 Methodology article
 Open Access
 Published:
Modifying the DPClus algorithm for identifying protein complexes based on new topological structures
BMC Bioinformatics volumeÂ 9, ArticleÂ number:Â 398 (2008)
Abstract
Background
Identification of protein complexes is crucial for understanding principles of cellular organization and functions. As the size of proteinprotein interaction set increases, a general trend is to represent the interactions as a network and to develop effective algorithms to detect significant complexes in such networks.
Results
Based on the study of known complexes in protein networks, this paper proposes a new topological structure for protein complexes, which is a combination of subgraph diameter (or average vertex distance) and subgraph density. Following the approach of that of the previously proposed clustering algorithm DPClus which expands clusters starting from seeded vertices, we present a clustering algorithm IPCA based on the new topological structure for identifying complexes in large protein interaction networks. The algorithm IPCA is applied to the protein interaction network of Sacchromyces cerevisiae and identifies many well known complexes. Experimental results show that the algorithm IPCA recalls more known complexes than previously proposed clustering algorithms, including DPClus, CFinder, LCMA, MCODE, RNSC and STM.
Conclusion
The proposed algorithm based on the new topological structure makes it possible to identify dense subgraphs in protein interaction networks, many of which correspond to known protein complexes. The algorithm is robust to the known high rate of false positives and false negatives in data from highthroughout interaction techniques. The program is available at http://netlab.csu.edu.cn/bioinformatics/limin/IPCA.
Background
In the postgenomic era, one of the most important issues is to systematically analyze and comprehensively understand the topology of biological networks and biochemical progress in cells. Protein complexes can help us to understand certain biological progress and to predict the functions of proteins. As John Donne pointed out, no protein is an island entire of itself or at least, very few proteins are. Most proteins seem to function within complicated cellular pathways, interacting with other proteins either in pairs or as components of larger complexes [1, 2].
Various methods have been used to detect protein complexes. Largescale massspectrometric studies in Saccharomyces cerevisiae provide a compendium of protein complexes that are considered to play a key role in carrying out yeast functionality [3, 4]. Although vastly informative, such methods offer information only on the composition of a protein complex at a given time and developmental or environmental condition [5]. Repeated individual purifications coupled with each of these complexes could offer a more precise picture [6, 7], but such approaches on a large scale are unavailable at present. Affinity purification techniques using mass spectrometry provide a particularly effective approach to identifying protein complexes [1]. These highthroughput techniques have been used to perform large scale proteinprotein interaction screens in the yeast Saccharomyces cerevisiae [3, 4, 8â€“11]. Sharan et al. [12, 13] developed a probabilistic model for protein complexes in a single species and a model for the conservation of complexes between two species. Based on the assumption that proteins in the same pathway are typically present or absent in a genome as a group, Pellegrini et al. [14] detected the conserved complexes across two species. Methods based on integrated multiple information (e.g. functional annotations for proteins, protein structures, gene expression, et al.) have been proposed [5, 15]. Dezso et al. [5] believe that the cellular role and the essentiality of a protein complex may largely be determined by a small group of protein subunits that display a high mRNA coexpression pattern, belong to the same functional class, and share the same deletion phenotype and cellular localization. However, the relation between protein interactions and gene coexpressions may be very complicated with a high rate of false positive in the protein interaction data generated by highthroughput methods.
At present, a general trend is to represent the proteinprotein interactions as a graph and to apply suitable graph algorithms to extract necessary information [16]. There have been several graph clustering approaches proposed to detect protein complexes, including SPC (Super paramagnetic clustering) [17], RNSC (Restricted Neighborhood Search Clustering) [18], MCODE (Molecular Complex Detection) [19, 20], DPClus [16], LCMA (Local Clique Merging Algorithm) [21], CFinder [22], and STM (Signal Transduction System) [23]. For later comparisons of our proposed algorithm with these algorithms, we give a brief description and discussion on each of these algorithms.
SPC [17] is a hierarchical clustering algorithm that simulates a ferromagnetic model with physical properties subject to fluctuation at nonzero temperature. Algorithm SPC identifies vertices belonging to a highly connected subgraph. However, as a disadvantage, SPC is sensitive to noisy data [24]. In fact, to our knowledge, all methods of predicting proteinprotein interactions cannot avoid yielding a nonnegligible amount of noise (false positives).
RNSC [18] is a costbased clustering algorithm, which partitions the vertices of a graph into clusters based on a cost function that is assigned to each partitioning. It starts from an initial random solution and iteratively moves a vertex from one cluster to another to decrease the total cost of clusters. It ends up when some moves have been reached without decreasing the cost function. RNSC is a randomized algorithm and its results depend heavily on the quality of the initial clustering.
MCODE [19, 20] is a densitybased local search algorithm that operates in three stages: vertex weighting, complex prediction, and optionally postprocessing. First, it assigns a weight to each vertex based on its local neighborhood density. Then, it seeds a complex with the highest weighted vertex and recursively moves outward from the seed vertex. A new vertex is added to the complex if its weight is larger than a given threshold. Then, it filters or adds proteins in the clusters by certain connectivity criteria. However, MCODE cannot guarantee that the predicted clusters are highly connected to each other, since the highly weighted vertices may not be highly connected to each other.
Most importantly, SPC, RNSC and MCODE cannot generate overlapping protein complexes, and require that each vertex belong to one specific cluster. In practice, a protein may be involved in multiple complexes and have more than one biological function. For example, in the CYGD database [25], the ratio of the number of proteins in known protein complexes over the sum of the sizes of these complexes is 2750/8932. Therefore, it is practically important to develop algorithms that identify overlapping protein complexes. The DPClus [16] clustering algorithm is based on density and periphery tracking and can detect both nonoverlapping clusters and overlapping clusters. To generate overlapping clusters, DPClus extends the nonoverlapping clusters by adding their neighbors in the original graph (rather than in the remaining graph). It starts at a highest weighted vertex and grows gradually by adding vertices from its neighbors. It uses two parameters, density d_{ k }and cluster property cp_{ nk }. A vertex added to a cluster must satisfy two conditions: 1) its addition does not cause the density d_{ k }of the cluster to fall below a given threshold d_{ in }; and 2) its cp_{ nk }is larger than another given threshold cp_{ in }.
LCMA [21] generates overlapping clusters based on local clique merging. It first locates local cliques for each vertex of the graph then merges the detected local cliques according to their affinity to form maximal dense subgraphs.
CFinder [22] is a tool of detecting overlapping clusters based on the Clique Percolation Method (CPM) [26]. CPM defines a protein complex as a union of all kcliques that can be reached from each other through a series of adjacent kcliques (two kcliques are adjacent if they share exactly k  1 vertices). Results of CFinder are highly correlated to the value of the parameter k. Larger values of k correspond to smaller subgraphs of higher density.
STM [23] models protein interaction networks as dynamic signal transduction systems, and demonstrates the signal transduction behavior of perturbations by proteins statistically. STM allows overlapping of output clusters and identifies clusters of large size, arbitrary shape, and low density. However, unexpected huge clusters may also be generated in its postprocess of merging.
In this paper, we propose a clustering algorithm, which follows the general framework of the algorithm DPClus [16] but is based on a new topological structure of complexes. By a careful study of the structures of known complexes, we discover that most complexes have a very small diameter and a very small average vertex distance. Also observing that vertex distance along would not precisely determine the desired complex structures, we propose a new topological structure of complexes that is the combination of vertex distance and subgraph density. Following the general approach of expanding clusters started with seeded vertices, as what DPClus did, we develop an algorithm IPCA for detecting protein complexes based on the new topological structure. We apply the algorithm IPCA to the protein interaction network of yeast, and identify many wellknown protein complexes. We compare IPCA with the six competing previous methods DPClus, CFinder, LCMA, MCODE, RNSC and STM. The clusters generated by each method are compared to the known protein complexes. The results of the comparisons show that much more experimentally determined complexes are recalled by IPCA than by other six methods. In addition, IPCA is robust against the high rate of false positives and false negatives in the protein interaction networks. Thus, the algorithm IPCA can be used to identify new protein complexes in protein interaction networks of various species and provide references for biologists in their research on protein complexes.
Before we present our algorithm, we would like to discuss the difference between our algorithm IPCA and the previously proposed algorithm DPClus [16]. The algorithm IPCA follows the general approach of cluster expanding based on seeded vertices, as what DPClus did. However, the rules of IPCA for expanding clusters and weighting vertices are somewhat different from that of DPclus especially they target a different topological structure for the resulted clusters. In particular, the algorithm DPClus identifies subgraphs that satisfy a density condition (i.e., d_{ k }) and certain cluster connectivity property (i.e., cp_{ nk }), while the algorithm IPCA looks for subgraph structures that have a small diameter (or a small average vertex distance) and satisfy a different cluster connectivitydensity property (i.e., IN_{ vK }). Also, the algorithm IPCA computes the vertex weights only once, based on the original input graph. On the other hand, once a new cluster is identified, the algorithm DPClus removes the cluster and recomputes the vertex weights based on the remaining subgraph. We believe that our approach is biologically more meaningful: the selection of a seeded vertex for a cluster is based on vertex weights, which should be measured by the original protein network because the cluster is a dense structure in the original network. On the other hand, a remaining subgraph in the process of DPClus may have lost some useful biological information because the algorithm recomputes the vertex weights based on the remaining subgraph. A byproduct of our approach is that our algorithm is more efficient because it avoids the recomputation.
The proposed algorithm
A protein interaction network is represented as an undirected simple graph G(V, E) with proteins as vertices and protein interactions as edges. Previous works [16â€“21, 27] have revealed that protein complexes in a protein interaction network generally correspond to dense regions (dense subgraphs, or simply clusters). Most densitybased clustering algorithms, such as DPClus [16], first generate a seed vertex and extend from the seed vertex by adding new vertices. The performance of such algorithms depends heavily on the quality of the seeds and the criterion of extending, especially the latter.
In this section, we propose a new extending model by analyzing the topology of the complexes in the protein interaction network of Saccharomyces cerevisiae. The protein interaction network of Saccharomyces cerevisiae is downloaded from MIPS (Munich Information Center for Protein Sequences) database [28]. We remove all the selfconnecting interactions and repeated interactions. The final network includes 4546 yeast proteins and 12319 interactions. The average clustering coefficient of the final network is 0.4, the network diameter is 13, and the average vertex distance is 4.42. We also collect from the MIPS database protein complexes annotated for Sacchromyces cerevisiae [28]. There are 216 manually annotated complexes that consist of two or more proteins. The largest complex contains 81 proteins, the smallest complex contains 2 proteins, and the average size of all the complexes is 6.31. For each protein complex, we analyze its topology in the network of Sacchromyces cerevisiae. Of the 216 protein complexes, 118 are connected (a protein complex is connected if there is a path connecting every pair of vertices in the complex). For a connected protein complex, its diameter is defined to be the maximum shortest path length between any pair of vertices in it. A nonconnected protein complex can be divided into connected components. Most of the nonconnected protein complexes can be divided into a large component and one or two separated proteins. For a nonconnected protein complex, we define its diameter to be the maximum diameter over all its connected components. We calculate the diameter and average shortest path length for each of the 118 connected protein complexes and for each of the 98 nonconnected protein complexes.
As shown in Table 1, 94.91% of the connected complexes have their diameter bounded by 2, and 99.15% of the connected complexes have their average shortest path length bounded by 2. There is only one connected complex in which the average shortest path length is larger than 2, which is 2.047. The average shortest path length of all the nonconnected complexes is bounded by 2.5, the longest one is 2.409. 93.88% of the nonconnected complexes have their average shortest path length bounded by 2, and 82.66% of the nonconnected complexes have their diameter bounded by 2. This fact matches the observation that the protein interaction networks have the smallworld property [29, 30]. The analysis on the statistical data shows that the length of the shortest path between each pair of vertices in most of the complexes is bounded by 2. With this important observation, we believe that the length of the diameter and the average length of shortest paths are important topological parameters for detecting protein complexes.
In the following discussion, we denote by SP(K) the diameter of a graph K (i.e., the largest length of a shortest path between a pair of vertices in K), and by ASP(K) the average length of all the shortest paths between each pair of vertices in K. Since the discussions for SP(K) and ASP(K) are similar, our discussion will be mainly focused on SP(K).
As shown in Figure 1, graphs with the same diameter can have very different topologies. To distinguish different topologies of graphs with the same diameter, we need another control parameter. For a dense graph, a vertex is connected to most of the vertices in the graph. On the other hand, in a sparse graph a vertex may be connected to only a few vertices in the graph. We introduce a new concept to measure how strongly a vertex v is connected to a subgraph K: the interaction probability IN_{ vK }of a vertex v to a subgraph K, where v âˆ‰ K, is defined by
where m_{ vK }is the number of edges between the vertex v and K, and n_{ K }is the number of vertices in K. We discuss the relationship between the parameter IN_{ vK }and the two parameters d_{ K }and cp_{ vK }introduced in the algorithm DPClus [16]. According to [16], the density d_{ K }of a subgraph K is defined as d_{ K }= 2m_{ K }/(n_{ K }(n_{ K } 1)), where m_{ K }is the number of edges in the subgraph K, and the cluster property cp_{ vK }of a vertex v with respect to the subgraph K is defined as cp_{ vK }= m_{ vK }/(d_{ K }n_{ K }). By the expressions, our parameter IN_{ vK }is similar to the parameter cp_{ vK }(differing by a factor of d_{ K }). Moreover, the following theorem shows that the parameter IN_{ vK }is also closely related to the density d_{ K }.
Theorem. Let K be a subgraph of a graph G. If for every vertex v in K, we have IN_{ vK' }â‰¥ t_{0}, where K' = K  v and t_{0} is a fixed constant, then d_{ K }â‰¥ t_{0}.
PROOF. By the conditions given in the theorem, for all vertices v in K, we have (where K' = K  v, m_{ vK' }is the number of edges between v and K', and n_{ K' }is the number of vertices in the subgraph K'),
IN_{ vK' }= m_{ vK' }/n_{ K' }â‰¥ t_{0}
Therefore, m_{ vK' }â‰¥ n_{ K' }t_{0}. Add this over all vertices v in K, and note that âˆ‘_{vâˆˆK}m_{ vK' }= 2m_{ K }, where m_{ K }is the total number of edges in K, and that n_{ K' }= n_{ K } 1, we have
This gives d_{ K }= 2m_{ K }/(n_{ K }(n_{ K } 1)) â‰¥ t_{0}, and proves the theorem. â–¡
By the above theorem, a lower bound on the parameter IN_{ vK' }for every vertex v in a subgraph K will also provide a lower bound on the density d_{ K }of the subgraph K. Next we show that the parameter IN_{ vK }can help distinguishing subgraph structures that are indistinguishable by their diameters. Consider the two graphs in Figure 1 again. Although both graphs have diameter 2, for all vertices v in the first graph, the value IN_{ vK' }is 4 = 5; while for five of the six vertices in the second graph, the value IN_{ vK' }is 1/5 (where we define K' = K  v).
Our algorithm IPCA looks for complex structures whose topological structure is controlled by the two parameters SP(K) and IN_{ vK }. More specifically, we look for complex structures whose diameter is controlled by the parameter SP(K) and whose density and cluster property are controlled by the parameter IN_{ vK }.
Definition 1. Let T_{ in }be a threshold ranging between 0 and 1, let d be a positive integer, and let K be a subgraph. A vertex v âˆ‰ K is a (K, T_{ in }, d)vertex if the following two conditions are satisfied (where K + v denotes the subgraph induced by K and v):

1.
IN_{ vK }â‰¥ T_{ in }; and

2.
The (SP â‰¤ d)Version: SP(K + v) â‰¤ d (or The (ASP â‰¤ d)Version: ASP(K + v) â‰¤ d)
Note that there are actually two versions for the definition of a (K, T_{ in }, d)vertex in terms of condition 2: one uses the condition SP(K + v) â‰¤ d (i.e., the (SP â‰¤ d)Version), and the other uses the condition ASP(K + v) â‰¤ d (i.e., the (ASP â‰¤ d)Version).
Our clustering algorithm IPCA that extends clusters based on (K, T_{ in }, d)vertices is given in Figure 2. If the algorithm uses the (SP â‰¤ d)Version in the conditions in the definition, we will say that "the algorithm uses SP â‰¤ d". Similarly, if the algorithm uses the (ASP â‰¤ d)Version in the conditions in the definition, we will say that "the algorithm uses ASP â‰¤ d".
The algorithm IPCA can be divided into four major parts: Weighting Vertex, Selecting Seed, Extending Cluster, and Extendjudgment. The input to the algorithm is an undirected simple graph.
Weighting Vertex
Our procedure of vertex weighting is the same as that of the DPClus algorithm [16]. For an input graph G = (V, E), we assign the weight of an edge [u, v] to be the number of neighbors shared by the vertices u and v. We define the weight of each vertex to be the sum of the weights of its incident edges. After all vertices are assigned weights, however (this is different from DPClus), we also sort in nonincreasing order the vertices by their weights and store them in a queue S_{ q }(vertices of the same weight are ordered in terms of their degrees). The complexity of calculating edge weights and vertex weights is O(VE), and the complexity of sorting all vertices by their weights is O(V log V).
Selecting Seed
The notion that vertex weight is a good measure for selecting seeds has been adopted by DPClus [16] and MCODE [19]. Here, we also pick the highest weighted vertices as the seeds. Our procedure proceeds as follows. We pick the first vertex in the queue S_{ q }and use it as a seed to grow a new cluster. Once the cluster is completed, all vertices in the cluster are removed from the queue S_{ q }and we pick the first vertex remaining in the queue S_{ q }as the seed for the next cluster. There are a number of important differences between this seed selection procedure and the one used in the DPClus algorithm [16]. First, our procedure computes the vertex weight for each vertex only once, based on the original graph; while the DPClus algorithm recomputes the vertex weight for each remaining vertex after a cluster is removed, based on the remaining subgraph. We feel that our approach is biologically more meaningful because a complex is a dense structure in the original protein network thus its seed vertex should be measured in terms of the original graph. On the other hand, a remaining subgraph G' during the process of DPClus may have lost certain biological information (e.g., interactions between the vertices in G' and vertices not in G' have been removed). Computationally, our approach also has the advantage of being more efficient. Moreover, our approach has also helped for generating overlapping complexes. This is because the vertices of the generated clusters are only removed from the queue S_{ q }, but not from the original graph G.
Extending Cluster
A cluster K is extended by adding vertices recursively from its neighbors according to the priority. The priority of a neighbor v of K is determined by the value IN_{ vK }. This procedure is similar to the one proposed in DPClus [16], except that we do not use "finetuning" to sort the neighbors. Whether a high priority vertex v is added to the cluster is determined by the Extendjudgment test that tests if v is a (K, T_{ in }, d)vertex. Only when the candidate vertex v is a (K, T_{ in }, d)vertex, can it be added to the cluster. Once the new vertex v is added to the cluster, the cluster is updated, i.e., the neighbors of the new cluster are reconstructed and the priorities of the neighbors of the new cluster are recalculated, and the algorithm goes recursively with the new cluster. The complexity of generating a candidate vertex from the neighbors of the cluster K is O(n_{ K }n_{N(K)}+ n_{N(K)}log n_{N(K)}), where n_{N(K)}is the number of neighbors of K.
Extendjudgment
Whether a candidate vertex v is added to a cluster K is determined by the two conditions given in Definition 1. First, we calculate the value IN_{ vK }. The vertex will not be added to the cluster if the value IN_{ vK }is less than T_{ in }. If the vertex v passes this test, then depending on whether using SP â‰¤ d or ASP â‰¤ d, the algorithm computes the diameter of the graph K + v or the average length of the shortest paths between pairs of vertices in K + v, and compares the value with the parameter d. If the computed value is bounded by d, then the vertex v is added to the cluster. If the vertex v fails any of these tests, then the next highest priority neighbor of the cluster is tested, and so on. If all neighbors fail the tests, then the cluster cannot be further extended, and a complete cluster is formed whose vertices are removed from the queue S_{ q }. In this paper, d = 2 is used according to our previous analysis. The complexity of testing whether a candidate vertex is added to a cluster is O({n}_{K}^{2}).
We remark that our algorithm IPCA guarantees that no two generated clusters would be the same: a seed vertex v for a new cluster is selected such that v does not belong to any of the previously constructed clusters. In fact, any two clusters constructed by the algorithm IPCA should be expected to be sufficiently different. To see this, let C_{2} be a cluster seeded at v_{2} that is constructed after a cluster C_{1}. If the two clusters C_{1} and C_{2} are largely overlapping, then intuitively, the vertex v_{2} is closely and densely connected to many vertices in C_{1}. Thus, during the construction of the cluster C_{1}, the vertex v_{2} would have a large chance to be included in C_{1} and would have not become a seed for the later cluster C_{2}.
The time complexity of the entire algorithm IPCA depends on the number and the size of predicted clusters. The running time of IPCA is given in the next section.
Results and Discussion
The protein interaction network of Sacchromyces cerevisiae is downloaded from the MIPS database [31]. After removal of all the selfinteractions and repeated interactions, the final network includes 4546 proteins and 12319 interactions. We apply the proposed algorithm IPCA to this network. In the following subsections, we discuss the effect of the value T_{ in }on clustering, compare the predicted clusters with the known complexes, evaluate the significance of the predicted clusters, and analyze the robustness and efficiency of the algorithm IPCA. We will also compare the algorithm IPCA to six competing previous methods for their performance of identifying protein complexes. The comparisons are also performed on protein interaction networks and random networks.
The effect of T_{ in }on clustering
To understand how the value of T_{ in }influences the outcome of the clustering, we generate 18 sets of clusters by using SP â‰¤ 2 and ASP â‰¤ 2 with T_{ in }= 0.1, 0.2,..., 0.9 from the protein interaction network of yeast. The effect on the predicted clusters with different T_{ in }is given in Figure 3. Figure 3(a) shows that the total number of the predicted clusters is increasing as T_{ in }increases. However, in Figure 3(b), there is a abrupt decrease at T_{ in }= 0.5. This is probably caused by the Hub structures in the protein interaction network. When T_{ in }= 0.5, these Hub structures are decomposed into complexes that consist of only 2 proteins.
Figure 3(c) shows that the size of the biggest cluster is decreasing as T_{ in }increases. The same trend of the average size of the predicted clusters is shown in Figure 3(d). With the increasing of T_{ in }, the probability of neighbors added to the cluster is decreasing. Thus, the size of the predicted clusters is also decreasing. From Figure 3, we can see that there is almost no difference for the clusters generated by using SP â‰¤ 2 or by using ASP â‰¤ 2 when T_{ in }â‰¥ 0.5. More and larger clusters are generated by using ASP â‰¤ 2 than by using SP â‰¤ 2 with the same T_{ in }when it is smaller than 0.5.
Comparison with the known complexes
To evaluate the effectiveness of the algorithm IPCA for detecting protein complexes, we compare the predicted clusters produced by the algorithm with known protein complexes in MIPS yeast complex database [32]. There are 216 manually annotated complexes considered as the gold standard data that each consists of two or more proteins. Here, we use the same scoring scheme used in [16, 19] to determine how effectively a predicted cluster (Pc) matches a known complex (Kc). The overlapping score OS(Pc, Kc) between a predicted cluster Pc and a known complex Kc is calculated by the following formula:
where V_{ Pc }âˆ© V_{ Kc } is the size of the intersection set of the predicted cluster and the known complex, V_{ Pc } is the size of the predicted cluster and V_{ Kc } is the size of the known complex. A known complex Kc that has no proteins in a predicted cluster Pc has OS(Pc, Kc) = 0 and a known complex Kc that perfectly matches a predicted cluster Pc has OS(Pc, Kc) = 1. A known complex and a predicted cluster are considered as a match if their overlapping score is equal to or larger than a specific threshold. The numbers of matched known complexes with respect to different overlapping score threshold (from 0 to 1 with a 0.1 increment) are shown in Figure 4. The best matching result is obtained when T_{ in }= 0.9 for both SP â‰¤ 2 and ASP â‰¤ 2. There are 165 known complexes matched when the overlapping score threshold is 0.2. There are 28 known complexes matched perfectly. When T_{ in }â‰¥ 0.5, the number of matched known complexes is almost the same for SP â‰¤ 2 and ASP â‰¤ 2. When T_{ in }â‰¤ 0.5, the number of matched known complexes is larger for SP â‰¤ 2 than for ASP â‰¤ 2. The probability that a known complex is matched perfectly by a cluster in which proteins are picked up randomly is determined by the size of the network and the known complex. The probability that a known complex with size = 3 matches perfectly by a cluster selected randomly in the yeast network used in this paper is 6.39 * 10^{11}. It is very obvious that more known complexes matched by the predicted clusters implies that the algorithm is more effective to detect complexes. Sensitivity and specificity are two important aspects to estimate the performance of algorithms for detecting protein complexes. Sensitivity is the fraction of the truepositive predictions out of all the true predictions, defined by the following formula:
where TP (true positive) is the number of the predicted clusters matched by the known complexes with OS(Pc, Kc) â‰¥ 0.2, and FN (false negative) is the number of the known complexes that are not matched by the predicted clusters. Specificity is the fraction of the truepositive predictions out of all the positive predictions, defined by the following formula:
where FP (false positive) equals the total number of the predicted clusters minus TP. According to the assumption in [19], a predicted cluster and a known complex are considered to be matched if OS(Pc, Kc) â‰¥ 0.2. Here, we also use 0.2 as the matched overlapping threshold.
Another integrated method, called the fmeasure, has been used in [21, 20], which is defined as follows:
As pointed out in [21, 20], the fmeasure of each method can only be taken as a comparative measure rather than its real values, because the reference set MIPS is incomplete and some predicted clusters that may be true complexes could be regarded as false positives (FP) if they do not match with the known complexes. Nevertheless, it is still reasonable to consider a method more effective if it detects more known complexes. The coverage of the known complexes is defined by the following formula:
where N_{ Kc }is the total number of known complexes. The sensitivity, the specificity, the f measure, and the coverage of the clusters generated by the algorithm IPCA using different parameters are shown in Table 2. The sensitivity of the clusters generated by IPCA is about 0.8 and very close to 0.9 when T_{ in }> 0.5. The value TP is 4 times more than the value FN, which implies that the clusters generated by IPCA are reliable. The specificity of an algorithm represents the real positive proportion of all the predicted clusters. As shown in Table 2, the specificity of the algorithm IPCA is larger than 0.1, but smaller than 0.2. The low specificity is probably because of the incompleteness of the known complexes. The fmeasure takes into account of both the sensitivity and the specificity, and is determined by the larger one. In this experiment, the fmeasure is mostly influenced by the sensitivity. The sensitivity is about 1.6 ~ 1.8 times of the specificity. The coverage of the clusters generated by IPCA increases with the increasing of T_{ in }. Especially, an obvious increase appears when T_{ in }â‰¥ 0.5. In Table 2, we can observe that the sensitivity, the specificity, the fmeasure, and the coverage of the clusters generated by IPCA using SP â‰¤ 2 are slightly larger than those generated by IPCA using ASP â‰¤ 2.
Comparison of protein interaction networks and random networks
To evaluate whether the clusters generated by the algorithm IPCA from the protein interaction network are biologically significant, we experiment the algorithm on the protein interaction network of yeast and on a random network of a similar structure. The random network, which has the same size and the same degree distribution as the yeast network, is obtained by shu^{2}ing the edges between vertices in the yeast network. More clusters are generated from the random network than from the yeast network, and the clusters generated from the random network have less proteins than those generated from the yeast network. Figure 5 shows the size distributions of the clusters generated by IPCA using T_{ in }= 0.6 from the yeast network and from the random network. As shown in the Figure, the predicted clusters identified in the yeast network are in various sizes from 2 to 25, while those in the random network are in various size from 2 to 10. Many small clusters are detected in the random network. To evaluate whether all these small clusters in the random network are significant, we compare them with the known complexes. As shown in Figure 6, while there are more than 100 known complexes matched by the predicted clusters identified in the yeast network when the overlapping score threshold is larger than 0.2, there are almost no known complexes matched by the predicted clusters identified in the random network when the overlapping score threshold is larger than 0.2. This result shows that the random network destroys the biological intrinsic character in the protein interaction network, though it has the same degree distribution as the original yeast network.
Comparison of IPCA and other methods
Since there have been protein complexes that were experimentally determined, a good protein complexes detecting algorithm should identify these known complexes as many as possible. Table 3 shows the numbers of known complexes matched to the clusters generated by IPCA and by other six previous known methods: DPClus [16], CFinder [22], LCMA [21], MCODE [19], RNSC [18], and STM [23].
Though the method DPClus can generate clusters with two modes (nonoverlapping and overlapping), it does not include the overlapping option at present. We generated by ourselves ten sets with nonoverlapping mode using DPClus. Another ten sets with overlapping mode were provided by an author (Md AltafUIAmin) of DPClus. When CP_{ in }= 0.5 and D_{ in }= 0.9, DPClus gets the best matching results. Since it is more significant to detect overlapping protein complexes, we consider the best matching results generated by DPClus with overlapping mode.
The method CFinder gets the best matching results by setting k = 3.
The method LCMA gets the best matching results by setting NA = 0.4.
For the method MCODE, there are 840 parameter combinations, and most of them have similar matching results. The method MCODE gets the best matching results when the "haircut" option is not used and when VWP = 0.1 and Fluff = 0.
The method RNSC gets the best matching results when the number of the predicted clusters is about 1200. The authors of the method STM have shown that the method gets the best performance when the merge threshold value is 1.0. However, a huge cluster that includes 4358 proteins and 85 clusters whose sizes vary from 2 to 7 are generated when the merge threshold value is 1.0. Of all the 86 clusters, only 4 clusters match known complexes with the overlapping sore OS â‰¥ 0.2. Thus, we consider the matching results of the clusters not merged for the method STM.
As shown in Table 3, the clusters generated by IPCA match more known complexes than all six other methods for all different overlapping score thresholds. When OS > 0.2, the number of matched known complexes by IPCA is about 1.2 times more than that by DPClus, LCMA and RNSC. With the same overlapping score threshold, the number of matched known complexes by IPCA is about 2.7 times more than that by CFinder, and about 4.2 times more than that by MCODE and STM. When OS = 1, there are no more than 10 matched known complexes by CFinder, MCODE, RNSC, and STM. On the other hand, there are 28 known complexes matched perfectly by the clusters of IPCA.
The results show that IPCA outperforms all these six previous methods on the performance of identifying protein complexes.
As an additional and interesting example, we compare the performance of IPCA and DPClus for generating large complexes. Since small complexes (e.g., complexes consisting of no more than 2 proteins) have been represented in the proteinprotein interaction data, one may be more interested in identifying large complexes. For this, we compare the results generated by IPCA and that generated by DPClus. The comparisons are given in Table 4. As shown in the table, the large clusters (consisting of 3 or more proteins) generated by IPCA match more known complexes than that generated by DPClus for all different overlapping score thresholds. Let N be the number of total known complexes that are matched by any generated clusters (consisting of 2 or more proteins) and N_{â‰¥3} be the number of known complexes matched by the generated clusters consisting of at least 3 proteins. Then N_{â‰¥3}/N of IPCA is 62.8% and that of DPClus is 58.1%. That is, the clusters generated by IPCA match more known large complexes than that by DPClus.
Function Enrichment Analysis
In order to detect the functional characteristics of the predicted clusters, we compare the predicted clusters with known functional classification. The Pvalue based on hypergeometric distribution is often used to estimate whether a given set of proteins is accumulated by chance. It has been used as a criteria to assign each predicted cluster a main function [18, 16]. Here, we also calculate Pvalue for each predicted cluster and assign a function category to it when the minimum Pvalue occurrs. The Pvalue is defied as follows.
where N is the total number of vertices in the network, C is the size of the predicted cluster, F is the size of a functional group, and k is the number of proteins of the functional group in the predicted cluster. The functional classification of proteins used in this paper was collected from the MIPS Functional Catalog (FunCat) database. FunCat [33] is an annotation scheme of treelike structure for the functional description of proteins. There are up to 6 levels of increasing specificity and 1360 functional categories in FunCat. We obtained 443 clusters with size â‰¥ 6 when using T_{ in }= 0.6 and obtained 132 clusters with the same size when using T_{ in }= 0.9. All these predicted clusters with size â‰¥ 6 by using T_{ in }= 0.6 and T_{ in }= 0.9 match well with the known functional categories with Pvalue < 0.001. As the fact that proteins in the same complex are of similar function, we predicted 7 previously uncharacterized proteins in the predicted clusters generated by T_{ in }= 0.9 and predicted 50 previously uncharacterized proteins in the predicted clusters generated by T_{ in }= 0.6. For example, the unknown function protein YOR264w is included in a 7member cluster, of which six are the cytoskeleton/structural proteins. Thus, we can predict that the function unknown protein YOR264w is also a cytoskeleton/structural protein. All the clusters of size â‰¥ 6 generated by T_{ in }= 0.9 and T_{ in }= 0.6 and their main function annotations are given in an additional file 1.
The uncharacterized proteins in these clusters are also given in the additional file 1. As the incompleteness of the function annotation, we can also predict new membership for the known complexes and predict new functions for known proteins. As shown in additional file 2, the main function of a 10member cluster is splicing (11.04.03.01). Seven proteins of the cluster are related to splicing. Other three proteins in the cluster without the function of splicing are all related to mRNA processing (splicing, 5', 3'end processing), which is a higher level of splicing. Thus, we can deduce that the three proteins involved in mRNA processing may be members of the splicing complexes.
Robustness Analysis
In this analysis, we evaluated the robustness of the algorithm IPCA to various levels of graph alterations. Since all the methods of PPIs (ProteinProtein Interactions) prediction are known to yield a nonnegligible amount of noise (false positives) and to miss a fraction of existing interactions (false negatives) [24], we tested the robustness of IPCA to false positive by adding edges randomly and to false negatives by removing edges randomly. Proportions of edges (0%, 10%, 20%, 30%,..., 90% and 100%) were added to the yeast protein interaction network randomly, and the same proportions (except that of 100%) of edges were removed from the yeast network randomly. It should be expected that the false positives would not randomly contribute to the formation of dense subgraphs, and that the number of matched known complexes does not decrease fast with the increasing of false negatives. Figure 7 displays the impact of edge addition and removal on the results of the algorithm IPCA. As one can see, IPCA is barely affected by addition of up to 100% edges. It is also affected faintly by removal of up to 50% edges. It starts to drop perceivably from 60%, and a fast drop starts from 80%. However, there are still 93 known complexes matched to the predicted clusters (T_{ in }= 0.9 and T_{ in }= 0.6) when 80% edges are removed. The analysis strongly shows that the algorithm IPCA is very robust against the high rate of false positives and false negatives in protein interactions.
Efficiency Analysis
All experiments in this paper are implemented on a PC with 1.7 GHz processor and 512 M RAM. Table 5 shows the running time of the algorithm IPCA when it generates clusters using different T_{ in }values from the yeast network, which consists of 4546 proteins and 12319 interactions. The longest running time is 64 seconds when T_{ in }= 0.1, and the shortest running time is 10 seconds when T_{ in }= 0.9. The running time is decreasing as T_{ in }increases. This is because the probability that proteins added to the clusters is decreasing with the increasing of T_{ in }, and because the candidate selection and the judgement whether a candidate can be added to a cluster are time consuming. As a comparison, under the same test environment, the running time of the method DPClus to generate clusters with nonoverlapping mode from the same yeast protein interaction network is about 20 minutes.
Conclusion
It is believed that identification of protein complexes is useful to explain certain biological progress and to predict functions of proteins. In this paper, we proposed a new topological structure for protein complexes and developed an algorithm IPCA to identify protein complexes in large protein interaction networks based on the new topological structure. Interaction networks are represented by undirected simple graphs and we generate predicted clusters in the networks by using seed selection and local search. The seeds in the networks are calculated only once, which has reduced the running time of the algorithm effectively. Two parameters, SP(K) (or ASP(K)) and IN_{ vK }, are used that reflect the statistics of topological structures of the networks. As the accumulation of new complexes and proteinprotein interactions, the thresholds of the parameters SP(K) (or ASP(K)) and IN_{ vK }can be changed easily for generating different types of clusters. Moreover, the algorithm IPCA can generate overlapping protein complexes, which is consistent with the fact that many of the known protein complexes are overlapping. Interesting questions for further research include how many functions a protein can have, how many processes a protein can participate in, and how heavily two protein complexes should overlap with each other.
We applied the algorithm IPCA to the protein interaction network of Sacchromyces cerevisiae. Many wellknown complexes were found in the protein interaction network. We predicted the functions for uncharacterized proteins and predicted new functions for the known proteins by minimizing the Pvalues of the predicted clusters. We tested the robustness of our algorithm by adding and removing edges in the network randomly. The results have shown that our algorithm is robust against the high rate of false positives and false negatives in the protein interaction networks. Our algorithm can thus be used to identify new protein complexes in protein interaction networks of various species and to provide references for biologists in their research on protein complexes.
Methods
The protein interaction data of Sacchromyces cerevisiae was collected from MIPS [31], represented as pairs of interacting proteins. First we removed selfinteractions and repeated interactions. The final network includes 4546 yeast proteins and 12319 interactions. We also collected from the MIPS database protein complexes annotated for Sacchromyces cerevisiae [32]. We discarded those consisting of only one protein and the final remaining 216 manually annotated complexes are considered as the gold standard data. The proposed algorithm IPCA has been implemented in C++.
References
Kumar A, Snyder M: Protein complexes take the bait. Nature 2002, 415: 123â€“124. 10.1038/415123a
Bauer A, Kuster B: Affinity purificationmass spectrometry:Powerful tools for the characterization of protein complexes. Eur J Biochem 2003, 270(4):570â€“578. 10.1046/j.14321033.2003.03428.x
Gavin A, Bosche M, Krause R, Grandi P, Marzioch M, Bauer A, Schultz J, Rick J, Michon A, Cruciat C, Remor M, fert C, Schelder M, Brajenovic M, Ruffner H, Merino A, Klein K, Hudak M, Dickson D, Rudi T, Gnau V, Bauch A, Bastuck S, Huhse B, Leutwein C, Heurtier M, Copley R, Edelmann A, Querfurth E, Rybin V, Drewes G, Raida M, Bouwmeester T, Bork P, Seraphin B, Kuster B, Neubauer G, SupertiFurga G: Functional organization of the yeast proteome by systematic analysis of protein complexes. Nature 2002, 415(6868):141â€“147. 10.1038/415141a
Ho Y, Gruhler A, Heilbut A, Bader G, Moore L, Adams S, Millar A, Taylor P, Bennett K, Boutilier K, Yang L, Wolting C, Donaldson I, Schandorff S, Shewnarane J, Vo M, Taggart J, Goudreault M, Muskat B, Alfarano C, Dewar D, Lin Z, Michalickova K, Willems A, Sassi H, Nielsen P, Rasmussen K, Andersen J, Johansen L, Hansen L, Jespersen H, Podtelejnikov A, Nielsen E, Crawford J, Poulsen V, Sorensen B, Matthiesen J, Hendrickson R, Gleeson F, Pawson T, Moran M, Durocher D, Mann M, Hogue C, Figeys D, Tyers M: Systematic identification of protein complexes in saccharomyces cerevisiae by mass spectrometry. Nature 2002, 415(6868):180â€“183. 10.1038/415180a
Dezso Z, Oltvai Z, Barabasi A: Bioinformatics Analysis of Experimentally Determined Protein Complexes in the Yeast Saccharomyces cerevisiae. Genome Res 2003, 13(11):2450â€“2454. 10.1101/gr.1073603
Frank J: Cryoelectron microscopy as an investigative tool: The ribosome as an example. Bioessays 2001, 23: 725â€“732. 10.1002/bies.1102
Abbott A: Proteomics: The society of proteins. Nature 2002, 417(6892):894â€“896. 10.1038/417894a
Gavin A, SupertiFurga G: Protein complexes and proteome organization from yeast to man. Curr Opin Chem Biol 2003, 7(1):21â€“27. 10.1016/S13675931(02)000078
Gavin A, Aloy P, Grandi P, Krause R, Boesche M, Marzioch M, Rau C, Jensen L, Bastuck S, Dmpelfeld B, Edelmann A, Heurtier M, Hoffman V, Hoefert C, Klein K, Hudak M, Michon A, Schelder M, Schirle M, Remor M, Rudi T, Hooper S, Bauer A, Bouwmeester T, Casari G, Drewes G, Neubauer G, Rick J, Kuster B, Bork P, Russell R, SupertiFurga G: Proteome survey reveals modularity of the yeast cell machinery. Nature 2006, 440(7084):631â€“636. 10.1038/nature04532
Krogan N, Peng W, Cagney G, Robinson M, Haw R, Zhong G, Guo X, Zhang X, Canadien V, Richards D, Beattie B, Lalev A, Zhang W, Davierwala A, Mnaimneh S, Starostine A, Tikuisis A, Grigull J, Datta N, Bray J, Hughes T, Emili A, Greenblatt J: Highdefinition macromolecular. composition of yeast RNAprocessing complexes. Molecular Cell 2004, 13: 225â€“239. 10.1016/S10972765(04)000036
Krogan N, Cagney G, Yu H, Zhong G, Guo X, Ignatchenko A, Li J, Pu S, Datta N, Tikuisis A, Punna T, PeregrnAlvarez J, Shales M, Zhang X, Davey M, Robinson M, Paccanaro A, Bray J, Sheung A, Beattie B, Richards D, Canadien V, Lalev A, Mena F, Wong P, Starostine A, Canete M, Vlasblom J, Wu S, Orsi C, Collins S, Chandran S, Haw R, Rilstone J, Gandi K, Thompson N, Musso G, Onge P, Ghanny S, Lam M, Butland G, AltafUl A, Kanaya S, Shilatifard A, O'Shea E, Weissman J, Ingles C, Hughes T, Parkinson J, Gerstein M, Wodak S, Emili A, Greenblatt J: Global landscape of protein complexes in the yeast Saccharomyces cerevisiae. Nature 2006, 440(7084):637â€“43. 10.1038/nature04670
Sharan R, Ideker T, Kelley B, Shamir R, Karp R: Identification of protein complexes by comparative analysis of yeast and bacterial protein interaction data. J Comput Biol 2005, 12: 835â€“846. 10.1089/cmb.2005.12.835
Hirsh E, Sharan R: Identification of conserved protein complexes based on a model of protein network evolution. Bioinformatics 2007, 23(2):170â€“176. 10.1093/bioinformatics/btl295
Pellegrini M, Marcotte E, Thompson M, Eisenberg D, Yeates T: Assigning protein functions by comparative genome analysis: protein phylogenetic profiles. Pro Natl Acad Sci 1999, 96: 4285â€“8. 10.1073/pnas.96.8.4285
Krause R, von Mering C, Bork P: A comprehensive set of protein complexes in yeast: Mining large scale proteinprotein interaction screens. Bioinformatics 2003, 19(15):1901â€“1908. 10.1093/bioinformatics/btg344
AltafUlAmin M, Shinbo Y, Mihara K, Kurokawa K, Kanaya S: Development and implementation of an algorithm for detection of protein complexes in large interaction networks. BMC Bioinformatics 2006, (7):207. 10.1186/147121057207
Spirin V, Mirny L: Protein complexes and functional modules in molecular networks. Pro Natl Acad Sci 2003, 100: 12123â€“12128. 10.1073/pnas.2032324100
King A, Przulj N, Jurisica I: Protein Complex Prediction via Costbased Clustering. Bioinformatics 2004, 20: 3013â€“3020. 10.1093/bioinformatics/bth351
Bader G, Hogue C: An Automated Method for Finding Molecular Complexes in Large Protein Interaction Networks. BMC Bioinformatics 2003, 4: 2. 10.1186/1471210542
Zhang S, Ning X, Liu H, Zhang X: Prediction of protein complexes based on protein interaction data and functional annotation data using kernel methods. In International Conference on Intelligent Computing. Lecture Notes in Computer Sciences (LNCS) Edited by: Perner P. 2006, 514â€“524.
Li X, Tan S, Foo C, Ng S: Interaction Graph Mining for Protein Complexes Using Local Clique Merging. Genome Informatics 2005, 16: 260â€“269.
Adamcsek B, Palla G, Farkas I, Derenyi I, Vicsek T: CFinder:locating cliques and overlapping modules in biological networks. Bioinformatics 2006, 22(8):1021â€“3. 10.1093/bioinformatics/btl039
Hwang W, Cho Y, Zhang A, Ramanathan M: A novel functional module detection algorithm for proteinprotein interaction networks. Algorithms Mol Biol 2006, 12: 1â€“24.
Brohee S, van Helden J: Evaluation of clustering algorithms for proteinprotein interaction networks. BMC Bioinformatics 2006, 7: 488. 10.1186/147121057488
GÃ¼ldener U, MÃ¼nsterkÃ¶tter M, KastenmuÃ¼ller G, Strack N, van Helden J, Lemer C, Richelles J, Wodak S, GarcÃaMartÃnez J, PÃ©rezOrtÃn J, Michael H, Kaps A, Talla E, Dujon B, AndrÃ© B, Souciet J, De Montigny J, Bon E, Gaillardin C, Mewes H: CYGD:the Comprehensive Yeast Genome Database. Nucleic Acids Res 2005, 33: D364D368. 10.1093/nar/gki053
Palla G, Derenyi I, Farkas I, Vicsek T: Uncovering the overlapping community structure of complex networks in nature and society. Nature 2005, 435: 814â€“818. 10.1038/nature03607
Gunsalus K, Ge H, Schetter A, Goldberg D, Han J, Hao T, Berriz G, Bertin N, Huang J, Chuang L, Li N, Mani R, Hyman A, Snnichsen B, Echeverri C, Roth F, Vidal M, Piano F: Predictive models of molecular machines involved in Caenorhabditis elegans early embryogenesis. Nature 2005, 436: 861â€“865. 10.1038/nature03876
Mewes H, Amid C, Arnold R, Frishman D, Guldener U, Mannhaupt G, Munsterkotter M, Pagel P, Strack N, Stumpflen V, Warfsmann J, Ruepp A: MIPS: analysis and annotation of proteins from whole genomes. Nucleic Acids Res 2004, 32: D41â€“4. 10.1093/nar/gkh092
del Sol A, O'Meara P: Smallworld network approach to identify key residues in proteinprotein interaction. Proteins 2004, 58(3):672â€“682. 10.1002/prot.20348
del Sol A, Fujihashi H, O'Meara P: Topology of smallworld networks of proteinprotein complex structures. Bioinformatics 2005, 21(8):1311â€“1315. 10.1093/bioinformatics/bti167
Website title[ftp://ftpmips.gsf.de/yeast/PPI]
Website title[ftp://ftpmips.gsf.de/yeast/catalogues/complexcat/]
Ruepp A, Zollner A, Maier D, Albermann K, Hani J, Mokrejs M, Tetko I, Guldener U, Mannhaupt G, Munsterkotter M, H M: The FunCat:a functional annotation scheme for systematic classification of proteins from whole genomes. Nucleic Acids Res 2004, 32: 5539â€“5545. 10.1093/nar/gkh894
Acknowledgements
The authors would like to thank Md AltafUIAmin, Y. Shinbo, K. Mihara, K. Kurokawa, and S. Kanaya for their kindly sharing the tool of DPClus, providing the experiment results for the overlapping mode, and discussions on the protein complexes data. The authors are also thankful to G. Bader and C. Hogue for their sharing the tool of MCODE, to B. Adamcsek, G. Palla, I. Farkas, I. Derenyi, and T. Vicsek for their publicity of CFinder, to A. King, N. Przulj, and I. Jurisica for their sharing the source code of RNSC, and to W. Hwang, Y. Cho, A. Zhang, and M. Ramanathan for their sharing the source code of STM. The authors also thank the anonymous reviewers for their helpful and constructive suggestions.
This research is supported in part by the National Science Foundation of China under Grant No. 60433020.
Author information
Authors and Affiliations
Corresponding author
Additional information
Authors' contributions
ML developed and implemented the clustering algorithm. JC and JW supervised the work and contributed to the problem formulation and paper writing. BH and GC developed the program for generating random graphs and testing the robustness of IPCA. The manuscript was written by ML. All authors read and approved the final manuscript.
Electronic supplementary material
12859_2008_2383_MOESM1_ESM.xls
Additional file 1: Pvalues for the predicted clusters with size â‰¥ 6 generated using T_{ in }= 0.9 and T_{ in }= 0.6. The data provided represent the statistical analysis of the predicted clusters. Pvalue is calculated for each predicted cluster and a function category is assigned to it when the minimum Pvalue occurs. When T_{ in }= 0.9, there are 132 clusters (size â‰¥ 6) generated by IPCA. When T_{ in }= 0.6, there are 443 clusters (size â‰¥ 6) generated by IPCA. (XLS 168 KB)
12859_2008_2383_MOESM2_ESM.doc
Additional file 2: Functional annotation for a predicted cluster of 10 proteins. This file provides a cluster which is composed of ten proteins: YGL173c, YOL149w, YBL026w, YCR077c, YJR022w, YER112w, YER146w, YDR378c, YNL147w, and YLR438ca. The functional annotations for each protein in the cluster are listed in this file. (DOC 47 KB)
Authorsâ€™ original submitted files for images
Below are the links to the authorsâ€™ original submitted files for images.
Rights and permissions
Open Access This article is published under license to BioMed Central Ltd. This is an Open Access article is distributed under the terms of the Creative Commons Attribution License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Li, M., Chen, Je., Wang, Jx. et al. Modifying the DPClus algorithm for identifying protein complexes based on new topological structures. BMC Bioinformatics 9, 398 (2008). https://doi.org/10.1186/147121059398
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/147121059398
Keywords
 Random Network
 Protein Interaction Network
 Vertex Weight
 Dense Subgraph
 Average Short Path Length