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Originally published In Press as doi:10.1074/jbc.M409072200 on November 30, 2004

J. Biol. Chem., Vol. 280, Issue 12, 11683-11695, March 25, 2005
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Candidate Metabolic Network States in Human Mitochondria

IMPACT OF DIABETES, ISCHEMIA, AND DIET*{boxs}

Ines Thiele{ddagger}§, Nathan D. Price§, Thuy D. Vo§, and Bernhard Ø. Palsson, Serves on the scientific advisory board of Genomatica Inc.§

From the §Department of Bioengineering, University of California, San Diego, California 92093-0412 and the {ddagger}Ecole Supérieure de Biotechnologie de Strasbourg, 67412 Strasbourg, France

Received for publication, August 9, 2004 , and in revised form, October 25, 2004.


   ABSTRACT
TOP
 ABSTRACT
INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 REFERENCES
 
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.


   INTRODUCTION
TOP
 ABSTRACT
 INTRODUCTION
MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 REFERENCES
 
The emergence of genomic, metabolic, and proteomic data has facilitated the reconstruction of genome-scale metabolic networks. Various network reconstructions have been carried out in recent years for microorganisms such as Escherichia coli (1), Saccharomyces cerevisiae (2), Helicobacter pylori (3), Haemophilus influenzae (4), and Geobacter sulferreducans.1 Most recently, proteomics data (6, 7) were utilized to reconstruct the metabolic network of the human cardiac mitochondrion (8).

Constraint-based analyses have proven to be valuable for studying genome-scale metabolic networks. The constraint-based approach is based on the fact that cellular networks are constrained to operate within boundaries set by physico-chemical constraints (mass conservation, directional flow, enzymatic capacity, etc). The imposition of constraints corresponds to a mathematical definition of a solution space within which all feasible solutions lie. For example, the steady-state flux space contains all feasible steady-state flux distributions for a biochemical network. Methods for flux analysis within the constraint-based framework include flux balance analysis (9), network-based pathway analysis (10–13), and more recently, uniform random sampling of the steady-state flux space (14–16). These constraint-based modeling procedures have been successful in predicting metabolic phenotypes in various model (1, 2, 17–19) and infectious (3, 4) microorganisms. Previous constraint-based modeling studies have been used to identify optimal metabolic network states (20–23), calculate a range of potential cellular objectives consistent with an experimentally measured state (24), compute minimal necessary reaction (or gene) sets (25, 26), and quantify network redundancy (27, 28) and robustness (29). Constraint-based modeling has also proven valuable in predicting phenotypes such as optimal growth rates (21), lethality of gene knock-outs (30, 31), effects of gene additions and deletions (32), and the endpoints of adaptive evolutions (22, 33). The complete description of currently available constraint-based analysis methods used to compute the properties of genome-scale network reconstructions has recently been reviewed in detail (35).

In this study, uniform random sampling was used to calculate candidate steady-state flux distributions in the human cardiac mitochondrion under different sets of constraints representing various physiological conditions. Experimental data from many literature sources were integrated as physico-chemical constraints of the mitochondrial metabolic network. Constraints based on these experimental data were applied to segment the steady-state flux space defined by mass-balance constraints. This segmentation resulted in a characterization of all feasible steadystate flux distributions, termed candidate steady-state flux distributions, that occurred under each specified condition (see Fig. 1). Candidate steady-state flux distributions were determined for the following cases: (i) normal physiological condition, (ii) diabetic condition, (iii) ischemic conditions, and (iv) two types of diets. In addition, the effect of currently used and potential therapeutic approaches on mitochondrial metabolism was studied. Taken together, the approach used herein allows us to assess the implications of many different experimental measurements within the context of an integrated metabolic network.



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  		FIG. 1.
Schematic representation of the analysis of candidate metabolic network states under normal and disease conditions. A, the stoichiometric matrix (S) accounts for known metabolic reactions associated with the human mitochondrion as columns (r) and metabolites as rows (x). B, the null space of S contains the allowable steady-state fluxes. By applying constraints (Vmin, Vmax) to reaction, uptake, and secretion rates based on experimental data, the range of allowable steady states of the metabolic network consistent with these experimental data can be generated. C, additional or modified constraints based on experimental measurements in patients identify the candidate network states under the disease conditions.

 

   MATERIALS AND METHODS
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
RESULTS
 DISCUSSION
 REFERENCES
 
Definition of Steady-state Flux Space
The biochemical reactions comprising the reconstructed metabolic network were represented by a stoichiometric matrix, S (m x n), where m was the number of metabolites and n was the number of reactions. All reactions within the network were mass-balanced such that S·v = 0, where v was a flux vector (36, 37). The constraints for each reaction had the form {alpha}i ≤ vi ≤ {beta}i, where {alpha}i and {beta}i represented the lower and upper limits of the corresponding reaction flux. The {alpha}i values for irreversible reactions were set to zero, whereas {beta}i values were usually set either to measured uptake rates for transport reactions or to the Vmax of the corresponding enzymes. Fluxes that satisfy all of these stated constraints lie in the steady-state flux space.

Content of the Reconstructed Mitochondrial Network
The metabolic network of the human cardiac mitochondrion has been recently reconstructed (8). This network contained 189 reactions, 230 metabolites, and 29 exchange reactions. Exchange reactions are used to describe the metabolites available for the system and do not correspond to actual biochemical reactions (see Ref. 8 for complete reaction list). To study the effects of diabetes and diets, both of which are generally correlated with high concentrations of ketone bodies in the blood, the previously published mitochondrial metabolic network was expanded to include ketone body degradation. We added one enzymatic reaction ((R)-3-hydroxybutanoate:NAD+ oxidoreductase, EC 1.1.1.30 [EC] , (38, 39)) and six transport reactions (supplemental data Table S2a) to the previous reconstruction. These reactions added five more metabolites to the network (supplemental data Table S2b). The reconstruction was done using the software package SimPheny® (Genomatica Inc., San Diego, CA). This updated model is available for download at systemsbiology.ucsd.edu/organisms/.

In general, incomplete knowledge about a biological system results in gaps and dead-end metabolites that are only consumed or only produced by the reconstructed network. Due to the mass-balance constraints, reactions involving these metabolites must have zero net flux at steady state. These "unused" reactions were identified under the four conditions studied and removed from the S matrix prior to the sampling calculations. We identified 39 unused reactions out of the 224 reactions in the network (supplemental data Table S3). The resulting network contained 185 reactions, including 23 exchange reactions, and 235 metabolites (121 mitochondrial, 89 cystolic, and 25 extracellular). The null space of the corresponding stoichiometric matrix had 21 dimensions.

Sampling of the Steady-state Flux Space
Sampling the steady-state flux space was performed using a random walk algorithm (artificial centering hit-and-run, ACHR)2 as described by Kaufmann and Smith (40). The algorithm involves three steps. The first step requires the identification of an initial point within the solution space. We found this initial point by reducing each of the maximum constraints and increasing each of the minimum constraints by a small value and then calculating a candidate solution within these new constraints using linear programming. This procedure ensured that the initial point was chosen within the solution space, thus avoiding the computational difficulties that arise when the initial point lies at the extremity of the solution space. The second step of the ACHR algorithm calculates "warm-up" points from the initial point using several iterations of a basic hit-and-run algorithm (40). These warm-up points were stored as columns of a matrix W, and an approximate center, s, was calculated. The third step of the ACHR calculates the sample points. The direction for the next iteration from a sample point xm was chosen by randomly taking one point y out of the matrix W and applying the direction vector of y and s () to xm. At each iteration, the newly calculated point, xm+1, was substituted randomly into W in the place of a previously calculated point. The approximate center was also recalculated after each iteration. This last step was repeated until a desired number of sample points was reached. In practice, this approach allowed the distribution of points to converge to a uniform distribution much faster than the standard hit-and-run algorithm does (40). In each sampling procedure, 500,000 randomly distributed points were computed with 100 iterations between each point. The algorithm was implemented in Matlab® (MathWorks Inc., Natick, MA) with Lindo® (LINDO Systems Inc., Chicago) as the linear programming solver.

Verification of the Sampling Procedure
The same solution space was sampled five times using different randomly chosen initial points to verify that the calculated distribution of points was independent from the starting point. The resulting distributions were compared with one another to ensure that no difference was observed. In addition, we sampled the steady-state flux space of the red blood cell model (41) with the ACHR algorithm and evaluated our results with the previously published results from an elimination algorithm to confirm that both algorithms (15) led to the same distributions.

Metabolic Constraints Applied under Different Conditions
A literature search for available in vivo measurements in the human heart was performed. The aim was to segment the steady-state solution space by using measured values for exchange reactions as specific constraints for each of the four conditions studied. Following the convention described in Schilling et al. (42), substrate uptake rates were assigned negative flux values, whereas efflux rates were assigned positive values during computation. In the text, however, we will distinguish between uptake and efflux rates, but we will use positive values to describe both. All fluxes used in this study were calculated in µmol/min/g of proteins in accordance with the units commonly found in the literature for flux measurements.

Normal Physiological Conditions—The normal physiological condition we describe here represents the metabolism of a human resting heart. Constraints were applied on the uptake and efflux rates of metabolites according to the normal physiologic state of the mitochondrion. We applied a positive minimum constraint on the demand for ATP (DM ATP) to represent the minimal energy required for the maintenance of the cell in its normal physiological state. The required ATP level for ion homeostasis was set at ~26% of the total ATP production (43). We used an ATP production rate of 30 µmol/min/g of proteins, which was taken from measurements in the working dog heart (44), because the corresponding value for humans could not be found. Therefore, the lower bound of DM ATP was set to 7.5 µmol/min/g of proteins. The hexadecanoate (n-C16:0) uptake rate was measured to be 1 µmol/min/g of proteins (45). This value was used to approximate the uptake rates of other fatty acids based on their observed distributions in mammalian cardiomyocytes: 53% for octadecanoate (n-C18:0), 16% for hexadecanoate (n-C16:0), and 7% for octadecenoate (n-C18:1) and for octadecynoate (n-C18:2) (43). The fatty acids eicosanoate (n-C20:4) and docosanoate (n-C22:6) could not be detected in these experiments (43); we assumed that each of these fatty acids made up 2% of the total fatty acid uptake. These values allowed us to calculate the upper and lower limits for each fatty acid uptake rate (Table I). The lower bound and upper bound constraints were set by taking 25% variation around these experimentally measured values. The uptake rate of lactate was constrained to the same upper bound as that defined for glucose, as both substrates are consumed at approximately equivalent rates under normal physiological conditions (46). The uptake of the ketone bodies, acetoacetate and (R)-3-hydroxybutanoate, were allowed, but the secretion of ketone bodies was set to zero since the heart does not normally produce and export ketone bodies as the liver does (47). In addition, the maximum uptake rates for these ketone bodies were set to be very small (0.001 µmol/min/g of proteins) since the plasma concentration of circulating ketone bodies is less than 0.1 mM at normal physiological conditions (47). Table I summarizes these constraints and their references.


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  		TABLE I
Constraints for the normal physiological condition These constraints were applied for all conditions in this study unless otherwise specified. See "Materials and Methods" for details. Empty field signifies an unconstrained Vmax value. Asterisks (*) indicate a restricted ketone body uptake due to their low blood concentrations. The symbols (c) and (e) in the Abbreviation column stand for cytosol and extracellular, respectively. All the values shown are in units of µmol/min/g of proteins.

 
Diabetic Conditions—Diabetic conditions are characterized by a lack of, or insensitivity to, insulin. This condition results in an unregulated and increased fatty acid uptake in mitochondria via the carnitine-palmitoyl-transferase (CPT-I) shuttle (48, 49). The mitochondrial fatty acid uptake flux through CPT-I was increased in the model to reflect its activity under diabetic conditions. No quantitative experimental values could be found on how much these uptake rates increase, but a significant increase of free cellular fatty acids in the myocardium was observed in diabetic mice (50). Therefore, the lower bound constraints of the fatty acid uptake rates were increased to 80% of the maximum allowed flux values under normal physiological conditions (Table II, supplemental data Table S4a). However, it was not possible to increase the minimum uptake rates for eicosonoate (n-C20:4) and docosonoate (n-C22:6) without elimination of all possible network steady states. The maximum glucose uptake was reduced to 75% of its normal rate (51) based on the decreased number of glucose transporters (GLUT4) in the cardiomyocyte membrane under insulin insufficiency or insensitivity (48, 49). Another effect of diabetes is that the plasma ketone body concentration may be as high as 25 mM (52); thus, the upper limits on uptake rates of ketone bodies defined under normal conditions were removed. It should be noted, however, that although the constraints on ketone body uptake were removed in the diabetic condition, the resulting maximum allowable fluxes through the ketone body transporter under diabetic conditions were only 1.4% of that under normal physiological conditions. This reduction in ketone body uptake was due to the increased fatty acid uptake rate in the diabetic condition, thus systemically limiting ketone body transport. To simulate a high influx of ketone bodies, the minimum uptake rates were set to be 10% of the maximum possible flux of each ketone body under this condition. These new constraints represented the salient features of changed substrate supply observed in the diabetic cardiomyocyte (supplemental data Table S4a).


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  		TABLE II
Schematic representation of constraints applied under four different conditions

 
Ischemic Conditions—Severe ischemia leads to at least a 70% reduction in local blood flow, and thus, a consequent decrease in oxygen supply. An ischemic condition was simulated by a 75% reduction of the maximum oxygen uptake rate defined under normal physiological conditions (Table II). In addition, the effects of two therapeutic approaches suggested in the literature on mitochondrial metabolism for ischemia were studied (53, 54). One therapy raises the amount of ATP available for metabolic and non-metabolic tasks by increasing the rate of glycolysis. This therapy seeks to alleviate the damage during and after an ischemic event by administering glucose, insulin, and potassium (GIK) (53). Alternately, one may combine the first approach with an increased ketone body uptake. This latter treatment was based on the observation that ketone bodies had a positive effect on the "metabolic activity" in the case of acute insulin deficiency (54). These two approaches were implemented in silico by (a) increasing the maximum glucose uptake rate to 2 µmol/min/g of proteins and (b) increasing the lower bound constraint on the uptake rates of both types of ketone bodies (supplemental data Table S4b).

Dietetic Conditions—To simulate the consequence of a low fat-high glucose diet, the maximum uptake rate of glucose was increased to 1.5 µmol/min/g of proteins, whereas the maximum uptake rates of all types of fatty acids were decreased to 30% of their corresponding maximum uptake rate under normal conditions (Table II, supplemental data Table S4c). No quantitative experimental data were found in the literature for these changes in glucose and fatty acid uptake under this dietetic condition, but we assumed that such changes must occur due to the change in nutrient supply (55, 56). A similar assumption was used to simulate conditions corresponding to changed nutrition during the high fat-low glucose diet (56, 57). The glucose uptake rate was reduced to 30% of its normal value, whereas the minimum uptake rate of the fatty acids was increased to 70% of the normal maximum uptake rate (Table II, supplemental data Table S4c). The maximum uptake rates of each fatty acid were not constrained. In addition, constraints on the lower bound of uptake rates of ketone bodies were applied, whereas their maximum uptake rates were not constrained. This reflects the increased blood level of ketone bodies produced from the excess fatty acids in the liver.

Correlation between Reactions in the Network
The pairwise correlations between all reaction fluxes were calculated using MATLAB® (MathWorks Inc.). The correlation coefficients were calculated four times for each condition by using different sets of uniform random samples. The maximum variation between any pairwise correlation coefficient between the four calculated sets was 0.0168.


   RESULTS
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
DISCUSSION
 REFERENCES
 
Monte-Carlo sampling of the steady-state flux space was used to analyze candidate metabolic network states of the cardiac mitochondrion. Segmentation of the solution space to determine these network states (Fig. 1) was performed using experimental measurements taken under 1) normal physiological conditions (heart at rest), 2) diabetes, 3) ischemia, and 4) two different types of diets. Results corresponding to these different conditions were computed by applying constraints on exchange reactions and, in a few cases, intracellular reactions (Table I, supplemental data Table S4a-c). The substrate uptake rates were based on experimental data taken from the literature; a variation of 25% around these values was set for upper and lower constraints on reaction fluxes (see "Materials and Methods") to account for experimental errors. The segmentation of the steady-state flux space based on these data allowed us to characterize the feasible flux distributions for different metabolic states in mitochondria.

Candidate Steady-state Flux Distributions
Normal Physiological Conditions—The physiological condition of the network was defined by constraining substrate uptake rates using relevant experimental data from the literature. The resulting range of allowable metabolic network states represents all possible steady-state flux distributions in the reconstructed metabolic network that were consistent with measured fluxes under normal physiological conditions. Fig. 2A and B, panels a–h, black line, shows the allowable steady-state flux distributions for selected metabolic reactions of the mitochondrial network under normal conditions. Flux distributions for all reactions can be found in the supplemental data (Fig. S1). Each histogram corresponds to the range of possible steady-state flux values allowed for a reaction in the network. Peak values represent the most probable flux values within the distribution. In general, four shapes of flux distribution can be distinguished: a) right peak (e.g., Fig. 2A and B, panels g, black line, Ex H+), b) left peak (e.g., Fig. 2A and B, panels h, black line, Ex urea), c) central peak (e.g. CRNtim, supplemental data Fig. S1), and d) broad peak (e.g. Fig. 2A and B, panels a, black line, DM ATP) (15).



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  		FIG. 2.
Metabolic network states under normal physiological conditions (black line) and disease conditions (colored lines). A, histograms (50 bins) of uniform random samples of the steady-state flux space are shown for 1) normal physiological conditions (black lines), 2) diabetic conditions (pink), 3) diabetic conditions with the effect of normalization of glucose (green), and 4) diabetic conditions with the effect of normalization of ketone body uptake (blue). B, histograms (50 bins) of uniform random samples of the steady-state flux space are shown for 1) normal physiological conditions (black lines), 2) ischemic conditions (red), 3) ischemic conditions with the effect of increased glucose uptake (GIK therapy) (blue), and 4) ischemic conditions with GIK and added capacity for ketone body uptake (green). The abbreviations used are for the metabolic functions are: DM ATP, cellular ATP consumption; DM Heme, heme biosynthesis pathway; DM Phospholipids, phospholipid biosynthesis pathway. The abbreviations used for network reactions are: EX L-Lac, L-lactate exchange; Ex O2, oxygen exchange; Ex H+, proton exchange; Ex Urea, urea exchange. The exchange reactions transport metabolites across the network boundary. A negative rate on an exchange flux corresponds to an influx, whereas the positive flux indicates an efflux of the metabolite.

 
Similar to a previous mitochondrial study (8), we considered three metabolic demand reactions: ATP production, heme biosynthesis, and phospholipid biosynthesis (Fig. 2A and B, panels a–c, black line). The ATP demand (DM ATP), the flux distribution of which showed a broad peak, corresponds to the ATP consumption by the cell. The histograms of DM Heme (Fig. 2A and B, panel b, black line) and of heme biosynthesis reactions (supplemental data Fig. S1) showed that the most probable flux values were small (close to zero). The biosynthesis of mixed phospholipids (cardiolipin, phosphatidylethanolamine, and phosphatidylcholine), represented by DM Phospholipids, showed a relatively wide range of highly probable flux values (Fig. 2A and B, panels c, black line). These phospholipids are mainly used for the maintenance of the mitochondrial membranes.

Under normal physiological conditions, lactate is usually consumed rather than produced in the cardiac tissue (46). This observation is consistent with the computed flux distributions of lactate production and exchange (Fig. 2A and B, panels d, black line, supplemental data Fig. S1). That lactate is consumed rather than produced also agrees with the calculated high oxygen uptake rates (Fig. 2A and B, panels d and e, black line) since an increase in lactate production mainly occurs under oxygen-restricted conditions. The urea cycle removes harmful ammonia from the cell; the activity of urea transporter reflects the ammonium production rate in the network (Fig. 2A and B, panels h, black line).

The computed flux values for each reaction were compared with available in vivo flux measurements in human cardiomyocytes. If data from humans were unavailable, we used selected data from different mammalian cardiac mitochondria (Table III). The experimentally measured fluxes for transport and intracellular reactions fell within the computed steadystate flux space. In some cases, the computed most probable flux values offered a good estimate for the measured values (NADH2-u10m, complex I of the electron transport chain, and ASPGLUm, aspartate-glutamate shuttle). The enzymatic activities measured in vivo were different for each metabolic state; Table III summarizes the conditions under which the measurements were done. The agreement of the sampling results with experimental data shows that by applying constraints on exchange reactions of a reconstructed mitochondrial metabolic network, the calculated flux values were physiologically relevant.


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  		TABLE III
Comparison between experimentally measured flux rates compared to computed flux ranges The "computed most probable flux" corresponds to the peak of the histogram made from the uniform random samples of the steady-state flux space. Lower and upper bounds (Vmin and Vmax) were determined by linear programming. All the values shown are in units of µmol/min/g of proteins.

 
Diabetic Conditions—One primary consequence of insulin deficiency or insensitivity is that mitochondrial fatty acid uptake becomes unregulated (48, 49). The effects of diabetes on mitochondrial metabolism were assessed by applying constraints reflecting (i) unregulated fatty acid uptake via carnitine-palmintoyl-transferase-I (CPT-I) shuttles, (ii) decreased glucose consumption due to the reduced number of glucose transporters in cell membrane, and (iii) increased ketone body uptake due to their higher concentration in the blood (for details, see "Materials and Methods"). These additional constraints altered the shape of the steady-state solution space, resulting in a set of candidate network states that were consistent with conditions observed in diabetic patients. One treatment for diabetes is to increase the blood sugar concentration, and as a consequence, increase the uptake of glucose by cardiomyocytes. We also investigated the effect of a restored normal glucose uptake rate under conditions of unregulated mitochondrial fatty acid uptake by returning the constraints on glucose uptake to their values in normal conditions. Additionally, the contribution of increased ketone body uptake (58) to the network was also studied by returning the constraints on ketone body uptake to their levels under normal physiological conditions. Thus, probability distributions for individual fluxes were compared for four cases (Fig. 2A): 1) normal physiological (black), 2) diabetic (red), 3) diabetic with normal glucose uptake (green), and 4) diabetic with normal glucose and ketone body uptake (blue). Results from these four conditions showed that the increased mitochondrial fatty acid uptake led to smaller ranges of allowed flux values for most reactions, thus reducing network flexibility dramatically. Although adjusting ketone body and/or glucose uptake to normal physiological values led to higher flexibility in reaction fluxes, the effect was minimal. The flux distribution of the metabolic function "cellular ATP consumption" (DM ATP) only differed slightly from that in the normal physiological condition. Interestingly, the network appeared to maintain a high flux value for this demand function, whereas minimizing fluxes through most ATP-consuming reactions under all conditions investigated in this study (supplemental data Fig. S2). Due to the high mitochondrial fatty acid breakdown, the flux distribution of oxygen uptake was increased, and hence, its flexibility was very restricted (Fig. 2A, panel e).

One interesting finding from our results was that the flux through the mitochondrial pyruvate dehydrogenase (PDHm) enzyme was significantly restricted by network stoichiometry when the fatty acid uptake was increased (Fig. 2A, panel f). Many studies have tried to identify factors that affect the inhibitory mechanism of PDHm (48, 49) under conditions such as diabetes (57); this study showed that an increase in cellular fatty uptake flux forced a significantly lower flux through PDHm as a direct consequence of the overall network stoichiometry (Fig. 2A, panel f). In silico predictions of changes in metabolic function in the diabetic condition were compared with those reported in the literature (Table IV). For most of the mitochondrial functions, the results from this study were consistent with experimental observations.


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  		TABLE IV
Qualitative properties for diabetes and ischemia from literature compared with model results OxPhos, oxidative phosphoryation; FA, fatty acid; KB, ketone bodies. —, no information, qualitative or quantitative, was found.

 
Ischemic Conditions—Severe ischemia leads to an undersupply of oxygen, and consequently of energy, for cardiomyocytes. Cell damage during and after an ischemic event is associated with reduced energy production, decreased contractile work, and increased acidosis due to increased glycolysis. We investigated the effect of decreased oxygen supply (25%) on the network.

Candidate flux distributions under ischemic conditions were compared with those computed for the normal physiological condition (Fig. 2B, supplemental data Fig. S3). The maximal ATP production (DM ATP) under ischemic conditions did not differ significantly from what was found for diabetic conditions. However, the shape of the steady-state flux space changed significantly due to the changed maximal oxygen uptake rate. Both the maximally allowed and the most probable flux values were reduced in most reactions since ATP was the most under-supplied metabolite in ischemic conditions (Fig. 2B, supplemental data Fig. S3). Exceptions were reactions of the phospholipid biosynthesis pathway (Fig. 2B, panel c), where the most probable flux values were increased. This result could be explained by the fact that the oxidation of fatty acids decreased due to oxygen restriction, whereas the minimum fatty acid uptake rates were unchanged. The network balanced the nonoxidized free mitochondrial fatty acids by converting them into phospholipids (Fig. 2B, panel c).

Overall, the changed tendency of the flux distributions of the network is comparable with the reported changes observed in ischemic patients (Table IV). This suggests that the in silico candidate states presented here are good predictions to the changes in the mitochondria metabolic activities responding to ischemia.

Effects of Therapeutic Approaches for Ischemia—There have been multiple approaches in ischemia therapy to reduce the negative effects of reperfusion and minimize tissue damage; two of these therapies were investigated in this study. The first approach, GIK infusion (53), was simulated by increasing the maximal glucose uptake rate. Fig. 2B (blue line) shows the effect on the metabolic network under oxygen-restricted conditions. The candidate flux distributions differ only slightly from those of the ischemic condition, suggesting that this therapeutic approach may not be effective. The main goal of this therapy is to increase ATP available for contractile work, but neither the ATP consumption (DM ATP) nor the fluxes through ATP-consuming reactions increased as a result. However, the effluxes of protons and lactate were shifted higher, corresponding to two well known side effects of this therapy. These side effects often lead to further damage during reperfusion. The second approach in ischemia therapy is based on the observation that ketone body oxidation produces a higher ATP yield per oxygen molecule than does glucose oxidation but requires less oxygen than fatty acid breakdown. We tested whether or not the administration of ketone bodies in combination with GIK would increase the available ATP and reduce the proton and lactate production. The resulting network states differed only slightly from those of GIK alone, with the flux distribution of lactate and proton production shifted to a higher flux value range (Fig. 2B, panels a–h, green line). Overall, the administration of ketone bodies alone did not lead to significantly different results (data not shown).

An alternative therapeutic approach proposed in the literature to treat ischemia is to stimulate the activity of PDHm (59). The rationale for this approach is that higher flux through PDHm should result in a higher glytolytic flux and therefore a reduced lactate level (59). However, our calculations predict that a stimulation of PDHm could only lead to a slightly higher steady-state flux through this reaction (Fig. 2B, panel f) due to stoichiometric constraints. The maximum possible flux through PDHm under the ischemic condition was 17% of the maximum flux under normal physiological conditions. These data suggest that the increase of PDHm activity may not prove to be an effective mode of therapy unless there are relevant undiscovered metabolic pathways.

Effect of Dietary Restrictions—The effects of two types of diets on cardiac energy metabolism were investigated. The two types of diets studied were: 1) ketonic diet, corresponding to high fat-low glucose diet, and 2) high carbohydrates diet, corresponding to low fat-high glucose diet. The results of the sampling of the steady-state flux space for the high fat-low glucose diet (Fig. 3, black line) and the low fat-high glucose (dashed line) were compared with those in the normal physiological condition (gray line). In general, these results indicate that the low fat-high glucose diet maintained greater flexibility in most metabolic reactions as compared with the high fat-low glucose diet. The latter diet showed a similar profile to that of diabetic conditions.



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  		FIG. 3.
Metabolic network states under dietetic conditions. Histograms (50 bins) of uniform random samples of the steady-state flux space are shown for normal physiological conditions, high fat-low glucose dietetic conditions, and low fat-high glucose dietetic conditions. The abbreviations used for metabolic functions are: DM ATP, cellular ATP consumption; DM Heme, heme biosynthesis pathway; DM Phospholipids, phospholipid biosynthesis pathway. The abbreviations used for network reactions are: Ex L-Lac, L-lactate exchange; Ex O2, oxygen exchange; Ex H+, proton exchange; Ex Urea, urea exchange; FAOX, fatty acid oxidation reaction for the six different types of fatty acids. The exchange reactions transport metabolites across the network boundary. A negative rate on an exchange flux corresponds to an influx, whereas the positive flux indicates an efflux of the metabolite.

 
The {beta}-oxidation activity of the six types of fatty acids is shown in Fig. 3 (panels i–l). Naturally, the low fat-high glucose diet reduced fatty acid uptake rates. However, we found that fatty acid oxidation had approximately the same most probable flux values as those under normal physiological conditions, but with a reduced range of allowable flux values (dashed line). In comparison, the high fat-low glucose diet led to an increased oxidation of octadecenoate (n-C18:1), hectadecanoate (n-C16:0), and octadecanoate (n-C18:0) that are mainly used for ATP production. The fluxes of fatty oxidation for docosanoate (n-C22:6), eicosanoate (n-C20:4), and octadecynoate (n-C18:2) were close to zero. The ranges of flux distributions of the six fatty acid oxidation reactions were generally restricted to high flux values. Consequently, the most probable flux of oxygen uptake was increased to the maximum allowed flux value (Fig. 3, panel e).

Correlation among the Network Reactions
We defined correlated reaction sets (Co-sets) to be sets of reactions that have perfectly correlated fluxes at steady state (R2 = 1) (60). Co-sets can be calculated using linear programming (26), network-based pathways (25, 61), or Monte-Carlo sampling methods (15). However, the Monte-Carlo sampling approach also allows the calculation of the pairwise correlation coefficients between all reaction fluxes, yielding an R2 value anywhere between 0 and 1. A high correlation coefficient between two reactions suggests a high degree of dependence between these reactions.

Correlations under Normal Physiological Conditions—By grouping perfectly correlated reactions, 151 reactions were grouped into 33 Co-sets (R2 = 1.0) and 34 reactions in single reaction Co-sets (Fig. 4, supplemental data Table S5). We referred to the single reaction Co-sets by the reaction abbreviation, whereas the multiple reaction Co-sets were named after their main subsystem or metabolites. Fig. 4 (gray lines) illustrates the high correlation among the Co-sets. There are 42 pairs of Co-sets with correlation coefficients (R2) between 0.90 and 0.99, and 26 pairs with correlation coefficients (R2) between 0.80 and 0.89. Only one Co-set (Co-set 14, Citrulline/Ornithine) did not correlate (R2 < 0.006) with any other reaction or Co-set (supplemental data Table S6). This Co-set contained three reactions: CITRtm, ORNt3m, and ORNt4m. The ORNt4m reaction exchanges cellular ornithine for mitochondrial citrulline, whereas CITRtm and ORNt3m transport both citrulline and ornithine. These three reactions form a futile cycle, through which no net flux is physiologically possible. The Co-sets Ketone bodies I (Co-set 11) and Ketone bodies II (Co-set 19) were also not correlated with the rest of the network, but they had a correlation coefficient of 0.70 with each other. It is likely that the low uptake rate of ketone bodies to the network led to the lack of correlation of these two reactions with the rest of the network. Under normal physiological conditions, the Glycolysis Co-set was only weakly correlated to the rest of the fluxes through the metabolic network (R2 < 0.014). One reason for this lack of correlation may be that glycolysis contributes only a small fraction of the overall ATP production under these conditions.



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  		FIG. 4.
Map of Co-sets. Reactions that are highlighted with the same color of boxes belong to perfectly correlated reaction subsets (Co-sets). Correlations between the Co-sets (boxes) with R2 ≥ 0.8 are indicated with dotted lines. Reactions in circles represent the single reaction Co-set. Reactions that are unused in all candidate flux maps are marked with gray slashes. The numbers in parentheses in the color legend give the number of reactions for the corresponding Co-set. A larger version of this figure is available at systemsbiology.ucsd.edu/organisms/.

 
It should be noted that Co-sets were also identified in our previous mitochondrial study (8). The Co-sets reported in that study identified correlated reactions among flux distributions that were optimal for a defined metabolic function. In the present study, the Co-sets were calculated among all flux distributions that were feasible with respect to the imposed constraints; they may or may not be optimal for any particular metabolic task. Interestingly, the resulting Co-sets calculated in both studies were mostly similar, with some exceptions such as "aerobic metabolism" (8), where the Co-sets were only identified in one study but not the other.

Correlations under Diabetic Conditions—Segmentation of the steady-state solution space based on experimental measurements corresponding to diabetic conditions altered both the probability distributions of individual fluxes and the intercorrelation between Co-sets. This effect was most significant for Glycolysis Co-set 3, where reactions were only weakly correlated with those in other Co-sets (R2 ≤ 0.014) calculated for normal conditions. For diabetic conditions, however, the Glycolysis Co-set was more correlated with the following Co-sets: Heme (Co-set 24, R2 = 0.61), TCA I (Co-set 15, R2 ~ 0.4), TCA II (Co-set 24, R2 ~ 0.4), Mal-Asp Shuttle I (Co-set 26, R2 ~ 0.6), Mal-Asp Shuttle II (Co-set 27, R2 ~ 0.6), and proton exchange (R2 = 0.78). Nevertheless, the Glycolysis Co-set remained uncorrelated with PDHm (R2 ~ 0.08). Other Co-sets showed reduced correlations with each other. For example, the high correlation of ATPase (ATPS4m) with the H2O transport Co-set (Co-set 31) and mitochondrial oxygen transport (O2tm) were reduced under diabetic conditions (from R2 > 0.97 to R2 ~ 0.8 for each). These results illustrate the network-wide consequences of the changes in substrate supply.

Correlations under Ischemic Conditions—Analyses of the correlation between the flux distributions under ischemic conditions revealed the same Co-sets as found under normal physiological conditions. Correlations between the Co-sets did change somewhat, but the changes were not as dramatic as what had been observed under diabetic condition. The Glycolysis Co-set (Co-set 3) became more correlated with pyruvate kinase (R2 = 0.27) and with EX_h(e) (R2 = 0.46). The L-lactacte (Co-set 16) was also more correlated with the other Co-sets of the network, such as heme (Co-set 2, R2 = 0.79), and with the complex II of the respiratory chain (SUCD3-u10m) (R2 = 0.80).

Correlations under Dietetic Conditions—Although the low fat-high glucose diet produced only small changes between the Co-sets as compared with those in the normal physiological condition, the high fat-low glucose diet led to a large number of changes in the correlation between Co-sets. For example, Glycolysis (Co-set 3) became highly correlated with the rest of the Co-sets of the network, especially with TCA I (Co-set 15), TCA II (Co-set 24), Heme (Co-set 2), and the complex II of the respiratory chain (SUCD3-u10m) under high fat/low glucose dietetic conditions. Fig. 5 illustrates the changes in the correlation of the Co-sets between the high fat-low glucose dietetic and normal physiological conditions.



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  		FIG. 5.
Changes in correlation between reaction fluxes in high fat-low glucose diet as compared with the normal physiological condition. Panels A and B are enlargement of boxes A and B in the top plot. Boxes A and B highlight the reaction pairs that show (A) a low correlation under normal physiological conditions (R2 < 0.3) and a high correlation under high fat-low glucose dietetic conditions (R2 > 0.7) or (B) a high correlation in normal physiological condition (R2 > 0.7) and a low correlation in the dietetic condition (R2 < 0.3).

 

   DISCUSSION
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
REFERENCES
 
This study illustrated that uniform random sampling of the steady-state flux space can lead to further understanding of network properties in healthy and disease states. Experimental data were used to constrain and segment the steady-state flux solution space defined by the stoichiometry of the reconstructed mitochondrial network. Key results are as follows: 1) the application of constraints on the exchange reactions resulted in predictions of intracellular fluxes that were consistent with in vivo measurements; 2) reactions of the metabolic network were grouped in Co-sets, defining functional network modules; 3) network stoichiometry constraints were sufficient to explain the observed inhibitory effect on pyruvate dehydrogenase flux under diabetic conditions; 4) oxygen became highly restricted under diabetic conditions due to the increased mitochondrial fatty acid uptake, which might explain the higher probability for cardiomyopathies in diabetic patients; and 5) both types of diet, high fat-low glucose and low fat-high glucose, reduced the flexibility of the metabolic network.

The calculated intracellular fluxes were compared with reported experimental data (Tables III and IV) to evaluate the predictive power of the model. Monte-Carlo sampling of candidate metabolic network states provided a tool for investigating the network-wide consequences of altered substrate supply in disease states. Conditions such as ischemia, diabetes, or a high fat-low glucose diet led to a reduced range of feasible flux values, and therefore, a loss of flexibility of the network (Figs. 2AB, and 3). This reduced flexibility makes mitochondrial metabolism more sensitive to disturbances such as changes in oxygen levels or a higher ATP demand.

Metabolic reactions were grouped into Co-sets that contain reactions with perfectly correlated steady-state fluxes. This perfect correlation implies that if one of these steady-state fluxes is known, the steady-state flux through each of the other reactions in the set can be unambiguously predicted. These Co-sets form unbiased modules (60) that act as functional units of the network. In addition to the perfectly correlated Co-sets, for the human cardiac mitochondrion, 37 functional units with correlation coefficients of at least 0.85 could be identified (supplemental data Table S6). These units demonstrate the high interconnectivity of the network. Knowledge about these modules can enable more efficient experimental design by reducing the number of redundant experiments.

In this study, we also showed that if the cellular or mitochondrial fatty acid uptake was increased, the activity of PDHm was necessarily reduced to almost zero as a direct consequence of stoichiometric constraints (Figs. 2A, panel f, and 3, panel f). Given insufficient oxygen supply, the maximum possible flux through PDHm was only 17% of its maximal value under normal physiological conditions (Fig. 2B, panel f). It has been thought that the increase in fatty acids indirectly inhibits the activity of PDHm by the activation of a number of enzymes (48, 49, 62). Interestingly, although the mitochondrial model we used here did not account for regulatory elements, a reduced PDHm activity under diabetic and ischemic conditions was observed nonetheless. This observation indicates that PDH inhibition can also be explained as a direct result of stoichiometric constraints.

The oxygen uptake rate was one of the most restrictive constraints imposed on the network under diabetic conditions due to the higher rate of fatty acid oxidation caused by unregulated mitochondrial fatty acid uptake. Fatty acids are the main source of energy in cardiomyocytes (48, 49), but an overload of fatty acids leads to an inability of the network to handle disturbances in oxygen supply (Fig. 2A, panel e). Oxygen is not only used for metabolic reactions but is also essential for contractile work of the heart muscle. Therefore, further restriction of oxygen, as what may occur in an ischemic event, for a diabetic patient could result in reduced contractile work as well as in harmful accumulation of fatty acids in plasma and mitochondrion. It was found that an increase in glucose alone as a treatment did not result in higher ATP production, nor did the additional decrease in ketone body uptake restore the flexibility in steady-state fluxes through the metabolic network (Fig. 2A). These results suggest that only a decrease of fatty acid uptake could re-establish network flexibility. Such a decrease could be achieved by administering insulin or a stimulator for malonyl-CoA production since both are inhibitors of carnitine-palmitoyl-transferase I. The reduction of fatty acid uptake seems to be essential to reduce the risk of (i) insufficient oxygen supply, (ii) lipid toxicity, and (iii) changes in the mitochondrial membrane composition due to increased phospholipid production. The latter is thought to be crucial in apoptosis, which is induced by changes in membrane composition (63). This idea may explain why diabetic patients have twice the risk for heart failure as patients with other cardiovascular diseases (48).

Both types of diet studied herein, high fat-low glucose and low fat-high glucose, reduced the flexibility of the metabolic network as compared with the normal physiological condition (Fig. 3). The high fat-low glucose diet resulted in reduced heme biosynthesis activity. Heme is used for the synthesis of cytochrome c, and therefore, for the function of the electron transport chain. The high fat-low glucose diet also led to higher activity of phospholipid biosynthesis as a result of excess fatty acids. It is likely that the excess production of phospholipids and fatty acids would result in their accumulation in the cell and/or mitochondrion. The second diet studied, low fat-high glucose, led to an increase in lactate production, an increased activity of the urea cycle, and a decrease in phospholipid biosynthesis (Fig. 3). Both types of diets are thought to increase the risk for cardiovascular diseases (56, 64). The results suggest that these diets lead to profound changes in the energy metabolism of the cardiac mitochondria, which might result in cell damage and heart failure. The ranges of steady-state fluxes were less flexible under these dietetic conditions as compared with those in the defined normal physiological condition. This reduced flexibility makes the tissue more sensitive to changes in oxygen supply during an ischemic event.

In summary, this study has shown that the metabolism of the human cardiac mitochondrion can be studied by sampling the steady-state flux space of the reconstructed network. This method allows for the unbiased assessment of candidate metabolic network states that are consistent with experiment data. The availability of reactions for protein complex formation or regulatory elements in models such as the human cardiac mitochondria will enable better predictions and further information about protein and regulatory network interactions. In silico methods have a high potential to increase knowledge of network interactions and metabolic changes in disease states.


   FOOTNOTES
 
* This work was supported by grants from the Thieles, Bourse d'Alsace, France and a grant from the National Science Foundation (NSF/BES-01-20363). The costs of publication of this article were defrayed in part by the payment of page charges. This article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact. Back

{boxs} The on-line version of this article (available at http://www.jbc.org) contains supplemental figures and tables. Back

To whom correspondence should be addressed: Dept. of Bioengineering, 9500 Gilman Dr. 0412, La Jolla, CA 92093-0412; Tel.: 858-534-5668; Fax: 858-822-3120; E-mail: palsson{at}ucsd.edu.

1 D. Lovley, unpublished results. Back

2 The abbreviations used are: ACHR, artificial centering hit-and- run; DM ATP, demand for ATP; CPT-1, carnitine-palmitoyl-transferase I; GIK, glucose, insulin, and potassium; PDHm, mitochondrial pyruvate dehydrogenase; TCA, tricarboxylic acid; Co-set, correlated reaction set. Back


   ACKNOWLEDGMENTS
 
We acknowledge Jan Schellenberger and Markus Herrgard for assistance in implementing the sampling algorithm, and Marc Abrams for a critical reading of this manuscript.



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