Most cellular processes are the result of a cascade of events mediated by proteins that act in a cooperative manner. Proteins combine into macromolecular complexes to execute essential tasks, such as replication, transcription, protein transport or metabolic reaction catalysis. Proteins can therefore be viewed as elementary building blocks of these molecular machines. Moreover, protein complexes can share components: proteins can be reused and participate to several complexes. Identifying protein complexes and the way they share components hence appears as an essential step in describing cellular biology on a molecular basis.

Several technologies for detecting protein interactions such as yeast two-hybrid (Y2H) and protein-complex purifications (PCP) have recently been scaled-up to high-throughput level and have generated large-scale protein-protein interaction datasets 1234. Up to now, methods for analyzing such datasets have mainly been based on clustering techniques. They have been applied to assign protein function by inference from the biological context as given by their interactors 5, and to identify complexes as dense regions of the network 67. Such approaches, in general, do not aim to reveal the detailed structure within and between the detected regions. The logical organization into shared and specific components, and its representation, remains elusive.

The phenomenon of shared components, that is, proteins or groups of proteins occurring in different complexes, is fairly common. A shared component may be a small part of many complexes, acting as a unit that is constantly reused for its function. It may also be the main part of the complex, for example in a family of variant complexes that differ from each other by distinct proteins that provide functional specificity. It is important to capture and properly represent the modularity of protein-protein interaction networks by identifying the shared components and the way they are arranged to generate complexes.

Protein-protein interaction networks are classically represented by graphs with proteins as nodes and physical interactions represented by edges connecting the nodes. Here, we introduce a novel method to elucidate and represent the logical organization of protein-protein interaction networks by using the graph-theory notion of module and the related idea of modular decomposition. Following a brief description of the concept, we first verify the method on known complexes and then interpret a large-scale protein-protein interaction network around the transcription factor NFκB.

The relationship between proteins identified via PCP and yeast two-hybrid (Y2H) methods is of a different nature (PCP in this instance comprises both the TAP-MS method (tandem affinity purification with mass spectrometric identification) used by Gavin et al. 10 and HMS-PCI (high-throughput mass spectrometric protein complex identification) as defined by Ho and colleagues 3). TAP, for example, identifies multiprotein complexes, whereas Y2H systems 11 detect direct physical interactions between two proteins. Hence, edges in the corresponding networks symbolize different physical relationships between proteins. Therefore, with a different semantic given to the graph, modular decomposition has a different meaning. As we are interested in protein complexes, we focus our analysis on the PCP context.

A PCP experiment starts with the selection of a protein, called the bait. Purification of the bait results in the co-purification of proteins that co-occur in at least one complex with the bait protein. We assume in a first approach the datasets to be complete, that is, with all proteins in a given network systematically selected as baits. The dataset can be represented by a graph with proteins as nodes and an edge between proteins A and B if, and only if, there is at least one complex containing both A and B (Figure 3a).

It is important to note that a complex appears as a clique, that is, a fully connected sub-part of the network (Figure 4). However, the converse is not true: not every clique in the network necessarily derives from an existing complex. For example, three connected proteins can be the outcome of a single trimer, three heterodimers or combinations thereof (Figure 3b). On the basis of network analysis one cannot discriminate between these theoretical options. Moreover, the stoichiometry of complex constituents, that is, the respective number of copies of the same protein in the assembly cannot be inferred from PCP experiments. We therefore disregard stoichiometry issues and deal with the multiple options by adopting a parsimonious solution that embeds all possibilities: we consider the largest possible complex, which appears as a maximal clique in the graph. Finding maximal cliques is the basis for algorithms of protein-complex computation based on protein-protein interaction networks 67.

Modular decomposition provides an instruction set to deliver all maximal cliques of a graph. In particular, when the decomposition has only series and parallels, the maximal cliques are straightforwardly retrieved by traversing the tree recursively from top to bottom. When encountered, a series module acts as a product: the maximal cliques are all the combinations made up of one maximal clique from each 'child' node. A parallel module acts as a sum: the set of maximal cliques is the union of all maximal cliques from the 'child' nodes. In the following, particular examples illustrate the application of this method.

Modular decomposition provides a comprehensive representation of the logical rules in the cooperation of the component proteins. In the tree the leaves are proteins; the root represents the whole network. In between, each node of the tree is a module that is a sub-part of its parent. The label of a node gives the nature of the relationship between its direct children.

Proteins or modules in a parallel module can be seen as alternatives (Figure 3c). If A is neighbor of B and C, which are not neighbors of each other, then A can belong to a complex together with either B or C, but not with both at the same time. B and C define a parallel module and thus are alternative partners in a complex with their common neighbor A. This situation corresponds to a logical 'exclusive OR' (also noted XOR) between B and C. Proteins or modules in parallel do not interact but can perform closely related biological functions.

Proteins or modules in a series module can be seen as potentially combined in any way (Figure 3b). If A is neighbor of B and C, which in turn are also neighbors of each other, then A can belong to a complex together with B or C, or with both at the same time. This situation corresponds to a logical 'OR' between B and C. The parsimonious solution restricts to the simplest model where the three proteins combine into a single complex. With a parsimonious solution, the series module interprets as a logical 'AND' between B and C. One can think of a series module as a unit: a set of proteins or modules that function together. A prime is a graph where neither of these cases occurs (Figure 3d).

The following examples illustrate how modular decomposition can reveal the combinatorial assembly of complexes from interaction networks.

Series modules reveal the presence of sub-complexes, that is, groups of proteins that function as single units in several complexes. Examples of such modules are found in RNA polymerases, which are protein complexes that synthesize RNA on DNA templates. RNA polymerase I synthesizes rRNA, polymerase II mRNA, and polymerase III many small RNAs such as tRNA.

The three polymerases involve a total of 31 proteins (Figure 5b). Modular decomposition defines the organization into shared and specific subunits and the logical rule set to derive the three enzyme complexes. Like PP2A, the catalytic unit is represented by alternative variants. For RNA polymerases, however, these alternative units consist of a multiprotein sub-complex (modules 4, 6 or 7). The two proteins Rpc19 and Rpc40 in module 3, shared by polymerase I and III, correspond to alternative variants of Rpb3 and Rpb11 in polymerase II, a relationship that is detected by sequence homology. The two proteins also form a sub-complex in the three-dimensional structure 131415.

A series module containing five proteins at the root of the tree captures proteins that are common to all RNA polymerases. Interestingly, these proteins are scattered over the surface of the catalytic complex. For one of them, Rpc10, there is evidence that it acts as a bridging component between the sub-complex Rpb3/Rpb11 and the catalytic sub-complex. It is conceivable that those shared proteins all serve a comparable adaptor role to other cellular structures and might therefore be under too strong evolutionary constraints to allow divergence.

The organization of a network cannot always be summarized by series and parallel modules only. In complex interaction arrangements, prime modules emerge in the decomposition and can be interpreted as irreducible backbones of the network, as we illustrate with a selection of complexes involved in chromatin remodeling and the transcriptional machinery.

We consider a network of 50 proteins that define the chromatin-remodeling complexes RSC and SWI/SNF; the general transcription factor complex TFIIF; the general transcription factor complex TFIID, which is responsible for promoter recognition; and the mediator complex that mediates signals to RNA polymerase II 16171819.

Modular decomposition (Figure 5c) of the network identifies six series modules as elementary units of the networks: the proteins specific to the RSC, SWI/SNF, TFIID, TFIIF, and the mediator complex (modules 2, 4, 6, 7 and 8 respectively), and Arp7 and Arp9 (module 3), which are common to the two chromatin-remodeling complexes. The three series modules specifically interacting with Anc1 are then embedded as alternatives into a parallel module (module 5). The root of the tree is a prime module, which summarizes the rather complex picture of the network.

Anc1, whose specific role in the respective complexes remains unclear in the literature, appears as the common point in the SWI/SNF complex and the transcription-related complexes. As with the shared components of the RNA polymerases, Arp7, Arp9 and Anc1 are rather peripherally arranged in the individual complexes. Arp7 and Arp9 have recently been proposed to compose a heterodimeric sub-complex that cooperates with DNA-bending proteins to facilitate chromatin remodeling and complex-complex interactions 20.

The structure of the prime reflects the progression from chromatin remodeling to transcription. Chromatin remodeling appears to come in two contexts: with mRNA transcription (module 4, SWI/SNF-specific proteins); and not reflected in this network (module 2, RSC-specific proteins that have no further connections here except for the shared components in module 3). Module 5 contains the elements responsible for promoter recognition (module 6), the mediator (module 8), and a general transcription factor (module 7). Anc1 links the chromatin-remodeling part to the transcriptional machinery.

As illustrated by these three examples, modular decomposition defines modules in a molecular-interaction network. Acting as a factorization principle, it helps to represent protein-protein interaction networks in a condensed and structured manner. Although the module representation can be helpful to derive hypotheses from interaction networks, interpretation requires particular attention for experimental datasets. Nodes in the network are not necessarily equivalent, as usually not all of the corresponding proteins have also been tested as baits in the underlying experiments.

We applied modular decomposition to the analysis of a large experimental dataset based on a systematic study of the human TNF-α/NFκB signal transduction pathway 2122. In this study, 32 proteins implicated in TNF-α/NFκB signaling had been selected as baits. Purification of these bait proteins resulted in a high confidence protein interaction network displaying 221 interactions involving 131 proteins.

In this experimental setup, not all proteins have been selected as baits. In such a network a pair of two non-neighbor proteins occurs in two situations. If at least one protein is bait, there is experimental evidence for the absence of the interaction. If none is bait, there is just no information about their interaction. We decided to apply the stringent spoke model convention 23, that is, not to infer an edge between two proteins if they are not bait, even if they occur in the same purification. To distinguish the case of unobserved interactions we flag the proteins in the network as well as in the tree differently for baits and non-baits. Because of the consequent lack of edges, the modular decomposition holds fewer series and more parallel modules. The interpretation for series still holds; however, the interpretation for prime and parallel modules is not as stringent as described above if containing unbaited proteins. Alternatives, in particular, are not exclusive if the interaction between two proteins has never been tested. The tree shows a current view of the network that can change with experimental evidence for additional interactions.

Modular decomposition has been applied to the TNF-α/NFκB dataset 21, resulting in a tree with 20 modules (see Additional data file 1). The root is labeled parallel, joining the different isolated parts of the network. Most of the proteins selected for purification are direct descendants of a prime module together with 16 parallel modules that group mainly new interactors that had consistently and specifically been identified by the same bait proteins. Using the known annotations of the proteins, we observed that those modules group proteins of common biological processes. For instance, the majority of the proteins of the module specific to relB are members of the SWI/SNF chromatin-remodeling complex. The HSP90/CDC37 chaperone complex is grouped in a module sharing nine common interactors of the pathway. Hence, the modules derive a consistent grouping of the newly identified interactors as a basis for biological understanding and interpretation.

Modular decomposition of the whole network contains a large prime module, reflecting the epistatic order of functionally distinct units in the cascade that cannot be further compressed simply by ANDs and ORs (see Additional data file 1). Nevertheless, modular decomposition can be used to further investigate local zones of functionally related proteins in the transduction pathway. We illustrate this strategy on the central knot of the pathway: the NFκB system.

The transcription factor NFκB is the convergence point for several signal transduction pathways activated by various stimuli, including TNF-α, IL-1β, bacterial lipopolysaccharide (LPS) and the T-cell receptor. The prevailing model is that NFκB is a dimer composed of a DNA-binding subunit and a transcriptional activator subunit. In the absence of stimulation, NFκB dimers are sequestered in the cytoplasm by inhibitors called IκB. After stimulation, an activated IKK complex phosphorylates IκBs, earmarking them for proteasome-mediated degradation. Following IκB degradation, NFκB translocates to the nucleus to activate gene transcription. Each of the proteins and complexes mentioned above exists in several variants. The existence of combinatorial variants could explain why this central knot can transmit different upstream signals resulting in distinct downstream outputs. In the following we use modular decomposition of NFκB members to investigate this hypothesis.

The NFκB family consists of five structurally related members: relA, relB, c-rel, NFκB1/p50 and NFκB2/p52. To analyze the NFκB variants, we first considered the networks of interactions among those five proteins selected as baits. Modular decomposition of the experimental TAP network characterizes the existence of complex variants (Figure 6). In line with the prevailing hypothesis we detect mutually exclusive usage of NFκB1/p50 and NFκB2/p52, which occur in a parallel module. Surprisingly, we detect the transcriptional activator subunit relA in all purifications, indicating that relA is complexed with all other members. As noted above, we cannot infer a direct physical interaction or the stoichiometry from such PCP experiments. Therefore, this finding suggests either that relA can form dimeric interactions with c-rel and relB, or the presence of higher-order tetrameric complexes involving at least two transcriptional activator subunits that are conjoined via either of the two DNA-binding subunits. Modular decomposition gives the rule of all potential tetramers, namely any combination where neither NFκB1/p50 and NFκB2/p52 nor c-rel and relB coexist. RelA appears with a central role in the tree. It is a necessary component of any potential tetramer formed of different NFκB dimers.

Next we analyzed the direct neighbors of NFκB members to obtain functional insight into the role of the various distinct NFκB complexes (Figure 7a). Here, the added proteins are not baits and therefore parallel modules are alternative, but not necessarily exclusive alternatives. Some proteins are specific to individual NFκB members. For instance, distinct members of the importin family of nuclear facilitator proteins have been co-purified with distinct NFκB subunits. In resting cells, KPNA2 and KPNA6 are identified only with relB. This is in line with the observation that relB is constitutively nuclear. Upon stimulation by TNF-α, a further member, KPNA3, co-purifies specifically with p50 21, reflecting that p50 only translocates in response to TNF-α. Beside modules of interactors that are specific to individual proteins, others consist of shared interactors. In this context two modules are worth mentioning. One contains two IκB proteins (IκB-α and IκB-β) and ABIN2, which have been identified with all NFκB members except relB. ABIN2 had previously been implicated in TNF-α signaling 24. Interestingly, overexpression of ABIN2 also inhibits TNF-α-induced activation of NFκB. However, the molecular mechanism has remained elusive. As ABIN2 appears in a module with IκB-α and IκB-β this suggests that ABIN2 exerts a function in directly modulating distinct NFκB complexes that do not contain relB, either by facilitating IκB-mediated retention in the cytosol or alternatively through an IκB-independent retention mechanism. The other module worth mentioning contains IKKα and Cot/Tpl2, which are common interactors of NFκB1/p50, NFκB1/p52 and relA. Cot/Tpl2 is a MAP kinase known as an agonist of NFκB, in particular by being implicated in the proteolysis of the precursor of p50, NFκB1/p105 25.

Therefore, modular decomposition delivers one module of inhibitors and one module of activators of NFκB. Relations within and between those two modules can be further investigated by selecting their member proteins as baits to capture the pattern of interactions among those proteins (Figure 7b). Confirming previous reports, we show here that IKKα binds constitutively to IκB-α and IκB-β. Interestingly, Cot/Tpl2 as bait co-purifies with ABIN2, but we did not observe any interactions between IκB-α or IκB-β and ABIN2, nor between IKKα and Cot/Tpl2. From this observation we hypothesize that Cot/Tpl2 may perhaps modulate ABIN2 in a manner akin to the action of IKK on the IκBs. As no directionality can be derived from interaction networks, ABIN2 may alternatively be a modulator of Cot/Tpl2.

Experimental procedures for TAP-tagged purification of complexes in the TNFα/NFκB signaling pathway are described in 21.

We followed the description of a practical algorithm for modular decomposition of graphs 37. We give free access to our implementation 38.