Genome-scale metabolic models have become a fundamental tool for examining metabolic principles. However, metabolism is not solely characterized by the underlying biochemical reactions and catalyzing enzymes, but also affected by regulatory events. Since the pioneering work of Covert and co-workers as well as Shlomi and co-workers it is debated, how regulation and metabolism synergistically characterize a coherent cellular state. The first approaches started from metabolic models, which were extended by the regulation of the encoding genes of the catalyzing enzymes. By now, bioinformatics databases in principle allow addressing the challenge of integrating regulation and metabolism on a system-wide level. Collecting information from several databases we provide a network representation of the integrated gene regulatory and metabolic system for Escherichia coli, including major cellular processes, from metabolic processes via protein modification to a variety of regulatory events. Besides transcriptional regulation, we also take into account regulation of translation, enzyme activities and reactions. Our network model provides novel topological characterizations of system components based on their positions in the network. We show that network characteristics suggest a representation of the integrated system as three network domains (regulatory, metabolic and interface networks) instead of two. This new three-domain representation reveals the structural centrality of components with known high functional relevance. This integrated network can serve as a platform for understanding coherent cellular states as active subnetworks and to elucidate crossover effects between metabolism and gene regulation.
So far, metabolic processes and gene regulatory events are typically considered individually in system-level investigations. However, ample evidence exists that the majority of cellular processes involves both, metabolism and gene regulation, and thus requires their joint examination [1]. One of the best-investigated individual examples in Escherichia coli (E. coli) is the phosphoenolpyruvate–carbohydrate phosphotransferase system (PTS) which is responsible for import and phosphorylation of sugars [2]. Additionally, the PTS is involved in the regulation of the import process depending on the available carbohydrate mixtures in the growth medium. By carbon catabolite repression and inducer exclusion, primarily the uptake of a preferred carbon source to be metabolized, such as glucose, is selected from other carbohydrates present in the growth medium. In order to understand the underlying principles, not only the effects of both ‘layers’, metabolism and regulation, need to be taken into account, but also their interface [3]. On a more qualitative level, the importance of the interface of metabolism and gene regulation can be illustrated by having a closer look at their most prominent representatives, namely, enzymes and metabolic transcriptional regulators. Both examples are proteins and can be thought of as a component type organizing the interplay of genes and metabolic reactions (Fig 1). For enzymes the connection is straightforward: The majority of metabolic reactions can only take place if the corresponding genes of the catalyzing enzymes are expressed. These genes, in turn, are often involved in regulatory processes, especially if they are associated with central biochemical reactions. In contrast, metabolic transcriptional regulators can be illustrated by looking at transcription factors, the probably best-investigated transcriptional regulators. Some of them require the binding of a metabolite to be active and are therefore called metabolic transcriptional regulators. In the context of the integrative view discussed here, it is noteworthy that only the interaction with a metabolic component enables their functionality as gene expression regulators. Conventional reconstructions of E. coli’s metabolism as well as of its gene regulation thoroughly describe the process itself but usually lack information on interacting elements of the other biological system. While there are numerous genome-scale metabolic reconstructions available [4–9], only a few large-scale transcriptional regulatory networks exist that are mainly based on the information from RegulonDB [10]. First attempts to integrate both cellular processes started from metabolic reconstructions which were expanded by regulatory genes and stimuli of the associated encoding metabolic genes [11,12]. Both studies started from the metabolic model of [5] and include 104 regulatory genes and 583 regulatory rules regulating approximately 50% of the metabolic genes. In this manner, the close proximity of regulatory events was captured but more far-reaching and global effects, e. g. , self-contained regulatory dynamics among genes, could not be considered. Further approaches examine the regulatory processes of the metabolic network based on the aforementioned pioneering attempts [13,14]. For this purpose, the information about regulatory events was assembled in terms of Boolean rules as a variant of Boolean network models. More recently, [15] introduced a method called probabilistic regulation of metabolism, a new variant of regulatory flux balance analysis, i. e. , the class of approaches behind some of the pioneering integrative models discussed above [11,12]. The state of the many variants of integrating regulatory information into flux-balance analysis models has been reviewed in [16] and [17]. The necessity of achieving such data integration, even on the network level, has recently been discussed in [18]. To a certain degree all these studies consider the regulation of metabolism but only cover the proximity rather than a genome scale. A very recent example of a successful topological characterization of biological networks, in order to understand the interplay of gene regulation and metabolism, is the analysis of gene regulatory—metabolic feedback loops [19]. In [19] joint representations of gene regulatory and metabolic networks have been compiled for two organisms, E. coli and B. subtilis. These network representations are then analyzed with a focus on the hierarchical structure of the resulting network and on feedback loops between gene regulation and metabolism. These feedback loops are then further characterized in terms of their potential impact (defined as the number of genes downstream of the transcription factor receiving the feedback from metabolism), their possible function (e. g. , in the processing of environmental information) and other properties [19]. In this way, the authors obtain insight in the algorithmic features of the interface between gene regulation and metabolism. Understanding the interplay of metabolism and gene regulation will help to gain insight in cellular, system-wide responses such as to changing environmental conditions. Here, we present the database-assisted reconstruction of an integrative E. coli network capturing metabolic as well as regulatory processes. The attribution of network components (in terms of individual vertices) to the metabolic and regulatory domains, as well as the protein interface enables the further characterization of the network in terms of its modular organization, its path statistics and the vertex centrality. In particular, we formulate a new measure by evaluating domain-traversing paths, in order to quantitatively assess the role of components in the interface domain and thus identify cross-systemic key elements contributing to both regulatory and metabolic processes. In all cases, these topological assessments highlight system components and functional subsystems, which are well known for their biological relevance, thus emphasizing the predictive power of network topology. Employing observations on the topological (structural, network-architectural) level, in order to identify components in the system of particular functional relevance has a long tradition in network biology (and in network science in general). The main results of our investigation are: We present an integrated network representation of gene regulation and metabolism of E. coli and illustrate how it is a promising starting point for the structural investigation of system-wide phenomena. In particular, the network perspective suggests the explicit consideration of a protein interface between the genetic and metabolic realms of the cell. Employing network metrics we (1) argue that a three-domain partitioning is architecturally and functionally plausible, and (2) show that prominent components of the network according to the structural investigation tend to be of evident biological importance. Especially, the evaluation of possible paths through the interface domain of the network reconstruction yields well-known functional subsystems. The overlap of structural and biological relevance here suggests that a careful analysis of such a structural model can guide biological investigations by focusing on a limited number of structurally outstanding components. This network model can also serve as a starting point for a range of topological analyses with methods developed in statistical physics (see, e. g. , [20] for a recent review). Summarizing, in contrast to the separate analyses of (e. g. , the metabolic or gene regulatory) subsystems, we expect that the integrative network model shown here will draw the attention to system-wide feedback loops not contained in the individual subsystems and to different roles of individual components, which become only visible from the perspective of interdependent networks.
By now, the dramatic growth of bioinformatics databases [21], both in content and in diversity, allows addressing the challenge of integrating regulation and metabolism on a system-wide level. We devised a semi-automated framework to integrate information from EcoCyc database [22] and RegulonDB [10] into a network for E. coli including major cellular processes, from metabolic processes via protein modifications to a variety of regulatory events (see Methods). Networks are an efficient data structure for integrating this wealth of information [23–25]. In this way, the vast amount of data contained in the bioinformatics databases provide an ‘architectural embedding’ for metabolic-regulatory networks and guides subsequent steps of model refinement and validation. We augmented and validated the resulting network based on existing reconstructions of metabolic [6,8, 26–28] as well as of gene regulatory processes [10]. The integrative E. coli network constructed here comprises the three major biological components, genes, proteins, and metabolites, as well as the metabolizing reactions summing up to more than 12,000 components. Represented as a graph, the network has seven types of vertices depicting the major biological components (Fig 2, Table A in S1 Text) and seven different types of edges including two types of encoding associations, i. e. , transcription and translation processes, four types describing the associations within biochemical reactions, and one type summarizing regulatory relations (Table B in S1 Text). Two small annotated biological examples are shown in Fig B in S1 Text. The graph representation facilitates the mapping of reactions and their catalyzing enzymes, as both are depicted as vertices. In contrast, metabolic systems are often represented as hypergraphs to illustrate the Boolean ‘AND’ association of reaction educts and the fixed stoichiometric ratio of the involved metabolites. Those aspects are assigned explicitly as edge properties in the graph representation. For the purpose of measuring the propagation of perturbations through the network, for example, the following logical assignments are helpful (see [29] for details on these definitions): Besides the associations of reaction educts, the encoding relations of protein complexes are of Boolean ‘AND’ type, termed conjunct links. On the contrary, associations representing isoforms of protein subunits, isoenzymes as well as reaction products are implemented by Boolean ‘OR’ links, called disjunct. The third linkage type, regulation, covers approximately 7,300 regulatory associations, i. e. , transcriptional, translational as well as metabolic ones (Table C in S1 Text). The comparison with existing models reveals that the presented integrative network is a comprehensive representation of the metabolic and regulatory processes in E. coli. The very first approach of embedding metabolic processes in the regulatory context of [11], the iMC1010 model, started from a metabolic model which was extended by the regulation of the encoding genes of the catalyzing enzymes. For the purpose of determining the overlap of the integrative metabolic-regulatory network and the iMC1010 model, transport reactions as well as the artificial biomass reaction have been disregarded and, moreover, only unique metabolites (neglecting compartmentation) have been taken into account. Else, the different levels of details of the transport systems such as PTS as well as of the compound compartmentation would render a correct mapping impossible. Overall, the iMC1010 model is covered by our model to more than 89% (Fig 2, see Table D in S1 Text, column 3). To assess the coverage of E. coli’s metabolic processes, the embedded metabolic processes of the integrative E. coli network have been associated to the ones of an established E. coli metabolic reconstruction, namely the iAF1260 model from [6]. About 67% of the involved biochemical reactions, compounds and genes could be mapped directly (see Table D in S1 Text, column 4). Particularly, these two thirds capture almost all biologically relevant components in terms of in silico viability. Using flux balance analysis for simulating the biomass production capacity of the iAF1260 model and taking the overlap with mapped components of the integrative E. coli network revealed that for the default medium setup approximately 75% of the essential reactions (to yield 1% biomass) are covered by the integrative E. coli network. Analogous to the metabolic processes, the coverage of E. coli’s gene regulation has been determined using the transcriptional regulatory network from RegulonDB [10]. This model has been assembled in a similar fashion but is accounting only for transcription factors and their regulated genes. With a coverage of more than 98%, the transcription-related regulatory processes are considered as completely recorded in the integrative E. coli network (see Table D in S1 Text, column 5). Apart from that, for this assessment of overlap a comparison of regulatory processes associated with RNA translation as well as metabolic regulatory events is not possible since the RegulonDB transcriptional regulatory network does not consider protein and metabolic interaction processes. The most conspicuous links between metabolic and gene regulatory processes are metabolic transcription factors, i. e. , gene expression regulators binding metabolites, and metabolic genes, i. e. , genes with significant and coordinated response on the metabolic level such as encoding enzymes. Intuitively, the interface is considered so far as the direct interactions of metabolic elements and gene regulatory elements, and the integrative E. coli network can be partitioned into metabolic and regulatory domain (MD—RD). However, by examining those interactions in more detail the topological role of proteins becomes apparent. Regarding the metabolic transcription factors, the respective metabolite binds to a protein and this metabolite-protein complex then subsequently regulates the gene expression. In the case of metabolic genes, ultimately the respective gene encodes a protein which either by itself or as a complex serves as an enzyme. In line with this, the interface of metabolic and gene regulatory processes should be considered as the series of interactions of metabolites and genes, respectively, with proteins and subsequent protein modifications. Thus, the interface does not only comprise interactions (edges) but also components (vertices), and the integrative E. coli network will in the following be divided into a metabolic domain, a protein interface and a regulatory domain (MD—PI—RD). In the next section, the plausibility of the three-domain partition (and the set of biologically motivated rules devised to create it) will be assessed in comparison to the likewise proposed two-domain (MD—RD) representation. The integration of metabolic and regulatory events allows us to determine the key elements of E. coli, especially those beyond the individual processes. In particular, the functional three-domain partition facilitates to recover network components (in terms of individual vertices) of evident biological relevance, e. g. , by means of simple centrality measures. In the following, two different aspects of centrality have been examined [40]: degree centrality depicting the direct linkage of a vertex, and betweenness centrality which can be thought of as the participation of a vertex in the network flow [41]. Starting with the prominent local vertex structure, the so-called hubs (here, vertices with a total degree larger than 50), it is noticeable that they are primarily compounds and proteins, in particular protein complexes and appear in all three domains (see Table H in S1 Text, columns 3–5). In the metabolic domain, hubs include trivial compounds such as H+ and H2O and, so-called, currency metabolites, e. g. , ATP, NAD (P) H and coenzyme A, while hubs of regulatory processes are obviously global regulators which characteristically exhibit a remarkably strong asymmetry of in-degree and out-degree. Particularly, well-known transcriptions factors top this list such as FNR (fumarate and nitrate reduction) [42], Fis (factor for inversion stimulation) and H-NS (histone-like nucleoid structuring protein) [43]. As stated above, hubs predominantly occur in metabolic and gene regulatory domain while only a few are affiliated to the protein interface. However, it was not to be expected to identify cross-systemic elements solely based on their degree. To assess/detect cross-systemic key elements an extended approach of degree centrality has been used that additionally accounts for the domain boundaries. The intra-domain degree fraction ξ, also termed embeddedness [44], denotes the ratio of the internal degree of a vertex, within a domain, and the total degree in the network. This measure very clearly distinguishes between, on the one hand, metabolic and regulatory hubs which show intra-domain degree fractions ξ > 0. 87 (except one single compound with ξ = 0. 185) and hubs in the interface which in contrast have ξ ≤ 0. 06 (see Table H in S1 Text, last column). Thus, while metabolic and regulatory hubs are embedded in their respective domains, hubs in the protein interface are mainly connected to vertices in the neighboring domains. In total, seven hubs show a significant low intra-domain degree fraction pointing to their prevalent interactions with the other two domains (Fig A in S1 Text and Table K in S1 Text, column 5). Six of them are affiliated to the protein interface exhibiting numerous interactions with the regulatory domain. Their linkages to the metabolic domain become visible when considering their composition, in case of the protein complexes, and their modes of action, respectively. The former involve the four protein-compound complexes Crp-cAMP (cyclic-AMP receptor protein binding cyclic-AMP) [31,45,46], DksA-ppGpp (dnaK suppressor binding guanosine 3’-diphosphate 5’-diphosphate) [47–49], NsrR-NO (nitrite-sensitive repressor binding nitric oxide) [50–52] and Lrp-Leu (leucine-responsive regulatory protein binding leucine) [53–55] whose naming schemes already indicate the metabolic link. The latter, namely, protein complex Cra (catabolite repressor activator) and protein monomer Lrp (leucine-responsive regulatory protein) form in the presence of appropriate metabolites, i. e. , fructose 1,6-bisphosphate/fructose 1-phosphate and leucine, complexes affecting their regulatory effect. The remaining hub is the metabolic-domain vertex representing guanosine 5’-diphosphate 3’-diphosphate (ppGpp). Besides its special domain-affiliation among the low intra-domain degree hubs, ppGpp acts as an important regulator of both, metabolism and transcriptional processes. More precisely, it regulates several enzyme activities as well as numerous transcription initiations by allosterically binding to RNA polymerase. So far, we demonstrated that the protein interface of the E. coli network reconstruction acts as a bridging module between regulatory and metabolic domain enabling their interaction and communication. Therefore, we expect the betweenness centrality to directly highlight vertices from the interface. Indeed, ten out of the top-25-ranked (still including currency metabolites) vertices are from the interface (see Table I in S1 Text, column 5), while overall the interface only accounts for about 18% of the vertices of the network. Especially, the already mentioned protein-compound complexes Crp-cAMP and DksA-ppGpp are among these compounds. In general, currency metabolites and trivial compounds (see above) as well as global regulators are among the central components with respect to betweenness. Apart from that, biochemical reactions building up and/or breaking down these metabolites and proteins as well as the other involved reactants pertain to the most betweenness-central components. Component association to functional systems allows to assess the systemic feature and by considering the corresponding network affiliation to depict the candidates for cross-systemic key elements. In this manner the network analysis allows us to detect the central role of Crp-cAMP, Lrp-Leu and ppGpp on purely topological grounds, as each component is the focus of such a functional system with high betweenness. Additionally among the top-ranked vertices with respect to betweenness centrality are five further cross-systemic components which are assigned to the protein interface, namely, phosphorylated PhoB (PhoB-P), Fur-Fe2+, and three outer membrane proteins (Omp), OmpC, OmpE and OmpF (Table I in S1 Text). The former two components are transcription factors and therefore acting in the gene regulatory domain, while at the same time they are protein complexes binding a metabolic small molecule depicting the connection to the metabolic processes. The latter three, the outer membrane porins, form hydrophilic channels, enabling non-specific diffusion of small molecules across the outer membrane [56–58]. In this role these proteins represent the most obvious connections of gene regulatory and metabolic domain—their encoding genes are highly regulated while the porins enable numerous metabolic transport reactions. By focusing on the connecting domain of gene regulation and metabolism, the two centrality measures reinforce the key role of further cross-systemic elements. Considering the protein interface-induced subgraph both centralities point out the vertices that top the list of the above-discussed downwards traversing paths (Table J in S1 Text). In more detail, both major systems contributing to the downwards traversing paths are represented each by three vertices, namely, PTS and RNR system (Fig 7, panels A and B). Having a look at the intra-domain degree fraction, which put the focus on protein interface vertices as described above, additionally highlights a representative of the upwards traversing path system NtrBC (Fig 7, panel C), as the second non-hub (Table K in S1 Text). This corroborates the predictions from the traversing paths and, thus, shows that our new topological measure reveals cross-systemic elements which otherwise only stand out under detailed scrutiny of a large amount of biological information.
With their balance of structural detail and functional simplicity, network models are capable of revealing organizational principles, which are hard to recognize on a smaller systemic scale (e. g. , by analyzing individual pathways) or in functionally richer system representations (e. g. , in dynamical models). One purpose of the network provided here is to enable work at the interface of statistical physics and systems biology, where the rich toolbox of complex network analysis is employed to identify functionally relevant non-random features of such biological networks. The recent work of [69], for example, showed that network structure can reveal, whether an enzyme is susceptible rather to genetic knockdown or pharmacologic inhibition. While in the present study, the network measures do not distinguish between different kinds of vertices or links, the rich biological meta data concerning the different biological roles of the components could be translated into distinct vertex and edge classes. In our own investigation [29] we used this fact to study, in a further example of such an interdisciplinary effort, the balance of robustness and sensitivity in the interdependent network of gene regulation and metabolism, based on the reconstructed network provided here. In general, we expect that our network reconstruction can serve as a relevant data resource for the application of methods from the analysis of multiplex [70] and other multilayer networks [20,71]. Recently, there has been a growing interest in the properties of these systems, especially in the presence of explicit interdependencies between vertices [70,72]. In contrast to monoplex networks interdependent networks can show a qualitatively different robustness against failures, i. e. , cascading failures leading to a sudden system breakdown at a critical initial attack size [73,74]. The case of different vertex types (as opposed to different edge types) has been considered, for example, in the context of secure communication in a network where eavesdroppers control sets of vertices [75]. On a general level, analyzing statistics of paths with respect to the network’s large-scale structure, like the domain-traversing paths used here, might prove useful for the evaluation of other networks that show (possibly more than one) interface-like features. In summary, the analysis of network topology allows to determine key system components in the integrative E. coli network. In line with expectations, trivial compounds as well as currency metabolites showed up regardless of the measure that has been applied. In addition, further obvious components including several global regulators were identified. More striking is the detection of components and systems which solely emerge when analyzing specifically the interface. These hidden elements are associated to two of the biologically well-investigated functional subsystems, PTS and NtrBC. Both well-established and newly designed measures of the interface point out the same subsystems, and even the analysis of the entire network discloses components indirectly related to these hidden subsystems. Apart from trivial and currency metabolites, every detected key element of the entire network contributes to some extent to the downwards and/or upwards interface. This unlooked-for cross-systemic property is reflected either in the complex composition, the intra-domain degree fraction, the proximity to key systems, and/or the interplay with regulatory and metabolic processes. The biological relevance of these components supports their detection and reinforces the predictive power of the novel traversing path measure. In general, we believe that the presented integrative E. coli network allows further investigations of the interplay of metabolism and gene regulation which will provide insights into cellular, system-wide responses.
First, relevant information of the database has been extracted and arranged (Algorithm 1 in Fig 9). For each regulatory process, the respective source and target were specified and converted to match one of the vertex types (‘regulation. dat’, file name of the EcoCyc-archive). To this end, the transcript units were separated into promoter, genes and terminator (if applicable), and the regulatory processes were multiplied per comprising gene. Moreover, each regulating RNA has been translated into its encoding gene to meet the vertex types. In case of the metabolic processes, the reaction educts and products as well as the catalyzing enzymes have been assembled and converted to match one of the vertex groups, the respective educt and product stoichiometry have been assigned and the reaction compartmentation and reversibility have been assessed (‘reactions. dat’). Thereby, as cell compartments the periplasmic space, the inner membrane, and the cytosol have been taken into account and reversible reactions have been split up. Second, vertex candidates have been validated (‘reactions. dat’, ‘compounds. dat’, ‘proteins. dat’, ‘genes. dat’, ‘rnas. dat’) and divided into reaction, compound, protein monomer, protein-protein complex, protein-compound complex, protein-RNA complex, and gene. In doing so, generic terms such as DIPEPTIDES have been substituted (‘classes. dat’) and double annotations, e. g. , CPD-15709 and FRUCTOSE-6P have been decoded. Thereupon, the compositions and the encoding genes of the assembled proteins have been gathered and matched to the vertex groups and the respective logical operation and stoichiometry have been annotated (‘protcplxs. col’). Based on the validated vertex lists, the regulatory and metabolic processes have been updated whereby each process was removed with at least one unidentified vertex resulting in the final edge lists. Fig C in S1 Text provides a flowchart of this algorithmic procedure. With the validated vertex and edge lists the graph has been assembled and its largest weakly connected component has been extracted. The three domain partition MD—PI—RD (Table 1 and A in S1 Text) as well as the two-domain partition are implemented as vertex properties affiliation and metabolic. The initial categorization of both partitions is based on the vertex type ‘reaction’ which is denoted as purely metabolic and interface-related if all educts and products are compounds and proteins, respectively. Mixed educt and product types demand further clarification later on. Similarly, non-ambiguous vertices of type ‘compound’, ‘protein’ and ‘gene’ are affiliated based on the affiliation of their neighbor vertices. This means that, if the influential adjacent vertices have the same affiliation, the vertex will be assigned to the same or its assignment needs a detailed consideration. In this way, genes and proteins that are not involved in any regulatory process can be assigned to the metabolic domain (MD). Subsequently, deferred vertex affiliations are resolved iteratively based on their neighbor vertices affiliation until no further vertex affiliations can be assigned. The final ambiguous vertices, in total 13 of 12868, are assigned as interface vertices since they cannot be uniquely assigned to metabolic or regulatory domain. Algorithms 2A and 2B in Figs 10 and 11 show the detailed domain affiliation process of a vertex. Moreover, the mapping to the E. coli model of [11] has been annotated which integrates the metabolic network iJR904 published by [5] and the transcription regulatory events related to the encoding genes of the catalyzing enzymes. To this end, genes, proteins, metabolites as well as biochemical reactions of the metabolic model have been mapped to the EcoCyc database (release 20. 0), in a first step automatically based on their identifier and the resulting dictionaries have been manually curated. As the EcoCyc database does not account for compartmentation of compounds and reactions as well as for exchange reactions, unique metabolites and internal reactions have been considered resulting in a coverage of more than 93%. By additionally disregarding internal transport reactions a coverage of 96. 5% can be achieved (Table 1). Integrating the manually curated Covert dictionaries, each vertex has attributed (1) a unique identifier, according to the EcoCyc identifier but also indicating the compartment, (2) a unique type reference, (3) a unique assignment of the model components from [11], if applicable, and (4) the affiliations of the two- and three-domain partition. Furthermore, vertices of types gene and reaction have (5a) a name assigned, the Blattner ID and the EC number, if applicable. The remaining vertices have additionally (5b) a compartment assigned, where cytosol (c), extracellular space (e), periplasmic space (p), inner membrane (i), outer membrane (o) and membrane in general (m) were taken into account. Similarly, each edge of the network has the attribute (1) type, specifying the connected vertices, and the corresponding (2) stoichiometry, where zero is assigned if not applicable or ambiguous. For edges depicting regulatory processes the stoichiometry actually denotes the mode of regulation, namely activation (+, 1) inhibition (−, −1) or combined (0). These edges additionally have assigned (3) an identifier, according to the EcoCyc identifier and (4) a name, specifying the regulation type. All other edge types can be classified as either representing conjunct or disjunct links in the sense that all or solely one incoming link is required for functionality (Table B in S1 Text). The fully annotated integrative reconstruction of E. coli’s metabolic and regulatory processes is provided as a graph representation in S1 File. The following measures have been used in the assessment of the graph partitioning scheme. Inter-module edge fraction c. Given the set of vertices with the domain label D, edges connecting these vertices to a vertex of a different label are considered external, while edges between vertices of the same label are internal. We call c D = # (external edges of D) # (external + internal edges of D) the inter-module edge fraction of domain D. Network modularity M. denotes the degree to which a given partition divides the network in highly connected groups, modules, which are comparably sparsely connected among each other. Therefore, the intra-module links are counted against the total degree of the module vertices (Eq 1), M = ∑ j = 1 N M (L (v M j, w M j) L G - (deg (v M j) 2 L G) 2) (1) with NM − # of modules, LG − # of links of graph G, L (v M, w N) = ∑ v ∈ M ∑ w ∈ N link (v, w), deg (v M) = ∑ v ∈ M k v. Here kv is the degree of vertex v and link (v, w) denotes an undirected edge between vertices v and w. Note that this formulation of modularity (taken from [30]) coincides with the definition from [76]. A traversing path connects the regulatory and the metabolic domains via the protein interface, specifically, a traversing path of length k is of the form [ (u, v 1), (v 1, v 2), … (v k - 1, w) ] (2) where the vertices u and w are from the regulatory and the metabolic domain (and vice versa) and the vertices vi are distinct and part of the protein interface. Starting from the set of edges directly at the intersection of two domains iteratively the vertex successors of the interface domain as well as the final, first successor in the third domain have been determined (Algorithm 3 in Fig 12). The key elements of the integrative E. coli network have been determined based on two graph properties. Degree Centrality DC. is a local centrality measure and denotes the total number of in- and out-going edges of a vertex, (Eq 3), D C (v) = k v = k v in + k v out. (3) Here, the vertices with a total degree greater than 50 are termed hubs (see the degree distribution in Fig D in S1 Text). By additionally accounting for the domain boundaries, the intra-domain degree fraction ξ (also termed embeddedness [44]) has been defined as ratio of internal degree, within domain D, and total degree of a vertex, (Eq 4), ξ D (v) = k v int k v = 1 k v ∑ w ∈ D (A v w + A w v) (4) where A denotes the adjacency matrix of the graph. Betweenness Centrality BC. describes the impact on the flux through the network, under the assumption that the transfer follows the shortest paths. In particular, it quantifies the fraction of shortest paths between all pairs of vertices which involve the designated vertex (Eq 5), B C (v) = ∑ s ≠ v ≠ t ∈ V σ s t (v) σ s t (5) where σst is the number of all shortest-paths between the vertices s and t while σst (v) yields the number of these paths that run through v [41].
Networks—the compact representation of systems in terms of nodes and links—are an efficient data structure for biological information. They also allow us to establish relationships between network structure and dynamical function and thus hold the potential of implementing a systems-level view on biological processes. Using the formal language of networks and careful manual curation, we unite information from a range of publicly available databases, in order to provide a metabolic-regulatory network model for the gut bacterium Escherichia coli, which allows us to provide novel topological characterizations of system components based on their positions in the entire network. From the network representation we derive a new partition of the system into three network domains, one predominantly associated with gene regulation, a second, which covers all metabolic processes and a third domain, containing protein interactions and serving as an interface between the two other domains. This has consequences for the topological prediction of the biological relevance of the system components. We discuss specific examples, where this new three-domain representation reveals the structural centrality of components with known high functional relevance. This integrated network can serve as a platform for understanding biological phenomena jointly mediated by gene regulation and metabolism and thus provide insight relevant for genetic and metabolic engineering.
protein metabolism
gene regulation
metabolic processes
regulatory proteins
metabolic networks
enzymology
dna-binding proteins
network analysis
enzyme metabolism
transcription factors
enzyme chemistry
computer and information sciences
transcriptional control
proteins
enzyme regulation
gene expression
biochemistry
genetics
biology and life sciences
metabolism
A system-wide network reconstruction of gene regulation and metabolism in Escherichia coli