Path finding methods accounting for stoichiometry in metabolic networks
© Pey et al.; licensee BioMed Central Ltd. 2011
Received: 1 March 2011
Accepted: 27 May 2011
Published: 27 May 2011
Graph-based 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 path-finding approaches via mixed-integer linear programming. This major advance at the modeling level results in improved prediction of topological and functional properties in metabolic networks.
The use of graph theory in the analysis of biological networks has been extensive in the past decade . Particularly, in metabolic networks different relevant topics have been examined using the rich variety of graph-theoretic 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].
Graph-based 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. .
Importantly, graph-based 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 path-finding methods is that they neglect reaction stoichiometry and there is, therefore, no guarantee that any path found can operate in sustained steady-state.
The steady-state condition requires the definition of the boundary of the metabolic network under study. Metabolites inside the boundary of the network, typically called internal metabolites , 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 steady-state 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. , which (unsuccessfully) tested the ability of path-finding methods to answer the question as to whether (or not) fatty acids can be converted into sugars. Klamt et al.  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 graph-theoretical sense, so no nodes revisited) from a source metabolite to a target metabolite able to operate in sustained steady-state. In essence, flux paths incorporate reaction stoichiometry into traditional path-finding 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 mixed-integer linear programming. We present below details as to our mathematical optimization model for determining K-shortest flux paths between source and target metabolites.
Results and discussion
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 Scr be the stoichiometric coefficient associated with metabolite c (c = 1,...,C) in reaction r (r = 1,...,R). As usual in the literature , 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 steady-state. 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 zero-one (binary) variable uij, where uij = 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 path-finding 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 . 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 (D-Glc). 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 D-Glc and glucose 6-phosphate (G6P), and PEP and Pyr.
Let dijr be a binary (0/1) coefficient establishing whether (or not) there exists an effective carbon exchange between input metabolite i (Sir < 0) and output metabolite j (Sjr > 0) in reaction r. If , so there is no reaction involving metabolites i to j in carbon exchange, then uij is also fixed to zero.
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 steady-state. As will be shown below, to do this, it is required to find a steady-state flux distribution able to involve the path. We here introduce variables and constraints needed to define the steady-state flux space.
In addition, it guarantees that vr is non-zero if zr = 1. Here we have scaled fluxes so that the maximum flux is M and the minimum (non-zero) flux is 1. This does not constitute an issue if we consider M sufficiently large.
Current path-finding approaches deal with this situation indirectly, namely by removing computed paths involving a reaction and its reverse.
Equations 5 to 8 define the steady-state 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 uij = 1), then at least one reaction r containing this arc in carbon exchange (so dijr = 1) is forced to be active. By forcing zr to be 1 there will be a non-zero 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 non-zero flux (to satisfy the requirements of steady-state, Equation 5) but without any of their input/output metabolites being involved in the CFP.
This section is organized as follows. By means of several well-documented examples, we first illustrate the biochemical relevance of particular constraints in our CFP approach. We then carry out a side-by-side comparison of our CFP approach with current path-finding approaches.
As shown in the 'Mathematical model' section, the path-finding strategy used in our CFP approach is based on using arcs involving effective carbon exchange and imposing the reversibility constraint, Equation 8. In this sub-section we illustrate the importance of these factors and show that a path-finding 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 sub-section ('Effect of stoichiometry'). Therefore, for this analysis, Equations 5 and 6 were ignored.
Effective carbon exchange
Path-finding methods typically split reversible reactions into two irreversible steps. In contrast to current approaches , 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 D-Glc to Pyr in E. coli, which is the Entner-Doudoroff pathway, as shown in the left-hand side of Figure 3b. When we applied our CFP approach from D-Glc to Pyr without including Equation 8, we obtained the path in the right hand-side of Figure 3b (D-Glc→AcGlc-D→AcCoA→L-Mal→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 D-glucose O-acetyltransferase (GLCATr: D-Glc + AcCoA ↔ AcGlc-D + CoA). By adding Equation 8 this path is removed from the solution space and our CFP approach directly obtains the Entner-Doudoroff pathway.
In order to analyze the performance of any path-finding method, it is usual in the literature to evaluate its ability in recovering well-known metabolic pathways. For this purpose, we used a database of 40 reference E. coli (metabolic) pathways previously discussed in Planes and Beasley  (these 40 pathways are listed in Additional file 2).
The input metabolic graph was built from the genome-scale metabolic network of E. coli . 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 sub-section 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 path-finding 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 ); 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 . Finding K-shortest 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 path-finding 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. , based on the RPAIR database , 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. , 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. , and recently revisited in Blum and Kohlbacher , and Heath et al. . 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 genome-scale metabolic networks, especially for those from the Biochemical Genetic and Genomic (BiGG) database , which we are using here. For completeness, we will include results for the most recent approach , denoted as atom mapping-based strategy. Results were extracted from the web service (named AtomMetaNetWeb) available from Kavraki's lab .
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 graph-theoretical level) cannot perform in steady-state and therefore are not biologically meaningful. We then repeat the side-by-side 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
Side-by-side comparison with stoichiometry
We repeated the side-by-side comparison previously presented in Figure 4 for path-finding methods when stoichiometry is considered. Similarly, we used the 40 E. coli metabolic pathways discussed in Planes and Beasely , and the E. coli metabolic network in Feist et al. .
As we previously showed above (Figure 4) that our CFP approach (without considering stoichiometry, Equations 5 and 6) outperforms existing path-finding 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.  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).
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 path-finding 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 sub-section with the pathway analysis of Pyr-OAA metabolism.
PEP, Pyr and OAA are important metabolites whose underlying inter-conversions control the carbon flux distribution in bacteria . The performance of the PEP-Pyr-OAA 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 CO2 and requires OAA to operate.
We calculated the 100 shortest CFPs in both scenarios using the metabolic network presented in Feist et al. . 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 non-meaningful 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 mapping-based 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.
Graph-based 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 path-finding 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 path-finding 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 Pyr-OAA metabolism, which emphasizes the importance of the steady-state 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 path-finding 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 path-finding approach. Our public release of the manually curated E. coli database incorporating effective carbon exchange information (based on BiGG  and the work of Feist et al. ) 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 mixed-integer 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 path-finding approaches.
Our analysis of CFPs in aerobic Pyr-OAA 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  and generating flux modes for the bypass via PPCoA . In addition, we found an alternative bypass to the TCA cycle via L-Arg. 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 path-finding 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' sub-section define a mixed-integer linear problem and, algorithmically, such problems are solved by linear programming-based 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 64-bit, 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 K1.4. This implies that the computation time of CFPs grows only as a low power of the number of paths (K) sought.
Biochemical Genetic and Genomic
carbon flux path
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.
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