Flux-balance analysis of mitochondrial energy metabolism: consequences of systemic stoichiometric constraints

Mitochondrial metabolism is a critical component in the functioning and maintenance of cellular organs. The stoichiometry of biochemical reaction networks imposes constraints on mitochondrial function. A modeling framework, flux-balance analysis (FBA), was used to characterize the optimal flux distributions for maximal ATP production in the mitochondrion. The model predicted the expected ATP yields for glucose, lactate, and palmitate. Genetic defects that affect mitochondrial functions have been implicated in several human diseases. FBA can characterize the metabolic behavior due to genetic deletions at the metabolic level, and the effect of mutations in the tricarboxylic acid (TCA) cycle on mitochondrial ATP production was simulated. The mitochondrial ATP production is severely affected by TCA-cycle mutations. In addition, the model predicts the secretion of TCA-cycle intermediates, which is observed in clinical studies of mitochondriopathies such as those associated with fumarase deficiency. The model provides a systemic perspective to characterize the effect of stoichiometric constraints and specific metabolic fluxes on mitochondrial function.     

Candidate metabolic network states in human mitochondria. Impact of diabetes, ischemia, and diet

The human mitochondrial metabolic network was recently reconstructed based on proteomic and biochemical data. Linear programming and uniform random sampling were applied herein to identify candidate steady states of the metabolic network that were consistent with the imposed physico-chemical constraints and available experimental data. The activity of the mitochondrion was studied under four metabolic conditions: normal physiologic, diabetic, ischemic, and dietetic. Pairwise correlations between steady-state reaction fluxes were calculated in each condition to evaluate the dependence among the reactions in the network. Applying constraints on exchange fluxes resulted in predictions for intracellular fluxes that agreed with experimental data. Analyses of the steady-state flux distributions showed that the experimentally observed reduced activity of pyruvate dehydrogenase in vivo could be a result of stoichiometric constraints and therefore would not necessarily require enzymatic inhibition. The observed changes in the energy metabolism of the mitochondrion under diabetic conditions were used to evaluate the impact of previously suggested treatments. The results showed that neither normalized glucose uptake nor decreased ketone body uptake have a positive effect on the mitochondrial energy metabolism or network flexibility. Taken together, this study showed that sampling of the steady-state flux space is a powerful method to investigate network properties under different conditions and provides a basis for in silico evaluations of effects of potential disease treatments.     

Multi-Target Analysis and Design of Mitochondrial Metabolism

Analyzing and optimizing biological models is often identified as a research priority in biomedical engineering. An important feature of a model should be the ability to find the best condition in which an organism has to be grown in order to reach specific optimal output values chosen by the researcher. In this work, we take into account a mitochondrial model analyzed with flux-balance analysis. The optimal design and assessment of these models is achieved through single- and/or multi-objective optimization techniques driven by epsilon-dominance and identifiability analysis. Our optimization algorithm searches for the values of the flux rates that optimize multiple cellular functions simultaneously. The optimization of the fluxes of the metabolic network includes not only input fluxes, but also internal fluxes. A faster convergence process with robust candidate solutions is permitted by a relaxed Pareto dominance, regulating the granularity of the approximation of the desired Pareto front. We find that the maximum ATP production is linked to a total consumption of NADH, and reaching the maximum amount of NADH leads to an increasing request of NADH from the external environment. Furthermore, the identifiability analysis characterizes the type and the stage of three monogenic diseases. Finally, we propose a new methodology to extend any constraint-based model using protein abundances.     

Isotopomer analysis of myocardial substrate metabolism: a systems biology approach

The increasing accessibility of mass isotopomer data via GC-MS and NMR technology has necessitated the use of a systematic and reliable method to take advantage of such data for flux analysis. Here we applied a nonlinear, optimization-based method to study substrate metabolism in cardiomyocytes using (13)C data from perfused mouse hearts. The myocardial metabolic network used in this study accounts for 257 reactions and 240 metabolites, which are further compartmentalized into extracellular space, cytosol, and mitochondrial matrix. Analysis of the perfused mouse heart showed that the steady-state ATP production rate was 16.6 +/- 2.3 micromol/min . gww, with 30% of the ATP coming from glycolysis. Of the four substrates available in the perfusate (glucose, pyruvate, lactate, and oleate), exogenous glucose forms the majority of cytosolic pyruvate. Pyruvate decaboxylation is significantly higher than carboxylation, suggesting that anaplerosis is low in the perfused heart. Exchange fluxes were predicted to be high for reversible enzymes in the citric acid cycle (CAC), but low in the glycolytic pathway. Pseudoketogenesis amounted to approximately 50% of the net ketone body uptake. Sensitivity analysis showed that the estimated flux distributions were relatively insensitive to experimental errors. The application of isotopomer data drastically improved the estimation of reaction fluxes compared to results computed with respect to reaction stoichiometry alone. Further study of 12 commonly used (13)C glucose mixtures showed that the mixtures of 20% [U-(13)C(6)] glucose, 80% [3 (13)C] glucose and 20% [U-(13)C(6)] glucose, 80% [4 (13)C] were best for resolving fluxes in the current network.     

The principle of flux minimization and its application to estimate stationary fluxes in metabolic networks.

Cellular functions are ultimately linked to metabolic fluxes brought about by thousands of chemical reactions and transport processes. The synthesis of the underlying enzymes and membrane transporters causes the cell a certain 'effort' of energy and external resources. Considering that those cells should have had a selection advantage during natural evolution that enabled them to fulfil vital functions (such as growth, defence against toxic compounds, repair of DNA alterations, etc.) with minimal effort, one may postulate the principle of flux minimization, as follows: given the available external substrates and given a set of functionally important 'target' fluxes required to accomplish a specific pattern of cellular functions, the stationary metabolic fluxes have to become a minimum. To convert this principle into a mathematical method enabling the prediction of stationary metabolic fluxes, the total flux in the network is measured by a weighted linear combination of all individual fluxes whereby the thermodynamic equilibrium constants are used as weighting factors, i.e. the more the thermodynamic equilibrium lies on the right-hand side of the reaction, the larger the weighting factor for the backward reaction. A linear programming technique is applied to minimize the total flux at fixed values of the target fluxes and under the constraint of flux balance (= steady-state conditions) with respect to all metabolites. The theoretical concept is applied to two metabolic schemes: the energy and redox metabolism of erythrocytes, and the central metabolism of Methylobacterium extorquens AM1. The flux rates predicted by the flux-minimization method exhibit significant correlations with flux rates obtained by either kinetic modelling or direct experimental determination. Larger deviations occur for segments of the network composed of redundant branches where the flux-minimization method always attributes the total flux to the thermodynamically most favourable branch. Nevertheless, compared with existing methods of structural modelling, the principle of flux minimization appears to be a promising theoretical approach to assess stationary flux rates in metabolic systems in cases where a detailed kinetic model is not yet available.     

Parametric analysis of metabolic fluxes of α-amylase and protease-producing Bacillus subtilis

Parametric analysis was applied for a metabolic flux model for the fed-batch culture of Bacillus subtilis producing recombinant α-amylase and protease. The metabolic flux model was formulated as a linear programming problem consisting of 49 reactions (decision variables) and 50 metabolites (equality constraints). This study was aimed to determine the response of the metabolic fluxes and objective function value of minimizing the difference between ATP consumption and ATP production (ATP balance). With regard to intracellular metabolite accumulation, the objective function value was least sensitive to variation in succinate and most sensitive to variation in malate. Amongst the variations in the accumulation rates of extracellular metabolites, the objective function value was least sensitive to variation in glutamate and most sensitive to variation in starch hydrolysis and triglyceride synthesis. A 10% variation in metabolite accumulation rates caused a maximum of 13.8% variation (standard error = 3.8%) in the objective function value.     

Metabolic adaptation of Escherichia coli during temperature-induced recombinant protein production: 2. Redirection of metabolic fluxes

The impact of temperature-induced synthesis of human basic fibroblast growth factor (hFGF-2) in high-cell-density cultures of recombinant Escherichia coli was studied by estimating metabolic flux variations. Metabolic flux distributions in E. coli were calculated by means of a stoichiometric network and linear programming. After the temperature upshift, a substantially elevated energy demand for synthesis of hFGF-2 and heat shock proteins resulted in a redirection of metabolic fluxes. Catabolic pathways like the Embden-Meyerhof-Parnas pathway and the tricarboxylic acid (TCA) cycle showed significantly enhanced activities, leading to reduced flux to growth-associated pathways like the pentose phosphate pathway and other anabolic pathways. Upon temperature upshift, an excess of NADPH was produced in the TCA cycle by isocitrate dehydrogenase. The metabolic model predicted the involvement of a transhydrogenase generating additional NADH from NADPH, thereby increasing ATP regeneration in the respiratory chain. The influence of the temperature upshift on the host's metabolism was investigated by means of a control strain harboring the "empty" parental expression vector. The metabolic fluxes after the temperature upshift were redirected similarly to the production strain; the effects, however, were observed to a lesser extent and with different time profiles.     

Quantification of metabolism in Saccharomyces cerevisiae under hyperosmotic conditions using elementary mode analysis

Yeast metabolism under hyperosmotic stress conditions was quantified using elementary mode analysis to obtain insights into the metabolic status of the cell. The fluxes of elementary modes were determined as solutions to a linear program that used the stoichiometry of the elementary modes as constraints. The analysis demonstrated that distinctly different sets of elementary modes operate under normal and hyperosmotic conditions. During the adaptation phase, elementary modes that only produce glycerol are active, while elementary modes that yield biomass, ethanol, and glycerol become active after the adaptive phase. The flux distribution in the metabolic network, calculated using the fluxes in the elementary modes, was employed to obtain the flux ratio at key nodes. At the glucose 6-phosphate (G6P) node, 25% of the carbon influx was diverted towards the pentose phosphate pathway under normal growth conditions, while only 0.3% of the carbon flux was diverted towards the pentose phosphate pathway during growth at 1?M NaCl, indicating that cell growth is arrested under hyperosmotic conditions. Further, objective functions were used in the linear program to obtain optimal solution spaces corresponding to the different accumulation rates. The analysis demonstrated that while biomass formation was optimal under normal growth conditions, glycerol synthesis was closer to optimal during adaptation to osmotic shock.     

Combining pathway analysis with flux balance analysis for the comprehensive study of metabolic systems

The elucidation of organism-scale metabolic networks necessitates the development of integrative methods to analyze and interpret the systemic properties of cellular metabolism. A shift in emphasis from single metabolic reactions to systemically defined pathways is one consequence of such an integrative analysis of metabolic systems. The constraints of systemic stoichiometry, and limited thermodynamics have led to the definition of the flux space within the context of convex analysis. The flux space of the metabolic system, containing all allowable flux distributions, is constrained to a convex polyhedral cone in a high-dimensional space. From metabolic pathway analysis, the edges of the high-dimensional flux cone are vectors that correspond to systemically defined "extreme pathways" spanning the capabilities of the system. The addition of maximum flux capacities of individual metabolic reactions serves to further constrain the flux space and has led to the development of flux balance analysis using linear optimization to calculate optimal flux distributions. Here we provide the precise theoretical connections between pathway analysis and flux balance analysis allowing for their combined application to study integrated metabolic function. Shifts in metabolic behavior are calculated using linear optimization and are then interpreted using the extreme pathways to demonstrate the concept of pathway utilization. Changes to the reaction network, such as the removal of a reaction, can lead to the generation of suboptimal phenotypes that can be directly attributed to the loss of pathway function and capabilities. Optimal growth phenotypes are calculated as a function of environmental variables, such as the availability of substrate and oxygen, leading to the definition of phenotypic phase planes. It is illustrated how optimality properties of the computed flux distributions can be interpreted in terms of the extreme pathways. Together these developments are applied to an example network and to core metabolism of Escherichia coli demonstrating the connections between the extreme pathways, optimal flux distributions, and phenotypic phase planes. The consequences of changing environmental and internal conditions of the network are examined for growth on glucose and succinate in the face of a variety of gene deletions. The convergence of the calculation of optimal phenotypes through linear programming and the definition of extreme pathways establishes a different perspective for the understanding of how a defined metabolic network is best used under different environmental and internal conditions or, in other words, a pathway basis for the interpretation of the metabolic reaction norm.     

What do mitochondrial diseases teach us about normal mitochondrial functions...that we already knew: threshold expression of mitochondrial defects

This paper shows how metabolic control analysis (MCA) can help to explain two important features of mitochondrial diseases: (i) the existence of a threshold in the expression of the complex deficiencies on the respiratory flux or on ATP synthesis, i.e. the fact that it is necessary to have a large complex deficiency in order to observe a substantial decrease in these fluxes; (ii) the tissue specificity, i.e. the fact that all tissues are not affected, even if the complex deficiency is present in all of them. We also show the limits of MCA, particularly when considering the in vivo situation. However, MCA offers a new way to consider mitochondrial diseases. The fact that fluxes only slightly change, when a complex is affected, is done at the expense of great changes in intermediate metabolite concentrations; intermediate metabolites situated upstream from the deficient complex are more reduced, leading to a greater generation of free radicals. This could bring an explanation for the diseases observed in conditions where the mitochondrial rate of ATP synthesis is only slightly affected.     

Estimating Metabolic Fluxes Using a Maximum Network Flexibility Paradigm

MotivationGenome-scale metabolic networks can be modeled in a constraint-based fashion. Reaction stoichiometry combined with flux capacity constraints determine the space of allowable reaction rates. This space is often large and a central challenge in metabolic modeling is finding the biologically most relevant flux distributions. A widely used method is flux balance analysis (FBA), which optimizes a biologically relevant objective such as growth or ATP production. Although FBA has proven to be highly useful for predicting growth and byproduct secretion, it cannot predict the intracellular fluxes under all environmental conditions. Therefore, alternative strategies have been developed to select flux distributions that are in agreement with experimental “omics” data, or by incorporating experimental flux measurements. The latter, unfortunately can only be applied to a limited set of reactions and is currently not feasible at the genome-scale. On the other hand, it has been observed that micro-organisms favor a suboptimal growth rate, possibly in exchange for a more “flexible” metabolic network. Instead of dedicating the internal network state to an optimal growth rate in one condition, a suboptimal growth rate is used, that allows for an easier switch to other nutrient sources. A small decrease in growth rate is exchanged for a relatively large gain in metabolic capability to adapt to changing environmental conditions.ResultsHere, we propose Maximum Metabolic Flexibility (MMF) a computational method that utilizes this observation to find the most probable intracellular flux distributions. By mapping measured flux data from central metabolism to the genome-scale models of Escherichia coli and Saccharomyces cerevisiae we show that i) indeed, most of the measured fluxes agree with a high adaptability of the network, ii) this result can be used to further reduce the space of feasible solutions iii) this reduced space improves the quantitative predictions made by FBA and contains a significantly larger fraction of the measured fluxes compared to the flux space that was reduced by a uniform sampling approach and iv) MMF can be used to select reactions in the network that contribute most to the steady-state flux space. Constraining the selected reactions improves the quantitative predictions of FBA considerably more than adding an equal amount of flux constraints, selected using a more naïve approach. Our method can be applied to any cell type without requiring prior information.AvailabilityMMF is freely available as a MATLAB plugin at: http://cs.ru.nl/~wmegchel/mmf.     

A hybrid kinetic and constraint-based model of leaf metabolism allows predictions of metabolic fluxes in different environments

While flux balance analysis (FBA) provides a framework for predicting steady-state leaf metabolic network fluxes, it does not readily capture the response to environmental variables without being coupled to other modelling formulations. To address this, we coupled an FBA model of 903 reactions of soybean (Glycine max) leaf metabolism with e-photosynthesis, a dynamic model that captures the kinetics of 126 reactions of photosynthesis and associated chloroplast carbon metabolism. Successful coupling was achieved in an iterative formulation in which fluxes from e-photosynthesis were used to constrain the FBA model and then, in turn, fluxes computed from the FBA model used to update parameters in e-photosynthesis. This process was repeated until common fluxes in the two models converged. Coupling did not hamper the ability of the kinetic module to accurately predict the carbon assimilation rate, photosystem II electron flux, and starch accumulation of field-grown soybean at two CO₂ concentrations. The coupled model also allowed accurate predictions of additional parameters such as nocturnal respiration, as well as analysis of the effect of light intensity and elevated CO₂ on leaf metabolism. Predictions included an unexpected decrease in the rate of export of sucrose from the leaf at high light, due to altered starch–sucrose partitioning, and altered daytime flux modes in the tricarboxylic acid cycle at elevated CO₂. Mitochondrial fluxes were notably different between growing and mature leaves, with greater anaplerotic, tricarboxylic acid cycle and mitochondrial ATP synthase fluxes predicted in the former, primarily to provide carbon skeletons and energy for protein synthesis.     

Uniform sampling of steady-state flux spaces: means to design experiments and to interpret enzymopathies

Reconstruction of genome-scale metabolic networks is now possible using multiple different data types. Constraint-based modeling is an approach to interrogate capabilities of reconstructed networks by constraining possible cellular behavior through the imposition of physicochemical laws. As a result, a steady-state flux space is defined that contains all possible functional states of the network. Uniform random sampling of the steady-state flux space allows for the unbiased appraisal of its contents. Monte Carlo sampling of the steady-state flux space of the reconstructed human red blood cell metabolic network under simulated physiologic conditions yielded the following key results: 1), probability distributions for the values of individual metabolic fluxes showed a wide variety of shapes that could not have been inferred without computation; 2), pairwise correlation coefficients were calculated between all fluxes, determining the level of independence between the measurement of any two fluxes, and identifying highly correlated reaction sets; and 3), the network-wide effects of the change in one (or a few) variables (i.e., a simulated enzymopathy or fixing a flux range based on measurements) were computed. Mathematical models provide the most compact and informative representation of a hypothesis of how a cell works. Thus, understanding model predictions clearly is vital to driving forward the iterative model-building procedure that is at the heart of systems biology. Taken together, the Monte Carlo sampling procedure provides a broadening of the constraint-based approach by allowing for the unbiased and detailed assessment of the impact of the applied physicochemical constraints on a reconstructed network.     

System-level insights into yeast metabolism by thermodynamic analysis of elementary flux modes

One of the most obvious phenotypes of a cell is its metabolic activity, which is defined by the fluxes in the metabolic network. Although experimental methods to determine intracellular fluxes are well established, only a limited number of fluxes can be resolved. Especially in eukaryotes such as yeast, compartmentalization and the existence of many parallel routes render exact flux analysis impossible using current methods. To gain more insight into the metabolic operation of S. cerevisiae we developed a new computational approach where we characterize the flux solution space by determining elementary flux modes (EFMs) that are subsequently classified as thermodynamically feasible or infeasible on the basis of experimental metabolome data. This allows us to provably rule out the contribution of certain EFMs to the in vivo flux distribution. From the 71 million EFMs in a medium size metabolic network of S. cerevisiae, we classified 54% as thermodynamically feasible. By comparing the thermodynamically feasible and infeasible EFMs, we could identify reaction combinations that span the cytosol and mitochondrion and, as a system, cannot operate under the investigated glucose batch conditions. Besides conclusions on single reactions, we found that thermodynamic constraints prevent the import of redox cofactor equivalents into the mitochondrion due to limits on compartmental cofactor concentrations. Our novel approach of incorporating quantitative metabolite concentrations into the analysis of the space of all stoichiometrically feasible flux distributions allows generating new insights into the system-level operation of the intracellular fluxes without making assumptions on metabolic objectives of the cell.     

Sensitivity Analysis of Flux Determination in Heart by H218O -provided Labeling Using a Dynamic Isotopologue Model of Energy Transfer Pathways

To characterize intracellular energy transfer in the heart, two organ-level methods have frequently been employed: inversion and saturation transfer, and dynamic labeling. Creatine kinase (CK) fluxes obtained by following oxygen labeling have been considerably smaller than the fluxes determined by saturation transfer. It has been proposed that dynamic labeling determines net flux through CK shuttle, whereas saturation transfer measures total unidirectional flux. However, to our knowledge, no sensitivity analysis of flux determination by oxygen labeling has been performed, limiting our ability to compare flux distributions predicted by different methods. Here we analyze oxygen labeling in a physiological heart phosphotransfer network with active CK and adenylate kinase (AdK) shuttles and establish which fluxes determine the labeling state. A mathematical model consisting of a system of ordinary differential equations was composed describing enrichment in each phosphoryl group and inorganic phosphate. By varying flux distributions in the model and calculating the labeling, we analyzed labeling sensitivity to different fluxes in the heart. We observed that the labeling state is predominantly sensitive to total unidirectional CK and AdK fluxes and not to net fluxes. We conclude that measuring dynamic incorporation of into the high-energy phosphotransfer network in heart does not permit unambiguous determination of energetic fluxes with a higher magnitude than the ATP synthase rate when the bidirectionality of fluxes is taken into account. Our analysis suggests that the flux distributions obtained using dynamic labeling, after removing the net flux assumption, are comparable with those from inversion and saturation transfer.     

Interpretation of metabolic flux maps by limitation potentials and constrained limitation sensitivities

Two new concepts, "Limitation Potential" and "Constraint Limitation Sensitivity" are introduced that use definitions derived from metabolic flux analysis (MFA) and metabolic network analysis (MNA). They are applied to interpret a measured flux distribution in the context of all possible flux distributions and thus combine MFA with MNA. The proposed measures are used to quantify and compare the influence of intracellular fluxes on the production yield. The methods are purely based on the stoichiometry of the network and constraints that are given from irreversible fluxes. In contrast to metabolic control analysis (MCA), within this approach no information about the kinetic mechanisms are needed. A limitation potential (LP) is defined as the reduction of the reachable (theoretical) maximum by a measured flux. Measured fluxes that strongly narrow the reachable maximum are assumed to be limiting as the network has no ability to counterbalance the restriction due to the observed flux. In a second step, the sensitivity of the reduced maximum is regarded. This measure provides information about the necessitated changes to reach higher yields. The methods are applied to interpret the capabilities of a network based on measured fluxes for a L-phenylalanine producer. The strain was examined by a series of experiments and three flux maps of the production phase are analyzed. It can be shown that the reachable yield is drastically reduced by the measured efflux into the TCA cycle, while the oxidative pentose-phosphate pathway only plays a secondary role on the reachable maximum.