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Application of the metabolic control theory to the study of the dynamics of substrate cycles.

Substrate cycles are ubiquitous structures of the cellular metabolism (e.g. Krebs cycle, fatty acids beta-oxydation cycles, etc...). Moiety-conserved cycles (e.g. adenine nucleotides and NADH/NAD, etc...) are also important. The role played by such cycles in the metabolism and its regulation is not clearly understood so far. However, it was shown that these cycles can generate multistationarity (bistability), irreversible transitions, enhancement of sensitivity, temporal oscillations and chaotic motions (Hervagault & Canu, 1987; Hervagault & Cimino, 1989; Reich & Sel'kov, 1981; Ricard & Soulié, 1982). [formula: see text] Fig. 1: Scheme of the open binary substrate cycle under study. The substrate S is converted into P with a net rate v2. Substrate P is converted in turn into S with a net rate v3. Step v2 is inhibited by excess of the substrate, S. In addition, the cycle operates under open conditions, that is zero-order input of S at rates alpha 0(v1) and first order outputs of S and P at rates alpha S and alpha P(v4), respectively. The metabolic control theory (see also Fell, 1990), which shows how a metabolic network reacts to small perturbations in the vicinity of a steady state, and is formulated with the so-called "control coefficients", was applied to such a cycle in order to get a better knowledge on the importance of each step at the regulatory point of view. The behaviour of a binary substrate cycle (fig. 1) in which one of the enzymes may be subjected to inhibition by excess of its substrate (v2) was studied theoretically.(ABSTRACT TRUNCATED AT 250 WORDS)

Models, Biological

Theoretical analysis of the flux control properties of a substrate cycle.

Substrate cycles are able to increase the flux control of a non-equilibrium reaction in a wide range of situations related to the effects of the metabolites involved in the cycle on the reaction producing them. No limit exists for that amplification if appropriate conditions are attained if those effects are positive. In all cases, the ratio between the rate of the reverse reaction and the net flux through the pathway plays an important role in defining the final amplification.

Enzymes

Fiber-type-related differences in the enzymes of a proposed substrate cycle.

A substrate cycle between citric acid cycle (CAC) intermediates isocitrate and 2-oxoglutarate, involving NAD+- and NADP+-linked isocitrate dehydrogenase (NAD-IDH and NADP-IDH, respectively) and mitochondrial transhydrogenase (H+-Thase), has recently been proposed. This cycle has been hypothesized to enhance mitochondrial respiratory control by increasing the sensitivity of NAD-IDH to its modulators and allowing for enhanced increases in flux through this step of the CAC during periods of increased ATP demand. The activities of the enzymes comprising the substrate cycle: NAD-IDH, forward and reverse NADP-IDH, and forward and reverse H+-Thase, along with the activity of a marker of mitochondrial content, citrate synthase (CS) were measured in mitochondria isolated from rabbit Type I and Type IIb muscles and in whole muscle homogenates, representing the various fiber types, from rats. In isolated rabbit muscle mitochondria, NAD-IDH had significantly higher (1.6 x ) activity in white muscle while forward NADP-IDH, forward and reverse H+-Thase, and CS all had significantly higher (1.2-1.6 x ) activities in red muscle. There was no difference in reverse NADP-IDH between fiber types. Similarly, in rat whole muscle enzyme activities normalized to CS, NAD-IDH had significantly higher activity in fast-twitch glycolytic (FG) fibers, while forward NADP-IDH and forward H+-Thase had significantly higher activities in slow-twitch oxidative (SO) fibers. These results suggest that differences in the activities of the substrate cycle enzymes between skeletal muscle fiber types could contribute to differences in respiratory control due to differential cycling rates and/or loci of control.

Animals

Substrate cycles in metabolic regulation and in heat generation.

1. The presence of substrate cycles in tissues has been demonstrated by direct isotope methods in recent years. This demonstration has provided the impetus for a reappraisal of the roles of substrate cycling in metabolic regulation and in heat production. These aspects of substrate cycling are discussed in this paper. The relationship between near-equilibrium reactions and substrate cycles is emphasized, since this provides a basis for the derivation of a function describing in precise quantitative terms the factors governing the amplification provided by substrate cycles in metabolic regulation. Some examples of the roles of substrate cycles in providing sensitivity in metabolic regulation are described. The importance of substrate cycling in heat generation in the flight muscle of the bumble-bee and in brown adipose tissue is discussed in detail. 2. We point out that the two possible roles of cycling, heat production and amplification, are intimately linked so that they must be discussed together. It is proposed that variable rates of substrate cycling may be possible so that, for short periods of time. sensitivity can be maximal without excessive heat generation. Variable rates over the long term may be involved in weight control, and the control of such variability in cycling rates may be impaired in obese subjects. Finally, the possibilities that substrate cycles provide explanations for the specific dynamic action of food and for alcoholic and accidental hypothermia are raised.

Adipose Tissue

Effect of severe burn injury on substrate cycling by glucose and fatty acids.

Increases in metabolic rate and core temperature are common responses to severe injury. We have investigated the hypothesis that these responses are due to increases in substrate cycling. A substrate cycle exists when opposing, nonequilibrium reactions catalyzed by different enzymes are operating simultaneously. At least one of the reactions must involve the hydrolysis of ATP. Thus, a substrate cycle both liberates heat and increases energy expenditure, yet there is not net conversion of substrate to product. In studies in volunteers (n = 18) and in patients with severe burns who were in a hypermetabolic state (n = 18), we used stable-isotope tracers to quantify substrate cycling in the pathways of glycolysis and gluconeogenesis and a cycle involving the simultaneous breakdown and synthesis of stored triglyceride (triglyceride-fatty acid cycle). The total rates of triglyceride-fatty acid and glycolytic-gluconeogenic cycling were elevated in the patients by 450 and 250 percent, respectively (P less than 0.01). An infusion of propranolol in the patients greatly reduced triglyceride-fatty acid cycling but did not affect gluconeogenic-glycolytic cycling. We conclude that increased substrate cycling contributes to the increased thermogenesis and energy expenditure following severe burns and that the increased triglyceride-fatty acid cycling is due to beta-adrenergic stimulation.

Adolescent

Steady-state properties of a model ternary substrate cycle: theoretical predictions.

Numerous ternary substrate cycles are metabolically operative in vivo. The relative concentrations of the interconverted substrates are generally correlated with different physiological states. These cycles often include reversible and/or substrate-inhibited enzymic steps. The switch between one steady state (metabolic state) and another may be the consequence of either the effect of an exogeneous metabolite or signal, or the alteration of a cycle internal parameter. The interpretation of results obtained with currently designed experiments on substrate cycles seldom take into account the very dynamic and regulatory properties inherent in the cyclic and often autocatalytic nature of the pathway. In the present report, the various dynamic properties of a model ternary substrate cycle, bounded by moiety conservation, are investigated. Three situations with increasing complexity are considered: (i) the three enzymes are michaelian and catalyse irreversible steps; (ii) one of the enzymic steps is reversible; and (iii) one step is subjected to a destabilizing factor, i.e. inhibition by excess of substrate. The behavior(s) of the whole cycle is mainly controlled by four parameters, that is, ST, the total concentration of the substrate pool, and the three enzyme maximal velocities, VMi (i = 1,2,3). As ST (= S1 + S2 + S3) is constant, the Si steady-state concentrations (stable or not) can be represented in barycentric coordinates in a triangle (simplex). This convenient representation allows us to predict the different states of the system when one enzyme maximal activity is varied. The steady-state concentration dependencies as a function of one or several parameters may be either monostable (possibility of zero-order ultrasensitivity) or bistable (with or without reversible transitions). The physiological and experimental relevances of these observations are emphasized.

Enzyme Stability

Substrate cycles: their metabolic, energetic and thermic consequences in man.

The role of substrate cycles in metabolic control was first indicated over ten years ago, but the recent interest in such cycles has developed from the direct demonstration by isotopic techniques of their existence in various tissues. I propose that substrate cycles form part of a logical series of biochemical mechanisms that exist to increase the sensitivity of non-equilibrium reactions to changes in concentrations of metabolic regulators. The possible importance of such cycles for provision of precise metabolic regulation in the tissues of the normal subject and the trained athlete is proposed. Furthermore, cycling may provide a mechanism by which hormones can change the magnitude of response in a tissue to a given metabolic signal, without interfering in the biochemistry of the basic control mechanism. It is, however, possible to extend the role of cycling to heat generation and thus to controlled energy loss by an organism. Heat generation by substrate cycles may be important as an acute mechanism for maintaining the body temperature in man in response to a sudden decrease in the environmental temperature; alcoholic hypothermia would be explained by inhibition of substrate cycling in the liver, and accidental hypothermia in the elderly could be explained by decreased capacity of substrate cycles with age. If heat generated by the cycles is rapidly lost to the environment, the expenditure of energy to maintain this heat loss could explain, in part, the physiological phenomena of the thermic response to food and the oxygen debt which is always observed after exercise. Finally, the energy expended in these ways could be part of a general biochemical mechanism for maintenance of the correct body weight; a decrease in the capacity of substrate cycles might be one factor involved in the development of obesity.

Adenosine Triphosphate

Proton-translocating transhydrogenase and NAD- and NADP-linked isocitrate dehydrogenases operate in a substrate cycle which contributes to fine regulation of the tricarboxylic acid cycle activity in mitochondria.

H(+)-transhydrogenase (H(+)-Thase) and NADP-linked isocitrate dehydrogenase (NADP-ICDH) are very active in animal mitochondria but their physiological function is only poorly understood. This is especially so in the case of the heart and muscle, where there are no major consumers of NADPH. We propose here that H(+)-Thase and NADP-ICDH have a combined function in the fine regulation of the activity of the tricarboxylic acid (TCA) cycle, providing enhanced sensitivity to changes in energy demand. This is achieved through cycling of substrates by NAD-linked ICDH, NADP-linked ICDH and H(+)-Thase. It is proposed that NAD-ICDH operates in the forward direction of the TCA cycle, but NADP-ICDH is driven in reverse by elevated levels of NADPH resulting from the action of the transmembrane proton electrochemical potential gradient (delta p) on H(+)-Thase. This has the effect of increasing the sensitivity to allosteric modifiers of NAD-ICDH (NADH, ADP, ATP, Ca2+ etc), potentially giving rise to large changes in the net flux from iso-citrate to alpha-ketoglutarate. Furthermore, changes in the level of delta p resulting from changes in the demand for ATP would, via H(+)-Thase, shift the redox state of the NADP pool and this, in turn, would lead to a change in the rate of the reaction catalysed by NADP-ICDH and hence to an additional and complementary effect on the net metabolic flux from isocitrate to alpha-ketoglutarate. Other consequences of this substrate cycle are, (i) the production of heat at the expense of delta p, which may contribute to thermoregulation in the animal, and (ii) an increased rate of dissipation of delta p (leak).

Animals

Futile substrate cycles in the glycolytic pathway of boar and rat spermatozoa and the effect of alpha-chlorohydrin.

In boar spermatozoa incubated with 0.1 mM-glucose about 20 nmol glucose were converted to lactate and CO2 and the rate of futile substrate cycling between glucose and glucose 6-phosphate was about 6 nmol/10(8) spermatozoa/30 min. Futile cycling was increased in the presence of 0.05 or 1 mM-alpha-chlorohydrin but not to an extent sufficient to account for the rapid decline in ATP concentration observed under these conditions. These estimates include a substantial rate of fructose formation from fructose phosphates. The addition of 10 mM-L-lactate plus 1 mM-pyruvate protected the spermatozoa against the effect of alpha-chlorohydrin and glucose on the ATP concentration but increased futile substrate cycling. Substrate cycling between fructose 6-phosphate and fructose 1,6-bisphosphate could not be measured in boar spermatozoa but in rat spermatozoa its rate (nmol/10(8) spermatozoa/30 min) was about 10 under control condition and about 25 in the presence of 1 mM-alpha-chlorohydrin. This increase was insufficient to account for the decline in ATP concentration. In both species futile substrate cycling consumed a significant proportion of the ATP synthesis during lactate production but only about 5% of that produced in the oxidation of glucose to acetyl carnitine and CO2.

Adenosine Triphosphate

Fructose-6-phosphate substrate cycling and hormonal regulation of gluconeogenesis in vivo.

The possible role of the hepatic fructose-6-phosphate substrate cycle (phosphofructokinase, fructose-1,6-diphosphatase) in the rapid hormonal regulation of gluconeogenesis was investigated in vivo in fasted normal and adrenalectomized rats after administration of [3-3H, U-14C]- or [3-3H, 6-14C]glucose. The plasma glucose 3H/14C ratio was used as an index of substrate cycling because the amount of 3H loss from liver hexose phosphates is determined by the extent of cycling. PFK and FDPase activities limit 3H loss during gluconeogenesis and glycolysis, respectively. Glucagon-stimulated hepatic glucose production is always accompanied by increased substrate cycling, i.e., increased FDPase and PFK activities. The high PFK activity may be a secondary event due possibly to elevated cellular fructose-6-phosphate levels. Decreased substrate cycling, i.e., lowered FDPase activity, always accompanies the depressed hepatic glucose production that occurs during hyperglycemia. Glucagon has no effect on substrate cycling in adrenalectomized rats that are insensitive to the hormone. The in vivo experiments presented provide evidence, although indirect, that glucagon administration results in changes in the fructose-6-phosphate substrate cycle in a living animal. Whether these changes are primary regulatory events or occur secondarily to hormone actions elsewhere is not known.

Adrenal Glands

The effect of substrate cycling on the ATP yield of sperm glycolysis.

The conversion of glucose to lactate via the Embden-Meyerhof pathway yields a maximum of 2 mol of ATP per mol of glucose degraded with lesser amounts of ATP obtained if metabolic intermediates are removed for biosynthesis or if substrate cycling occurs during carbohydrate degradation. Bull sperm are an ideal test system for a quantitative estimate of ATP yield because they contain enzymes necessary for potential substrate cycling and do not carry out significant biosynthesis. Experiments utilized specifically labeled glucose and fructose to test for substrate cycling between glucose in equilibrium glucose-6-PO4, fructose in equilibrium fructose-6-PO4, and fructose-6-PO4 in equilibrium bisphosphate. Flux through the pathway (low = less than or equal to 0.05; moderate = 0.1; high = greater than 0.5 mol of carbohydrate consumed per h/10(8) cells) was altered by changing incubation temperature and/or by adding metabolic effectors. The data (in situ rates of kinases and phosphatases and metabolite crossover plots) obtained under these conditions were used to establish if the generally accepted regulatory enzymes (hexokinase and phosphofructokinase) approach the overall flux through the glycolytic pathway, i.e. that of a classical kinetically limiting rate-determining step. In summary: (a) At low flux, the rates of phosphorylation greatly exceeded overall flux. This resulted in extensive substrate cycling at all points and a net ATP yield of less than or equal to 0 mol/mol of glucose initially phosphorylated. (b) At moderate flux, rate of hexokinase approached that of overall flux through the glycolytic pathway but "excessive" phosphofructokinase activity led to substrate cycling between fructose-6-PO4 and fructose 1,6-bisphosphate and resulted in a low net ATP yield (0-0.6 mol/mol of glucose). (c) At high flux, rates of phosphofructokinase and hexokinase approached that of the overall flux. Minimal substrate cycling occurred between fructose-6-PO4 and fructose 1,6-bisphosphate, and the net ATP yield approached 1.8 mol/mol of glucose. These experiments established that the theoretical stoichiometric ATP yield of the pathway was rarely achieved during these incubations because of extensive substrate cycling. Indeed, for cauda epididymal bull sperm isolated and incubated in vitro, the glycolytic pathway serves to generate lactate, an intermediate product that is further metabolized in the mitochondria to yield the ATP that is associated with degradation of exogenous carbohydrate.

Adenosine Triphosphate

Sensitivity of pathway rate to activities of substrate-cycle enzymes: application to gluconeogenesis and glycolysis.

In a study of metabolic regulation, it is frequently useful to consider the degree to which an enzyme can influence the rate of its pathway. The most productive expression of rate-controlling influence is the fractional change in pathway rate per fractional change in enzyme activity (called control strength or sensitivity coefficient). We have developed a system for considering how a substrate-cycle enzyme's control strength depends on its flux and reaction order and on related features of other enzymes of its pathway. We have applied this system to the gluconeogenic pathway of rat liver and the glycolytic pathway of bovine sperm, where enough fluxes and reaction orders have been published to allow valid estimates of several control strengths. In normal fed animals where gluconeogenesis is slow and unidirectional substrate-to-product and product-to-substrate fluxes are comparable, all substrate-cycle limbs have very high and similar control strengths regardless of their flux rates and positions in the pathway. The activity of a step affects all substrate-cycle control strengths similarly as it affects unidirectional end-to-end fluxes relative to net rate. Control strengths of non-substrate-cycle enzymes are negligible compared to those of substrate cycles. In fasting animals, on the other hand, where unidirectional Pyr----Glc flux is much greater than Glc----Pyr flux, upstream enzymes (near Pyr) have a regulatory advantage over downstream enzymes (near Glc). In this circumstance, control strength of each substrate-cycle enzyme is inversely related to rate limitingness between its substrate and the pathway substrate. Because the Pyr/PEP cycle is significantly rate limiting, the control strength of the Pyr----PEP limb is much greater than that of pyruvate kinase and all downstream enzymes. In the glycolytic pathway of bovine sperm, strong product inhibition of hexokinase detracts greatly from its rate limitingness and control strength, which are very small despite its position at the beginning of the pathway and its large free energy. Because the glucose-transport-hexokinase segment is not rate limiting, phosphofructo 1-kinase has almost as much control strength as it would have as the first enzyme of the pathway, and because the F6P/FDP cycle is only moderately rate limiting, Fru-1,6-P2ase and enzymes further downstream have substantial control strengths. When glycolysis is accelerated by stimulation of phosphofructo 1-kinase, control strength shifts from phosphofructo-1-kinase and all downstream enzymes to the transporthesokinase segment.(ABSTRACT TRUNCATED AT 400 WORDS)

Animals

Substrate cycling between triglyceride and fatty acid in human adipocytes.

Substrate cycles in metabolism require energy and generate heat, and they may be involved in thermogenesis. We have studied one such cycle between triglyceride and fatty acid in isolated human adipocytes using a nonisotopic technique. In the absence of added hormone, and with 5 mmol/L (90 mg/dL) glucose in the incubation medium, lipolysis and fatty acid reesterification coexisted such that 40 +/- 4% (mean +/- SEM) of the fatty acid produced was cycled back into triglyceride. In 51 individual subjects the range was from 0% to 100%. Both lipolysis and the quantity of fatty acid recycled correlated positively with cell volume (P less than .001 and P less than .005, respectively). Norepinephrine (10(-6) mol/L) alone (33 experiments) increased lipolysis 3.1-fold, and reduced the percentage of fatty acid reesterified. Cycling was similar to that in the basal state. Lipolysis was inhibited 46% by postabsorptive levels of insulin alone (18 experiments), but the proportion of fatty acid reesterified increased such that the quantity cycled back into triglyceride was similar to that observed in the basal state. In the presence of both norepinephrine and insulin (18 experiments), lipolysis was increased by 58% while 31 +/- 4% of the fatty acid released was reesterified. In consequence, the quantity of fatty acid cycled back into triglyceride increased 2.1-fold. Increasing the insulin level fivefold or the medium glucose concentration to 20 mmol/L produced no further increase in the quantity of fatty acid reesterified. A substrate cycle exists, therefore, between triglyceride and fatty acid in human adipose tissue, and its activity is modified by norepinephrine and insulin.(ABSTRACT TRUNCATED AT 250 WORDS)

Adipose Tissue

Fructose-6-phosphate substrate cycling and glucose and insulin regulation of gluconeogenesis in vivo.

The question whether glucose or insulin regulates gluconeogenesis by effecting changes in the fructose-6-phosphate (F-6-P) substrate cycle (phosphofructokinase (PFK), fructose-1,6-diphosphatase (FDPase)) was investigated in vivo in fasted normal rats using [3-3H,U-14C]- or [3-3H,6-14C]glucose. The plasma glucose 3H/14C ratio was used as an index of substrate cycling because 3H loss from the liver hexose phosphate pool is limited by the activities of PFK and FDPase during gluconeogenesis and glycolysis, respectively. The 3H/14C ratio was corrected where necessary for glucose or insulin-induced changes in reincorporation of 14C from C-6 to C-1-3 of plasma glucose. A glucose infusion producing hyperglycemia and insulinemia was accompanied by decreased hepatic glucose production and diminished F-6-P substrate cycling, i.e., decreased FDPase activity. When insulin was infused along with glucose to produce high plasma insulin levels and avoid hypo- or hyperglycemia, the 3H/14C decay rate did not change, suggesting that the hormone does not influence basal rates of gluconeogenesis or PFK or FDPase activities. These in vivo results suggest that increased blood glucose levels inhibit gluconeogenesis and depress F-6-P substrate cycling. Whether these cycle changes constitute primary regulatory actions of glucose or occur secondarily to other metabolic events resulting from excess hexose (e.g., increased glycogen synthetase activity) cannot now be concluded.

Animals

Hormonal control of substrate cycling in humans.

Recent studies have established the existence of substrate cycles in humans, but factors regulating the rate of cycling have not been identified. We have therefore investigated the acute response of glucose/glucose-6P-glucose (glucose) and triglyceride/fatty acid (TG/FA) substrate cycling to the infusion of epinephrine (0.03 microgram/kg.min) and glucagon. The response to a high dose glucagon infusion (2 micrograms/kg.min) was tested, as well as the response to a low dose infusion (5 ng/kg.min), with and without the simultaneous infusion of somatostatin (0.1 microgram/kg.min) and insulin (0.1 mU/kg.min). Additionally, the response to chronic prednisone (50 mg/d) was evaluated, both alone and during glucagon (low dose) and epinephrine infusion. Finally, the response to hyperglycemia, with insulin and glucagon held constant by somatostatin infusion and constant replacement of glucagon and insulin at basal rates, was investigated. Glucose cycling was calculated as the difference between the rate of appearance (Ra) of glucose as determined using 2-d1- and 6,6-d2-glucose as tracers. TG/FA cycling was calculated by first determining the Ra glycerol with d5-glycerol and the Ra FFA with [1-13C]palmitate, then subtracting Ra FFA from three times Ra glycerol. The results indicate that glucagon stimulates glucose cycling, and this stimulatory effect is augmented when the insulin response to glucagon infusion is blocked. Glucagon had minimal effect on TG/FA cycling. In contrast, epinephrine stimulated TG/FA cycling, but affected glucose cycling minimally. Prednisone had no direct effect on either glucose or TG/FA cycling, but blunted the stimulatory effect of glucagon on glucose cycling. Hyperglycemia, per se, had no direct effect on glucose or TG/FA cycling. Calculations revealed that stimulation of TG/FA cycling theoretically amplified the sensitivity of control of fatty acid flux, but no such amplification was evident as a result of the stimulation of glucose cycling by glucagon.

Adult

Substrate cycling between 5-amino-4-imidazolecarboxamide riboside and its monophosphate in isolated rat hepatocytes.

AICA (5-amino-4-imidazolecarboxamide)-riboside is taken up by isolated rat hepatocytes and converted by adenosine kinase (ATP:adenosine 5'-phosphotransferase, EC 2.7.1.20) into AICAR (ZMP), an intermediate of the de novo synthesis of purine nucleotides. We investigated if, in these cells, a cycle analogous to the adenosine-AMP substrate cycle operates between AICAriboside and ZMP. When 50 microM ITu, an inhibitor of adenosine kinase, was added to hepatocytes that had metabolized AICAriboside for 30 min, the concentration of ZMP decreased immediately. This was mirrored by a reincrease of AICAriboside. Rates of the ITu-induced decrease of ZMP and the increase of AICAriboside, calculated at different concentrations of ZMP, were first order, up to the highest concentration of ZMP (approx. 5 mumol/g of cells). Dephosphorylation of ZMP added to crude cytosolic extracts of rat liver displayed hyperbolic kinetics, with a Vmax of 0.65 mumol/min per g protein and an apparent Km of 5 mM, and was markedly inhibited by Pi, an inhibitor of IMP-GMP 5'-nucleotidase (5'-ribonucleotide phosphohydrolase, EC 3.1.3.5). We conclude that hepatocyte ZMP is continuously dephosphorylated, most likely by IMP-GMP 5'-nucleotidase, into AICAriboside, which is rephosphorylated into ZMP by adenosine kinase. Substrate cycling was also shown to occur between other nucleoside analogs and their phosphorylated counterparts.

Aminoimidazole Carboxamide

On the analysis of substrate cycles in large metabolic systems.

The simultaneous operation of paired, opposing reactions (substrate cycles) or parallel reactions (dual pathways) with seeming wastage of ATP is widespread in cellular metabolism. Analysis of such "futile" pathways has hitherto been limited to loci with only two or three interconnecting fluxes. We introduce here a method that allows straightforward analysis of more complex systems. The method involves the linear superposition of "fundamental" modes, one or more of which may be energetically wasteful. Decomposition of a flux pattern into such modes allows computation of the amount of free energy "wasted" at any locus. Appropriate normalizations of energy wastage yield a number of indices useful for assessing the energetic impact of futile pathways on the cell and for comparing the degree of regulation of substrate cycles or dual pathways under different metabolic conditions. This approach is applied to steady-state flux data obtained in the protozoan Tetrahymena pyriformis and in isolated rat hepatocytes under a variety of conditions.

Animals