Including metabolite concentrations into flux balance analysis: thermodynamic realizability as a constraint on flux distributions in metabolic networks
© Hoppe et al; licensee BioMed Central Ltd. 2007
Received: 16 January 2007
Accepted: 01 June 2007
Published: 01 June 2007
In recent years, constrained optimization – usually referred to as flux balance analysis (FBA) – has become a widely applied method for the computation of stationary fluxes in large-scale metabolic networks. The striking advantage of FBA as compared to kinetic modeling is that it basically requires only knowledge of the stoichiometry of the network. On the other hand, results of FBA are to a large degree hypothetical because the method relies on plausible but hardly provable optimality principles that are thought to govern metabolic flux distributions.
To augment the reliability of FBA-based flux calculations we propose an additional side constraint which assures thermodynamic realizability, i.e. that the flux directions are consistent with the corresponding changes of Gibb's free energies. The latter depend on metabolite levels for which plausible ranges can be inferred from experimental data. Computationally, our method results in the solution of a mixed integer linear optimization problem with quadratic scoring function. An optimal flux distribution together with a metabolite profile is determined which assures thermodynamic realizability with minimal deviations of metabolite levels from their expected values. We applied our novel approach to two exemplary metabolic networks of different complexity, the metabolic core network of erythrocytes (30 reactions) and the metabolic network iJR904 of Escherichia coli (931 reactions). Our calculations show that increasing network complexity entails increasing sensitivity of predicted flux distributions to variations of standard Gibb's free energy changes and metabolite concentration ranges. We demonstrate the usefulness of our method for assessing critical concentrations of external metabolites preventing attainment of a metabolic steady state.
Our method incorporates the thermodynamic link between flux directions and metabolite concentrations into a practical computational algorithm. The weakness of conventional FBA to rely on intuitive assumptions about the reversibility of biochemical reactions is overcome. This enables the computation of reliable flux distributions even under extreme conditions of the network (e.g. enzyme inhibition, depletion of substrates or accumulation of end products) where metabolite concentrations may be drastically altered.
Sequencing the whole genomes in conjunction with high-throughput analyses of mRNA, protein and metabolite profiles  has paved the way for a fast reconstruction of metabolic networks . For a quantitative assessment of metabolic fluxes, Palsson and co-workers have developed a theoretical approach commonly referred to as flux balance analysis (FBA) . This method relies on the hypothesis that the most likely distribution of stationary fluxes in the network has to be optimal with respect to a feasible optimization criterion linking the fluxes with cellular functions. In most applications of FBA the fluxes have been determined to maximize a specific network output as the production of biomass [4–6]) or the production of ethanol in yeast . Whereas the maximization of biomass production appears to be a reasonable objective of the cellular metabolism of rapidly growing and replicating primitive cells such as bacteria, the flux distribution in complex eukaryotic cells is governed by a larger variety of cellular functions that have to be met simultaneously. Therefore, the principle of flux-minimization was proposed as a more general optimization criterion for FBA [8–10]. The extension of FBA outlined in this paper will be tested by choosing as flux objective both the maximization of biomass and the minimization of internal fluxes at given output fluxes. Flux distributions predicted by FBA are hypothetical because they depend essentially upon the choice of the flux evaluation criterion used. Therefore, to increase the reliability of FBA results, one has to seek for strategies to include additional biochemical knowledge into FBA. One way is to include measured flux rates as further side constraints. However, flux measurements – except for exchange reactions that deliver metabolites into the external space – are still difficult and costly to perform as they require determining labeled isotopomers in a time-dependent manner [11–13]. Another possibility to increase the credibility of flux balance calculations is to include some basic thermodynamics of the reactions and transport processes constituting the network. The thermodynamic consensus rule dictates that a positive net flux through a reaction implies a negative corresponding change of the Gibb's free reaction energy and vice versa. Based on this fundamental criterion one may check whether given flux directions conflict with known Gibb's free energy changes. This allows to identify putative regulatory sites in the network [14, 15] or to decide on the reversibility/irreversibility of reactions [16–20]. However, regarding flux distributions predicted by constrained optimization methods as FBA it is desirable to judge their feasibility not only post-hoc but to include thermodynamic constraints on flux directionalities directly into their calculation [21–23]. In our previous work [8, 9] this was accomplished by weighting negative (backward) fluxes with the thermodynamic equilibrium constant of the reaction. The rationale behind this empirical weighting procedure is to impede reversing the direction of a reaction (such that the change of Gibb's free energy has the opposite sign than under standard conditions) with increasing value of the thermodynamic equilibrium constant. However, this way of mixing the costs for the maintenance of metabolic fluxes with the thermodynamic 'costs' for reversing the direction of a reaction in one and the same objective function is questionable for two reasons. First, the concentrations of metabolites in a cell differ significantly from 1 M so that the actual free energy changes of biochemical reactions may considerably differ from their standard values. Second, increasing or decreasing the concentration of the reactants to an extent that enables reversal of the flux direction might occur in the cell by regulations that does not cause much real 'costs' in terms of the production of more enzyme and of used external resources. One way to overcome this shortcoming of our previous approach [8, 9] is to incorporate fulfillment of the thermodynamic consensus rule as additional side constraint into the calculation of the flux distribution. Such an approach was recently outlined by Henry et al. . These authors studied the range of metabolite concentrations that is still compatible with a thermodynamically feasible flux distribution in a genome-scale network of E. coli under conditions of optimal bacterial growth. Here we go one step further to include information on metabolite concentrations directly into the calculation of the flux distribution. Our algorithm considers the optimization of two different objectives: On one hand a functionally optimal and thermodynamically feasible flux distribution is demanded an on the other hand the calculated metabolite concentrations are required to deviate as little as possible from set-point values prescribed on the basis of biochemical knowledge. In the following we outline the method and provide applications to two different metabolic networks: (i) the energy- and redox metabolism of red blood cells for which a detailed kinetic model has been established  thus allowing to check the feasibility of our method and (ii) the large-scale genome-based metabolic network of Escherichia coli iJR904  which has already been subjected to FBA in several studies [23, 26, 27].
The directionality of the net flux of a chemical reaction and the change of Gibb's free energy are related to each other by the consensus rule
sgn(v) = -sgn (ΔGr), (1)
Owing to the first law of thermodynamics the values of the standard Gibb's free energy changes are not independent from each other but have to obey the principle of micro-reversibility dictating the sum of standard free energy values in a closed system to be zero. In several flux balance studies [16, 17, 29–32] this criterion has been referred to as generalization of Kirchhoff's loop law which [see Additional file 3]. The problem is that experimentally determined values for the changes of standard Gibb's energies are not consistent with the principle of micro-reversibility per se because of experimental errors. Therefore, we add correction terms (forming the vector E) to all observed values of standard Gibb's energy changes and determine minimal corrections necessary to assure the principle of micro-reversibility. The corresponding optimization problem reads
where ||E|| is the 2-norm of the vector E, and are hypothetic Gibb's free energy changes of formation. is then used as the modified vector of standard Gibb's energy changes fulfilling the condition of micro-reversibility.
Constraints on metabolite concentrations
Metabolic network models may contain reactions which are simplified in a way that reactants are dropped from the reaction formula. For example, the oxidation of glutathione (GSH) to glutathione disulfide (GSSG) is usually written as an overall reaction 2GSH → GSSG. Actually, this reaction should read 2GSH + R-OOH → GSSG + R-OHH2O where R-OOH represents a large group of not further specified hydroperoxides that can be detoxified by the glutathione system. For these lumped reactions it is impossible to give a realistic value. For other reactions a value is simply not known (e. g. for 37 reactions in E. coli ). For such reactions the consensus rule (eq. 1) is not applied.
Setting up the constrained optimization problem
In FBA, formulation of the optimization problem requires to define the following three elements: (i) a physiologically meaningful scoring function to evaluate flux distributions, (ii) the steady-state conditions for all internal metabolites valid for the time-scale of interest (e. g. the time-scale of growth) and (iii) further constraints taking into account biochemical knowledge as, for example, maximal enzyme capacities limiting the flux rates  or thermodynamic constraints on flux directions as those discussed above. The steady-state condition can be formulated as
S' V = 0 (9)
is a vector of ranges defined by the hard concentration bounds. V is the vector of flux rates and v j is the j-th component of V. λ1, λ2 ∈ ℝ+ are empirical factors weighting the relative contribution of the various penalty scores relative to the scoring function of fluxes. (For our computations we have chosen λ1 = 100, λ2 = 0.01 putting a lower weight to the attainment of set-point concentration values than to the restriciton of the metaboite concentration values to physiologically feasible soft bounds.) n is the number of reactions, and for any 1 ≤ j ≤ n, d j is a binary variable. α is set to a positive number which is larger than any possible flux value and larger than any possible Gibb's energy value, and it can easily be shown that the constraints 0 ≤ v j + αd j ≤ α and 0 ≤ - + αd j ≤ α are equivalent to v j ≠ 0 → sgn(v j ) = -sgn (). Intentionally, for a zero flux through a reaction the change of Gibb's free energy is not constrained because it might be substantially different from zero if the corresponding enzyme is missing or inhibited. The optimization problem corresponds to a mixed integer (boolean) linear program with quadratic scoring function.
We call a flux distribution obtained by solving the above optimization problem (eq. 10) thermodynamically realizable and refer to it in the following as TR-fluxmin, i.e. t hermodynamically r ealizable flux-min imized solution. If the maximization of biomass is used as flux objective, the sum of internal fluxes appearing in the objective function is replaced by the negative biomass production rate.
Application to a metabolic network of human erythrocytes
Using the same values of standard Gibb's free reaction energy changes as used in the kinetic model and putting the set-point values of the metabolite concentrations to the 'observed' ones, the TR-fluxmin solution of the optimization problem turns out to be identical with the flux-minimized solution determined by our previous approach . A detailed description of the model and the solution mentioned below can be found in the supplements [see Additional file 1].
Inspection of alternative flux distributions
The three alternative flux distributions obtained in the perturbation studies differ in the uptake flux of glucose (1.50495 mmol/h for the unperturbed network; 1.5015 mmol/h and 1.4972 mmol/h for the two alternative flux distributions) and the fluxes in the non-oxidative pentose phosphate pathway converting ribulose-5-phosphate (Ru5P) into fructose-5-phosphate (Fru6P) and glyceraldehyde 3-phosphate (GraP). The reference solution predicts this pathway to proceed in forward direction thereby forming 20.7 μ mol/h ribulose-5-phosphate. In the second solution (glucose uptake 1.5015 mmol/h) this pathway is not used at all whereas in the third flux distribution (glucose uptake 1.4972 mmol/h) it is used in backward direction producing 25.8 μ mol/h ribulose-5-phosphate. Interestingly, the latter flux distribution is also obtained for the unperturbed network if the maximization of biomass production used as flux criterion. Notably, all three different flux distributions obtained as solution of the minimization problem (eq. 10) for randomly altered thermodynamic parameters and set-point concentrations are feasible from the kinetic view point, i.e. the kinetic model of the erythrocyte metabolism yields a stable stationary solution.
Effect of external concentrations
Our algorithm allows assessing how the predicted flux distributions are affected by changes in the concentration of external metabolites. In vivo, such a situation may occur if some essential fuels for the cellular metabolism are depleted, for example, due to a reduced blood flow through vessels with severe atherosclerotic stenoses, or some end products of the cellular metabolism accumulate because of a reduced excretion capacity of the body. For example, in case of strong physical exercise the concentration of lactate in human blood may rise to values as high as 19.5 mM (in blood plasma) respectively 7.0 mM (in erythrocyte cytoplasm)  indicating that the lactate production by the anaerobic skeletal muscle clearly exceeds its rate of re-conversion to glucose in the liver and its utilization rate in the heart muscle. To investigate the consequences of such high blood lactate levels for the metabolism of red cells we calculated thermodynamically realizable flux distributions at gradually increasing concentration of external lactate. For all metabolites except external lactate, the hard bounds were put to ± 25% and the soft bounds to ± 10% deviation from of the 'observed' concentration values. For external lactate concentrations up to a critical value of 12.4 mM our algorithm predicted a thermodynamically realizable flux distribution. For concentrations higher than 12.4 mM no stationary flux distribution solution was found. Increasing gradually the concentration of external lactate up to the critical value of 12.4 mM, the concentrations of pyruvate, NAD+ and NADH tended towards the hard bounds to ensure the flux through the lactate dehydrogenase (EC:184.108.40.206) to be directed towards formation of lactate. Our find of a metabolic threshold effect with respect to blood lactate levels corresponds well with clinical observations. At lactate levels higher than 4 mM a reduced deformability of erythrocytes is observed, which may account for the exercise-induced arterial hypoxemia occuring in athletes . Decreasing deformability of erythrocytes is a clear indication for a severely perturbed metabolism of the cell.
Application to a metabolic network of E. coli
To check the applicability of our algorithm to genome-scale metabolic networks comprising hundreds of reactions and metabolites, we performed the same type of analysis as described above with respect to the metabolic network iJR904 of the bacterium E. coli reconstructed by Palsson and co-workers . In this model a minimal medium composed of glucose, ammonium, sulfate, oxygen, phosphate is sufficient for growth according to the biomass creation formula associated with the model. Experimental flux data for E. coli has been determined by Emmerling et al.  which correspond to 17 internal fluxes of the iJR904 network (using the projection of Segre et al.  onto the iJE660a network of E. coli .) The thermodynamic properties of the iJR904 network, consisting of 659 metabolites and 931 reactions, have been analyzed previously [14, 15, 20, 23, 32]. Since experimentally determined Gibb's free energies are available only for a minor fraction of reactions [20, 39] we use computed values given by Henry et al. .
These values were obtained by a slightly modified version of the group contribution method [40, 41]. Physiological concentration ranges were available for 22 internal metabolites (given in Kümmel et al. ) and 10 external metabolites (given in Henry et al. ). For the other metabolites generic concentration bounds were used based on typical cellular concentration ranges reported in the literature: 20 μ M-0.5 mM (soft bounds), 5 μ M-2 mM (hard bounds). Further details of the model are given in the supplement [see Additional file 2].
In a third calculation we again omitted the condition of thermodynamic realizability from the optimization algorithm but instead used the heuristic classification of reactions into reversible and irreversible ones as outlined in  (case: 'biomass, heuristic irreversibilities', data points symbolized by green diamonds). The obtained flux distribution also yielded a reasonably good concordance between predicted and observed flux values. Notably, this 'classical' variant of FBA gave no better predictions of the observed fluxes than the TR-solution obtained with our algorithm. This qualifies our method as a valuable flux predictor for large-scale networks without the need to apply heuristic rules for the assignment of flux directionalities.
Using the same perturbation analysis as outlined above for the erythrocyte network we investigated the impact of alterations in the values of the Gibb's free standard energies on the predicted flux distributions. Such an analysis is of importance as the values of standard Gibb's free energy changes computed by the group contribution method may generally exhibit a large degree of uncertainty .
Discussion & Conclusion
Quantitative evaluation of genome-scale metabolic models by means of FBA is becoming more and more appealing because it works without knowledge of the kinetics and regulation of the underlying enzymes and membrane transporters. However, the outcome of FBA is rather hypothetical because it relies on plausible but hardly provable optimality principles that are thought to govern metabolic flux distributions.
Therefore, a challenge for computational systems biology lies in the incorporation of all biochemical knowledge that is obtainable at genome-scale (which is not the case for enzyme kinetics). One important restriction of fluxes in the network arises from thermodynamics. Reactions associated with a decrease of free energy larger than 30 kJ/mol are generally thought to be irreversible. This condition can be used as an additional constraint on feasible flux distributions. Kümmel and co-workers  have recently developed an algorithm that – based on thermodynamics, network topology and heuristic rules – automatically assigns reaction directions in metabolic models such that the reaction network is thermodynamically feasible with respect to the production of energy equivalents. However, an a priori distinction between reversible and irreversible reactions may become problematic under extreme conditions, e.g. depletion of substrates or accumulation of intermediates due to inhibition of enzymes, where metabolite concentrations may drastically change thus allowing to reverse reactions that are normally designated to be irreversible.
For example, under hypoxic conditions, the cellular concentration of oxygen may become so low that the respiratory chain – usually thought to carry electrons from hydrogen to oxygen in a strictly irreversible manner – may indeed operate in the reverse direction, i.e. reducing NAD to NADH2 . Thus, it is necessary to replace the rigid priori classification of reactions into reversible and irreversible ones by a more flexible constraint that assures the flux directions to be compatible with the change of Gibb's free energies, exhibiting a wide range of values depending on the actual metabolite concentrations. An important step into this direction was recently made by Henry et al.  who included the thermodynamic consensus rule as additional side constraint into FBA. In their study, they investigated the range of metabolite concentrations that allow in a genome-scale network of E. coli the realization of a specific flux distribution assuring optimal bacterial growth. In contrast to this approach, the algorithm proposed in this work aims at employing reliable information on metabolite concentrations to restrain the solution space of FBA.
Hence, depending on reported ranges of metabolite concentrations, our algorithm may yield different flux distributions. In other words, in our approach we do not ask for metabolite concentrations that are compatible with a given flux distribution but in contrast ask for the flux distribution that is compatible with a given metabolite profile. As demonstrated for two exemplary metabolic networks, even if no measurements of metabolite concentrations are available restriction of concentrations to physiologically feasible ranges alone allows the prediction of reliable flux distributions if no a priori assumptions are made on the reversibility of the reactions. As demonstrated for the erythrocyte network, our approach may provide valuable information about alterations in the external conditions of a cell that may result in a metabolic dysfunction. Of course, FBA cannot assess whether a stable steady state may exist at very high concentrations of external lactate because this is determined by kinetic regulation. Possibly the metabolite concentrations may vary even in a larger interval than imposed in our calculations. This problem can be addressed better by a comprehensive kinetic network model. Nevertheless, our method may provide valuable information on external conditions causing metabolic problems merely for thermodynamic reasons. Concerning the predictive capacity of our method it must be critically noted that – based on a comparison with a relatively small number of measured fluxes – the most reliable flux distribution for the E. coli network is still obtained if the directionality of fluxes is a priori defined based on biochemical conventions. This is obviously due to the fact that – owing to the lack of reliable experimental data – the soft bounds for the metabolite concentration used in our method have been too generously chosen. Indeed, enlarging systematically the physiologically feasible concentration ranges one eventually obtains a network without any constraint of flux directionalities. Hence, the usefulness of the proposed method essentially depends upon the availability of reliable information on values of free energy changes and metabolite concentrations. As long as this information is not available, the benefit of our method consists mostly in the generation of alternative flux distributions by varying the values of standard Gibb's free energy changes and/or in the physiologically relevant concentration ranges of metabolites. Applying such perturbation analysis to two networks of different complexity has provided evidence that the larger the network is, the more alternative flux distributions occur, even at relatively modest variation of energy values of about 5 kJ/mol. Inspection of such alternative flux distributions reveals critical reactions for which fluxes are largely undetermined by the FBA approach. In this respect, our method represents a useful complement to the thermodynamic evaluation method recently proposed by Kümmel et al. to identify putative regulatory sites by network-embedded thermodynamic analysis of metabolome data .
The work of AH was funded by the German Federal Government's Program "Systems Biology of Hepatocytes – HepatoSys".
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