Method Modular decomposition of protein-protein interaction networks

We introduce an algorithmic method, termed modular decomposition, that defines the organization of protein-interaction networks as a hierarchy of nested modules. Modular decomposition derives the logical rules of how to combine proteins into the actual functional complexes by identifying groups of proteins acting as a single unit (sub-complexes) and those that can be alternatively exchanged in a set of similar complexes. The method is applied to experimental data on the pro-inflammatory tumor necrosis factor-α (TNF-α)/NFκB transcription factor pathway. Background Protein complexes and their shared components 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 [1-4]. 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 [6,7]. 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 Published: 21 July 2004 Genome Biology 2004, 5:R57 Received: 23 April 2004 Revised: 19 May 2004 Accepted: 7 June 2004 The electronic version of this article is the complete one and can be found online at http://genomebiology.com/2004/5/8/R57 Genome Biology 2004, 5:R57 R57.2 Genome Biology 2004, Volume 5, Issue 8, Article R57 Gagneur et al. http://genomebiology.com/2004/5/8/R57 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. Modular decomposition of graphs Modules A graph is a formal framework for representing elements and their relations. Elements are represented as nodes and a link connects two nodes of elements in relation. Nodes connected by a link are said to be neighbors. In graph theory, a module is a set of nodes that have the same neighbors outside the module (Figure 1). Quotient Because elements of a module have exactly the same neighbors outside the module, one can substitute all of them for a representative node. In a quotient, all elements of the module are replaced by the representative node, and the edges with the neighbors are replaced by edges to the representative. Quotients can be iterated until the entire graph is merged into a final representative node. Iterated quotients can be captured in a tree, where each node represents a module, which is a subset of its parent and the set of its descendant leaves. Modular decomposition The modular decomposition is a unique, canonical tree of iterated quotients. Formal proofs as well as generalizations to structures other than graphs have been described by Möhring et al. [8,9]. The nodes of the modular decomposition are labeled in three ways (Figure 2): as series when the direct descendants are all neighbors of each other; as parallel when the direct descendants are all non-neighbors of each other; and by the structure of the module otherwise (the so-called prime module case). Modular decomposition derives an exact alternative representation of a graph as a tree of labeled nodes. Protein-protein interaction networks 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 [6,7]. 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 A graph and its modules Fi ure 1 A graph and its modules. By definition, a module is a set of nodes that have the same neighbors outside the module. In addition to the trivial modules {a},{b},. ,{g} and {a,b,c,. ,g}, this graph contains the modules {a,b,c}, {a,b},{a,c},{b,c} and {e,f}. a