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Institute of Molecular Systems Biology, ETH Zurich, Zurich, Switzerland
Correspondence: Address reprint requests to W. Wiechert, E-mail: wiechert{at}simtec.mb.uni-siegen.de.
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ABSTRACT |
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INTRODUCTION |
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). The aim of a MFA is the quantitative determination of steady-state fluxes in metabolic pathways of a given organism. The method is based on carbon-labeling experiments in which cells are fed with a 13C-labeled substrate. Due to metabolic activity, the labeled material is then distributed throughout the metabolic pathways until, eventually, the labeling state in the system equilibrates and can be measured in cellular compounds. The resulting labeling data is evaluated on the basis of a mathematical model of the carbon flow to estimate the intracellular fluxes. To establish the model, carbon atoms have to be traced through the network. The available methods together with a variety of applications have been reviewed in several recent articles (1–5).
The conceptual and mathematical platform for the evaluation of carbon-labeling experiments is well established (6–9). One basic requirement for all 13C MFA procedures is the stationarity of the metabolic network during the time taken by the labeling experiment. Consequently, the stoichiometric equations relating the intracellular fluxes under steady-state conditions are one cornerstone of any flux analysis model. The other building block for MFA is the balance equation system for the transport of labeled carbon atoms through the network. Combining both sets of equations with measurable fluxes (e.g., substrate uptake and product formation) and the labeling data, the intracellular fluxes can be estimated with a parameter fitting procedure. This formalism is described in detail in the literature (10,11). Several computational tools are available for 13C MFA (12,13).
In contrast to MFA, different methods for metabolic network analysis (MNA) explore the possible solution space of the stoichiometric equations (14) or optimal flux distributions with respect to certain criteria (15). Only recently, MNA has been combined with thermodynamic constraints derived from metabolite concentrations and standard Gibbs energies (16–18). It is shown in the following that 13C MFA yields additional relations between Gibbs energies and fluxes that can be used in the context of flux and network analysis.
Bidirectional reaction steps
Thermodynamic reversibility (in the context of chemical reactions) usually means that the net flux of the reaction can take both directions under different physiological conditions. But even in the case where one direction is strongly preferred, there may still be a simultaneous forward and backward flux present. In fact, any chemical reaction step proceeds in both directions at the same time, i.e., there is a permanent bidirectional exchange of metabolites between the substrate and product pools (19). However, in many cases, one flux direction can be neglected and the reaction is then considered as unidirectional. Otherwise, both forward and backward fluxes must be taken into account in 13C MFA.
One remarkable feature of 13C MFA is that not only the net fluxes of reaction steps can be determined but also the individual forward and backward fluxes of bidirectional reaction steps (19). This distinguishes 13C MFA from methods that are solely based on stoichiometry and/or thermodynamics and, consequently, are restricted to net fluxes. The flux in bidirectional reaction steps has already been quantified in some pioneering 14C studies (20,21). A prominent example of bidirectionality was given in Marx et al. (22) where large amounts of 13C-labeled material arrived at the pentose-5-phosphate pools via the bidirectionally operating transaldolase and transketolase steps of the pentose phosphate pathway, although the net flux was in the opposite direction.
The necessary inclusion of forward and backward fluxes in 13C MFA models introduces more degrees of freedom in addition to the unknown net fluxes. For this reason, on the one hand, exchange fluxes are often considered as an unwanted computational and statistical burden for the evaluation of 13C-labeling data (23). On the other hand, the resolution of both flux directions can, in some cases, give invaluable information on metabolic cycles suggesting gene knockouts for improving product formation (24,25). Generally, it has never been discussed in the context of MFA how exchange fluxes can be interpreted and what their biological meaning is. This is the purpose of the present article.
Exchange fluxes of elementary reaction steps
Consider an elementary bidirectional reaction S + T ... 
P + Q + ... with a specified nominal flux direction (meaning: left to right side). Elementary here means that the reaction proceeds in one single step governed by a mass action law. Particularly, the products are immediately formed from the substrates without any intermediate states. Clearly, this is an idealization of real reaction mechanisms. In the following, it is strictly distinguished between elementary reaction steps and reaction mechanisms proceeding in several elementary steps as, for example, any enzyme-catalyzed reaction (Fig. 1).
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0 and v
0. These two fluxes are unambiguously defined by the amount of molecules per time converted from substrates to products and vice versa. Here, it plays no role which substrate or product is taken to determine the fluxes, because all consumption and production rates are directly coupled by stoichiometry.
Although the two fluxes v, v are essential to formulate the carbon-labeling balances constituting the backbone of any 13C MFA model, they are hard to interpret when the results of a MFA have to be presented. For this reason, the flux pair (v, v) is equivalently described by the pair of net and exchange flux (vnet,vxch). The net flux is given by vnet = v – v, which has a clear physical meaning and is also the quantity used in classical stoichiometric MFA or in thermodynamic network formalisms (16,26,27). It can be positive or negative with the sign defined relative to the specified nominal direction of the reaction.
In contrast to the net flux, the exchange flux vxch characterizes the reaction bidirectionality and has no direct counterpart in classical network theories. This led to different definitions in the literature (19,28,29). Since, essentially, all these definitions can be transformed into each other, the most widely used definition is taken here. Precisely, vxch quantifies the amount of material flowing simultaneously in both directions of a reaction step:
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In the following thermodynamic analysis of exchange fluxes, it will turn out that it is more convenient to use the forward/backward flux quotient v/v to characterize bidirectionality. In the case vnet 0 this quotient is related to the net/exchange flux quotient by
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Thermodynamic nonequilibrium coefficients
The major aim of this contribution is to relate exchange fluxes to thermodynamic properties of reaction steps under physiological conditions. The thermodynamics of a (not necessarily elementary) reaction step 
is characterized by its equilibrium constant 
where 
G0' denotes the Gibbs energy of the reaction under standard conditions. Assuming standard mass action theory, it holds for the Gibbs energy G' under physiological conditions (for abuse of notation the same symbols are used both for the names of substance and their concentrations):
 | (3) |
The quantity 
will be henceforth called the thermodynamic nonequilibrium coefficient (30). It provides for the most compact representation of the following results. Using Eq. 3 all obtained relations can be easily translated to a Gibbs energy formulation. The coefficient is 1 if the reaction is operating in thermodynamic equilibrium. Clearly, because the Gibbs energies of a reaction sequence behave additively, the corresponding nonequilibrium coefficients behave multiplicatively, which will be frequently used in the following.
Related work
During the reviewing process, a second article was published which is concerned with the same topic from another viewpoint (31). Whereas the present article develops all results within the classical framework of mass action kinetics, similar relations between thermodynamic driving forces, and forward/backward fluxes could be proven there for general chemical processes not necessarily governed by mass action laws.
In contrast, the present contribution has a strong focus on the consequences of the exchange flux relations to the practice of MFA. Particularly, the relation between the forward/backward fluxes of multistep reaction mechanisms obtained from classical reaction kinetic formalisms, on the one hand, and from 13C-labeling experiments, on the other hand, is investigated. The conceptual difference between reverse fluxes in classical reaction kinetic theory and measured exchange fluxes in 13C MFA is elucidated.
Nevertheless, some of the results on Michaelis-Menten-like mechanisms and flux quotients near/far from equilibrium, can be found in both articles. Combining, the present contribution with the results from Beard and Qian (31) significantly extends the generality of most statements. Some of these generalizations are already given in Beard and Qian (31).
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MEASURED EXCHANGE FLUXES ARE NOT WELL DEFINED |
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 TOPABSTRACTINTRODUCTION MEASURED EXCHANGE FLUXES ARE... SOME COMPUTATIONAL RULESSOME EXAMPLESA GENERAL THEOREMPRACTICAL RELEVANCE OF THE...CONCLUSIONSUPPLEMENTARY MATERIALACKNOWLEDGEMENTSREFERENCES |
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Apparent fluxes
Any enzyme reaction mechanism constitutes a small subnetwork within a metabolic network. As an example, an extended Michaelis-Menten mechanism as shown in Fig. 1 a is taken (S, substrate; P, product; E, free enzyme; and M1,...,Mn are intermediate enzyme substrate/product complexes). All elementary reaction steps numbered 1,...,n+1 are considered to be reversible. The respective forward and backward fluxes 
are henceforth called the individual forward and backward fluxes of the reaction steps. These fluxes are well defined by mass action laws.
Generally, in 13C MFA, it is not possible to resolve the individual steps of an enzyme mechanism because no measurement information on the intermediate complexes can be gathered. Consequently, a complete reaction mechanism is replaced by one single (apparent) step:
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By performing an MFA based on the one-step model in Eq. 4, only one single pair (v, v) of forward and backward flux is measured. These fluxes are henceforth called the apparent fluxes. Apparent fluxes are the central concept of this article. They are of high practical relevance because they constitute the actually measured quantities in 13C MFA. Consequently, in the following, all relations between fluxes and thermodynamic quantities have to be formulated in terms of apparent fluxes.
Path dependency of apparent fluxes
Consider now a simple reaction mechanism involving two substrates S,T, which bind to the enzyme in this order and are converted to one product P (Fig. 1 c). Four elementary reaction steps are involved in this mechanism with the individual forward and backward fluxes 
Clearly, if a labeled carbon atom is traced through the reaction sequence, its origin can be both in S or T. If an atom of S is chosen to determine the apparent fluxes, it will be involved in all four elementary reactions. In contrast, an atom of P will see only the steps 2–4.
This difference has dramatic consequences as can be shown with an extreme case: Assume that the first step is unidirectional (i.e., 
) while the others are strongly exchanging. Consequently, there is a high exchange of atoms between P and T whereas it is impossible for any carbon atom to proceed backward from P to S. 13C MFA based on S or T labeling then yields inconsistent results because the same reaction seems to be both unidirectional and bidirectional depending on the chosen carbon atom.
In other words, the apparent exchange flux depends on the path an isotope takes when it travels through the network. This property will henceforth be denoted as path dependency of apparent fluxes. Interestingly, this phenomenon is well known from tracer studies in enzyme kinetics (32,33). However, its consequences have never been discussed in the context of 13C MFA, although other possible pitfalls of MFA have been recognized (34).
If path dependency has a strong relevance in practice, then the currently used network models for MFA must be changed. Precisely, each enzymatic reaction with multiple substrates must be resolved into its elementary steps. This would introduce so many additional exchange flux parameters that MFA runs into strong identifiability problems. Consequently, a major goal of the following analysis is to quantitate the path-dependency effect to judge its practical relevance.
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SOME COMPUTATIONAL RULES |
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Reaction networks and isotope-labeling networks
The term reaction-network is used here with its common biochemical meaning. It consists of elementary reaction steps with their respective reactands. Reactions with many substrates and products are possible. This is quite different with an isotope-labeling network. Choosing a single substrate for isotope labeling, the corresponding labeling network is obtained by tracing this isotope through the reaction network. By doing this only one substrate and one product is involved in each reaction step, whereas other cosubstrates or coproducts are not recognized by the labeling. Moreover, some reactions may completely vanish because no label is involved. Omitting these reaction steps and all unlabeled cosubstrates and coproducts from the original reaction network, the labeling network emerges. It contains no bimolecular steps and, thus, can be described by a simple directed graph.
As an example, the bimolecular reaction mechanism from Fig. 1 d is discussed. Depending on the isotope chosen to be traced through the network, two different labeling networks arise (Fig. 2). In each case, one of the reaction steps of the original reaction network (Fig. 1 d) is not recognized by the label flow. Moreover, all unlabeled cosubstrates and coproducts (dashed lines in Fig. 2) are not a part of the labeling network. The resulting network is an ordinary graph with one-to-one edges.
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Exchange fluxes and thermodynamics
Based on this preliminary understanding the rules for network analysis can now be given. The rather technical proofs can be found in Appendix A in the Supplementary Material.
Sequential composition rule
Consider two sequential reaction steps 
in an isotope-labeling network with individual forward and backward fluxes 
Assume that a carbon atom is traced through both reactions. Then it holds for the apparent and individual fluxes of the sequence:
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Parallel composition rule
Consider two parallel reaction steps 
in an isotope-labeling network with individual forward and backward fluxes Assume that the same isotope is traced through both reactions, arriving at the same position in T. Then it holds for the apparent and individual fluxes of the parallel composition:
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is determined by kinetic properties of the reactions and, thus, cannot be explained within thermodynamic categories.
Elementary exchange flux rule
For any elementary reaction step obeying the mass action law the following relation between forward/backward flux quotient and nonequilibrium coefficient holds:
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SOME EXAMPLES |
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General Michaelis-Menten mechanism
For the general reversible Michaelis-Menten mechanism shown in Fig. 1 a, the analysis proceeds as follows:
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An example with a strict inequality
It is tempting to claim that the exchange flux rule holds for any unimolecular enzymatic reaction sequence with net reaction S P. However, the example from Fig. 1 b shows that this is not the case. Here, the substrate binding reaction is preceded by the binding of a metabolic cofactor T that might, for example, be ATP. Likewise, the unbinding of the product is succeeded by the unbinding of a cofactor Q (e.g., ADP). Since cofactor carbon atoms are never traced in MFA, application of the rules with S and P now yields the result
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An example with path dependency
The bimolecular example from Fig. 1 c used above for demonstrating path dependency of exchange fluxes is now discussed. The two substrates S,T bind sequentially until they are fused to the product P. Two different labeling paths S P and T P are possible:
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P indicates the path relative to which the apparent fluxes are calculated.
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1 makes the difference and, thus, it precisely quantitates the path-dependency effect. Depending on this factor, the exchange fluxes measured through the two different paths will be more or less different. Path dependency completely vanishes if the first reaction step is in equilibrium.
An example with parallel reaction steps
Consider now the six-step bimolecular reaction mechanism shown in Fig. 1 d, where two substrates S, T can bind in random order until they are fused to the product P. This scheme contains two parallel reaction sequences 1, 2 and 3, 4.
(here all reaction steps must be taken into account) and Eq. 3, the exchange flux result is
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1 is the mixing coefficient between the upper 1,2 branch and the lower 4 branch of the carbon flow. 
):
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3 and 1/1. They stem from the unrecognized reaction steps in the two labeling paths. A lower bound for the path-dependency effect can be calculated from
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A more realistic lower bound might be obtained by assuming = 1/2, = 1/2, which should approximately hold in typical enzyme mechanisms. It then holds that
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A multisubstrate multiproduct example
As a more complex example, consider the mechanism shown in Fig. 1 e with two substrates and two products each binding and unbinding in random order. Four different labeling networks are now possible depending on the substrate that is labeled and the product where the label arrives. However, due to the symmetrical nature of the system it is sufficient to analyze just one combination. If an S atom is traced through the network and arrives at P, the result is
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Mechanisms with inhibition steps
Enzymes with inhibition are discussed as a last example. Inhibition poses no problem when the binding of an inhibitor (EX EXI) reversibly inactivates an enzyme or an enzyme substrate complex (which is the usual assumption). Such a reaction constitutes a dead end in the reaction network. Consequently, the net flux of the reaction is zero and it holds v/v = 1. On the other hand, the enzyme complex and its inhibited state are in equilibrium, i.e., = 1. It follows that inhibition steps leading to dead ends need not be considered when reaction mechanisms are analyzed.
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A GENERAL THEOREM |
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,33) are proper, and all their labeling networks are reducible. Some few transport mechanisms can also be described in this way (36).
Exchange fluxes of proper enzyme mechanisms
Generalizing the concepts introduced with the examples from the last section, a general theorem can now be stated:
Exchange flux theorem
Consider any proper reaction mechanism. Then it holds for any reducible labeling network derived from this mechanism that
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is the nonequilibrium coefficient of the overall mechanism and R is the coefficient of the (elementary) reaction step of the mechanism. The rather technical proof of this theorem is given in Appendix B in the Supplementary Material. Generally, the lower bound is only reached in special, rather unrealistic situations. On the other hand, if all binding and unbinding steps as well as enzyme state changes are in a rapid equilibrium, the upper bound is reached. In this case, the path-dependency effect vanishes completely.
Exchange fluxes of proper enzyme mechanisms
It should be pointed out that the theorem does not apply to any possible reaction mechanism, i.e., only a subset of all enzyme and transport mechanisms can be analyzed in this way. Exceptions essentially occur if the mechanism is not reducible by applying the parallel and sequential composition rules, or if the mechanism is not proper. This occurs, for example, if a labeling path contains more than one reaction or transport step. In this case, it can even hold v/v > .Some examples are given in Appendix B in the Supplementary Material.
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PRACTICAL RELEVANCE OF THE THEOREM |
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–39).
Relevance of the path-dependency effect
Typically, binding steps operate closer to the equilibrium than reaction steps. Since, for proper mechanisms, is always a product of individual nonequilibrium coefficients including the reaction step (i.e., 
), it can be concluded that the flux ratio v/v is at least in the order of magnitude of . If, moreover, all binding steps and state transformations are close to equilibrium, it will hold v/v 
. On the other hand, if there are unrecognized nonequilibrium state transformations present in the mechanisms (Fig. 1 b), it is impossible to reach the upper bound .
In the practice of MFA, it is well known that the sensitivity of the measured labeling state, with respect to exchange fluxes, is very low. This means that the precise exchange values need not be known to obtain a consistent result. Fortunately, this strongly relaxes the precision requirements for apparent flux quotients. As long as the quotients of different paths are in the same order of magnitude, the path-dependency effect will have little practical relevance. This, finally, provides a rescue for the common procedure in 13C MFA, because one (approximate) exchange flux parameter is sufficient to describe the flow of labeled material through all possible paths.
Potential applications
The theorem can be applied in four different ways:
). To this end, all these data must be available and a rough estimate of /R must be available. G0' of the reactions in vivo from estimated net and exchange fluxes and measured concentrations in the case r . In contrast, the current estimates of in vivo Gibbs energies rely on empirical corrections for intracellular conditions (40). G0' data. In contrast to the net fluxes which are strongly constrained by stoichiometry, every exchange flux in the model per se is unknown. Application of the theorem will greatly reduce the computational complexity of MFA methods. ).
Statistical considerations
The statistical quality of the results obtained by one of these applications should be briefly addressed. Clearly, it will depend on the quality of the flux, metabolite concentration, and thermodynamic data:
). However, it is possible to design special carbon-labeling experiments that produce useful confidence intervals for some chosen exchange fluxes in the focus of interest (25). ,42). One of the major problems here is a proper calibration and the potential loss of metabolites in the sample preparation process. Since concentrations appear as quotients on the right side of the theorem, this error is tendentially reduced. ,43,44).
Reactions far from or close to equilibrium
There is a long-lasting discussion on the biological meaning of metabolic reaction steps close to and far from equilibrium. Generally, reaction steps with a large Gibbs energy are supposed to have a regulatory function (16,31,35). Using Eqs. 2 and 3, the exchange flux theorem can be reformulated as
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), then it must hold vnet >> vxch. This justifies the common assumption that reactions operating far from equilibrium can be assumed unidirectional in MFA. G' 0, vnet > 0). In this case, it will hold vnet << vxch, and high exchange fluxes (relative to the net flux) must be expected. G' 0), it follows vnet 0. In this situation the relation between net and exchange flux becomes singular and vxch is no more determined by vnet and G'. In this near equilibrium operation regime the exchange flux is dominated by the enzyme kinetic parameters. A kinetic analysis reveals that an exchange flux is still present when the net flux is zero (see Appendix C in the Supplementary Material).
Exchange fluxes in central metabolism
For several sets of biological reactions in the central metabolism the question of uni- or bidirectionality played an important role in the development of MFA. It is still under discussion which reaction steps can be assumed unidirectional and when both reaction directions have to be considered. The exchange flux theorem now supplies an instrument to decide which decision in appropriate under physiological conditions. A detailed analysis of the thermodynamic driving forces in vivo of all reactions for an Escherichia coli network has recently been undertaken in Kümmel et al. (16). Based on these results recommendations for the choice of the network model used in 13C MFA are derived in Appendix D in the Supplementary Material.
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CONCLUSION |
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A common assumption in early enzyme kinetic theories was a rapid equilibrium of binding steps, whereas modern formalisms just need a steady-state assumption for the reaction network. In the case of rapid binding equilibria, the exchange flux theorem reduces to an equality. Fortunately, exchange fluxes are, in practice, only determined up to an order of magnitude. This allows us to release the rapid equilibrium condition to the requirement that the reaction step should share a significant part of the overall reaction energy. Some few examples of known reaction velocity constants from literature (32,33) support that this will be the case for the majority of enzymes.
At the same time, the analysis revealed a conceptual problem of 13C MFA, which is the path dependency of measured exchange fluxes. Looking closer, it turned out that path dependency is the deeper reason why the exchange flux theorem for enzymes does not yield an exact equality. On the one hand, this effect is a fundamental limitation for the precision of 13C MFA. On the other hand, it could be shown that the quantity of this effect is not significant in most practical applications. However, being rigorous, the precise conditions for path independency must still be checked for every individual enzyme.
The theorem is proven for the class of proper reducible enzyme or transport mechanisms. This covers many mechanisms commonly published in text books. However, there are still mechanisms which do not belong to this category. It has to be investigated in the future how far the theorem can be generalized. One important step has already been taken in Beard and Qian (31) by generalizing the elementary exchange flux rule to arbitrary mechanisms not necessarily governed by mass action laws.
Although some basic results were already available in the 1970s it took until now that its value for MFA has been recognized. These developments are obviously driven by the recent experimental progress in 13C MFA, quantitative metabolomics, and network thermodynamics that make the theorem practical. Clearly, the practical application in various situations will be the next step in research.
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SUPPLEMENTARY MATERIAL |
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ACKNOWLEDGEMENTS |
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The project was partly funded by the German Ministry of Education and Research (BMBF) (SysMAP Project), the Fond der Chemischen Industrie, and the ETH Zurich.
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FOOTNOTES |
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Submitted on October 23, 2006; accepted for publication May 18, 2007.
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REFERENCES |
|---|
2. Szyperski, T. 1998. 13C -NMR, MS and metabolic flux balancing in biotechnology research. Q. Rev. Biophys. 31:41–106.[CrossRef][Medline]
3. Klapa, M. I., and G. Stephanopoulos. 2000. Metabolic flux analysis. In Bioreaction Engineering—Modeling and Control. K. Schügerl and K. H. Bellgardt, editors. Springer, New York.
4. Christensen, B., and J. Nielsen. 2000. Metabolic network analysis. A powerful tool in metabolic engineering. Adv. Biochem. Eng. Biotechnol. 66:209–231.[Medline]
5. Sauer, U. 2006. Metabolic networks in motion: 13C-based flux analysis. Mol. Syst. Biol. 2:52.[Medline]
6. Wiechert, W., and M. Wurzel. 2001. Metabolic isotopomer labeling systems. Part I: global dynamic behavior. Math. Biosci. 169:173–205.[CrossRef][Medline]
7. Schmidt, K., M. Carlsen, J. Nielsen, and J. Villadsen. 1997. Modeling isotopomer distribution in biochemical networks using isotopomer mapping matrices. Biotechnol. Bioeng. 55:831–840.[Medline]
8. van Winden, W., J. J. Heijnen, and P. J. T. Verheijen. 2002. Cumulative bondomers: a new concept in flux analysis from 2D [13C,1H] COSY data. Biotechnol. Bioeng. 80:731–745.[CrossRef][Medline]
9. Antoniewicz, M. R., J. K. Kelleher, and G. Stephanopoulos. 2007. Elementary metabolite units (EMU): a novel framework for modeling isotopic distributions. Metab. Eng. 1:68–86.
10. Möllney, M., W. Wiechert, D. Kownatzki, and A. A. de Graaf. 1999. Bidirectional reaction steps in metabolic networks. Part IV: optimal design of isotopomer labeling experiments. Biotechnol. Bioeng. 66:86–103.[CrossRef][Medline]
11. Fischer, E., N. Zamboni, and U. Sauer. 2004. High-throughput metabolic flux analysis based on gas chromatography-mass spectrometry derived 13C constraints. Anal. Biochem. 325:308–316.[CrossRef][Medline]
12. Wiechert, W., M. Möllney, S. Petersen, and A. A. de Graaf. 2001. A universal framework for 13C metabolic flux analysis. Metab. Eng. 3:265–283.[CrossRef][Medline]
13. Zamboni, N., E. Fischer, and U. Sauer. 2005. FiatFlux—a software for metabolic flux analysis from 13C-glucose experiments. BMC Bioinformatics. 6:209.[CrossRef][Medline]
14. Schilling, C. H., S. Schuster, B. O. Palsson, and R. Heinrich. 1999. Metabolic pathway analysis: basic concepts and scientific applications in the post-genomic era. Biotechnol. Prog. 15:296–303.[CrossRef][Medline]
15. Kauffman, K. J., P. Prakash, and J. S. Edwards. 2003. Advances in flux balance analysis. Curr. Opin. Biotechnol. 14:491–496.[CrossRef][Medline]
16. Kümmel, A., S. Panke, and M. Heinemann. 2006. Putative regulatory sites unraveled by network-embedded thermodynamic analysis of metabolome data. Mol. Syst. Biol. 2:2006–2034.
17. Beard, D. A., E. Babson, E. Curtis, and H. Qian. 2004. Thermodynamic constraints for biochemical networks. J. Theor. Biol. 228:327–333.[CrossRef][Medline]
18. Henry, C. S., M. D. Jankowski, L. J. Broadbelt, and V. Hatzimanikatis. 2006. Genome-scale thermodynamic analysis of Escherichia coli metabolism. Biophys. J. 90:1453–1461.
19. Wiechert, W., and A. A. de Graaf. 1997. Bidirectional reaction steps in metabolic networks, Part I: modeling and simulation of carbon isotope labeling experiments. Biotechnol. Bioeng. 55:101–117.[Medline]
20. Crawford, J. M., and J. J. Blum. 1983. Quantitative analysis of flux along the gluconeogenic, glycolytic and pentose phosphate pathways under reducing conditions in hepatocytes isolated from fed rats. Biochem. J. 212:595–598.
21. Rabkin, M., and J. J. Blum. 1985. Quantitative analysis of intermediary metabolism in hepatocytes incubated in the presence and absence of glucagon with a substrate mixture containing glucose, ribose, fructose, alanine and acetate. Biochem. J. 225:761–786.[Medline]
22. Marx, A., A. A. de Graaf, W. Wiechert, L. Eggeling, and H. Sahm. 1996. Determination of the fluxes in central metabolism of Corynebacterium glutamicum by NMR spectroscopy combined with metabolite balancing. Biotechnol. Bioeng. 49:111–129.[Medline]
23. Yang, T. H., E. Heinzle, and C. Wittmann. 2005. Theoretical aspects of 13C metabolic flux analysis with sole quantification of carbon dioxide labeling. Comput. Biol. Chem. 29:121–133.[CrossRef][Medline]
24. Petersen, S., A. A. de Graaf, L. Eggeling, M. Möllney, W. Wiechert, and H. Sahm. 2000. In vivo quantification of parallel and bidirectional fluxes in the anaplerosis of Corynebacterium glutamicum. J. Biol. Chem. 275:35932–35941.
25. Petersen, S., E. von Lieres, W. Wiechert, and A. A. de Graaf. 2002. A multi-scale approach for the predictive modeling of metabolic regulation. In Metabolic Engineering in a Post Genomic Era. H. V. Westerhoff, editor. Horizon Scientific Press, Norfolk, UK.
26. Vallino, J. J., and G. Stephanopoulos. 1993. Metabolic flux distribution in Corynebacterium glutamicum during growth and lysine overproduction. Biotechnol. Bioeng. 41:633–646.[Medline]
27. Varma, A., and B. O. Palsson. 1994. Metabolic flux balancing: basic concepts, scientific and practical use. Bio/Technol. 12:994–998.[CrossRef]
28. Schuster, R., S. Schuster, and H.-G. Holzhütter. 1992. Simplification of complex kinetic models used for the quantitative analysis of nuclear magnetic resonance or radioactive tracer studies. J. Chem. Soc. Faraday Trans. 88:2837–2844.[CrossRef]
29. Wittmann, C., and E. Heinzle. 2001. Modeling and experimental design for metabolic flux analysis of lysine-producing Corynebacteria by mass spectrometry. Metab. Eng. 3:173–191.[CrossRef][Medline]
30. Fell, D. 1996. Understanding the Control of Metabolism. Portland Press, London, UK.
31. Beard, D. A., and H. Qian. 2007. Relationships between thermodynamic driving force and one-way fluxes in reversible processes. PloS. 1:e144.
32. Segel, I. H. 1975. Enzyme Kinetics: Behavior and Analysis of Rapid-Equilibrium and Steady-State Enzyme Systems. Wiley, New York.
33. Cornish-Bowden, A., and C. W. Wharton. 1988. Principles of Enzyme Kinetics. Oxford University Press, Oxford, UK.
34. van Winden, W., P. Verheijen, and S. Heijnen. 2001. Possible pitfalls of flux calculations based on 13C-labeling. Metab. Eng. 3:151–162.[CrossRef][Medline]
35. Rolleston, F. S. 1973. A theoretical background to the use of measured concentrations of intermediates in study of the control of intermediary metabolism. In Current Topics in Cell Regulation. B. L. Horecker and E. R. Stadtman, editors. Academic Press, New York.
36. Hill, T. L. 1977. Free Energy Transduction in Biology. Academic Press, New York.
37. Buchholz, A., R. Takors, and C. Wandrey. 2001. Quantification of intracellular metabolites in Escherichia coli K12 using liquid chromatographic-electrospray ionization tandem mass spectrometric techniques. Anal. Biochem. 295:129–137.[CrossRef][Medline]
38. van Winden, W. A., J. C. van Dam, C. Ras, R. J. Kleijn, W. M. van Gulik, and J. J. Heijnen. 2005. Metabolic-flux analysis of Saccharomyces cerevisiae CEN.PK113–7D based on mass isotopomer measurements of 13C-labeled primary metabolites. FEMS Yeast Res. 5:559–568.[CrossRef][Medline]
39. Nöh, K., K. Grönke, B. Luo, R. Takors, M. Oldiges, and W. Wiechert. 2007. Metabolic flux analysis at ultra short timescale: isotopically nonstationary 13C-labeling experiments. J. Biotechnol. In press.
40. Alberty, R. A. 2003. Thermodynamics of Biochemical Reactions, John Wiley, New York.
41. Wiechert, W., C. Siefke, A. A. de Graaf, and A. Marx. 1997. Bidirectional reaction steps in metabolic networks. Part II: flux estimation and statistical analysis. Biotechnol. Bioeng. 55:118–135.[Medline]
42. Luo, B., K. Grönke, R. Takors, C. Wandrey, and M. Oldiges. 2007. Simultaneous determination of multiple intracellular metabolites in glycolysis, pentose phosphate pathway and TCA cycle by liquid chromatography-mass spectrometry. J. Chromatogr. A. 1147:153–164.[CrossRef][Medline]
43. Goldberg, R. N., Y. B. Tewari, and T. N. Bhat. 2004. Thermodynamics of enzyme-catalyzed reactions—a database for quantitative biochemistry. Bioinformatics. 20:2874–2877.
44. Maskow, T., and U. von Stockar. 2005. How reliable are thermodynamic feasibility statements of biochemical pathways? Biotechnol. Bioeng. 92:223–230.[CrossRef][Medline]
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