Path finding methods accounting for stoichiometry in metabolic networks
 Jon Pey^{1},
 Joaquín Prada^{1},
 John E Beasley^{2}Email author and
 Francisco J Planes^{1}Email author
DOI: 10.1186/gb2011125r49
© Pey et al.; licensee BioMed Central Ltd. 2011
Received: 1 March 2011
Accepted: 27 May 2011
Published: 27 May 2011
Abstract
Graphbased methods have been widely used for the analysis of biological networks. Their application to metabolic networks has been much discussed, in particular noting that an important weakness in such methods is that reaction stoichiometry is neglected. In this study, we show that reaction stoichiometry can be incorporated into pathfinding approaches via mixedinteger linear programming. This major advance at the modeling level results in improved prediction of topological and functional properties in metabolic networks.
Background
The use of graph theory in the analysis of biological networks has been extensive in the past decade [1]. Particularly, in metabolic networks different relevant topics have been examined using the rich variety of graphtheoretic concepts, ranging from topological properties [2–5], evolutionary analysis [6–8], pathway analysis [9–13], transcriptional regulation [14–16], functional interpretation of 'omics' data [17–20] and prediction of novel drug targets [21–23].
Graphbased methods start by converting the metabolic network into an appropriate graph. Different representations are possible here: i) metabolite graphs, where nodes are metabolites and arcs represent reactions linking an input and output metabolite; ii) reaction graphs, in which nodes are reactions and arcs represent intermediate metabolites shared by reactions; iii) bipartite graphs, where nodes are reactions and metabolites, while arcs link metabolites to reactions (for substrates) and reactions to metabolites (for products). Note here that each type of graph can be either directed or undirected. A deeper introduction to such graphs can be found in Deville et al. [24].
Importantly, graphbased methods rely on the definition of connectivity based on paths, that is, two nodes in the graph are connected (or not) depending upon whether (or not) we have a path linking them. This definition of connectivity is debatable, however, particularly when it is claimed that such a path is a competent metabolic pathway, as recently discussed [25–27]. In this context, the major criticism raised as to pathfinding methods is that they neglect reaction stoichiometry and there is, therefore, no guarantee that any path found can operate in sustained steadystate.
The steadystate condition requires the definition of the boundary of the metabolic network under study. Metabolites inside the boundary of the network, typically called internal metabolites [28], must be in stoichiometric balance. Balancing does not apply to metabolites outside the boundaries of the system (external metabolites), which are typically input/output metabolites and (sometimes) cofactors. In other words, for internal metabolites, their production and consumption (if possible) must be captured with the reactions in the network under study.
The steadystate condition and its underlying boundary definition are critical for the performance of any method for analyzing a metabolic network and ignoring it may provide misleading insights. A nice illustration of this is the one presented in the work of de Figueiredo et al. [25], which (unsuccessfully) tested the ability of pathfinding methods to answer the question as to whether (or not) fatty acids can be converted into sugars. Klamt et al. [29] also recently emphasized this issue for different biological networks.
Note here that elementary flux modes (and extreme pathways) represent a more general and elegant concept for metabolic pathways than paths [28, 30]. Their computation is, however, much more expensive in large metabolic networks than paths and, though different efforts have been made in this area [31–33], much research is still needed to make elementary flux modes a practical tool for the analysis of large metabolic networks.
Given the limitations discussed above, a novel theoretical concept termed flux paths is introduced here. A flux path is a simple path (in the graphtheoretical sense, so no nodes revisited) from a source metabolite to a target metabolite able to operate in sustained steadystate. In essence, flux paths incorporate reaction stoichiometry into traditional pathfinding methods [4, 7, 34, 35]. By means of this concept we show that the path structure of metabolic networks is substantially altered when stoichiometry is considered. In addition, we illustrate (with several examples) that flux paths offer new perspectives for the analysis of metabolic networks at the topological and functional levels. The determination of flux paths requires going beyond graph theory via mixedinteger linear programming. We present below details as to our mathematical optimization model for determining Kshortest flux paths between source and target metabolites.
Results and discussion
Mathematical model
Assume we have a metabolic network that comprises R reactions and C metabolites. Note here that reversible reactions contribute two different reactions to the metabolic network. For this reason we can regard all fluxes as taking positive values. Let S_{cr} be the stoichiometric coefficient associated with metabolite c (c = 1,...,C) in reaction r (r = 1,...,R). As usual in the literature [28], input metabolites have a negative stoichiometric coefficient, whilst output metabolites have a positive stoichiometric coefficient.
Suppose that we are concerned with finding a flux path from a source metabolite α to a target metabolite β. As mentioned above, a flux path is a simple path from the source metabolite α to the target metabolite β able to operate in steadystate. We present below our mathematical optimization model for flux paths.
Path finding constraints
We need to decide the arcs involved in the flux path from the source metabolite α to the target metabolite β. This fact is represented with a zeroone (binary) variable u_{ij}, where u_{ij} = 1 if the arc i→j linking metabolite i (i = 1,...,C) to metabolite j (j = 1,...,C) is active in the flux path, 0 otherwise.
Deletion of arcs from the metabolic graph is standard practice in pathfinding methods [4, 7, 34, 35]. We removed arcs not involving an effective carbon exchange. Carbon exchange is indeed essential for metabolic purposes. For this reason, we henceforth use the term carbon flux paths (CFPs).
Note here that a similar criterion has been used in [35]. In this work, however, input and output metabolites can have any type of atom or atom groups in common. This criterion is illustrated in Figure 1b, where PEP donates a phosphate group to glucose (DGlc). The focus on carbon atoms makes our approach more restrictive, as observed in Figure 1c, which shows that there is only effective carbon exchange between DGlc and glucose 6phosphate (G6P), and PEP and Pyr.
Let d_{ijr} be a binary (0/1) coefficient establishing whether (or not) there exists an effective carbon exchange between input metabolite i (S_{ir} < 0) and output metabolite j (S_{jr} > 0) in reaction r. If , so there is no reaction involving metabolites i to j in carbon exchange, then u_{ij} is also fixed to zero.
Stoichiometric constraints
Equations 1 to 4 define a simple path that preserves carbon exchange in each of its intermediate steps. We need to guarantee that this path can work in sustained steadystate. As will be shown below, to do this, it is required to find a steadystate flux distribution able to involve the path. We here introduce variables and constraints needed to define the steadystate flux space.
In addition, it guarantees that v_{r} is nonzero if z_{r} = 1. Here we have scaled fluxes so that the maximum flux is M and the minimum (nonzero) flux is 1. This does not constitute an issue if we consider M sufficiently large.
Current pathfinding approaches deal with this situation indirectly, namely by removing computed paths involving a reaction and its reverse.
Equations 5 to 8 define the steadystate flux space for a particular metabolic network.
Linking path finding and stoichiometric constraints
Equation 9 ensures that if an arc i→j is active in the CFP (so u_{ij} = 1), then at least one reaction r containing this arc in carbon exchange (so d_{ijr} = 1) is forced to be active. By forcing z_{r} to be 1 there will be a nonzero flux associated with the reaction due to Equation 7. An important point to note from Equation 9 is that it allows reactions to be active even if they are not involved in the CFP. In other words reactions can be active with nonzero flux (to satisfy the requirements of steadystate, Equation 5) but without any of their input/output metabolites being involved in the CFP.
Objective function
Enumerating constraint
Validation
This section is organized as follows. By means of several welldocumented examples, we first illustrate the biochemical relevance of particular constraints in our CFP approach. We then carry out a sidebyside comparison of our CFP approach with current pathfinding approaches.
Pathfinding comparison
As shown in the 'Mathematical model' section, the pathfinding strategy used in our CFP approach is based on using arcs involving effective carbon exchange and imposing the reversibility constraint, Equation 8. In this subsection we illustrate the importance of these factors and show that a pathfinding approach incorporating them outperforms existing methods in the literature. For this analysis, the effect of stoichiometry is not considered, as is common in existing approaches. Its effect will be separately considered in detail in the next subsection ('Effect of stoichiometry'). Therefore, for this analysis, Equations 5 and 6 were ignored.
Effective carbon exchange
Reversibility
Pathfinding methods typically split reversible reactions into two irreversible steps. In contrast to current approaches [13], in our CFP approach we prevent two such irreversible steps from being active in the same path, as observed in Equation 8. To illustrate the importance of this constraint, we analyzed the shortest path from DGlc to Pyr in E. coli, which is the EntnerDoudoroff pathway, as shown in the lefthand side of Figure 3b. When we applied our CFP approach from DGlc to Pyr without including Equation 8, we obtained the path in the right handside of Figure 3b (DGlc→AcGlcD→AcCoA→LMal→Pyr). This solution has no biochemical meaning, since the first and second step in that path is a cycle involving the forward and backward step of the reversible reaction catalyzed by Dglucose Oacetyltransferase (GLCATr: DGlc + AcCoA ↔ AcGlcD + CoA). By adding Equation 8 this path is removed from the solution space and our CFP approach directly obtains the EntnerDoudoroff pathway.
Sidebyside comparison
In order to analyze the performance of any pathfinding method, it is usual in the literature to evaluate its ability in recovering wellknown metabolic pathways. For this purpose, we used a database of 40 reference E. coli (metabolic) pathways previously discussed in Planes and Beasley [37] (these 40 pathways are listed in Additional file 2).
The input metabolic graph was built from the genomescale metabolic network of E. coli [36]. We computed the 100 shortest CFPs between the source and target metabolites of each of the 40 reference pathways. As mentioned above, stoichiometric constraints are not considered in this subsection since the aim is to establish the effectiveness of carbon exchange when combined with reversibility in path finding. To compare the 100 shortest CFPs and the reference pathway, we used the recovery rate. Recovery is a 0/1 parameter, being 1 if a CFP fully matches with the reference pathway, 0 otherwise.
A similar analysis was conducted for existing pathfinding methods [4, 7, 34, 35]. These methods make use of different strategies to provide biochemical meaning to the computed paths. For comparison, we classified these strategies into different groups: the first strategy (denoted 'topology') involves the use of an unadjusted metabolic graph; the second strategy (denoted 'hubs') adjusts the metabolic graph by removing any arc involving a highly connected metabolite (hubs) [7, 34] (we took the list of hubs from Planes and Beasley [37]); the third strategy (denoted 'connectivity') assigns weights to metabolites according to their connectivity in an unadjusted metabolic graph, where connectivity is defined to be the number of reactions involving a metabolite [9]. Finding Kshortest paths is substituted here by finding K lightest paths, that is, the sum of weights of arcs involved in the path is minimized.
Note here that there are pathfinding strategies that use structural atomic mapping information. These approaches can be classified into two different groups. In the first group atomic mapping is used to build the metabolic graph, that is, an input metabolite is linked to an output metabolite in a given reaction if they share an atom mapping. In other words, an arc between a given pair of input/output metabolites exists if they have atoms in common in at least one reaction. The work of Faust et al. [35], based on the RPAIR database [38], is a reference example for these approaches. The effective carbon exchange strategy used in our CFP approach also falls into this group. However, it is slightly more restrictive than the approach presented in Faust et al. [35], since we exclusively focus on carbon atoms, that is, an arc between a given pair of input/output metabolites exists if they have carbon atoms in common in at least one reaction.
In the second group atomic mapping is used to guarantee that the pathway target metabolite involves at least one atom from the source metabolite. This concept was first introduced by Arita et al. [39], and recently revisited in Blum and Kohlbacher [40], and Heath et al. [41]. We are aware that this type of approach is, in theory, more restrictive than the effective carbon exchange strategy used in our CFP approach, since we guarantee effective carbon exchange between intermediates in the path, but not between the source and target metabolites. Tracing an atom from source to target metabolite, however, requires detailed knowledge of carbon atom mappings for each reaction. Though active research is being undertaken into this topic, more effort is still needed to release a fully curated and complete database for atomic mappings in genomescale metabolic networks, especially for those from the Biochemical Genetic and Genomic (BiGG) database [42], which we are using here. For completeness, we will include results for the most recent approach [41], denoted as atom mappingbased strategy. Results were extracted from the web service (named AtomMetaNetWeb) available from Kavraki's lab [43].
Finally, note that other works [9, 13] typically used the accuracy rate, instead of the recovery rate, for comparing the computed paths and reference pathways. We repeated the same analysis using this parameter. As observed in Additional file 2, a similar result to Figure 4 is obtained, which again shows that our CFP approach outperforms current methods.
Effect of stoichiometry
To illustrate the effect of stoichiometry, we first analyze a previously considered example from the literature, which emphasizes the fact that some paths (at the graphtheoretical level) cannot perform in steadystate and therefore are not biologically meaningful. We then repeat the sidebyside comparison presented in Figure 4 when stoichiometry is considered. To emphasize its importance, we examine how the connectivity structure of several metabolites is altered when stoichiometry is considered.
Stoichiometry and infeasible paths
Sidebyside comparison with stoichiometry
We repeated the sidebyside comparison previously presented in Figure 4 for pathfinding methods when stoichiometry is considered. Similarly, we used the 40 E. coli metabolic pathways discussed in Planes and Beasely [37], and the E. coli metabolic network in Feist et al. [36].
As we previously showed above (Figure 4) that our CFP approach (without considering stoichiometry, Equations 5 and 6) outperforms existing pathfinding methods, we here compare the performance of our CFP approach with and without Equations 5 and 6 so as to evaluate the effect of stoichiometry. For this purpose, we analyzed our CFP approach in two different scenarios, namely when we used a minimal medium based on glucose as a sole carbon source under oxic and anoxic conditions, respectively. See Additional file 3 for details.
It is important to note that the use of a specific minimal medium (as we do here) prevents some known metabolic pathways from functioning in E. coli due to stoichiometric constraints. For example, the tricarboxylic acid (TCA) cycle cannot work in anoxic conditions in E. coli. The ability to detect these false positives cannot be accomplished without the use of stoichiometry. In light of this, the definition of recovery (as used in Figure 4) is slightly modified here. Recovery rate is 1 if (under a given growth medium) the model recovers a feasible pathway or the model excludes from the solution space an infeasible pathway, 0 otherwise. For illustration, if our CFP approach (incorrectly) detects the TCA cycle in anoxic conditions, recovery would be zero. However, if our CFP approach correctly excludes the TCA cycle from the solution space, then recovery would be 1.
Connectivity analysis and stoichiometry
To emphasize the effect of stoichiometry, we examined the connectivity structure of oxaloacetate (OAA) in E. coli. OAA plays an important role in the regulation of carbon flux in most organisms. Again, for this study, we used the metabolic network presented in Feist et al. [36] and a minimal medium based on glucose as a sole carbon source and oxic conditions.
We determined CFPs from OAA to all reachable metabolites (obviously some metabolites may not be reachable via a CFP from OAA). In order to organize and compare the obtained results, we plotted a connectivity curve that shows the total number of connected metabolites when we move a specified number of reaction steps away from the source metabolite. To show the effect of stoichiometry, we plot the connectivity curves when stoichiometry is included (so including Equations 5 and 6) and when it is not included (so excluding Equations 5 and 6).
Application
It is usual to find K paths between a pair of key metabolites/reactions in, for example, the interpretation of 'omics' data [13, 20]. Current pathfinding methods do not take into account stoichiometric constraints for this analysis. In the analysis presented below we show that the resulting K functional paths are strongly dependent on stoichiometric constraints. This fact is illustrated in this subsection with the pathway analysis of PyrOAA metabolism.
PEP, Pyr and OAA are important metabolites whose underlying interconversions control the carbon flux distribution in bacteria [45]. The performance of the PEPPyrOAA node changes in different organisms and growth conditions. We focus here on the structure of CFPs from Pyr to OAA in E. coli in two different scenarios, namely in oxic and anoxic conditions. Pyr and OAA are linked by two fundamental metabolic processes. Firstly, Pyr (via PEP) can be carboxylated to OAA for the replenishment of TCA cycle intermediates or for anabolic purposes (for example, amino acid biosynthesis). This process is typically referred to as anaplerosis. In addition, Pyr and OAA are strongly related via the TCA cycle, which oxidizes carbon of Pyr to CO_{2} and requires OAA to operate.
We calculated the 100 shortest CFPs in both scenarios using the metabolic network presented in Feist et al. [36]. Again, we used the list of arcs presented in Additional file 1. In addition, we used a minimal medium based on glucose. See Additional file 3 for details as to the medium used.
Though the number of nonmeaningful paths has been substantially reduced, it can be observed in Figure 8 that they still exist  for example, different routes via CoA. These false positives do not arise from the lack of stoichiometric balancing, but due to carbon exchange constraints. Indeed, these routes exchange carbon atoms in each of their intermediate steps but do not exchange carbon atoms between Pyr and OAA. When the current limitations described above (in the discussion of atom mappingbased approaches) are addressed, such strategies may be an effective constraint to remove these false positives.
Finally, as observed in Figures 8 and 9, our CFP approach properly captures the metabolic changes induced when oxygen is removed from the medium. These changes cannot be captured if stoichiometric constraints are not considered, showing again the strength of our CFP approach.
Conclusions
Graphbased methods have been widely used for the analysis of metabolic networks, but suffer from the important weakness that reaction stoichiometry is neglected. In this paper we show that, using the novel concept of CFPs, reaction stoichiometry can be incorporated into pathfinding approaches, which constitute a clear progress over the state of the art at the methodological level.
Our results show that, when stoichiometry is incorporated into pathfinding methods, the resulting set of functional pathways is substantially altered, as observed in the analysis of the 40 reference pathways. This idea is also reflected in the analysis of aerobic and anaerobic PyrOAA metabolism, which emphasizes the importance of the steadystate condition and its underlying boundary definition for the analysis of metabolic networks. In addition, connectivity analysis revealed important differences when stoichiometry was considered, as we illustrated with regard to a number of metabolites. In summary, CFPs open new avenues for analyzing metabolic networks at the topological and functional levels and constitute a major advance.
Though the incorporation of stoichiometry into a pathfinding method is the main feature of our work, our CFP approach focuses on paths involving effective carbon exchange in each of their intermediate steps. The results we have presented confirm the relevance of this strategy when analyzing metabolic networks using a pathfinding approach. Our public release of the manually curated E. coli database incorporating effective carbon exchange information (based on BiGG [42] and the work of Feist et al. [36]) represents a valuable dataset available for the scientific community, which can be used for further analysis.
It is important to mention that our CFP approach is formulated as a mixedinteger linear program, which cannot be solved using classical algorithms from graph theory and requires a branch and bound approach. Computational experience shows that the determination of CFPs is not expensive, namely in the order of milliseconds. This fact makes our approach an effective tool for addressing other relevant questions previously addressed by pathfinding approaches.
Our analysis of CFPs in aerobic PyrOAA metabolism allowed us to detect several bypasses to the TCA cycle. Some of these bypasses have been recently reported using a different pathway analysis technique, namely elementary flux patterns for the bypass via the GABA shunt [48] and generating flux modes for the bypass via PPCoA [31]. In addition, we found an alternative bypass to the TCA cycle via LArg. This novel pathway is currently theoretical (it should be treated with caution) and requires experimental validation; however, it shows the capability of our CFP approach to generate new hypothesis.
Finally, despite much debate in the field comparing the performance of pathfinding methods and stoichiometric methods [25, 27, 49], this article shows that both approaches can work in a synergic fashion so as to explore the huge complexity in cellular metabolism.
Materials and methods
Equations 1 to 11 presented in the 'Mathematical model' subsection define a mixedinteger linear problem and, algorithmically, such problems are solved by linear programmingbased tree search. Modern software packages to perform this task, such as ILOG CPLEX, which we used, are well developed and highly sophisticated. ILOG CPLEX was run in a Matlab environment version 7.5 (R2007b).
The computation of the shortest CFP and the 100 shortest CFPs took us (on average) 300 ms and 2.5 minutes, respectively, on a 64bit, 2.00 GHz PC with 12 Gb RAM. Analysis using regression indicated that, over the range of K values examined (up to K = 250), the total time for computing the K shortest CFPs was (approximately) proportional to K^{1.4}. This implies that the computation time of CFPs grows only as a low power of the number of paths (K) sought.
Abbreviations
 AcCoA:

acetylCoA
 AcGlcD:

6acetylDglucose
 AKG:

2oxoglutarate
 AST:

Arginine succinyltransferase
 BiGG:

Biochemical Genetic and Genomic
 CDP:

cytidinediphosphate
 CFP:

carbon flux path
 CoA:

coenzyme A
 CoQ:

coenzyme Q
 DGlc:

glucose
 Fum:

fumarate
 G6P:

glucose 6phosphate
 GABA:

gammaaminobutyric acid
 GLCATr:

Dglucose Oacetyltransferase
 GLX:

Glyoxylate
 PA120:

phosphatidic acid
 HCO3:

bicarbonate
 ICit:

isocitrate
 LArg:

arginine
 LMal:

acetylmaltose
 OAA:

oxaloacetate
 PEP:

phosphoenolpyruvate
 PPCoA:

propionylCoA
 Pyr:

pyruvate
 SUCArg:

SuccinylLarginine
 SUCC:

Succinate
 SUCCoA:

succinylcoenzyme A
 SUCOAS:

succinylCoA synthetase
 TCA:

tricarboxylic acid.
Declarations
Acknowledgements
The work of JPe was supported by the Basque Government. The authors would like to thank two anonymous referees for their valuable comments improving the manuscript.
Authors’ Affiliations
References
 Aittokallio T, Schwikowski B: Graphbased methods for analysing networks in cell biology. Brief Bioinform. 2006, 7: 243255. 10.1093/bib/bbl022.PubMedView ArticleGoogle Scholar
 Jeong H, Tombor B, Albert R, Oltvai ZN, Barabasi AL: The largescale organization of metabolic networks. Nature. 2000, 407: 651654. 10.1038/35036627.PubMedView ArticleGoogle Scholar
 Ma HW, Zeng AP: The connectivity structure, giant strong component and centrality of metabolic networks. Bioinformatics. 2003, 19: 14231430. 10.1093/bioinformatics/btg177.PubMedView ArticleGoogle Scholar
 Arita M: The metabolic world of Escherichia coli is not small. Proc Natl Acad Sci USA. 2004, 101: 15431547. 10.1073/pnas.0306458101.PubMedPubMed CentralView ArticleGoogle Scholar
 Montanez R, Medina MA, Sole RV, RodriguezCaso C: When metabolism meets topology: Reconciling metabolite and reaction networks. Bioessays. 2010, 32: 246256. 10.1002/bies.200900145.PubMedView ArticleGoogle Scholar
 Fell DA, Wagner A: The small world of metabolism. Nat Biotechnol. 2000, 18: 11211122. 10.1038/81025.PubMedView ArticleGoogle Scholar
 Wagner A, Fell DA: The small world inside large metabolic networks. Proc Biol Sci. 2001, 268: 18031810. 10.1098/rspb.2001.1711.PubMedPubMed CentralView ArticleGoogle Scholar
 Yamada T, Bork P: Evolution of biomolecular networks: lessons from metabolic and protein interactions. Nat Rev Mol Cell Biol. 2009, 10: 791803. 10.1038/nrm2787.PubMedView ArticleGoogle Scholar
 Croes D, Couche F, Wodak SJ, van Helden J: Inferring meaningful pathways in weighted metabolic networks. J Mol Biol. 2006, 356: 222236. 10.1016/j.jmb.2005.09.079.PubMedView ArticleGoogle Scholar
 Arita M: Metabolic reconstruction using shortest paths. Simulation Practice Theory. 2000, 8: 109125. 10.1016/S09284869(00)000069.View ArticleGoogle Scholar
 Rahman SA, Advani P, Schunk R, Schrader R, Schomburg D: Metabolic pathway analysis web service (Pathway Hunter Tool at CUBIC). Bioinformatics. 2005, 21: 11891193. 10.1093/bioinformatics/bti116.PubMedView ArticleGoogle Scholar
 Planes FJ, Beasley JE: Path finding approaches and metabolic pathways. Discrete Appl Mathematics. 2009, 157: 22442256. 10.1016/j.dam.2008.06.035.View ArticleGoogle Scholar
 Faust K, Dupont P, Callut J, van Helden J: Pathway discovery in metabolic networks by subgraph extraction. Bioinformatics. 2010, 26: 12111218. 10.1093/bioinformatics/btq105.PubMedPubMed CentralView ArticleGoogle Scholar
 Patil KR, Nielsen J: Uncovering transcriptional regulation of metabolism by using metabolic network topology. Proc Natl Acad Sci USA. 2005, 102: 26852689. 10.1073/pnas.0406811102.PubMedPubMed CentralView ArticleGoogle Scholar
 Zelezniak A, Pers TH, Soares S, Patti ME, Patil KR: Metabolic network topology reveals transcriptional regulatory signatures of type 2 diabetes. PLoS Comput Biol. 2010, 6: e100072910.1371/journal.pcbi.1000729.PubMedPubMed CentralView ArticleGoogle Scholar
 Kharchenko P, Church GM, Vitkup D: Expression dynamics of a cellular metabolic network. Mol Syst Biol. 2005, 1: 2005.0016PubMedPubMed CentralView ArticleGoogle Scholar
 Antonov AV, Dietmann S, Wong P, Mewes HW: TICL  a web tool for networkbased interpretation of compound lists inferred by highthroughput metabolomics. FEBS J. 2009, 276: 20842094. 10.1111/j.17424658.2009.06943.x.PubMedView ArticleGoogle Scholar
 Antonov AV, Dietmann S, Mewes HW: KEGG spider: interpretation of genomics data in the context of the global gene metabolic network. Genome Biol. 2008, 9: R17910.1186/gb2008912r179.PubMedPubMed CentralView ArticleGoogle Scholar
 Jourdan F, Breitling R, Barrett MP, Gilbert D: MetaNetter: inference and visualization of highresolution metabolomic networks. Bioinformatics. 2008, 24: 143145. 10.1093/bioinformatics/btm536.PubMedView ArticleGoogle Scholar
 Cottret L, Wildridge D, Vinson F, Barrett MP, Charles H, Sagot MF, Jourdan F: MetExplore: a web server to link metabolomic experiments and genomescale metabolic networks. Nucleic Acids Res. 2010, 38: W132137. 10.1093/nar/gkq312.PubMedPubMed CentralView ArticleGoogle Scholar
 Rahman SA, Schomburg D: Observing local and global properties of metabolic pathways: 'load points' and 'choke points' in the metabolic networks. Bioinformatics. 2006, 22: 17671774. 10.1093/bioinformatics/btl181.PubMedView ArticleGoogle Scholar
 Fatumo S, Plaimas K, Mallm JP, Schramm G, Adebiyi E, Oswald M, Eils R, König R: Estimating novel potential drug targets of Plasmodium falciparum by analysing the metabolic network of knockout strains in silico. Infect Genet Evol. 2009, 9: 351358. 10.1016/j.meegid.2008.01.007.PubMedView ArticleGoogle Scholar
 Guimera R, SalesPardo M, Amaral LA: A networkbased method for target selection in metabolic networks. Bioinformatics. 2007, 23: 16161622. 10.1093/bioinformatics/btm150.PubMedPubMed CentralView ArticleGoogle Scholar
 Deville Y, Gilbert D, van Helden J, Wodak SJ: An overview of data models for the analysis of biochemical pathways. Brief Bioinform. 2003, 4: 246259. 10.1093/bib/4.3.246.PubMedView ArticleGoogle Scholar
 de Figueiredo LF, Schuster S, Kaleta C, Fell DA: Can sugars be produced from fatty acids? A test case for pathway analysis tools. Bioinformatics. 2008, 24: 26152621. 10.1093/bioinformatics/btn500.PubMedView ArticleGoogle Scholar
 de Figueiredo LF, Schuster S, Kaleta C, Fell DA: Response to comment on "Can sugars be produced from fatty acids? A test case for pathway analysis tools". Bioinformatics. 2009, 25: 33303331. 10.1093/bioinformatics/btp591.PubMedView ArticleGoogle Scholar
 Faust K, Croes D, van Helden J: In response to "Can sugars be produced from fatty acids? A test case for pathway analysis tools". Bioinformatics. 2009, 25: 32023205. 10.1093/bioinformatics/btp557.PubMedView ArticleGoogle Scholar
 Schuster S, Fell DA, Dandekar T: A general definition of metabolic pathways useful for systematic organization and analysis of complex metabolic networks. Nat Biotechnol. 2000, 18: 326332. 10.1038/73786.PubMedView ArticleGoogle Scholar
 Klamt S, Haus UU, Theis F: Hypergraphs and cellular networks. PLoS Comput Biol. 2009, 5: e100038510.1371/journal.pcbi.1000385.PubMedPubMed CentralView ArticleGoogle Scholar
 Schilling CH, Letscher D, Palsson BO: Theory for the systemic definition of metabolic pathways and their use in interpreting metabolic function from a pathwayoriented perspective. J Theor Biol. 2000, 203: 229248. 10.1006/jtbi.2000.1073.PubMedView ArticleGoogle Scholar
 Rezola A, de Figueiredo LF, Brock M, Pey J, Podhorski A, Wittmann C, Schuster S, Bockmayr A, Planes FJ: Exploring metabolic pathways in genomescale networks via generating flux modes. Bioinformatics. 2011, 27: 534540. 10.1093/bioinformatics/btq681.PubMedView ArticleGoogle Scholar
 Terzer M, Stelling J: Largescale computation of elementary flux modes with bit pattern trees. Bioinformatics. 2008, 24: 22292235. 10.1093/bioinformatics/btn401.PubMedView ArticleGoogle Scholar
 de Figueiredo LF, Podhorski A, Rubio A, Kaleta C, Beasley JE, Schuster S, Planes FJ: Computing the shortest elementary flux modes in genomescale metabolic networks. Bioinformatics. 2009, 25: 31583165. 10.1093/bioinformatics/btp564.PubMedView ArticleGoogle Scholar
 van Helden J, Wernisch L, Gilbert D, Wodak SJ: Graphbased analysis of metabolic networks. Bioinformatics Genome Analysis. 2002, 38: 245274.View ArticleGoogle Scholar
 Faust K, Croes D, van Helden J: Metabolic pathfinding using RPAIR annotation. J Mol Biol. 2009, 388: 390414. 10.1016/j.jmb.2009.03.006.PubMedView ArticleGoogle Scholar
 Feist AM, Henry CS, Reed JL, Krummenacker M, Joyce AR, Karp PD, Broadbelt LJ, Hatzimanikatis V, Palsson BO: A genomescale metabolic reconstruction for Escherichia coli K12 MG1655 that accounts for 1260 ORFs and thermodynamic information. Mol Syst Biol. 2007, 3: 121PubMedPubMed CentralView ArticleGoogle Scholar
 Planes FJ, Beasley JE: An optimization model for metabolic pathways. Bioinformatics. 2009, 25: 27232729. 10.1093/bioinformatics/btp441.PubMedView ArticleGoogle Scholar
 Kotera M, Hattori M, Oh M, Yamamoto M, Komeno T, Yabuzaki Y, Tonomura K, Goto S, Kanehisa M: RPAIR: a reactantpair database representing chemical changes in enzymatic reactions. Genome Informatics. 2004, 15: P062Google Scholar
 Arita M: In silico atomic tracing by substrateproduct relationships in Escherichia coli intermediary metabolism. Genome Res. 2003, 13: 24552466. 10.1101/gr.1212003.PubMedPubMed CentralView ArticleGoogle Scholar
 Blum T, Kohlbacher O: Using atom mapping rules for an improved detection of relevant routes in weighted metabolic networks. J Comput Biol. 2008, 15: 565576. 10.1089/cmb.2008.0044.PubMedView ArticleGoogle Scholar
 Heath AP, Bennett GN, Kavraki LE: Finding metabolic pathways using atom tracking. Bioinformatics. 2010, 26: 15481555. 10.1093/bioinformatics/btq223.PubMedPubMed CentralView ArticleGoogle Scholar
 Schellenberger J, Park JO, Conrad TM, Palsson BO: BiGG: a Biochemical Genetic and Genomic knowledgebase of large scale metabolic reconstructions. BMC Bioinformatics. 2010, 11: 21310.1186/1471210511213.PubMedPubMed CentralView ArticleGoogle Scholar
 AtomMetaNetWeb. [http://www.kavrakilab.org/atommetanetweb/#home]
 Eschenfeldt WH, Stols L, Rosenbaum H, Khambatta ZS, QuaiteRandall E, Wu S, Kilgore DC, Trent JD, Donnelly MI: DNA from uncultured organisms as a source of 2,5diketoDgluconic acid reductases. Appl Environ Microbiol. 2001, 67: 42064214. 10.1128/AEM.67.9.42064214.2001.PubMedPubMed CentralView ArticleGoogle Scholar
 Sauer U, Eikmanns BJ: The PEPpyruvateoxaloacetate node as the switch point for carbon flux distribution in bacteria. FEMS Microbiol Rev. 2005, 29: 765794. 10.1016/j.femsre.2004.11.002.PubMedView ArticleGoogle Scholar
 Fait A, Fromm H, Walter D, Galili G, Fernie AR: Highway or byway: the metabolic role of the GABA shunt in plants. Trends Plant Sci. 2008, 13: 1419. 10.1016/j.tplants.2007.10.005.PubMedView ArticleGoogle Scholar
 Lu CD: Pathways and regulation of bacterial arginine metabolism and perspectives for obtaining arginine overproducing strains. Appl Microbiol Biotechnol. 2006, 70: 261272. 10.1007/s002530050308z.PubMedView ArticleGoogle Scholar
 Kaleta C, de Figueiredo LF, Schuster S: Can the whole be less than the sum of its parts? Pathway analysis in genomescale metabolic networks using elementary flux patterns. Genome Res. 2009, 19: 18721883. 10.1101/gr.090639.108.PubMedPubMed CentralView ArticleGoogle Scholar
 Planes FJ, Beasley JE: A critical examination of stoichiometric and pathfinding approaches to metabolic pathways. Brief Bioinform. 2008, 9: 422436. 10.1093/bib/bbn018.PubMedView ArticleGoogle Scholar
 Baldoma L, Aguilar J: Metabolism of Lfucose and Lrhamnose in Escherichia coli: aerobicanaerobic regulation of Llactaldehyde dissimilation. J Bacteriol. 1988, 170: 416421.PubMedPubMed CentralGoogle Scholar
 Becker DJ, Lowe JB: Fucose: biosynthesis and biological function in mammals. Glycobiology. 2003, 13: 41R53R. 10.1093/glycob/cwg054.PubMedView ArticleGoogle Scholar
 Kanehisa M, Goto S, Hattori M, AokiKinoshita KF, Itoh M, Kawashima S, Katayama T, Araki M, Hirakawa M: From genomics to chemical genomics: new developments in KEGG. Nucleic Acids Res. 2006, 34: D354357. 10.1093/nar/gkj102.PubMedPubMed CentralView ArticleGoogle Scholar
 Keseler IM, BonavidesMartínez C, ColladoVides J, GamaCastro S, Gunsalus RP, Johnson DA, Krummenacker M, Nolan LM, Paley S, Paulsen IT, PeraltaGil M, SantosZavaleta A, Shearer AG, Karp PD: EcoCyc: a comprehensive view of Escherichia coli biology. Nucleic Acids Res. 2009, 37: D464470. 10.1093/nar/gkn751.PubMedPubMed CentralView ArticleGoogle Scholar
 Kornberg H: Krebs and his trinity of cycles. Nat Rev Mol Cell Biol. 2000, 1: 225228. 10.1038/35043073.PubMedView ArticleGoogle Scholar
 O'Donovan GA, Neuhard J: Pyrimidine metabolism in microorganisms. Bacteriol Rev. 1970, 34: 278343.PubMedPubMed CentralGoogle Scholar
 Xi H, Schneider BL, Reitzer L: Purine catabolism in Escherichia coli and function of xanthine dehydrogenase in purine salvage. J Bacteriol. 2000, 182: 53325341. 10.1128/JB.182.19.53325341.2000.PubMedPubMed CentralView ArticleGoogle Scholar
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