Bioinformatics Advance Access originally published online on March 7, 2007
Bioinformatics 2007 23(11):1378-1385; doi:10.1093/bioinformatics/btm065
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From structure to dynamics of metabolic pathways: application to the plant mitochondrial TCA cycle
1University Potsdam, Institute for Biochemistry and Biology, Karl-Liebknecht-Strasse 24-25, Haus 20, 2Max-Planck-Institute of Molecular Plant Physiology, Am Mühlenberg 1, 3Department of Chemical Engineering, Princeton University, Princeton, NJ 08544, USA, 4ICBM, University Oldenburg, 26111 Oldenburg, Germany
*To whom correspondence should be addressed.
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ABSTRACT |
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 TOP ABSTRACT 1 INTRODUCTION2 A MODEL OF...3 A STRUCTURAL KINETIC...4. EXTENDING THE MODEL5. SUMMARY AND CONCLUSIONSAPPENDIXACKNOWLEDGEMENTSREFERENCES |
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Motivation: Mitochondrial metabolism, dominated by the reactions of the tricarboxylic acid (TCA) cycle, is of vital importance for a wide range of metabolic processes. In particular for autotrophic tissue, such as plant leaves, the TCA cycle marks the point of divergence of anabolic pathways and plays an essential role in biosynthesis. However, despite extensive knowledge about its stoichiometric properties, the function and the dynamical capabilities of the TCA cycle remain largely unknown.
Methods and Results: Based on a recently proposed formalism, we investigate the dynamic and functional properties of the mitochondrial TCA cycle of plants. Starting with the structural properties, as described by the elementary flux modes of the system, we aim for the transition from structure to the dynamics of the TCA cycle. Using a parametric description of the system, encompassing all possible differential equations and parameter values, we detect and quantify regimes of different dynamic behavior. Optimizing the system with respect to dynamic stability, we demonstrate that maximal stability is associated with specific (relative) metabolite concentrations and flux values that are subsequently compared to the experimental literature. Our analysis also serves as a general example how to elucidate the transition from the structure to the dynamics of metabolic pathways.
Contact: steuer{at}agnld.uni-potsdam.de
Supplementary information: Supplementary data are available at Bioinformatics online.
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1 INTRODUCTION |
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TOPABSTRACT1 INTRODUCTION2 A MODEL OF...3 A STRUCTURAL KINETIC...4. EXTENDING THE MODEL5. SUMMARY AND CONCLUSIONSAPPENDIXACKNOWLEDGEMENTSREFERENCES |
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Over the last decade, significant advances in experimental high-throughput techniques have resulted in unprecedented information about the components that constitute cellular metabolism (Fernie et al., 2004; Sweetlove and Fernie, 2005). In particular, the genome-scale reconstruction of metabolic networks has revolutionized the ability to understand and engineer metabolic function along with the possibility to predict metabolic phenotypes and key aspects of network functionality (Famili et al., 2003; Schuster et al., 1999, 2000; Stelling et al., 2002).
However, the stoichiometric properties alone are not sufficient to define and predict the dynamical behavior of complex cellular systems (Ingram et al., 2006; Ronen et al., 2006). One of the primary challenges for post-genomic computational biology thus remains to bridge the gap between network structure on the one hand, and the resulting dynamic properties of metabolic systems on the other hand. Usually, the latter is accomplished by translating metabolic processes into a set of differential equations, requiring extensive additional (and often unavailable) information about the kinetic parameters and the specific functional form of the rate equations.
Recently, we have proposed an alternative formalism that allows to quantitatively evaluate the possible dynamics of metabolic systems without referring to any explicit set of differential equations (Steuer et al., 2006). Our approach is based upon a parametric representation of the Jacobian matrix of a metabolic system, such that each element of the Jacobian has a well-defined and straightforward interpretation in biochemical terms. Instead of focusing on a particular set of differential equations, this parametric representation allows to evaluate large ensembles of possible models, each restricted to comply with the available biochemical knowledge. In this way, it is possible to evaluate the stability with respect to perturbations, the existence of bifurcations and oscillatory regions as well as several other characteristic dynamic features of the system.
Utilizing this approach, we aim to investigate the robustness and possible dynamics of the mitochondrial tricarboxylic acid (TCA) cycle of plant leaves. In the first section, we outline the structural and stoichiometric properties of the system, making use of the concept of elementary flux modes (Schuster et al., 1999). Subsequently, after a brief synopsis of the mathematical background, we construct the parametric representation of the Jacobian matrix of the plant mitochondrial TCA cycle. In the following sections, we characterize the stability of the system with respect to perturbations and evaluate the existence and size of oscillatory regions. It is demonstrated that our method allows to identify specific biochemical conditions that result in an increased stability of the system. In the last section, we discuss our results and point out a general strategy to elucidate the transition from structure to dynamics of biochemical pathways.
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2 A MODEL OF THE TCA CYCLE |
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TOPABSTRACT1 INTRODUCTION2 A MODEL OF...3 A STRUCTURAL KINETIC...4. EXTENDING THE MODEL5. SUMMARY AND CONCLUSIONSAPPENDIXACKNOWLEDGEMENTSREFERENCES |
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In plants, the role of the TCA cycle differs fundamentally from that in heterotrophic organisms. Due to their autotrophic nature, plants synthesize their own respiratory substrates, mainly carbohydrates, which then serve as the main substrates for the TCA cycle. In addition to the synthesis of ATP, the TCA cycle marks a point of divergence of anabolic pathways and provides precursors for a number of biosynthetic processes, such as nitrogen fixation and the biosynthesis of amino acids (Hill, 1997).
Figure 1 depicts a simplified representation of the TCA cycle, which will be investigated in the following sections. As the main substrate, we consider phosphoenolpyruvate (PEP), metabolized via glycolysis and the oxidative pentose phosphate pathway. PEP is then either converted to pyruvate (Pyr) or to oxaloacetate (OAA), with a subsequent conversion to malate (Mal). Pyr and Mal enter the mitochondrion. In Figure 1, the export of metabolic intermediates for biosynthesis is restricted to citrate (Cit), in strict exchange of either OAA or Mal. Note that the TCA cycle in plants lacks the glyoxylate shunt. With Pyr as the only substrate, the TCA cycle is not able to catalyze a net synthesis of its metabolic intermediates (Hill, 1997).
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2.1 Structural analysis: elementary flux modes
Starting point of our analysis are the structural properties of the system, as described by the set of elementary flux modes depicted in Figure 2. An elementary flux mode (EFM) is defined as a minimal set of reactions that is consistent with a valid steady state, i.e. it defines a flux distribution that fulfills the stoichiometric mass-balance equation (Heinrich and Schuster, 1996; Schuster et al., 1999). The term ‘elementary’ refers to the non-decomposability of modes, i.e. an EFM cannot be represented by a combination of two or more smaller EFMs (Schuster et al., 1999, 2000).
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The TCA cycle, as depicted in Figure 1, gives rise to six elementary flux modes, corresponding to different modes of operation: EFM A: Pyr is imported into the mitochondrion. No metabolic intermediates are exported. Note that in this situation the cycle is subject to a mass-conservation relationship, i.e. the total amount of Mal, OAA, Cit, 2-OG and succinate is preserved. The cycle is thus not able to catalyze a net synthesis of its intermediates. However, in the presence of the NAD-malic enzyme (reaction
6 in Table 1), converting Mal to Pyr, this inevitably results in a depletion of metabolic intermediates and already indicates a potential dynamic instability of the pathway. EFM B: Mal substitutes for Pyr as substrate for the TCA cycle. In this case, Pyr and OAA are both synthesized from Mal. EFM C: As an alternative mode of operation, the TCA cycle provides Cit as a precursor for biosynthesis. The total amount of Cit and OAA within the mitochondrion is preserved. EFM D: Same as EFM C, only Mal substitutes for OAA uptake in exchange with Cit. Note that in this case, the presence of the NAD-malic enzyme again results in a depletion of metabolic intermediates. EFM E and F: Same C and D, but with Mal substituting for Pyr import. |
3 A STRUCTURAL KINETIC MODEL |
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TOPABSTRACT1 INTRODUCTION2 A MODEL OF...3 A STRUCTURAL KINETIC...4. EXTENDING THE MODEL5. SUMMARY AND CONCLUSIONSAPPENDIXACKNOWLEDGEMENTSREFERENCES |
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Given knowledge about the structural and stoichiometric properties of the TCA cycle, we now aim for the transition from structure to dynamics of the system. In particular, a steady-state flux distribution
that is consistent with the stoichiometric balance equation 
does not guarantee actual dynamic stability of the system. Taking dynamical properties of the system into account, we thus seek to elucidate the stability, robustness and possible dynamics of the pathway—based only on a minimal amount of additional information.
To this end, we have recently proposed a method that allows to assess the dynamic capabilities of a metabolic system without referring to any explicit set of differential equations (Steuer et al., 2006). Our approach is based on a decomposition of the Jacobian J of a metabolic system, consisting of m metabolites and r reactions, into a product of two matrices, such that the elements of each matrix have a well-defined and straightforward interpretation in biochemical terms.
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| (1) |
For a derivation of Equation (1), as well as a simple example, see the Appendix. Within our parametric representation of the Jacobian, the m x r dimensional matrix 
defines the time constants of the system. Given knowledge of the stoichiometric matrix N, a set of steady-state metabolite concentrations 
, and a steady-state flux distribution 
, the matrix 
is fully defined. The second term in Equation (1) defines the saturation or ‘effective kinetic order’ of each reaction with respect to the metabolites. For each element 
of the r x m dimensional matrix 
, we can specify a well-defined interval that covers all possible values of 
without referring to the explicit functional form of the rate equations.
Given the parametric decomposition of the Jacobian defined in Equation (1), the non-zero elements of both matrices span the associated parameter space and uniquely define the Jacobian at each possible point in parameter space. We emphasize that this generalized parameter space is equivalent to the more usual description in terms of Michaelis constants and maximal reaction velocities. A subsequent statistical evaluation of the Jacobian matrix then allows to assess several key dynamical properties of the system, such as the stability of steady states, relevant timescales (modal analysis) as well as the existence and location of possible bifurcations (Heinrich and Schuster, 1996; Steuer et al., 2006).
3.1 Application to the TCA cycle model
We now seek to apply our concept of structural modeling to the TCA cycle depicted in Figure 1. Starting with the simplest possible model, we first neglect the contribution of cofactors and consider only the main metabolic intermediates. The corresponding dynamical system consists of m = 9 variables and r = 13 reactions. To specify the Jacobian, we need (i) a set of steady-state metabolite concentrations 
; (ii) a steady-state flux distribution 
, determined by 
free parameters and (iii) a set of 14 saturation parameters 
specifying the degree of saturation of each reaction with respect to each of its substrates.
As a first approximation, all saturation parameters are set to unity 
, corresponding to a simple mass-action model of the system. Evaluating the resulting Jacobian for a large ensemble of randomly chosen flux and concentration values reveals that in this case the Jacobian is always rank deficient, i.e. the system does not allow for the existence of a dynamically stable steady state. Investigating this more closely, and testing for additional saturation parameters, reveals that the instability is resolved when the Mal dehydrogenase 5 is inhibited by its product OAA. Alternatively, the NAD-malic reaction 6 needs to be inhibited by its product Pyr. Indeed, as could be expected intuitively, the instability results from a possible imbalance between the rate of Pyr and OAA production within the mitochondrion. Since the Mal dehydrogenase is reversible, the instability is prevented in the actual biological system: Excess of OAA induces an increased flux from Mal towards Pyr. To account for this in the model, and to avoid the rank deficiency of the Jacobian, the parameters 
and 
are fixed to weak inhibition in the following.
Resuming the analysis for the modified ensemble of Jacobians, the next question relates to the dynamic stability of the TCA cycle. Following the approach described in (Steuer et al.,2006), we generate a large ensemble of possible models (Jacobians) by choosing the flux and concentration values from a random distribution. Subsequently, we select for those models that are maximally stable, i.e. those model that have a minimal largest real part 
within their spectrum of eigenvalues—corresponding to a fast response to perturbations. Within this constraint ensemble of maximally stable models, deviations in the distributions of individual parameters from their initial distribution indicate a relationship between these parameters and the stability of the model. Figure 3 compares the initial random distribution of all possible OAA concentrations to the distribution found within the maximally stable ensemble. As can be observed, models with maximal stability are associated with low concentrations of OAA. Similar results are obtained for Pyr, while all other concentration parameters show no, or only little, deviation in their distribution. The results for the remaining metabolites are summarized in Figure 4. Verifying this finding, we evaluate the distribution of the largest real part 
within the spectrum of eigenvalues, while fixing the steady-state concentrations of OAA and Pyr to low values. This results in a marked shift of 
towards negative values, as compared to the initial distribution for the unconstraint ensemble. See Figure 5 for a visualization. We thus can conclude that low concentrations of OAA and Pyr contribute significantly to an increased stability of the system, i.e. the return to the steady state after a perturbation is accelerated for lower concentrations of OAA and Pyr. Indeed, a previous study of plant mitochondria supports the assumption of a low OAA concentration in vivo (MacDougall and ap Rees, 1991). Though the concentration could not be detected directly, an equilibrium assumption of the Cit synthase would result in [OAA]0 
10–8 M, or about one molecule of OAA per mitochondrion (MacDougall and ap Rees, 1991). Thus, even when assuming that the Cit synthase is appreciably displaced from equilibrium, these results still indicates a rather low in vivo concentration.
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Jacobians with parameters drawn from a random distribution. Left plot: The initial distribution of steady-state OAA concentration for the unconstraint ensemble. Right plot: The distribution of steady-state OAA concentration for the constraint ensemble of Jacobians with a largest real part 
within the spectrum of eigenvalues (corresponding to 
of maximally stable models). Similar results are obtained for Pyr. As the absolute values of concentrations are not relevant, all concentrations are measured relative to cytosolic Pyr and are chosen equally distributed in 
. Steady-state flux values are chosen as random superpositions of flux modes, such that the total influx 
equals unity. All saturation parameters are 
, except 
and 
. A color version of this figure can be found in
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. The largest deviation is found for OAA and Pyr, indicating that these metabolites have the largest impact on the stability of the model. All concentrations are measured relative to Pyrcyt.
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within the spectrum of eigenvalues. For an ensemble of models (Jacobians) with low steady-state concentrations of OAA and Pyr, the distribution of 
is markedly shifted towards more negative values (red line), as compared to the unconstraint ensemble (blue line). A color version of this figure can be found in Extending to analysis of the distribution of metabolic fluxes, we now evaluate the elementary flux modes, depicted Figure 2, with respect to the dynamic stability of the system. When selecting for the maximally stable models among the ensemble of all possible models, most flux distributions attain a value close to the elementary modes EFM B and EFM F. Figure 6 shows a comparison of the contribution of the individual fluxmodes with respect to the unconstraint ensemble. Note that the fluxmodes B and F rely on Mal import only. This indicates that flux distributions relying on Mal import, rather than Pyr import, are favored with respect to their dynamic stability. Indeed, previous evidence shows that the capacity of the Pyr transporter is rather low, and may be insufficient under some circumstances (Hill, 1997). Furthermore, transgenic tobacco plants, lacking any detectable cytosolic Pyr kinase, were found to be indistinguishable from the wild type (Hill, 1997).
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, such that the total input flux equals unity. Shown is the ratio 
for each EFM. Within the ensemble of most stable models, the EFMs B and F are overrepresented. A color version of this figure can be found in 3.2 Bifurcations and oscillations
In addition to the stability of the steady states, the parameterized Jacobian of the TCA cycle allows to infer a variety of other dynamic properties of the system. In particular, not all models (Jacobians) of the unconstraint ensemble are dynamically stable. A small fraction of
3 x 10–3% of the random realizations exhibit a pair of eigenvalues with positive real parts and conjugate imaginary parts. Exploring the parameter space in the vicinity of these realizations reveals the existence of a Hopf bifurcation. Figure 7 depicts a corresponding bifurcation diagram.
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A Hopf bifurcation is usually associated with (at least transient) oscillatory behavior, as a pair of conjugate complex eigenvalues cross the imaginary axis. This is verified in Figure 7 using an explicit model of the TCA cycle based on differential equations (see Appendix). However, we emphasize that the existence of oscillatory dynamics can be deduced from our parametric representation of the Jacobian alone, without referring to any explicit kinetic model of the system. Furthermore, we note that the oscillations arise solely due to the stoichiometry of the system. As yet, all saturation parameters are set to unity and no allosteric interactions are included. Still, our analysis reveals the existence of an oscillatory region, confined to a small, but quantifiable, region in parameter space.
It should be noted that an analogous analysis of an explicit kinetic model, given in terms of the kinetic parameters of the reaction rates, is not trivial. Even when restricted to straightforward mass-action kinetics, involving only bilinear reaction terms, the steady state cannot be determined analytically. To evaluate possible bifurcations thus requires explicit numerical integration or other computationally costly methods. In contrast to this, our approach allows to directly specify the Jacobian of any desired steady-state solution without any further computational assistance.
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4. EXTENDING THE MODEL |
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TOPABSTRACT1 INTRODUCTION2 A MODEL OF...3 A STRUCTURAL KINETIC...4. EXTENDING THE MODEL5. SUMMARY AND CONCLUSIONSAPPENDIXACKNOWLEDGEMENTSREFERENCES |
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As yet, the analysis was restricted to non-saturable reaction rates with saturation parameters
. Extending the model, we now focus on a slightly different scenario, aiming to incorporate additional biochemical knowledge. In particular, we can make use of the fact that the in vivo flux distribution and the concentrations of many metabolic intermediates are known (see Appendix). Thus, instead of evaluating the Jacobian for all possible steady-state concentrations and flux values, we restrict the model to comply with a specific experimentally observed steady state. Within the parametric representation of the Jacobian, this can be straightforwardly achieved by setting the matrix 
constant.
In the following, by generating an ensemble of possible models (Jacobians) of the system at an observed steady state, we thus test for the stability and robustness of this specific state. All saturation parameters are allowed to take arbitrary values within their predefined interval, i.e. 
for any substrate of a reaction. The overall results agree qualitatively with the previous section: The vast majority of instances is dynamically stable, i.e. independent of the saturation (or non-saturation) of particular reactions, the observed operating point and flux values correspond to a stable steady state of the system— indicating a high robustness of the system with respect to the saturation of specific reactions. Only for a minor fraction of possible saturation parameters (
), the observed state is dynamically unstable. Again, we can select for the subset of maximally stable models and identify specific saturation parameters that contribute to an increased stability of the system (data not shown). However, a more interesting situation arises if additional regulation is included into the model. In particular, OAA is known to inhibit the succinate dehydrogenase (included within 4 in Table 1). Within the structural kinetic model, this gives rise to an additional saturation parameter 
, where n denotes a positive (integer) coefficient, specifying the maximal strength of the regulatory interaction. Figure 8 depicts the stability properties of the system at the observed state as a function of 
: For an increasing strength of the regulation, the size of the region in parameter space that is associated with dynamic instability rapidly increases. In such a situation, the system cannot be considered robust with respect to the saturation of the reactions. Instead, our analysis then allows to uncover specific conditions and requirements for the experimentally observed state to be stable, i.e. to be actually observable as a steady state of the system. Similar to the previous sections, we thus again compare the average saturation parameters 
within the subset of unstable models to the unconstraint ensemble. The results are summarized in Figure 9 for all saturation parameters. As can be observed, strong saturation of the Mal dehydrogenase (reaction 5) and low saturation of the NAD-malic reaction (reaction 6), both with respect to the substrate Mal, favors instability.
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of OAA on the succinate dehydrogenase. For each value of 
, an ensemble of Jacobians is generated and the distribution of the largest real parts 
is recorded. Left Plot: Shown is the median (solid black line) along with the 0.25 and 0.75 quantiles of the distribution of 
(with 
corresponding to instability). Right Plot: The percentage of dynamically instable models (
) as a function of 
. A color version of this figure can be found in
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within the subset of unstable models to the unconstraint ensemble (for 
). Strong saturation of 
(linear regime) and 
(strong saturation) the probability of finding 
decreases (
within 
instances, compared to 
in the unconstraint ensemble.). A color version of this figure can be found in |
5. SUMMARY AND CONCLUSIONS |
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TOPABSTRACT1 INTRODUCTION2 A MODEL OF...3 A STRUCTURAL KINETIC...4. EXTENDING THE MODEL5. SUMMARY AND CONCLUSIONSAPPENDIXACKNOWLEDGEMENTSREFERENCES |
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The TCA cycle is of vital importance for a large number of biosynthetic processes and plays an important role in the carbon metabolism of plants. Here, we have evaluated a model of the TCA cycle with respect to its dynamic capabilities. Starting with the structural properties, as described by the stoichiometry and the elementary flux modes, we have aimed for the transition from structure to dynamics of the pathway. In particular, structural properties alone are not sufficient to determine the dynamics and function of cellular systems (Ingram et al., 2006). Nonetheless, structural properties put constraints on the possible dynamics (Klipp et al., 2004; Liebermeister and Klipp, 2005) and dynamic properties seem to contribute to the large-scale organization of biological networks (Prill et al., 2005). Within the approach described here, we make use of the fact that the local linear approximation of a metabolic system, the Jacobian matrix, is already sufficient to determine essential dynamic properties of the system. Based on a recently proposed method, our approach consists of generating large ensembles of possible models (Jacobians) that are consistent with the network structure of the pathway. It is thus possible to evaluate the dynamical capabilities of a metabolic system without referring to any particular explicit set of differential equations.
With respect to the plant mitochondrial TCA cycle, this allows to identify ranges of steady-state flux values and metabolite concentrations for which the system is maximally stable. Furthermore, given an experimentally observed steady state, we can identify specific biochemical conditions—in terms of (normalized) saturation of reactions—that ensure the stability of the observed steady state. Though not yet conclusive, a comparison with experimental results indicates that these dynamic requirements for maximal stability hold under wild-type conditions.
As our approach relies only upon a minimal amount of additional information about the system, it is readily applicable to metabolic system of a realistic complexity. In particular when no, or only little, knowledge about the detailed reaction mechanisms is available, our methods allows to give a systematic and comprehensive evaluation of the generalized parameter space. As compared to explicit kinetic modeling, based on explicit differential equations, the computational costs are low. Nonetheless, we emphasize that within the generalized parameter space the evaluation of the Jacobian with respect to possible instability is exact—there is no approximation involved. Future investigations will aim to provide a more detailed model of the mitochondrial TCA cycle, including adjacent reactions and the interaction of the cycle with other compartments.
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APPENDIX |
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TOPABSTRACT1 INTRODUCTION2 A MODEL OF...3 A STRUCTURAL KINETIC...4. EXTENDING THE MODEL5. SUMMARY AND CONCLUSIONSAPPENDIXACKNOWLEDGEMENTSREFERENCES |
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Structural kinetic modeling
Our approach is based upon a generalized parameterization of metabolic networks and a subsequent decomposition of the Jacobian into a product of two matrices (Steuer et al., 2006). The time evolution of a metabolic system, consisting of m metabolites
and r reactions 
, is given as 
, where N denotes the stoichiometric matrix. Assuming the existence of a (not necessarily unique or stable) steady state 
with a steady-state flux distribution 
, the system of differential equations can be rewritten as
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| (2) |
To evaluate the dynamical capabilities of the metabolic system, we aim at a parametric representation of the Jacobian matrix. Using the normalized variables 
and the definitions given in Equation (2), the Jacobian with respect to the new variables x at the steady state 
is
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| (3) |
Within our approach, the non-zero elements of the matrices 
and 
define the new generalized parameter space of the system, i.e. we evaluate the possible dynamics of the system in terms of the new parameters 
and 
. Importantly, the elements of both matrices have a well-defined and straightforward interpretation in biochemical terms: The elements 
of the 
dimensional matrix 
are entirely specified by experimentally accessible quantities, such as average metabolite concentrations and flux values. The elements 
of the 
dimensional matrix 
are closely related to the elasticity coefficients of metabolic control analysis and measure the normalized degree of saturation of each reaction. For almost all biochemical rate functions, the saturation parameters 
are confined to well-defined intervals (Steuer et al.,2006): 
if the metabolite x is a substrate of the reaction µ. The limits 
and 
correspond to the linear regime (
) and full saturation (
), respectively. For cooperative and sigmoidal kinetics 
, with n denoting a positive integer. Allosteric activators and inhibitors are parameterized as 
and 
, respectively. For mathematical details and examples see also (Steuer et al., 2006).
A simple example
To briefly illustrate the generalized parameterization more clearly, we consider the simplest possible metabolic pathway, consisting of r = 2 reactions and m = 1 metabolite,
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with the stoichiometric matrix 
and the (as yet unspecified) vector of rate equations
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Within our approach, the pathway is described in terms of an average metabolite concentration S0, a steady-state flux value 
and the (normalized) saturation 
of reaction 2 with respect to its substrate S. Thus, according to Equation (2), the generalized parameter matrices are
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| (4) |
This parameterization has several advantages compared to the more usual description in terms of Michaelis constants and maximal reaction velocities: (i) The parameters are intuitively accessible: Even without detailed knowledge of the kinetic mechanisms, it is usually possible to define a physiologically plausible range for each parameter, i.e. to specify intervals 
, 
, and 
that define the physiologically admissible parameter space of the system. (ii) The parameterization is independent of the specific choice of the explicit functional form of the rate equations. All results obtained in terms of the generalized parameters hold for a large class of possible biochemical rate functions (Steuer et al., 2006). (iii) The generalized parameters define the corresponding Jacobian matrix at each point in parameter space. Thus, according to Equation (3),
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| (5) |
Once the parametric representation of the Jacobian is obtained, the possible dynamics of the system can be evaluated. In particular, the eigenvalues of the Jacobian matrix define the response of the system to (small) perturbations, possible transitions to instability and the existence of oscillatory regions within the parameter space. Moreover, by taking bifurcations of higher codimension into account, the existence of complex dynamics can be predicted (Steuer et al., 2006).
An explicit model of the TCA cycle
The numerical simulation shown in Figure 7 is based on an explicit model of the cycle. All reactions, except 5 and 6, were modeled as mass-action kinetics with 
. Given a desired steady state from the random ensemble, the kinetic parameters are 
. Product inhibition of Pyr and OAA on their production was incorporated as 
and 
, respectively. The parameters 
is determined as a function of the saturation parameter 
, 
. Analogously for 
. Note that the existence of the Hopf bifurcation does not depend on the particular explicit form of the rate equation.
Evaluation of the structural kinetic model
To evaluate the properties of the TCA cycle at a particular steady state (Figure 8), the (relative) flux distribution was chosen according to (Hanning and Heldt, 1993; Schwender et al., 2004). Metabolite concentration were based upon the measurements given in (Farre et al., 2001; MacDougall and ap Rees, 1999). In case no mitochondrial concentration was available, the respective concentration was replaced by its value in the cytosol.
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ACKNOWLEDGEMENTS |
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TOPABSTRACT1 INTRODUCTION2 A MODEL OF...3 A STRUCTURAL KINETIC...4. EXTENDING THE MODEL5. SUMMARY AND CONCLUSIONSAPPENDIXACKNOWLEDGEMENTSREFERENCES |
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RS acknowledges financial support by HWP 2004–2006 of the state Brandenburg. T.G. and B.B. acknowledge support from the German VW-Stiftung and SFB 555. Funding to pay the Open Access publication charges was provided by the Max Planck Society.
Conflicts of Interest: none declared.
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FOOTNOTES |
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Associate Editor: Alfonso Valencia
Received on August 20, 2006; revised on February 1, 2007; accepted on February 19, 2007
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TOPABSTRACT1 INTRODUCTION2 A MODEL OF...3 A STRUCTURAL KINETIC...4. EXTENDING THE MODEL5. SUMMARY AND CONCLUSIONSAPPENDIXACKNOWLEDGEMENTSREFERENCES |
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