Dynamic regulation of yeast glycolytic oscillations by mitochondrial functions

The glycolytic flux and the concentration of glycolytic intermediates in intact cells oscillate under some conditions (Chance et al. 1964; Hess and Boiteux, 1973; Shulmann, 1988). Possible mechanisms for the glycolytic oscillations have been elucidated. In cell-free extracts of yeast it was demonstrated that phosphofructokinase (PFK) is involved (Hess and Boiteux, 1971, 1973; Pye, 1973; Boiteux et al. 1975; Higgins et al. 1973; see Winfree, 1980, for a review). Modelling revealed that this involvement could result from allosteric effects at the PFK itself (Boiteux et al. 1975) or from the stoichiometric nature of glycolysis when one of the pathway products is used to prime the first reactions (Sel’kov, 1975; Cortassa et al. 1990). In most of the proposed mechanisms the adenine nucleotides play an important role (Hess and Boiteux, 1971; Sel’kov, 1975; Cortassa et al. 1990).

Although the previous models explain many of the observations made on the glycolytic oscillations in yeast, others remain unexplained as already pointed out (Pye, 1973). An unclear point was the possible effect on oscillations by the NAD⁺/NADH system (Pye, 1973). The redox couples NAD⁺/NADH and NADP⁺/NADPH play a central role in the metabolism of sugars by yeast (van Dijken and Scheffers, 1986). Therefore, important questions for physiology remain as to which factors control glycolytic oscillations in vivo. Such questions cannot be addressed in cell-free systems in the absence of important processes, e.g. such as those catalysed by the mitochondria.

Because of the important role of adenine and nicotinamide nucleotides in the dynamic behavior of glycolysis, we asked to what extent mitochondrial functions, such as those related to the regulation of the cytosolic ATP and NADH levels, control glycolysis dynamics in vivo. The main results obtained suggest that mitochondrial functions through the regulation of cytosolic ATP and NADH levels in turn regulate the dynamic behavior of glycolysis. In starved yeast cells, glycolysis probably operates near a Hopf bifurcation point, i.e. around which damped or sustained oscillations appear.

Saccharomyces cereviseae, strain X2180 (α/a, SUC2, mal melgal2 CUPI, diploid strain; Yeast Genetic Stock Center, Berkeley, California) was maintained on malt agar slopes. Cells were grown aerobically in batches at 30°C, pH 5, in defined medium (YNB). Glucose (1 %) served as a carbon source. Cells were harvested at mid-exponential growth, unless otherwise specified, washed twice with 0.1M phosphate buffer, pH 6.8, and resuspended in the same buffer at a concentration of 2×l0⁶ to 4×10⁶ cells per ml and starved for 3 h in a rotary shaker (120 revs min^-1) at 30°C. Preliminary studies showed that 3 h of starvation were optimal as judged from the amplitude of NAD(P)H oscillations. The respiratory-deficient mutants HR2 14^-, HR2 40^-and HR2 17^- obtained from Saccharomyces cerevisiae wild-type strain HR2 (Berden et al. 1988) were cultured under similar conditions to strain X2180 supplemented with 50 μgml^-1 histidine, leucine and uracil and 20 μgml^-1 tryptophan.

Intracellular pyridine nucleotide reduction was monitored with an Eppendorf fluorimeter at excitation wavelengths of 316–366 nm, emission being measured from 400 to 450 nm. Unless otherwise specified, all the experiments were performed at a cell concentration of 10⁷ cells per ml in a thermostated cuvette at 25 °C with stirring. Cells were kept on ice until assayed. Since experiments lasted from 2 to 4h, controls were performed by studying the oscillatory behavior as a function of time and no difference in the ability of cells to oscillate (amplitude and frequency) could be detected for up to 7 h while keeping the cells on ice.

All the fluorescence traces shown represent a typical result corresponding to duplicates or triplicates for each cell batch and for three independent experiments. Controls with 2-deoxy-o-glucose (Sigma, Grade H), a non-metabolized sugar, showed that the observed oscillations originated from glucose metabolism.

Yeast cell suspensions under the same conditions as in the fluorescence measurements were assayed for oxygen uptake with a Clark oxygen electrode (Yellow Springs Instruments, Yellow Springs OH, USA) in a magnetically stirred oxygraph vessel. At 25 °C, 495 μM of atomic oxygen is dissolved in air-saturated 0.1M phosphate buffer, pH 6.8. This value was used for the calculations of oxygen consumption by yeast cells.

Antimycin A (Sigma Chem. Co, St Louis, USA) from Streptomyces kitazawaensis; myxothiazol (Fluka Chemie AG, Buchs, Switzerland) from Myxobacterium myxococcus fulvus and carbonyl cyanide p-(trifluoromethyoxy)phenylhydrazone (FCCP; Aldrich Chemie N.V./S.A., Brussels) were dissolved in ethanol. For all the inhibitors, controls were run by adding the same amount of ethanol alone. The calculated percentages of respiratory inhibition were referred to the controls. Bongkrekic acid was used as an aqueous solution in 2 M NH4OH and its addition to the cell suspension did not provoke measurable pH changes.

Mitochondria of yeast cells (10⁷ cells ml^-1) were stained by incubation in 0.1M Tris-HCl buffer, pH 8, with 5/ZM DASPMI (Bereiter-Hahn, 1976; Bereiter-Hahn et al. 1983) for 30 min at room temperature. Fluorescence-labelled cells were incubated in the presence of glucose and inhibitors as described in each case and a sample of 6 μl was placed between a slide and coverslip and then sealed. Video films of the fluorescence-labelled cells were made with a Zeiss Universal photomicroscope equipped with an image intensifier (Videoscope, International Ltd, Washington DC) with a CCD camera (Grundig, Germany). Pictures of the cells subjected to different treatments were taken directly from the video films.

Glycolysis and the interactions with mitochondria (Scheme I) were modelled as described in Appendix A. The ordinary differential equations (ODEs) system (eqns (1) to (4)) was numerically integrated on an IBM PS2/80 using the program SCoP with an Adams method (Duke University, 1987). The analysis of the stability and bifurcation properties of the ODEs system (eqns (1) to (4)) were performed with AUTO (E. Doedle, 1986; Concordia University, Canada).

A quantitative model of glycolysis and its interactions with mitochondrial function was allied to the experimental work (Scheme I). This model is an elaboration of the one we developed previously (Cortassa et al. 1990; Aon and Cortassa, 1991) and describes realistic glycolytic oscillations. Oscillations reported in the literature for experiments with intact yeast cells and with yeast extracts could be simulated (Cortassa et al. 1990, and results not shown). The dynamic behavior of a metabolic pathway depends on the values of its kinetic and stoichiometric parameters. For the glycolytic model (ODEs, eqns (1) to (4)) a systematic analysis of this dependence was performed by the use of the program AUTO (see Appendix). Fig. 1A,B shows the results for a variation of the load represented by mitochondria (by varying the parameter K-p) and variation of the rate at which glucose entered the system (V_in) (Fig. IB). The continuous line refers to parameter values where the ensuing steady state is stable. In Fig. 1A the steady state concentration of ATP dropped as the work load increased, up to a work load of 1.1 mMs^-1. As the load was increased further, the ensuing steady state became unstable, which is indicated by the broken line in Fig. 1A. Such a transition from stable to unstable steady state is called a bifurcation. When K_P was taken slightly above its bifurcation value, the system did not relax to a single state but to a sustained oscillation (limit cycle). The point where that transition occurs is a Hopf bifurcation (HB). Variation of the rate at which glucose entered the system revealed two HB points (Fig. IB). Another change in dynamic behavior is given by the relaxation behavior around a steady state through damped oscillations (such transition is delimited by a small arrow and the HB on the continuous line in Fig. 1A,B).

Taken together, the results of Fig. 1 predict that (1) increasing the work load towards the glycolytic ATP-synthesizing machinery, or (2) decreasing or increasing the influx of glucose into the system, enhance the tendency of the system to exhibit oscillations. The former prediction suggests that elimination of mitochondrial oxidative phosphorylation so that the entire work load of producing the ATP needed for intracellular free-energy transduction befalls glycolysis, will cause oscillatory behavior. At moderate work loads, this should be noticeable as relaxation towards a steady state through damped oscillations. At extreme work loads, this might lead to sustained oscillations.

The activity of NADH reoxidizing processes would also affect the dynamics of glycolysis. According to model calculations, under conditions of high ATP load the following may be predicted: (1) high rates of NADH reoxidation provoke the appearance of sinusoidal damped oscillations; (2) a decrease in the rate of NADH reoxidation would tend to abolish oscillations; (3) at low rates of NADH reoxidation, if the ATP load is even increased the appearance of square type oscillations is possible.

The plausibility of model predictions was investigated by changing in different ways the cytosolic levels of ATP and NADH and looking for the appearance (approximating a Hopf bifurcation) or disappearance (further away from a Hopf bifurcation) of glycolytic oscillations (Fig. 1).

The hypothesis proposing that a drain of cytosolic ATP mediated by mitochondrial function would trigger glycolytic oscillations was experimentally tested by the use of respiratory inhibitors (Thierbach et al. 1981; von Jagow and Engel, 1981). Fig. 2A-C shows that the addition of myxothiazol or KCN after a pulse of glucose did give rise to glycolytic oscillations, in both the wild-type HR2 (not shown) and the X2180 strain, after a pulse of 20 mM glucose and KCN addition (10 mM) (Fig. 2). Oscillations were also triggered by a pulse of 20 mM glucose in the presence of myxothiazol, by increasing concentrations of KCN (from 5 to 15 HIM) after a pulse of 20 mM glucose or by 1 min preincubation with KCN (or myxothiazol: Fig. 2B) and subsequent glucose addition. The oscillations consisted of cycles of reduction and oxidation of the intracellular pool of pyridine nucleotides with periods of 45 s to 1 min and amplitudes of 0.8 mM or lower. Antimycin A only gave rise to quickly damped oscillations (Fig. 2C). KCN (10 DIM), myxothiazol (4UM) and antimycin (2;íM) inhibited >95%, 72% and 75% O2 consumption in yeast cells, respectively. Also, mutants in the bel complex (ubiquinokcytochrome c reductase) that were largely deficient in respiratory capacity exhibited oscillatory transients following pulses of glucose (Fig. 3 traces B and C). In a similar experiment, the wild-type HR2 did not show oscillations following pulses of glucose and reached a steady state value in NAD(P)H fluorescence after the third glucose pulse (Fig. 3, trace A). The S. cereviseae HR2 40⁻ and HR2 14⁻ mutants showed only 28% and 5%, respectively, of the wild-type strain’s respiratory capacity. The HR2 40⁻ mutant respiratory capacity (28%) was similar to the residual respiration of the wild-type HR2 and X2180 strains after treatment with 4 LIM myxothiazol.

Further insight into the regulatory mechanisms of glycolytic oscillations by mitochondrial activity was provided by experiments performed in the presence of bongkrekic acid (BKA), an inhibitor of the adenine nucleotide translocator in living cells (Gbelska et al. 1983; Lumbach et al. 1970). Starved yeast cells preincubated with BKA showed oscillatory transients after glucose addition (Fig. 4C). The same cells preincubated with BKA were still able to show damped oscillations after KCN and glucose additions (not shown). Yeast cells respiration was not affected by preincubation with BKA. The effect of BKA was concentration dependent; inhibitor concentrations in the range of 10–30 μg ml^-1 gave rise to oscillations. These results suggested that the adenine nucleotide translocator affects the oscillatory mechanism, probably through the control of the cytoplasmic-mitochondrial ATP/ADP exchange.

The disappearance of oscillatory behavior in the presence of FCCP (Fig. 4B) or FCCP plus BKA (Fig. 4D) indicated that the inner mitochondrial membrane Δμ_H plays a role in the triggering of glycolytic oscillations. Evidence for the uncoupling of mitochondrial membranes in vivo in one of the mutants (HR2 40⁻) (Fig. 3) and in the strain X2180 was obtained by fluorescent staining of mitochondria with dimethylaminostyryl-methylpyridin-iumiodine (DASPMI; Bereiter-Hahn, 1976) (Fig. 5). Mitochondria of strain X2180 stained with DASPMI (Fig. 5A,B) revealed the presence of an apparently continuous mitochondrial network or ‘mitochondrion’ (Davison and Garland, 1977; Skulachev, 1990). The wildtype strain and the 17⁻ HR2 mutant exhibited similar DASPMI distribution to that shown by the X2180 strain (Fig. 5A,B) in the presence of glucose (results not shown). In the presence of FCCP, DASPMI was uniformly distributed in the cytoplasm of strain X2180 (Fig. 5D). A similar effect of the uncoupler on DASPMI distribution in the wild-type HR2 strain and 17” HR2 mutant was observed (results not shown). Judging from the distribution of the dye, the mitochondria of the mutant HR40^- (without FCCP) appear not to have much Δμ_H (Fig. 5C). It must be pointed out that the respiratory-deficient mutant HR2 40” only showed 28% of the respiratory capacity when compared with the wild-type strain. The uncoupling effect provokes a release of the dye by annihilation of the transmembrane electric potential, since it is known that DASPMI is distributed in mitochondria according to the membrane potential (Bereiter-Hahn et al. 1983). Although a decrease in fluorescence was noticed, the distribution of DASPMI was not affected in strain X2180 metabolizing glucose after preincubation with BKA or additions of KCN or myxothiazol (results not shown) under the same conditions as described for Figs 2—4.

The model suggested that at low rates of NADH reoxidation and high ATP load, square-type oscillations could appear (see point (3) in Regulatory effects of mitochondrial function). This specific prediction was tested by inhibiting the alcohol dehydrogenase with pyrazole (Lieber et al. 1978) and the results are presented in Fig. 6. Square-like oscillations in NAD(P)H were obtained at 0.1 mM pyrazole (Fig. 6, trace B) while a tendency for the oscillations to become square-like appears already at 0.01 mM of the inhibitor (Fig. 6, trace C). Preincubation of starved yeast cells with pyrazole (0.01 to 0.1 HIM) did not affect the rate of O ₂ consumption.

A correlation between a continuous reduction of the NADH pool and de-energization of mitochondrial membranes was provided by the use of mutants in the bcl complex and FCCP. By carefully inspecting the kinetics of NAD(P)H in the respiratory mutants (Fig. 3, traces B,C) one may confirm that a continuous reduction in the pool of nicotinamide nucleotides occurs (at least in the interval of observation) while in the wild type a steady level of redox potential is attained (Fig. 3, trace A). The latter tendency of the intracellular NAD(P)H pool is taken as an indication of lower rates of NADH reoxidation. The mitochondria of the mutant HR2 40 ⁻ (without FCCP) appear to be de-energized as judged from the uniform distribution of DASPMI (Fig. 5C). When cells of strain X2180 were treated with FCCP (10 nM: 40% increase in the rate of O ₂ consumption with respect to the control), a tendency of the intracellular pool of NAD(P)H to become more reduced after a glucose pulse, could be verified (Fig. 4B). Under the same conditions, the uncoupler abolished the KCN-induced oscillations and uncoupled mitochondrial membranes as could be shown by fluorescence microscopy of DASPMI (Fig. 5D). Taken together, these results suggest that low rates of NADH reoxidation in the FCCP-treated X2180 strain or the HR2 40 ⁻ mutant are associated to de-energized mitochondrial membranes.

In the present work the regulation in vivo of the dynamics of glycolysis by mitochondrial activity in starved yeast cell suspensions was investigated. The main results obtained point to the existence of control exerted by mitochondrial functions through the cytosolic ATP and NADH levels on the appearance of glycolytic oscillations. Model calculations predict (Fig. 1) and experiments suggest (Figs 2–4, 6) that glycolysis in starved yeast cells may operate close to a Hopf bifurcation.

The model simulations describe qualitatively and quantitatively well the experimental results in the following aspects: (1) the amplitude, frequency and shape of the oscillations. By shape of the oscillations we mean their sinusoidal-, square-type or the damping of successive oscillatory cycles. Previous results obtained with a similar model of glycolysis gave good quantitative agreement with reported data in period, amplitude, substrate input rate and decrease in oscillatory frequency with substrate input (Cortassa et al. 1990). (2) The phase relationship between NADH and ATP was investigated by phase plane analysis (not shown) giving values that were close to reported experimental data (Hess and Boiteux, 1971). (3) The catabolic flux sustained by starved yeast cells (10⁷ cells ml^-1 harvested in the exponential phase of growth: A₅₄₀=2.2) was calculated to be 33 μMS ⁻¹. From model simulations, values of glycolytic fluxes that oscillated around 210 and 240 μM s^-1 were obtained at high and low energy loads, respectively, that are of the same order of magnitude of those experimentally determined. In starved yeast cells the drain of cytosolic ATP under the experimental, semianaerobic conditions reported here appears to be a key event triggering glycolytic oscillations (Figs 2–4). Essentially, mitochondria by modulating the level of cytosolic ATP, in turn modulate the approach of the dynamic behavior of glycolysis to an oscillatory region (Fig. 1). The experimental situation described in Fig. 2 (i.e. inhibition of mitochondrial oxidative phosphorylation) is reflected in Fig. 1A as an increase in the ATP load, namely a decrease in the steady state levels of cytosolic ATP.

Two regulatory levels appear to be implicated in the regulation of the cytoplasmic ATP pool by mitochondria: (1) the Δμ_H and (2) the adenine nucleotide translocator. Regulation of the ATP export to the cytosol by the adenine nucleotide translocator appears to be involved in the triggering of glycolytic oscillations (Fig. 4). Interestingly, the BKA effect appears to be rather specific, since O2 consumption of starved yeast cells was not altered in the presence of the inhibitor. The results presented (Figs 1, 2–4) strongly suggest that glycolysis dynamics in starved yeast cells operates very cl,ose to a Hopf bifurcation that is approached when the cytosolic ATP decreases.

When glycolysis approaches the oscillatory region at high ATP load, the rate of NADH reoxidation may trigger oscillations. Transient changes in the rate of NADH reoxidation by mitochondria triggered oscillations in model simulations (results not shown). Experimental evidence supporting predictions I and II (see section above: Regulatory effects of mitochondrial function) is presented in Fig. 2 (trace A), Fig. 3 (traces B,C) and Fig. 4B. A similar result to that obtained with the mutants (Fig. 3, traces B,C), i.e. a continuous reduction of the nicotinamide nucleotide pool and quickly damped oscillations after a third pulse of 2IBM glucose, was observed with cells of strain X2180 preincubated with 10 nM FCCP (Fig. 3 and results not shown). Two experimental observations suggest that de-energization of the mitochondria correlates with low rates of NADH reoxidation; (1) the HR2 40” mutant showed a low glycolytic flux (given by the steepness of the temporal change in NAD(P)H; Fig. 3) and a tendency to increase the intracellular pool of NAD(P)H after successive pulses of glucose (Fig. 3); (2) a similar pattern of behavior was shown by the X2180 strain when subjected to a pulse of glucose after preincubation with FCCP (compare the control (A) with the FCCP-treated cells (B) in Fig. 4). At high ATP drains, the rate of cytoplasmic NADH reoxidation by mitochondrial dehydrogenase (Alexander and Jeffries, 1990; Bruinenberg et al. 1985) and alcohol dehydrogenases provide another regulatory level of the dynamics of glycolysis. This additional regulatory level influences not only the approach of glycolysis to the oscillatory region (Fig. 1) but, once inside, it further regulates the shape of the oscillations (sinusoidal-, square-type: Fig. 6 and results not shown).

The metabolic features shown in Figs 3 and 4 (points (1), (2): see previous paragraph) correlated with unenergized mitochondrial membranes in starved yeast cells stained with DASPMI and metabolizing glucose (Fig. 5C,D). When mitochondrial membranes are unenergized, KCN-induced oscillations are quickly damped or tend to disappear as shown by the respiratory mutants (Fig, 3) and FCCP treatment (Fig. 4B), respectively. Taken together, these resulte suggest that low rates of NADH reoxidation in the FCCP-treated X2180 strain or the HR2 40” mutant are associated with de-energized mitochondrial membranes (Figs 4, 5). If the transmembrane electric potential controls the rate of NADH reoxidation through the outer mitochondrial membrane dehydrogenase deserves further investigation.

The glycolytic model and the interactions with mitochondrial activity (Scheme I) used in the present work to describe the experimental data are stoichiometric (Sel’-kov, 1975) with the non-linear kinetic mechanism given by the stoichiometries of ATP production in anaerobic glycolysis (Cortassa et al. 1990; Aon and Cortassa, 1991). In yeast extracts, the allosteric properties of phosphofructokinase were shown to be the source of oscillations (Hess and Boiteux, 1968; Higgins et al. 1973; Boiteux et al. 1975). The question arises as to whether in vivo the triggering of glycolytic oscillations can be also explained by allosteric effects on PFK. Experimental evidence now sheds doubt on the latter possibility: (1) the control of PFK under Pasteur effect conditions could not be ascribed to changes in any one particular effector but rather to contributions from a variety of effectors (Reibstein et al. 1986). Furthermore, Fru-2, 6-P₂ was the only effector that appreciably changed between the anaerobic and aerobic condition. These observations suggest that, in vivo, the allosteric properties of the PFK could hardly be manifested at least in aerobic-anaerobic transitions; (2) the present work shows that the autocatalytic mechanism given by the stoichiometry of glycolysis coupled to mitochondrial activity through the ATP and NADH cytoplasmic pools may constitute another regulatory level of glycolysis in vivo; (3) in yeast extracts the presence of a complex type of oscillations apart from sinusoidal, i.e. spike-like or square-type, was experimentally demonstrated (Hess and Boiteux, 1968; Hess and Boiteux, 1973). The stoichiometric model without taking into account allosteric effects was able to exhibit sinusoidal-, square- and spike-like oscillations (not shown); (4) great deal of experimental evidence supports the notion of several control steps in glycolysis apart from PFK (den Hollander et al. 1986; Reibstein et al. 1986). According to our results, the dynamics of glycolysis in starved yeast cells under semianaerobic conditions may attain oscillatory regions according to the following relevant physiological parameters: the rates of substrate input and substrate phosphorylation, the load induced by the non-glycolytic ATP-consuming processes and the rate of cytoplasmic NADH reoxidation. At present the possibility cannot be excluded that the effects of ATP/ADP and cellular redox states may run in part through allostericity of PFK and in part through the autocatalytic nature of the pathway.

Under growing conditions a main contribution to the NADPH balance in yeast cells is expected to occur through the functioning of the hexose monophosphate pathway (HMP) (Bruinenberg et al. 1985; van Dijken and Scheffers, 1986). Under those conditions a leak in hexose phosphate from glycolysis should be expected. The extent of the leak of phosphorylated intermediates may drive glycolysis into sustained oscillations at low substrate influx (Cortassa and Aon, 1991, unpublished data). However, since cell growth does not occur under the experimental conditions described in this work (Galazzo and Bailey, 1989), the NADPH turnover is expected to be very low or nonexistent.

Summarising, the results presented strongly suggest that mitochondrial functions are able to regulate in vivo the dynamics of glycolysis in starved yeast cells through cytoplasmic ATP and NADH levels. The regulation of the ATP and NADH pools in vivo by mitochondria may be achieved at various levels: the mitochondrial proton motive force through the regulation of the ATP synthetic or hydrolytic fluxes; the adenine nucleotide translocator presumably through the regulation of the rate of cytoplasmic-mitochondrial ATP/ADP exchange, and the mitochondrial dehydrogenases by regulating the rate of NADH reoxidation. Additionally, a further regulatory level in glycolysis when the cells are operating at high ATP loads may be provided by the alcohol dehydrogenase. Whether these potential regulatory interactions function in physiological transitions remains to be investigated.

The authors thank Dr J. A. Duine (Technical University of Delft, The Netherlands) for a generous gift of bongkrekic acid and Dr Bereiter-Hahn (J. W. Goethe Universitat, Frankfurt) for kindly providing us with DASPMI. Also the assistance in AUTO installation and helpful advice of Dr E. Doedle (Dept of Computer Science, Concordia University, Canada) and Mr R. Belleman (Biophysics Department, Netherlands Cancer Institute) are gratefully acknowledged. We are indebted to the National Biomedical Simulation Resource (Duke University, Durham, USA) for providing us with information about SCoP. M.A.A., S.C. and H.V.W. thank the Commission of European Communities and the Netherlands Organisation for the Advancement of Pure Research (NWO) for financial support.

The interactions of glycolysis and mitochondria are depicted in Scheme I. Glycolysis was modelled in two lumped steps. The first five steps (upper part) or preparatory phase (Lehninger, 1982) are represented by Glc → 1, with rate constant k₁, and I representing the pool of glyceraldehyde 3-phosphate. The second phase of glycolysis (lower part), is lumped in I → Pyr, with rate constant k₃. 2 and 4 in equation (2), below, are stoichiometric coefficients accounting for the two ATPs expended in the upper part of glycolysis and the four ATPs produced in the lower part of glycolysis.

Pyruvate does not appear explicitely as an ODE because covariation with NADH is assumed (eqns (4) and (11)). To the steps of glycolytic NADH production (k₃) and consumption (k₇) (Scheme I: heavy arrows), the cytosolic NADH reoxidation by the inner mitochondrial membrane dehydrogenase (&io, Scheme I: light arrow) (Alexander and Jeffries, 1990) was added to simulate transient behavior and was not included in the stability analysis. Additionally, the input rate of glucose (V_in) is considered to be constant.

The non-glycolytic ATP-consuming processes that represent the ATP load are captured by and the mitochondrial ATPase. The reactions describing either ATP synthesis (V_p2) or hydrolysis (V_p1) by the mitochondrial H⁺ pump were assumed to follow Michaelis-Menten kinetics as a function of ADP or ATP concentrations, respectively (Scheme I and eqns (2,12,13)). Mitochondrial respiration was only implicitly taken into account through Ap_H generation. Additional general properties of the present model have been described by Cortassa et al. (1990) and Aon and Cortassa (1991).

C_A, total concentration of adenine nucleotides ([ATP] + [ADP]); CN, total concentration of nicotinamide nucleotides ([NAD]+[NADH]); Δp°, reversal potential of the proton pump; Δμ_H, proton motive force; [Glc], intracellular concentration of glucose; [I], intracellular concentration of glycolytic intermediate(s) pool; k₁> k₃, k₅, k₇, k₉, rate constants; P_i, total intracellular concentration of inorganic phosphate; P_t, intracellular concentration of phosphate; V_in, input rate of glucose; ⁠, kinetic constants of the mitochondrial proton pump; V_P1, Vp2, fluxes of ATP hydrolysis and ATP synthesis; r, ratio between H⁺ conductance and membrane capacitance.