62.59% nodes by the MQ respectively. This fact shows the serious resolution liit proble of odularity Q, while that can not be observed on the sall networks such as the siulated networks using Newan s ethod. To test the perforance of MD and MQ in selecting the nuber of counities, we calculate the nuber of odules. The right plot of Fig. 1 shows the averaged nuber of odules on four different sizes of networks (NC = 20, 40, 60, 80) with respect to k out by MD and MQ respectively. We can see that MD perfors uch better than MQ. The MD can alost always identify the right nuber of odules in four different sizes of networks with k out 7. While MQ can not do that. For exaple, for 50 rando networks with NC = 60 and k out = 7, on an average 59.7 odules are identified by MD, while only about odules by the MQ. For the harder case (k out = 8), MD can sill do uch better than MQ. Actually, even for the easiest case k out = 2, MQ can not identify the right odules with odules for NC=80. This uncovers the underlying resolution liit just as pointed in [13]. In suary, the MD can recover the underlying counity structure ore often than the MQ by a sizable argin in the siulated odular networks. The odularity density D ore relies on local connectivity of a network and can uncover finer odular structure. While odularity Q ore relies on size and total links of a network and can lead to serious resolution liit. Moreover, the liit is ore serious as size of networks increasing. 3.2 Results on a PPI network The budding yeast S. cerevisiae PPI network was obtained fro the DIP database ( which contains huan-curated high-throughput and sall-scale binary interactions directly observed in experients, as well as binary interactions inferred fro high-confidence protein coplex data. We only considered non-self physical interactions and built the PPI network. The giant coponent of the PPI network is coposed of 2559 proteins linked by 7031 nonredundant interactions. In order to test the ability of SpeMD to extract coplexes fro the interaction network and copare it with other two ethods, we copared the detected odules to known coplexes in yeast as annotated by the Munich Inforation Center for Protein Sequences (MIPS) [15] using the P ol forula. We apply the SpeMD ethod to the yeast PPI network to detect functional odules. Totally, we obtain 279 protein odules of sizes fro 4 to 38 with λ = 0.6 (To extract statistically and biologically significant odules, we reove 48 odules with size 3). A coplete list of coplexes and odules with functional annotation is provided in Suppleentary Files. Figure 2 presents three such odules. For exaple, the second one

8 368 The 3rd International Syposiu on Optiization and Systes Biology YGR098C YDR311W YDR460W YPR056W YLR005W YDL108W YPR025C Figure 2: Exaples of odules which atch the MIPS coplexes with great significance. (A) A seven-eber odule atches with the SSL2-core TFIIH coplex when it is part of the nucleotide-excision repair factor 3 (NEF3) with (P ol = ). (B) A nine-eber odule atches with Golgi transport coplex which stiulates intra-golgi transport and is coposed of eight proteins (P ol = ). is a nine-eber odule which atches with Golgi transport coplex for stiulating intra-golgi transport with P ol = Coparison with MCL and MQ There has been any ethods for detecting network odules. The coparison of all the ethods are not an easy task. Here, we attept to copare the MD (SpeMD) with two types of classical ethods: MQ and MCL. Just as we have entioned, the odularity (Q) axiization based odule-detection ethod has been coprehensively applied in any fields including analysis of biological networks. Another ethod is the Markov Cluster algorith (MCL) which was developed by van Dongen [6]. The ethod siulates a flow on the network by calculating successive powers of the network adjacency atrix. In each iteration, an inflation step is applied to enhance the contrast between regions of strong or weak flow in the network. The process converges towards a partition of the network, with a set of high-flow regions separated by boundaries with no flow. The value of the inflation paraeter strongly influences the the size and nuber of the resulted odules. The odule size distribution of detected odules for each ethod on the PPI network have been shown in the left plot of Figure 3. The SpeMD and MCL both identify about (279 and 242) odules without extreely large clusters. The ajor trend generated by MD and MCL are both siilar to that of the coplexes in MIPS database, which suggest the definition of odularity density is reasonable (Note that the MIPS coplex is a cobination of hand-curated and experiental coplexes. They have soe overlap, so coplex curve is higher. But the trend is siilar). Unfortunately, the odule size distribution of MQ is very different fro the previous ones. The MQ only detect 21 odules with relative large size raging fro 39 to 263. As tested on the siulated networks, the MQ ethod is highly liited by the resolution proble. As to biological significance, the accuracy and separation are used for evaluating the correspondence between coplexes and odules fro each ethods [1]. Fro the right plot of Figure 3, we can easily see that the SpeMD and MCL have consistently better perforance than MQ. This eans the odularity density based partition ethod can

9 Exploring Modular Organization of Protein Interaction Networks 369 Nuber of odules MIPS MCL SpeMD MQ Accuracy Cluster wise separation Coplex wise separation Clustering wise separation Module size MQ MCL SpeMD Figure 3: Left figure: Module size distribution of different ethods and MIPS protein coplex with size > 3. Right figure: Perforance of different ethods on the PPI network. Four different easures including Accuracy, Cluster-wise separation, Coplex-wise separation, and Clustering-wise separation have been used. produce ore biologically significant odules than the odularity based ethod. And the new quality function ay becoe an evaluation index of odularity organization of networks. While MCL has no such evaluation function. 4 Discussion and Conclusion In suary, our ethod is very effective for uncovering odular organization in bioolecular networks. It provides an objective approach to explore the organization and interactions of biological processes. With the increasing aount of biological interaction data available, MD (SpeMD) can facilitate the construction of a ore coplete view of the coposition and interconnection of functional odules and the understanding of the organization of the whole cellular at syste level. We plan to autoate this algorith to copute functional odules for a large nuber of biological networks. We hope that related studies will benefit fro the present ethod coupled with the odularity density D. References [1] Broheé, S. and van Helden, J. (2006) Evaluation of clustering algoriths for protein-protein interaction networks. BMC Bioinforatics, 7, 488. [2] Newan, M.E. and Girvan, M. (2004) Finding and evaluating counity structure in networks. Phys. Rev. E 69, [3] Wang, Z. and Zhang, J. (2007) In search of the biological significance of odular structures in protein networks. PLoS Coput. Biol. 3, e107. [4] Caretta-Cartozo, C., De Los Rios, P., Piazza, F., et al. (2007) Bottleneck Genes and Counity Structure in the Cell Cycle Network of S. pobe. PLoS Coput. Biol. 3(6), e103. [5] Zhao, J., Yu, H., Luo, J.H., Cao, Z.W., Li, Y.X., (2006) Hierarchical odularity of nested bow-ties in etabolic networks. BMC Bioinforatics 7, 386. [6] van Dongen, S. Graph clustering by flow siulation. In PhD thesis Centers for atheatics and coputer science (CWI), University of Utrecht; [7] Friedel, C.C., Krusiek, J., Zier, R. (2008) Bootstrapping the interactoe: unsupervised identification of protein coplexes in yeast. RECOMB 2008, 4955, 3-16.

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