Many researchers hypothesize that subgraphs with different topological structures in PPI networks are factual protein complexes (Wang et al., 2010) such as density, k-clique, and core-attachment structures. Most of these methods are either global heuristic search, local heuristic search, or both. Meanwhile, some methods integrate topological and biological information to further improve the accuracy of detecting protein complexes.

Many local heuristic-based methods have been proposed to identify protein complexes. For instance, Altaf-Ul-Amin et al. (Altaf-Ul-Amin et al., 2006) developed DPClus, which generates clusters by ensuring density and checking the periphery of the clusters. Gavin et al. (Gavin et al., 2006) studied the organization of protein complexes, demonstrating that a protein complex generally contains a unique protein complex core and attachment proteins, called a core-attachment structure. Here, proteins in a protein complex core have relatively more reliable interactions among themselves. The attachment proteins are the surrounding proteins of the protein complex core to assist it in performing related functions (Lakizadeh et al., 2015). Wu et al. (Wu et al., 2009) proposed a classic protein complex discovery method (COACH) using the core-attachment structure. COACH first detects protein complex cores and then identifies its attachment proteins to form a whole protein complex. Peng et al. (Peng et al., 2014) designed a PageRank Nibble strategy to give adjacent proteins different probabilities with core-attachment structures and proposed WPNCA to predict protein complexes. Nepuse et al. (Nepusz et al., 2012) presented ClusterONE, which utilizes a demanding growth process to mine subgraphs with high cohesiveness that may be protein complexes. Recently, Wang et al. (Wang et al., 2020) presented a new graph clustering method using a local heuristic search strategy to detect static and dynamic protein complexes. These local heuristic methods have strong local searchability, but finding an optimal global solution is difficult.

Meanwhile, some global heuristic-based methods have been proposed to identify protein complexes. In 2009, Liu et al. (Liu et al., 2009) used an iterative method to weight PPI networks and developed a maximal clique-based method (CMC) to discover protein complexes from weighted PPI networks. Wang et al. (Wang et al., 2012) were inspired by the hierarchical organization of GO annotations and known protein complexes. Then they proposed OH-PIN, which is based on the concepts of overlapping M-clusters, λ-module, and clustering coefficients to detect both overlapping and hierarchical protein complexes in PPI networks. PC2P (Omranian et al., 2021) is a parameter-free greedy approximation algorithm casts the problem of protein complex detection as a network partitioning into biclique spanned subgraphs, which include both sparse and dense subgraphs. Although these global heuristic search methods have a strong global search ability, they require considerable time and computing resources.

Recently, some methods based on network embedding strategies have been used to detect protein complexes. DPC-HCNE (Meng et al., 2019) is a novel protein complex detection method based on hierarchical compressing network embedding and core-attachment structures. It can preserve both the local topological information and global topological information of a PPI network. CPredictor 5.0 (Yao et al., 2019) uses the network embedding method Node2Vec (Grover and Leskovec, 2016) to learn node feature vector representation and then calculates the node embedding similarity and the functional similarity between interacting proteins to construct the weight PPI networks. These methods illustrate that employing the network embedding method could improve the accuracy of protein complex identification.

It is well known that PPI networks contain many false-positive and false-negative interactions, i.e., noise. To overcome the noise of the PPI networks, some studies try to exploit biological information, such as gene expression data (Keretsu and Sarmah, 2016), gene ontology (GO) data (Wang et al., 2019; Yao et al., 2019), and subcellular localization data (Lei et al., 2018) to complement the interactions in PPI networks. CPredictor2.0 (Xu et al., 2017) effectively detects protein complexes from PPI networks, and first groups proteins based on functional annotations. Then, it applies the MCL algorithm to detect dense clusters as protein complexes. Zhang et al. (Zhang et al., 2016) calculated the active time point and the active probability of each protein and constructed dynamic PPI networks. Then a novel method was proposed based on the core-attachment structure. Zhang et al. (Zhang et al., 2019) proposed a novel method based on the core-attachment structure and seed expansion strategy to identify protein complexes using the topological structure and biological data in static PPI networks. ICJointLE (Zhang et al., 2019) is a novel method to identify protein complexes with the features of joint colocalization and joint coexpression in static PPI networks. NNP (Zhang et al., 2021) is a new method for recognizing protein complexes by topological characteristics and biological characteristics. Some methods (Zaki et al., 2013; Wang et al., 2019) are based on topological information to weight interactions in PPI networks. For example, PEWCC (Zaki et al., 2013) is a novel graph mining method that first assesses the reliability of the interactions and then detects protein complexes based on the concept of the weighted clustering coefficient. These methods have shown that the accuracy of protein complex identification can be significantly improved by integrating network topological structure and multiple biological information.

Several recent methods suggested that identifying protein complexes or community structures can be an optimization problem using network topology and protein attributes. For example, RNSC (King et al., 2004) attempts to find an optimal set of partitions of a PPI network graph by employing different cost functions for detecting protein complexes. RSGNM (Zhang et al., 2012) is a regularized sparse generative network model that adds another process that generates propensities into an existing generative network model for protein complex identification. EGCPI (He and Chan, 2016) formulates the problem as an optimization problem to mine the optimal clusters with densely connected vertices in the PPI networks to discover protein complexes. DPCA (Hu et al., 2018) formulates the problem of detecting protein complexes as a constrained optimization problem according to protein complexes’ topological and biological properties. In particular, it is an algorithm with high efficiency and effectiveness. GMFTP (Zhang et al., 2014) is a generative model to simulate the generative processes of topological and biological information, and clusters that maximize the likelihood of generating the given PIN are considered protein complexes. DCAFP (Hu and Chan, 2015) transforms the problem of identifying protein complexes into a constrained optimization problem and introduces an optimization model by considering the integration of functional preferences and dense structures. He et al. (He et al., 2019) introduced a novel graph clustering model called contextual correlation preserving multiview featured graph clustering (CCPMVFGC) for discovering communities in graphs with multiview features, viewwise correlations of pairwise features and the graph topology. VVAMo (He et al., 2021a) is a novel matrix factorization-based model for communities in complex network. It proposes a unified likelihood function for VVAMo and derives an alternating algorithm for learning the optimal parameters of the proposed model. In 2017, Zhang et al. (Zhang et al., 2017) proposed a new firefly clustering algorithm for transforming the protein complex detection problem into an optimization problem. IMA (Wang et al., 2021) is a novel improved memetic algorithm that optimizes a fitness function to detect protein complexes. These model optimization-based methods usually have more parameters and variables, and the parameter optimization process is time-consuming. However, these methods also have some significance for us to transform the identification of protein complexes into an optimization problem.

The methods mentioned above are either unsupervised learning-based or model optimization-based methods that identify protein complexes using predefined assumptions and determined models. Unsupervised learning-based methods do not need to resolve practical problems, such as insufficient feature extraction from known protein complexes, model selection, and model training. Those methods cannot utilize the information of known protein complexes, and they neglect some other topological protein complexes such as the ‘star’ mode and ‘spoke’ mode and so on. Generally, supervised learning-based methods first train a supervised learning model by extracting features, and then trained supervised learning models are used to search new protein complexes.

Many standard protein complex datasets have been obtained in recent years. Therefore, several supervised learning-based methods based on training regression or classification models are proposed to discover protein complexes from PPI networks. For example, Qi et al. (Qi et al., 2008) proposed a framework to learn the parameters of the Bayesian network model for discovering protein complexes. Yu et al. (Yu et al., 2014) presented a supervised learning-based method to detect protein complexes, which used cliques as initial clusters and selected a trained linear regression model to form protein complexes. Lei et al. (Shi et al., 2011) proposed a semisupervised algorithm, and trained a neural network model to detect protein complexes. ClusterEPs (Liu et al., 2016) estimated the possibility of a subgraph being a protein complex by emerging patterns (EPs). Dong et al.(Dong et al., 2018) provided the ClusterSS method, which integrates a trained neural network model and local cohesiveness function to guide the search strategy to identify protein complexes. Liu et al. (Liu et al., 2018) proposed a supervised learning method based on network embeddings and a random forest model for discovering protein complexes. Based on the decision tree, Sikandar et al. (Sikandar et al., 2018) presented a method using biological and topological information to detect protein complexes. Liu et al.(Liu et al., 2021) proposed a novel semisupervised model and a protein complex detection algorithm to identify significant protein complexes with clear module structures from PPI networks. Mei et al. (Mei, 2022) proposed a computational method that combines supervised learning and dense subgraph discovery to predict protein complexes. On the one hand, the accuracy of these detection methods based on semisupervised learning or supervised learning is limited due to the small training dataset. On the other hand, these methods only train a single type of learning model, so these models are not so generalizable and their learning ability has certain limitations.

Some existing studies show that graph neural networks (GNNs) methods can effectively learn graph structure and node features. For example, Kipf et al. (Kipf and Welling, 2016) presented a scalable approach for semisupervised learning on graph-structured data. The proposed graph convolutional network (GCN) model is based on an efficient variant of convolutional neural networks. It can encode both graph structure and node features in a way useful for semisupervised classification. In 2021, Zaki et al. (Zaki et al., 2021) introduced various GCN approaches to improve the detection of protein complexes. graph attention networks (GATs), which aggregate neighbor nodes through the attention mechanism, realize the adaptive allocation of weights of different neighbors, thus greatly improving the expression ability of GNN models. He et al. (He et al., 2021b) proposed a class of novel learning-to-attend strategies, named conjoint attentions (CAs) to construct graph conjoint attention networks (CATs) for GNNs. CAs offer flexible incorporation of layerwise node features and structural interventions that can be learned outside the GNNs to compute appropriate weights for feature aggregation. We will study the detection of protein complexes in PPI networks using GATs in the future.

In this paper, we used the four PPI networks for the experiments, i.e., Gavin (Gavin et al., 2006), Krogan core (Krogan et al., 2006), DIP (Xenarios et al., 2002), and MIPS (Güldener et al., 2006). The detailed properties of these PPI networks are shown in Table 1. Here, the self-interactions and duplicate interactions were eliminated.

We used two standard protein complexes that were constructed in the literature (Wang et al., 2020). Their properties are shown in Table 2. Here, standard protein complexes 1 consists of the known protein complexes from MIPS (Mewes et al., 2004), SGD (Hong et al., 2007), TAP06 (Gavin et al., 2006), ALOY (Aloy et al., 2004), CYC 2008 (Pu et al., 2009), and NEWMIPS (Friedel et al., 2009). Standard protein complexes 2 is also a combined protein complex dataset (Ma et al., 2017). It consists of the Wodak database (Pu et al., 2009), PINdb and GO complexes (Ma et al., 2017).

In this study, we used the GO-slim data for describing the functional similarity of interactions, which is available on the link: https://downloads.yeastgenome.org. Meanwhile, the gene expression data were obtained from https://www.ncbi.nlm.nih.gov/sites/GDSbrowser. Additionally, subcellular localization data was obtained from https://compartments.jensenlab.org/Downloads.

This work is a novel ensemble learning framework to identify protein complexes from PPI networks. The block diagram of the detection process is shown in Figure 1.

The framework of this method is outlined in Algorithm 1. The input to the algorithm is the PPI network, which produces a set of protein complexes as output. Our algorithm consists of five main steps. The first step is to construct a weighted PPI network by combining topological structure, gene expression data, GO annotation data, and subcellular location data in Line 2 (Constructing a weighted PPI network section). The second step is to design a protein complex core mining strategy to identify protein complex cores in the PPI networks (Mining protein complex cores section) in Line 3. The third step is first to construct feature vectors to describe the properties of known and false protein complexes in the PPI networks and train a voting regression model (Training a voting regression model section) to model and represent the protein complex based on supervised learning in Line 5. Then second, we define a quality function called structural modularity to describe the structural modularity of protein complexes. Then we combine the trained voting regression model and structural modularity to obtain an ensemble learning model in Line 6. In the fourth step, based on the ensemble learning model, we propose a graph heuristic search strategy (Forming protein complexes section) to extend each protein complex core for forming protein complexes from the PPI networks in Lines 7–14. Finally, we remove these redundant identified protein complexes in Line 15.

Some studies have confirmed that the performance of protein complex detection could be markedly enhanced when the weight of edges is considered (Keretsu and Sarmah, 2016; Lei et al., 2018). Meanwhile, integrating multiple data sources into a PPI network can strengthen the reliability of the PPI networks (Lei et al., 2018; Wang et al., 2020), which inspires us with confidence to give the weight for interactions. Moreover, a protein complex consists of proteins and interactions among themselves, and the proteins in the same protein complex are coexpressed and have a similar function and localization. Thus, we integrate multiple pieces of information, including gene expression data, protein localization data, and gene ontology data, to weight the interactions within the PPI networks.

According to the constructing a weighted PPI network section, the weight of interactions is weighted using multiple biological properties and its topological structure, so the higher weight the edge has, the more likely it is that two terminate proteins are inside the same protein complex (Wang et al., 2011; Li et al., 2012). Furthermore, the protein complex cores often correspond to dense subgraphs in PPI networks (Wu et al., 2009; Wang et al., 2019). The pseudocode of mining protein complex cores is presented in Algorithm 2.

First, for the edge (v, u), its weight is w(v, u), and its neighborhood graph is denoted as NG(v, u) = (V*, E*, W*), where V* = N _v ∪ N _u ∪ {v, u}. Furthermore, the average weighted degree of NG(v, u) is denoted as AWD(NG(v, u)) (Eq. 6): A W D N G v , u = 2 × ∑ s , t ∈ E * w s , t | V * | . (6)

Based on the analysis above, we propose a score function (Eq. 7) to score seed edges based on the weight of the edge w(v, u) and the average weighted degree of the neighborhood graph of the edge (Eq. 6) to select seed edges in Line 1. Then, we sort all edges in nonascending order based on the score function (see Eq. 7) in the PPI networks. Only edges whose score function is greater than the mean of the score function of all edges are queued into Q. Seed edges in Q will mine protein complex cores in Line 2.

As a result, the score function of edge (v, u) is defined as Eq. 7: S c o r e e d g e v , u = w v , u × A W D N G v , u . (7)

For an edge (v, u) ∈ E, its edge clustering coefficient (ECC(v, u)) is defined as the number of triangles to which (u, v) belongs, divided by the number of triangles that might potentially include (u, v), as shown in Eq. 8. E C C v , u = Z v , u min | deg v | , | deg u | . (8) where Z(v, u) denotes the number of triangles built on edge (v, u), and min(| deg(v)|, | deg(u)|) is the minimum degree of the two terminate proteins.

Initially, select the protein with the highest weight edge as the first seed edge (v, u), and create a protein complex core in Line 6, where neighbors of the complex core are added to both the weight of edge w(x, t) ≥ Avgedgesweight (Avgedgesweight is defined as Eq. 9) and ECC(x, t) is greater than the average edge clustering coefficient ECC of all edges (AvgweightECC), according to the closeness between the seed edge (v, u) and its neighbors in Lines 9–17. These two constraints can ensure that the proteins in the protein complex core are correlated in biological relations and closely connected in topological structure. The protein complex core is retained if it contains more than or equals two proteins in Lines 18–20. Meanwhile, the seed edge (including two terminate proteins) would be marked and cannot be used as the seed edge of another cluster in Lines seven and eight. We select the next edge with the highest weight where its two terminal proteins are not included before seed edges, and it is used to form the next protein complex core until the seed queue Q is empty in Lines 6–22. A v g e d g e s w e i g h t = ∑ v , u ∈ E w v , u | V | . (9)

CPredictor2.0 (Xu et al., 2017) is also employed to detect global protein complex cores. Here, CPredictor2.0 detects protein complexes using MCL and protein functional information. It first discovers clusters in each functional group using the Markov clustering algorithm and merges them with higher overlap. We use CPredictor2.0 to obtain global protein complex cores (CPrclusters) in Line 23. Next, we combine these local protein complex cores by a graph heuristic search method and global protein complex cores using the CPredictor2.0 method in Line 24.

Here, Algorithm 2 identifies the protein complex cores, which may have some redundant protein complex cores. For these redundant protein complex cores, we only keep one of them in the list of protein complex cores in Line 25.

Algorithm 2

Mining protein complex cores.

Based on the fact that a protein complex core and attachment proteins form a protein complex, we obtain some protein complex cores. Next, we extract the attachment proteins of each protein complex core and select reliable attachments cooperating with its protein complex core to form a protein complex. We design a graph heuristic search strategy for each protein complex core to extend the protein complex core to form a whole protein complex. First, it starts with a protein complex core, which iteratively inserts neighboring proteins into the protein complex core and then removes proteins from the protein complex core to search for a locally optimal cluster. In this paper, each protein complex core is subjected to a graph heuristic search strategy and an ensemble learning model to form a protein complex. The basic idea of a graph heuristic search strategy for a protein complex core is iteratively extended and corrected to form a protein complex by maximizing the score of the ensemble learning model (please see Obtaining an ensemble learning model section).

The pseudocode of the graph heuristic search strategy is shown in Algorithm 3, which consists of the following steps:

i Input a protein complex core.

Finally, we select the next protein complex core. Then we repeat this process using a graph heuristic search strategy (Algorithm 3) to extend the next protein complex core to form a protein complex until no seed edges remain. In the last step of the algorithm, some redundant protein complexes and protein complexes containing fewer than three proteins are discarded.

Algorithm 3

A graph heuristic search strategy

S _i is a standard protein complex in S, and D _j is a discovered protein complex D. Their neighborhood affinity score (NA(S _i , D _j )) (Brohee and Van Helden, 2006) can describe the similarity of two protein complexes S _i and D _j , and it is defined as Eq.15: N A S i , D j = | S i ∩ D j | 2 | S i | × | D j | . (15)

Generally, if NA(S _i , D _j ) is larger than or equal to 0.2, protein complexes S _i and D _j are regarded as matching protein complexes (Li et al., 2010).

Let N _sm be the number of standard protein complexes that match at least one detected protein complex, i.e., N _sm = |{s|s ∈ S, ∃d ∈ D, NA(s, d) ≥ ω}| and N _im be the number of detected protein complexes that match at least one standard protein complex, i.e., N _im = |{d|d ∈ D, ∃s ∈ S, NA(d, s) ≥ ω}|, where ω is a predefined threshold and is usually 0.20. Recall and precision are defined as r e c a l l = N s m | S | and p r e c i s i o n = N i m | D | , respectively. Finally, the F-measure is the compromise between precision and recall and is defined by Eq. 16: F − m e a s u r e = 2 × p r e c i s i o n × r e c a l l p r e c i s i o n + r e c a l l . (16)

Let T _ij be the number of proteins that are included in both standard protein complex S _i and detected protein complex D _j , and let N _i be the number of proteins that are included in standard protein complexes S. Meanwhile, Sn and PPV are calculated by S n = ∑ i = 1 | S | max j = 1 | D | T i j ∑ i = 1 | S | N i and P P V = ∑ j = 1 | D | max i = 1 | S | T i j ∑ j = 1 | D | ∑ i = 1 | S | T i j , respectively. As a result, the accuracy (ACC) is defined by Eq. 17: A C C = S n × P P V . (17)

We used the third metric, the maximum matching ratio (MMR) (Nepusz et al., 2012) based on the maximal one-to-one mapping between standard protein complexes and detected protein complexes. First, we need to construct a bipartite graph between S and D, and then each standard protein complex S _i ∈ S and detected protein complex D _j ∈ D are connected by the weight W(S _i , D _j ) edge. Next, we select disjoint edges from the bipartite graph to maximize the sum of their weights; Finally, the MMR is the sum of the weights of all selected edges divided by |S|, which is denoted by Eq. 18: M M R = ∑ i = 1 | S | max j N A S i , D j | S | . (18)

The coverage rate (CR) was used to assess how many proteins in the standard protein complexes could be covered by the identified complexes. When the standard protein complexes S and the detected protein complexes D are given, the |S|×|D| matrix T is constructed, where each element max{T _ij } is the most significant number of shared proteins between the ith standard protein complex, and the jth detected protein complex. The coverage rate is calculated by Eq. 19: C R = ∑ i = 1 | S | max T i j ∑ i = 1 | S | N i . (19) where N _i is the number of proteins in the ith standard complex.

Jaccard is the final method for measuring the clustering methods (Song and Singh, 2009). Here, a standard protein complex is S _i ∈ S, and a discovered protein complex is D _j ∈ D. Then, their Jaccard is J a c ( S i , D j ) = | S i ∩ D j | | S i ∪ D j | . For the discovered protein complex D _j , its Jaccard is J a c ( D j ) = m a x S i ∈ S J a c ( D i , S i ) . For a standard protein complex S _i , its Jaccard is J a c ( S i ) = m a x D j ∈ D J a c ( S i , D j ) . Then, for detected protein complexes D, the average of the weighted Jaccard is J a c c a r d D = ∑ D j ∈ D | D j | J a c ( D j ) ∑ D j ∈ D | D j | . Similarly, for the standard protein complexes S, its JaccardS is defined by J a c c a r d S = ∑ S i ∈ S | S i | J a c ( S i ) ∑ S i ∈ S | S i | . Finally, the Jaccard is calculated by Eq. 20: J a c c a r d = 2 × J a c c a r d D × J a c c a r d S J a c c a r d D + J a c c a r d S . (20)

In addition to these metrics to measure the performance of ELF-DPC, we investigated whether these identified protein complexes have biological significance by calculating the p-value. Generally, a detected protein complex possesses biological significance if its p-value is less than 0.01. In this paper, we used the fast tool LAGO (Boyle et al., 2004) to compute a p-value, and it is based on the hypergeometric distribution and Bonferroni correction. For more information about it, please refer to the literature (Boyle et al., 2004; Wang et al., 2019). The p-value is denoted as Eq. 21 p − v a l u e = 1 − ∑ i = 0 k − 1 F i N − F C − i N C , (21) where k is the number of functional group proteins in the protein complex, and N is the number of proteins in the PPI networks. F is the size of the functional group in the PPI networks. We assume that a discovered protein complex contains C proteins.