Kinetic Analysis of Tentoxin Binding to Chloroplast F1-ATPase
A MODEL FOR THE OVERACTIVATION PROCESS*

Jérôme SantoliniDagger §, Francis HarauxDagger , Claude SigalatDagger , Gwénaëlle MoalDagger , and François AndréDagger

From the  Protéines Membranaires Transductrices d'Energie (CNRS-URA 2096), DBCM-CEA Saclay, bâtiment 532, F-91191 Gif-sur-Yvette Cedex, France and Dagger  Section de Bioénergétique, Département de Biologie Cellulaire et Moléculaire, Commissariat à l'Energie Atomique-Saclay, F-91191 Gif-sur-Yvette Cedex, France

    ABSTRACT
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Abstract
Introduction
Procedures
Results
Discussion
References

The mechanism of action of tentoxin on the soluble part (chloroplast F1 H+-ATPase; CF1) of chloroplast ATP synthase was analyzed in the light of new kinetic and equilibrium experiments. Investigations were done regarding the functional state of the enzyme (activation, bound nucleotide, catalytic turnover).

Dialysis and binding data, obtained with 14C-tentoxin, fully confirmed the existence of two tentoxin binding sites of distinct dissociation constants consistent with the observed Kinhibition and Koveractivation. This strongly supports a two-site model of tentoxin action on CF1. Kinetic and thermodynamic parameters of tentoxin binding to the first site (Ki = 10 nM; kon = 4.7 × 104 s-1·M-1) were determined from time-resolved activity assays. Tentoxin binding to the high affinity site was found independent on the catalytic state of the enzyme.

The analysis of the kinetics of tentoxin binding on the low affinity site of the enzyme showed strong evidence for an interaction between this site and the nucleotide binding sites and revealed a complex relationship between the catalytic state and the reactivation process. New catalytic states of CF1 devoid of epsilon -subunit were detected: a transient overstimulated state, and a dead end complex unable to bind a second tentoxin molecule. Our experiments led to a kinetic model for the reactivation phenomenon for which rate constants were determined. The implications of this model are discussed in relation to the previous mechanistic hypotheses on the effect of tentoxin.

    INTRODUCTION
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Abstract
Introduction
Procedures
Results
Discussion
References

F0F1-ATP synthases are the purveyors of ATP in chloroplasts, mitochondria and bacteria. They are bound to energy-transducing membranes and couple the synthesis of ATP (through photophosphorylation of ADP) to the dissipation of a protonmotive force (1, 2). The enzyme consists of two discrete parts, F0 and F1, interconnected by a stalk. F0 is embedded in the membrane and behaves as a proton channel. The extrinsic part F1 bears the six nucleotide binding sites; three of them are catalytic sites for ATP synthesis (for a review, see Ref. 3). Depending on species, three or more different subunits compose the F0 moiety: a(1), b(2), and c(9-12) in Escherichia coli; a(1), b(1), b'(1), and c(9-12) in chloroplasts. In mitochondria, the F0 moiety bears additional subunits (4). The F1 part consists of five different subunits: alpha , beta , gamma , delta , and epsilon , with alpha (3), beta (3), gamma (1), delta (1), epsilon (1) as stoichiometry. It is proposed that the ATP synthase acts as a proton-driven motor (5-7): in chloroplasts and bacteria, subunit c and subunits gamma  and epsilon  (presumably linked to the c-crown) compose the rotor, while the extrinsic alpha 3beta 3 crown linked to the membranous a subunit by the delta  and b subunits acts as a stator (8, 9). The rotation relayed by gamma  would sequentially modify the three-dimensional structure of the three catalytic sites (10) and induce the ATP synthesis. Evidence for the rotation of gamma  relative to the alpha 3beta 3 crown has been presented in the case of ATP hydrolysis (11-14) and ATP synthesis (15). As an essential breakthrough, this model strictly correlates the cooperative mechanism of the enzyme to the rotation of the gamma -subunit and thus to the proton gradient dissipation.

Tentoxin (TTX)1 is a natural cyclic tetrapeptide (cyclo-(L-methyl-Ala1-L-Leu2-methyl-Delta ZPhe3-Gly4)) produced by several phytopathogenic fungi of the Alternaria genus (16-17), which induces chlorosis of many sensitive higher plants (18). Chlorosis seems to be a consequence of the inhibition of photophosphorylation. TTX indeed specifically inhibits ATP synthesis in chloroplasts of sensitive species (19) as well as ATP hydrolysis in isolated CF1 (20). In addition, TTX was shown to bind the extrinsic part F1 of the F0F1-ATP synthase (19). An interesting feature of this toxin is its dual effect: in vitro and at low concentrations (10-8 to 10-7 M), TTX inhibits ATP hydrolysis and synthesis either in isolated chloroplasts or in isolated CF1, while at high concentrations (10-5 to 10-4 M) it strongly stimulates ATPase activity of the isolated enzyme (21) and leads to a partial recovery of the coupled activity of membrane-bound ATP synthase (22).

The mechanism of TTX action, for the inhibition as well as for the reactivation, is still unknown. The number of binding sites involved in the reactivating effect remains controversial. It has been reported that CF1 binds two molecules of TTX on distinct sites (23) presumably located on beta -subunit, although this point remains obscure (24-26). Since the affinity of TTX for these two binding sites was found different, we suggested that they were respectively related to the inhibitory and stimulatory activities of TTX (23). Also, Mochimaru and Sakurai (27) have recently suggested the existence of a third very low affinity binding site, which would account for the reactivation process.

The affinity of TTX for the inhibitory site has been measured in various forms of the ATP synthase (20, 28), with limited efforts to relate it to the functional states of the enzyme (27). Rare investigations have been carried out on the interactions between the nucleotide binding sites and the TTX binding sites. TTX at inhibiting concentration does not interfere with the exchange of tightly bound nucleotide (29), while it seems to promote the ADP release at high concentrations (30). Nevertheless, the effect of the presence of nucleotides and, more generally, of the catalytic state of the enzyme on TTX binding to its low affinity site has never been studied.

In this report, we show strong evidence for the existence of only two TTX binding sites on chloroplast ATP synthase in the concentration range of in vitro assays, and we show that they directly account for the inhibitory and reactivating effects of TTX on the activity of CF1 and CF1-epsilon . In addition, thermodynamic and kinetic characteristics of TTX binding on the inhibitory site were thoroughly determined. No influence of the dynamic state of the enzyme on this inhibitory binding has been observed. As regards the reactivation process, kinetic experiments revealed the existence of two new dynamic forms of CF1-epsilon related to the presence of ATP: a transient overactivated state and a dead end complex unable to bind a second TTX molecule. We show that TTX binding to its reactivating site is strongly dependent on the presence of nucleotide on the enzyme. We propose a kinetic model that accounts for all of our results, of which most of the constants were determined.

    EXPERIMENTAL PROCEDURES
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Abstract
Introduction
Procedures
Results
Discussion
References

Enzyme Preparation-- The soluble chloroplast ATPase (CF1) was extracted and purified from spinach (Spinacia oleracea L.) leaves in the active form devoid of its inhibitory subunit epsilon , unless specified. The enzyme was stored at 5 °C in 34% ammonium sulfate buffer as described previously (31) at a protein concentration of 16 mg ml-1.

CF1-epsilon was activated by preincubation at 20 °C in 20 mM Tricine, pH 8, 3 mM DTT for at least 3 h. When CF1 was used, no activation was performed. Protein concentration was determined by UV absorption spectroscopy assuming for CF1-epsilon an extinction coefficient of 0.48 cm2 mg-1 at 278 nm (31).

Binding Experiments-- Samples of 500 µl of DTT-activated CF1-epsilon at various concentrations (10 nM to 10 µM) were dialyzed in SPECTRA/POR tubing (molecular weight cut-off = 6000-8000) for 24 h at 37 °C against 50 mM Tris-SO4, pH 8, buffer containing various concentrations of 14C-TTX (ranging from 10 nM to 10 µM). 14C-TTX was obtained as described previously (23) with a specific activity of 52 Ci mol-1. Equilibrium conditions were checked by measuring in small aliquots the inner and outer radioactivity at four different times during dialysis. Each counting was performed twice for 4 min using a Beckman LS 3801 Scintillation Counter. 14C cpm ranged from 500 ± 60 to 5 × 105 ± 200. Free TTX concentration was deduced from the radioactivity measured outside the dialysis tubing, and bound TTX concentration was calculated from the difference between the radioactivity measured inside and outside the dialysis tubing. No significant loss of activity occurred within 24 h in these experimental conditions. For TTX concentrations higher than 10 µM, 14C-TTX and CF1-epsilon were first equilibrated for 3 h at 37 °C. Bound and free TTX were then rapidly separated using a PD 10 Amersham Pharmacia Biotech chromatography column, and their concentrations were measured as indicated above. Binding curve was fitted using Microcal ORIGIN 5.0 (Microcal Software).

Kinetics Dialysis-- 500-µl samples of 14C-TTX at various concentrations were dialyzed at 37 °C in a low cut-off (Mr = 1000) SPECTRA/POR dialysis tubing against 20 ml of assay buffer (50 mM Tris-SO4, pH 8, 40 mM KHCO3, 0.18 mM MgSO4). Small aliquots were withdrawn from the outer buffer at different times, and the radioactivity was measured as described above and converted into TTX concentration. The volume of the dialysis tube, and thus the exchange surface, was nearly constant in each experiment. The outer TTX concentration was plotted as a function of time and fitted to Fick's law using Microcal ORIGIN 5.0. Control experiments were performed with 500 µM samples of ADP, diadenosine 5'-hexaphosphate, and Alcyan Blue, titrated by spectrophotometry.

Steady-state Activity Assays-- DTT-activated CF1-epsilon (10 nM) was incubated at 37 °C in 50 mM Tris-SO4, pH 8, buffer, in the presence of TTX at various concentrations for 4 h up to 12 h for the lowest concentrations. 5 min after the addition of 40 mM KHCO3 and 0.18 mM MgSO4, ATP hydrolysis was initiated by the addition of ATP (final concentration 1 mM). The reaction mixture was thermostatted at 37 °C, and aliquots of 10 µl were taken every 3.5 min (up to 15 min) and injected into a TSK DEAE 2SW 5-µm analytical high pressure liquid chromatography column for nucleotide determination. The nucleotides were separated by isocratic elution with 0.1 M KH2PO4, pH 4.3, 0.25 M NaCl, at a rate of 1.0 ml min-1. ADP concentration was determined from the area of the peak detected at 260 nm. The ATPase activities of CF1-epsilon in the presence of various concentrations of TTX were deduced from the plots of ADP concentration as a function of time and normalized to the control activity without TTX (Fig. 4).

Time-resolved ATPase Activity Assays-- For inhibition kinetic studies, DTT-activated CF1-epsilon (10 nM) was incubated for 10 min at 37 °C in a stirred reaction buffer in a spectrophotometer cuvette. The reaction buffer (50 mM Tris-SO4, pH 8, 40 mM KHCO3, 4 mM MgSO4, 1 mM phosphoenolpyruvate, 0.3 mM NADH, 0.1 mg/ml lactate dehydrogenase, 0.1 mg/ml pyruvate kinase) allowed us to couple ATP hydrolysis to NADH oxidation. The time response of this system, checked by ADP addition in the absence of TTX, was always significantly shorter than a few seconds. ATP hydrolysis was started by adding MgATP (final concentration 2 mM) and monitored by absorbance decrease at 340 nm.

In a first series of experiments (Fig. 3a), TTX was added at various inhibitory concentrations (from 5 nM to 5 µM) 3 min after the addition of MgATP. In another series of experiments (Fig. 3b), MgATP was added after the incubation of CF1-epsilon for various times with 75 nM TTX. The resulting kinetics were fitted to Equation 4.

For reactivation kinetic studies, DTT-activated CF1-epsilon (10 nM) was incubated either in the presence or in the absence of inhibitory concentration of TTX (500 nM). ATP hydrolysis was started by the addition of MgATP (final concentration 2 mM) and continuously monitored at 340 nm. In a first series of experiments, TTX was added at various reactivating concentrations (from 5 to 150 µM) to inhibited CF1-epsilon , 3 min after the addition of MgATP (Fig. 6a). Similar experiments were achieved by adding MgATP after the addition of reactivating concentrations of TTX (Fig. 6b). The decay phases of the kinetics of Fig. 6, a and b, were fitted to a primitive function of Equation 14. The resulting apparent rate constants kapp were fitted to Equation 13, which allowed the determination of k+, k-, and K. Other protocols are detailed in the text.

    RESULTS
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Abstract
Introduction
Procedures
Results
Discussion
References

Binding of 14C-Radiolabeled TTX to CF1-epsilon -- Fig. 1 shows the amount of 14C-TTX bound to CF1-epsilon , when equilibrated at 37 °C, as a function of the concentration of free 14C-TTX. The data were fitted to a model based on multiple independent binding sites. The fit clearly shows the existence of two binding sites of different affinities (Kd1 = 50 ± 20 nM; Kd2 = 80 ± 20 µM) in the concentration range investigated (between 10 nM and 1.2 mM). The two dissociation constants differ by about 3 orders of magnitude, and the binding of TTX to CF1-epsilon can therefore be considered to occur in a sequential mode.


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Fig. 1.   14C-tentoxin bound to DTT-activated CF1-epsilon as a function of free 14C-tentoxin. Conditions were as described under "Experimental Procedures." CF1-epsilon ranged from 10 nM to 10 µM (depending on TTX concentration). Nonlinear fitting is shown (solid line; see Equation 1) using a two-independent site model (Kd1 = 50 ± 20 nM; Kd2 = 80 ± 20 µM).

Monomeric Behavior of TTX at High Concentrations-- To validate the present analyses, which imply that TTX is monomeric in the reaction medium, we examined its rate of diffusion through a dialysis membrane of appropriate cut-off (see "Experimental Procedures"), at different concentrations. This diffusion indeed is expected to follow Fick's law only if molecules do not form concentration-dependent aggregates. Fig. 2 (inset) shows exponential kinetics of diffusion of 14C-TTX, as expected from Fick's law. The kinetic data, normalized to the equilibrium concentration, were identical for each initial concentration inside the dialysis tube between 1.8 and 500 µM. The initial rate of diffusion of 14C-TTX was thus found proportional to its internal concentration (Fig. 2, main panel). To check the dialysis membrane cut-off, we controlled the rate of diffusion of several molecules of different molecular masses, at internal concentrations of 500 µM. Their initial rates of diffusion were determined and plotted as a function of their concentration in Fig. 2. ADP (Mr = 427) diffused at the same rate as TTX (Mr = 414.5). Conversely, diadenosine 5'-hexaphosphate (Mr = 996) diffused at a 4-fold lower rate and Alcyan Blue (Mr = 1299) did not diffuse at all. This shows that TTX does behave as a monomer, even at 500 µM. This implies that the reactivation of CF1-epsilon by high TTX concentrations (Figs. 5 and 6) is not due to some artifact and should be interpreted as the specific binding of a second TTX molecule on its low affinity site.


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Fig. 2.   Diffusion rate of TTX through dialysis membrane as a function of TTX concentration. Inset, diffusion kinetics were determined for initial inner TTX concentrations between 1.8 () and 500 µM (black-square) as described under "Experimental Procedures." The profiles were fitted with the equation C/Cequilibrium = (1 - e-t/tau ). A fitted curve (solid line) is shown with tau  = 580 min () and tau  = 551 min (black-square). The main panel shows resulting rate constants plotted versus TTX concentration. In order to check the cut-off of the dialysis tubing, the same experiments were performed with 500 µM ADP (; Mr = 427), diadenosine 5'-hexaphosphate (AP6A) (; Mr = 996), and Alcyan Blue (black-triangle; Mr = 1299).

Kinetics of Inhibition of CF1-epsilon by TTX-- The rate of inhibition by TTX was investigated by injecting TTX to a solution of CF1-epsilon during ATP hydrolysis, continuously monitored by ADP-dependent NADH oxidation. Typical recordings of absorbance at 340 nm are displayed in the inset of Fig. 3a. The absorbance followed a single exponential decrease superimposed to a linear decay. Therefore, the experimental data were satisfactorily fitted to Equation 4. The apparent rate constant kapp of the exponential kinetics of inhibition was derived for each TTX concentration. As expected from a unique binding site model, kapp was found to depend linearly on TTX concentration (Fig. 3a). According to Equation 3, the slope of the straight line gives the forward rate constant kon of the binding of TTX to the inhibitory site. The value of kon at 37 °C was found to be 4.7 × 104 M-1 s-1, i.e. 7 times higher than the value estimated by Mochimaru and Sakurai (27) by experiments at room temperature.


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Fig. 3.   Kinetics of TTX binding to the high affinity site. Conditions were as described under "Experimental Procedures." a, apparent rate constant (kapp, bullet ) of ATPase inhibition as a function of TTX concentration. The forward binding rate constant (kon = 4.7 × 104 ± 2 × 103 s-1) was directly determined from the slope of the graph (Equation 5). Inset, typical time courses of ATP hydrolysis after the addition of TTX at 50 nM (1), 300 nM (2), and 1500 nM (3). Every kinetic profile was fitted to Equation 4. b, initial ATPase activity (bullet ) as a function of incubation time with TTX. Similar experiments are shown as in a but with 75 nM TTX added at various times before ATP. The activities were normalized to the control without TTX (5 ± 1 µmol of ATP/min/mg). Data were fitted to Equation 2, which gave kapp = 4 10-3 ± 10-3 s-1 and then kon = 6 × 104 ± 1.5 × 104 M-1 s-1 (Equation 3 neglecting koff). Inset, typical time courses of ATP hydrolysis after 1 min (1), 5 min (2), and 10 min (3) of incubation, fitted to Equation 4.

The rate of binding of TTX on the inhibitory site was also measured in the absence of ATP. CF1-epsilon was incubated in presence of 75 nM TTX for various times. Then ATP was added, the corresponding kinetics were recorded (Fig. 3b, inset), and the initial rate of ATP hydrolysis was determined. This rate was plotted as a function of the time of incubation with TTX (Fig. 3b). The kinetics of inhibition so obtained were fitted with a monoexponential function (Equation 2). Taking into account the TTX concentration and assuming that the koff is negligible, the derived kapp gave rise to a forward rate constant kon of 6 × 104 M-1 s-1. This value is close to that determined above in the presence of ATP.

Comparison of the Effects of TTX on the ATPase Activity of CF1 and CF1-epsilon -- Since the rate constant kon of TTX for the inhibitory site is not very high, long times of incubations are required to observe the full effect of low concentrations of TTX (below 100 nM), even at 37 °C. For this reason, we investigated the inhibitory effect of low concentrations of TTX under conditions similar to those previously reported (32), except that the time of incubation with TTX was raised in order to reach the binding equilibrium (see "Experimental Procedures"). The results are displayed in Fig. 4a. The data points were fitted to Equations 6 and 7, giving a Kd for the inhibitory site of about 8 nM. This Kd is lower than that previously determined (38 nM) for shorter incubation times (32). The resulting value of the rate constant of dissociation koff is 3.7 × 10-4 s-1. This value confirms that koff was negligible with respect to kon[T] above.


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Fig. 4.   ATPase activity of CF1 and CF1-epsilon as a function of TTX concentration. Conditions were as described under "Experimental Procedures." Activities were normalized to the control without TTX (5 ± 1 µmol of ATP/min/mg for DTT-treated CF1-epsilon and 250 ± 50 nmol of ATP/min/mg for CF1). a, experimental points (open circle ) were determined at equilibrium after the incubation of 5 nM of DTT-activated CF1-epsilon with various concentrations of TTX in the inhibitory range. The right part of the curve corresponds to the nonlinear fitting previously determined in the reactivatory range (32). The experimental data were fitted to Equations 6 and 7. Kd1 = 8 ± 1 nM; residual activity V1 = 5% of the control. b, experimental points (black-square) were obtained in the same conditions with 50 nM of untreated CF1. The left part (up to 1 µM of TTX) was fitted to Equations 6 and 7; Kd1 = 13 ± 1 nM; residual activity V1 = 3%. The right part (from 1 µM to 1 mM TTX) was fitted with Equation 8. See text for details.

The same protocol was used to investigate the effect of TTX on ATPase activity of native CF1 for concentrations ranging from 1 nM to 1 mM (Fig. 4b). The affinity of TTX for the inhibitory site of CF1 (Kd = 13 nM) was found almost identical to the one determined for CF1-epsilon . The lack of the epsilon -subunit and the DTT activation have no effect on the inhibition processes of TTX. Since the solubility of TTX is limited to 3.5 mM and the monomeric state of TTX is uncertain above 1 mM, the investigation of its effect on CF1 above 1 mM would be of little interest. Therefore, we only measured the reactivation of the ATPase for TTX concentrations up to 1 mM. Since no plateau of CF1 activity was reached, we were not able to precisely determine the dissociation constant for the second, loose site. Whatever it may be, the Kd varied from 700 µM to 2.7 mM for a reactivation level set from 5- to 20-fold the control. As a result, the binding of a second TTX molecule appears quite different between CF1 and CF1-epsilon .

Kinetics of Reactivation of ATPase Activity of CF1-epsilon by TTX Critically Depend on the ATP Preincubation Time-- We have investigated the kinetics of reactivation of CF1-epsilon by TTX. The enzyme was introduced in the spectrophotometric cuvette with TTX at 500 nM to fill the inhibitory site and in the presence of an enzymatic system coupled to NADH oxidation. In order to check the effect of nucleotides on the TTX-dependent reactivation process, the enzyme was incubated with ATP for various times (from 2 s to 10 min) before the addition of a reactivating concentration of TTX. As expected, the ATPase was inhibited by 500 nM of TTX (6-7% of the control). Reactivation was then initiated by adding 20 µM of TTX (Fig. 5). Surprisingly, a biphasic profile was observed for short preincubation time with ATP. It consisted of a fast rise followed by a slow monoexponential decay. This behavior progressively disappeared with increasing ATP incubation time to finally give rise to a simple monophasic reactivation.


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Fig. 5.   Kinetics of ATPase reactivation for different incubation times with ATP. Conditions were as described under "Experimental Procedures." CF1-epsilon (10 nM) was inhibited by 500 nM TTX and incubated for various times (9 s (1), 30 s (2), 60 s (3), 120 s (4), and 300 s (5)) with 2 mM MgATP. ATP hydrolysis was monitored after a 20 µM TTX addition. Instantaneous rates were obtained from the first derivative of ATP hydrolysis profiles. They were normalized to the control without TTX (6 ± 1 µmol of ATP/min/mg).

Rate of Reactivation of CF1-epsilon by High Concentrations of TTX Added after ATP Preincubation-- TTX-inhibited CF1-epsilon was incubated with ATP during 3 min (conditions leading to a monophasic profile of reactivation; Fig. 5), and high concentrations of TTX (5-150 µM) were then added. A progressive recovery of ATPase activity was observed. The kinetics of recovery of activity could be satisfactorily fitted to a monoexponential function (Fig. 6a). Surprisingly, the rate constant kapp was not a linear function of the TTX concentration. It was even found to decrease with TTX concentration (Fig. 6c). This behavior is not that expected from reactivation kinetically controlled by TTX binding on the low affinity site.


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Fig. 6.   Apparent rate constant of TTX binding to the loose site of CF1-epsilon as a function of TTX concentration. a, conditions were as described under "Experimental Procedures." CF1-epsilon (10 nM) was inhibited by 500 nM TTX and preincubated for 3 min with 2 mM MgATP. ATP hydrolysis was monitored after the addition of TTX (concentrations ranging from 5 to 150 µM). Instantaneous rates were obtained from the first derivative of ATP hydrolysis profiles. They were normalized to the control without TTX (6 ± 1 µmol of ATP/min/mg). Typical time courses are displayed for the following TTX concentrations: 150 (1), 100 (2), 75 (3), 30 (4), and 15 µM (5). kapp values were obtained by fitting each kinetic profile to Equation 2. b, same experiments as in a, but MgATP was added to the enzyme immediately after TTX (concentrations ranging from 5 to 150 µM). Typical time courses are represented for the following TTX concentrations: 150 (1), 100 (2), 60 (3), 25 (4), and 10 µM (5). kapp values were obtained by fitting each kinetic profile with Equation 2. c, apparent rate constants obtained from a and b as a function of TTX concentration. Data obtained with ATP preincubation (open circle ) were fitted to Equation 13 (k+ = 2 ± 1 × 10-2 s-1, k- = 1 ± 0.15 × 10-2 s-1, Kd = 10 ± 13 µM). The kapp value obtained without preincubation (black-square) was found to be almost constant (kapp = 8 ± 1 × 10-3 s-1).

Rate of Reactivation of CF1-epsilon by High Concentrations of TTX without ATP Preincubation-- We also checked the kinetics of reactivation obtained at different TTX concentrations, added together with ATP (Fig. 6b). The kinetics were this time clearly biphasic; the activity passed through a transient overactivation and then decayed to a plateau similar to the one obtained with long ATP preincubation (Fig. 6a). We did not try to analyze the fast rising phase, because its kinetic resolution was not good enough. However, the slow phase has been fitted to a monoexponential function. Contrary to what has been calculated in the kinetic analysis of Fig. 6a, the rate constant kapp was here independent of TTX concentration (Fig. 6c). Identical results were obtained when reactivating TTX was added to inhibited CF1-epsilon at different times before ATP (data not shown).

In other experiments, the kinetics of ATPase reactivation were investigated as in Fig. 6 by adding high concentrations of TTX, but directly to the active, noninhibited enzyme. The kinetics of reactivation also followed the biphasic mode already observed in Fig. 6b. Contrary to what was observed in the previous experiments, preincubation with ATP several minutes before TTX addition did not change the kinetics of activation (data not shown).

From these experiments, two different modes of reactivation can be discriminated on a kinetic basis. The first one, observed when TTX is added several minutes after ATP (Fig. 6a), is a monophasic rise of activity toward an equilibrium value. The second one is observed when ATP is added after or with TTX (Fig. 6b). It consists of a fast rise of activity followed by a slow monoexponential decay, leading to about the same equilibrium level as the first mode. Another major difference between these two modes of reactivation is the characteristic evolution of kapp as a function of TTX concentration.

Rapid Release of TTX from the Second Site in the Presence of ATP-- We have also observed that the release of TTX from the reactivatory site was very fast in the presence of ATP. CF1-epsilon was first incubated with TTX 50 µM at room temperature, for 10 min (a sufficient time for TTX to bind on the two sites), in the presence as well as in the absence of ATP. Then the sample was quickly 100-fold diluted in the reaction medium containing ATP, which keeps the high affinity site filled, and the ATPase reaction was immediately monitored by NADH spectrophotometry. Since no significant activity could be detected (data not shown), we concluded that in the presence of ATP, TTX is released from the low affinity site in a time shorter than the detection time (a few seconds).

    DISCUSSION
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Abstract
Introduction
Procedures
Results
Discussion
References

Two Binding Sites Completely Account for Inhibition and Reactivation of CF1-epsilon by TTX-- Two binding sites with different Kd values account for the inhibitory and reactivatory effects of TTX. The Kd values determined by 14C-TTX binding and kinetic experiments are compatible. They differ from those determined by Pinet et al. (23) through binding studies, probably because the conditions were critically different (buffer at 37 instead of 4 °C). We do not confirm or invalidate the existence of a third binding site of very low affinity recently proposed (27). However, we think that it should be considered very cautiously for different reasons. This third site was indeed revealed at TTX concentrations above 2 mM, where its monomeric character was not proved. Therefore, acquisition and modeling of few binding data in millimolar range, which led to an extrapolated Kd of 6.3 mM, seem very questionable. The authors also attribute the very strong level of reactivation (almost 2000% of the control, instead of the 200-300% generally obtained) to the filling of a third site. However, they used a nonactivated CF1, about 20-fold less active than DTT-treated CF1-epsilon . We confirmed this relative magnitude of reactivation of CF1 (Fig. 4b), but at 1 mM TTX the absolute activity of CF1 remains lower than the CF1-epsilon activity. Thus, this huge reactivation value may be due to the combination of the regular reactivation with a TTX-induced overcoming of the latent character of CF1 (21), and not to the existence of a third TTX-binding site on CF1.

The Dynamic State of CF1 Does Not Influence TTX Binding at the First Site-- We have quantitatively shown that TTX binding on the inhibitory site is identical, whether CF1 is activated or not. Mochimaru and Sakurai (27) also reported that partial trypsic digestion activation of CF1 had no effect on the rate of TTX binding on the low affinity site. Moreover, our results directly demonstrate that ATP addition and the catalytic turnover do not change the rate of TTX binding at the high affinity site. Consequently, there is no direct influence of the dynamic state of CF1 on the TTX binding on the first site. Conversely, it was previously shown that TTX binding on the inhibitory site modifies neither the affinity of isolated CF1 for ADP (25) nor the exchange of tightly bound nucleotide (29). All of these results may indicate that TTX does not inhibit CF1 by preventing its activation, normally achieved by the protonmotive force in the membrane-bound CF0CF1 and mimicked by various treatments in isolated CF1.

We also pointed out that the rate of exchange of TTX at its first binding site remains much lower than the turnover rate of the active enzyme; therefore, nonsaturating TTX concentrations lead to a mixture of fully inhibited and active enzymes. This explains why in thylakoids, partial inhibition by a low concentration of TTX did not change the Michaelis constant of ADP for ATP synthesis (33).

ATP Interferes in a Complex Way with TTX Binding at Its Second Site-- By contrast, ATP modifies the properties of the low affinity TTX binding site. We have developed a minimal model to account for all of our data. The main feature is the progressive change, during incubation with ATP, in the pattern of TTX-triggered reactivation (Fig. 5). It shifted from a biphasic kinetics with a transient overactivation (Fig. 6b) to a monotonous rise of activity (Fig. 6a). The simplest way to qualitatively explain this result is depicted in Scheme 1, where only enzymes bearing one or two molecules of TTX are represented.


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Scheme 1.   Three-state model of TTX binding to the low affinity site. T, tentoxin. ET and ET' are forms of the enzyme bearing one TTX molecule on the high affinity site. ET' is a dead end complex unable to bind a second TTX molecule. ETT is a complex bearing two TTX molecules, the only one that is active. Binding of ATP allows the transformation of ET into ET'. a, general case leading to biexponential kinetics. b, simplified case assuming quasi-equilibrium for TTX binding on the low affinity site and leading to monoexponential kinetics. In both cases, the top curve describes the theoretical time course of ATPase activity without ATP preincubation, and the bottom curve describes the theoretical time course of ATPase activity after ATP preincubation. c, predicted dependence of the apparent rate constant kapp of the exponential kinetics (case b) upon TTX concentration (Equation 13). See "Discussion" and "Appendix" for details.

In the presence of ATP, the ET form is slowly converted into a dead end complex, ET', unable to bind a second TTX molecule. The forward rate can be kinetically controlled either by ATP binding or by any further conformational change. TTX exchange at the low affinity site is much faster than the ATP-induced conversion (Scheme 1a). Thus, one may consider that this equilibrium is reached at any time and may restrict the analysis to the slow phase of the kinetics characterized by an apparent rate constant kapp (Scheme 1b, Equation 14). Without ATP, the ET complex can bind a second TTX molecule, leading to an ETT complex, the only state catalytically active.

When ATP is added to inhibited CF1-epsilon with or after reactivating concentrations of TTX, all of the enzyme is initially in equilibrium between the ET and ETT states. Therefore, a burst of activity is produced followed by a slow decay of activity until the equilibrium, involving the three forms, is reached. When inhibited CF1-epsilon is incubated with ATP a few minutes before the reactivating TTX addition, an equilibrium is reached between ET and ET'; after the TTX addition, the activity monotonously rises to the same equilibrium as before without giving rise to a transient overaccumulation of the ETT state. The kinetics observed in Fig. 6, a-b, are well described by this model, which also predicts that the slow apparent rate constant kapp decreases with TTX concentration (Scheme 1, bottom). This prediction conforms with the data obtained with preincubation of ATP, but not for those obtained without ATP preincubation for which the kapp was found almost independent on the TTX concentration (Fig. 6c). In order to account for all these results, this model has been refined into a final model depicted in Scheme 2. We kept the dead end complex, which is necessary to explain the kapp decrease in the case of ATP preincubation. But to explain the invariance of the kapp in the case of no ATP-preincubation, an irreversible and slow transformation of the enzymatic state (bearing either one or two toxins) has been introduced upstream from the states of Scheme 1, now included in the right part of Scheme 2. Moreover, to account for a transient overactivation observed even at saturating concentration of TTX (not shown), we had to introduce an overstimulated state.


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Scheme 2.   Multistate model of TTX binding to the low affinity site. Tentoxin binding sites are represented by notches. The left notch represents the inhibitory (high affinity) site, and the right notch shows the reactivatory (low affinity) site. When filled, they are, respectively, shaded in gray and black. For the loose site, the notch is an open triangle when TTX is slowly exchangeable and an open rectangle when TTX is quickly exchangeable. T, tentoxin. Left column, states of CF1-epsilon in the absence of ATP; right column, states of CF1-epsilon in the presence of ATP. State A is the active form of the enzyme catalyzing ATP hydrolysis; states I1, I2, and D are inhibited forms of the enzyme bearing one TTX molecule; states O and S are reactivated states bearing two TTX molecules. Two unidirectional arrows with a box, quasiequilibrium; single unidirectional arrow, irreversible conversion; two unidirectional arrows without a box, reversible conversion. Thermodynamic and kinetic constants indicated have been determined from the experiments. The catalytic activities normalized to the control (activity without TTX) are displayed inside the different complexes. See "Discussion" and "Appendix" for details.

In the absence of ATP, binding and release of TTX to and from the two sites are slow processes (TTX remains bound on these two sites during the rapid elution of CF1-epsilon on a column, data not shown). These facts were taken into consideration by introducing ATP-free enzymatic states (Scheme 2, left), well characterized by the binding studies (Fig. 1). However, since the addition of ATP generates in a rapid and irreversible way new enzymatic states characterized by fast TTX exchange rates on the loose site, the only states required to explain the kinetic experiments are the ATP-loaded states.

Let us consider the effect of TTX addition to inhibited CF1-epsilon . When ATP is added just before, simultaneously with, or after stimulatory concentrations of TTX to inhibited CF1-epsilon , the enzyme is first in rapid equilibrium between an overstimulated state (state O) and an inhibited state (state I). Then it experiences an irreversible and slow conversion into two states also in rapid equilibrium (named S and I2, respectively, and corresponding to the ETT and ET states described in Scheme 1). State I2 slowly equilibrates with the dead end state D (corresponding to ET' in Scheme 1). The global process gives rise to a slow decay of the activity, as observed in Fig. 6b, toward an equilibrium among three enzymatic forms. The kapp apparent for this decay (Fig. 6c) is about 0.5 min-1 (8 × 10-3 s-1). This value gives the forward rate constant of the O right-arrow S and I1 right-arrow I2 transitions. The rate constants are assumed to be the same for both conversions, in agreement with the fact that kapp does not depend on TTX concentration. Since the inhibited enzyme is subjected to the same slow conversion as the overstimulated one, this step does not seem to be directly correlated to the catalytic turnover.

After a long preincubation of the inhibited enzyme with ATP, the enzyme is in equilibrium between the inhibited state (state I2) and the dead end state (state D). The addition of stimulatory TTX induces, as in Scheme 1, a monotonous rise of the ATPase activity until the equilibrium between the stimulated, dead end, and inhibited enzymatic states is reached (Fig. 6a). The kapp of this monoexponential growth decreases with the TTX concentration (Fig. 6c, Equation 13). The data gave the Kd of TTX for the loose site and the forward and backward rate constants k+ and k- (Scheme 1). Due to the existence of the dead end complex, the Kd2 measured at equilibrium is only apparent and is a function of the true Kd, k+, and k- (Equation 15). From Fig. 6c and Equation 13, the true Kd is about 10 µM. Consequently, the apparent Kd2 should be about 30 µM (Equation 15), which is in accordance with the previous estimate in equilibrium (Kd2 = 40 µM; Ref. 32).

When TTX at reactivating concentrations was added directly to the active complex (without TTX on the tight site), the kinetic profiles did not depend on the preincubation of the enzyme with ATP (data not shown). It consisted in all cases in a fast overstimulation followed by a slow decay. This is easily explained if we assume that the active enzyme (state A) does not experience a transition like states I1 and O. After the addition of high concentrations of TTX, with or after ATP, the enzyme immediately goes, through the inhibited states, to the overstimulated state (state O). Then it goes slowly to the stimulated one (state S) and reaches an equilibrium between states D (dead end), S (stimulated), and C (inhibited).

Structural Considerations-- The presently available data do not give a clear cut structural explanation of our results. First, a high resolution structure of CF1 is still lacking. Second, there is only indirect evidence (24, 26) that beta -Asp-832 takes part in the TTX binding site. However, for illustrating the TTX effects in relation to the CF1 structure, we can take for granted that the TTX binding site is located in this region and that the structure of this well conserved domain is equivalent in CF1 and MF1, although MF1 is TTX-insensitive. A survey of the structure of MF1 (10) indicates that this region, located at an alpha /beta interface in the vicinity of the upper beta -barrel of the beta -subunit, corresponds to a pocket whose opening critically depends on the nucleotide occupancy. The alpha /beta pair comprising the adenosine triphosphate-loaded beta -subunit (chains B and F in Protein Data Bank structure 1cow) presents the most favorable conformation of this region for TTX binding. This domain is rich in aromatic residues. One of them, alpha -Tyr-2922, located in a homologous sequence, FYLH, is a good candidate to stack with the methyl-Phe((Z)Delta ) residue of TTX. We have shown that the characteristics of TTX binding on the first site do not depend on the catalytic state of the enzyme. This site is certainly not subjected to turnover-dependent changes of exposure. Considering a stepped rotative mechanism of CF1 (34), the probability for TTX to bind the most favored site is indeed the same in resting and dynamic states of the enzyme.

The asymmetry of the complex, and thus of the location of bound TTX, could also explain the differential effect of TTX on ATP synthesis and hydrolysis under single site conditions (29). In these conditions, the enzyme does not run a complete revolution during the experiment, and thus the effect of TTX could depend on the direction of rotation. In the direction of ATP synthesis, TTX could block the first step(s) of the rotation and would therefore inhibit the unisite catalysis. In the other direction, TTX could block more distal step(s) of the rotation, allowing the release of the ADP molecule, without apparent effect on the unisite hydrolysis. This hypothesis could also account for the ATP-dependent reactivatory behavior of TTX. In the presence of ATP, inhibited forms of CF1-epsilon would carry out a fraction of turn and thus modify the relative position of TTX. Therefore, once a first molecule of TTX is bound, the conformations of the alpha /beta sites and of the whole structure of CF1-epsilon would depend on the presence of ATP. This conformational variation would modify the binding properties of the second TTX molecule and explain why, in the presence of ATP, CF1-epsilon does not pass through the overstimulated form observed in the absence of ATP. The present discussion shows that the microscopic states, here revealed by kinetic experiments, should be taken into account in the further investigations of the catalytic mechanism of the ATP synthase.

    ACKNOWLEDGEMENTS

Thanks are due to Véronique Mary for the extraction of the spinach chloroplast F1-ATPase. 14C-TTX was provided by Drs. Jean-Marie Gomis and Jean-Pierre Noel (Service des Molécules Marquées, Commissariat à l'Energie Atomique-Saclay).

    FOOTNOTES

* This work was supported by Ministère de l'Enseignement Supérieur et de la Recherche Contract ACC-SV5 (interface Chimie-Physique-Biologie) no. 9505221 and by CNRS grant (Physique-Chimie du Vivant) no. 97N21/0122.The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

§ To whom correspondence should be addressed. Tel.: 33 1 69 08 44 32; Fax: 33 1 69 08 87 17; E-mail: santo{at}dsvidf.cea.fr.

The abbreviations used are: TTX, tentoxin or cyclo-(L-N-methyl-Ala1-L-Leu2-N-MeDelta ZPhe3-Gly4); CF1, chloroplast F1 H+-ATPase; CF1-epsilon , chloroplast F1 H+-ATPase devoid of epsilon -subunit; DTT, dithiothreitol; 14C-TTX, 14C-methyl-Phe((Z)Delta )-tentoxin.

2 In CF1 primary sequence numbering.

    APPENDIX

Kinetics of ATPase Inhibition by TTX

The binding of TTX to the high affinity site is described as follows,
E+<UP>T</UP> <LIM><OP><ARROW>⇄</ARROW></OP><LL>k<SUB><UP>off</UP></SUB></LL><UL>k<SUB><UP>on</UP></SUB></UL></LIM> E<UP>T</UP> (Eq. 1)
E is the active form, and ET is the inhibited one. When the toxin is in excess with respect to the enzyme, the rate of ATP hydrolysis as a function of time is as follows,
V<SUB>t</SUB>−V<SUB><UP>eq</UP></SUB>=(V<SUB>0</SUB>−V<SUB><UP>eq</UP></SUB>)e<SUP><UP>−</UP>k<SUB>app</SUB>t</SUP> (Eq. 2)
with
k<SUB>app</SUB>=k<SUB><UP>on</UP></SUB>[<UP>T</UP>]+k<SUB><UP>off</UP></SUB> (Eq. 3)
where V0 and Veq are, respectively, the initial and final rates of ATP hydrolysis.

The instantaneous concentration of ATP can be deduced from Equation 2
[<UP>ATP</UP>]<SUB>t</SUB>−[<UP>ATP</UP>]<SUB>0</SUB>=A(e<SUP><UP>−</UP>k<SUB>app</SUB>t</SUP>−1)+Bt (Eq. 4)
where
A=<FR><NU>V<SUB><UP>eq</UP></SUB>−V<SUB>0</SUB></NU><DE>k<SUB>app</SUB></DE></FR> <UP>and</UP> B=V<SUB><UP>eq</UP></SUB> (Eq. 5)
A, B, and kapp were derived by fitting the experimental data (Fig. 3a) to Equation 4. kon can be derived from kapp using Equation 3.

Inhibition by the First TTX Molecule at Equilibrium and Kd1 Determination

To fit the curve of activity (V) versus TTX concentration, we account here for the concentration of enzyme, which is not negligible, and for a possible activity of the ET complex. The activity is described as follows,
V=V<SUB>0</SUB>&agr;+V<SUB>1</SUB>(1−&agr;) (Eq. 6)
where alpha  is the fraction of the noninhibited enzyme, V0 is the activity of E, and V1 is the activity of ET. In a noncompetitive model, alpha  can be expressed as follows,
&agr;=<FR><NU>([E]<SUB>0</SUB>−K<SUB>d1</SUB>−[<UP>T</UP>]<SUB>0</SUB>)+<RAD><RCD>([E]<SUB>0</SUB>−K<SUB>d1</SUB>−[<UP>T</UP>]<SUB>0</SUB>)<SUP>2</SUP>+4[E]<SUB>0</SUB>K<SUB>d1</SUB></RCD></RAD></NU><DE>2[E]<SUB>0</SUB></DE></FR> (Eq. 7)
where Kd1 is the dissociation constant of the ET complex and [E]0 and [T]0 are the total concentrations of enzyme and TTX. For a noncompetitive inhibition, Kd1 is not substrate concentration-dependent.

Reactivation by the Second TTX Molecule at Equilibrium and Kd2 Determination

At high concentrations of TTX, the concentration of TTX-free enzyme is negligible. Therefore, the activity versus concentration curve is represented as follows,
V=<FR><NU>V<SUB>1</SUB>[E<UP>T</UP>]</NU><DE>[E<SUB>0</SUB>]</DE></FR>+<FR><NU>V<SUB>2</SUB>[E<UP>T</UP>]</NU><DE>[E<SUB>0</SUB>]</DE></FR>=<FR><NU>V<SUB>1</SUB></NU><DE>1+<FR><NU>[<UP>T</UP>]</NU><DE>K<SUB>d2</SUB></DE></FR></DE></FR>+<FR><NU>V<SUB>2</SUB></NU><DE>1+<FR><NU>K<SUB>d2</SUB></NU><DE>[<UP>T</UP>]</DE></FR></DE></FR> (Eq. 8)
where V1 is the activity of ET complex, V2 is the activity of ETT complex, and Kd2 is its dissociation constant. In the presence of a large excess of toxin, [T] can be taken as [T] = [T]0.

Three-state Kinetics of Reactivation by TTX at High Concentration (Scheme 1)

The following three states are considered: ET, the enzyme bearing one TTX molecule; ETT, the enzyme bearing two TTX molecules; and ET', a dead end complex, reversibly formed from ET in the presence of ATP and unable to bind a second TTX molecule. This system is displayed in Scheme 1.

Scheme 1b describes a rapid equilibrium between states ET and ETT, which is formalized by the equation,
K−<FR><NU>[E<UP>T</UP>][<UP>T</UP>]</NU><DE>[E<UP>TT</UP>]</DE></FR> (Eq. 9)
where K is the equilibrium dissociation constant.

Scheme 1b also displays a slow equilibrium between states ET and ET' characterized by the forward and backward rate constants k+ and k-. The evolution of the whole system is described by the following differential equation.
<FR><NU><UP>d</UP>([E<UP>T</UP>]+[E<UP>TT</UP>])</NU><DE><UP>d</UP>t</DE></FR>−<UP>−</UP>k<SUP><UP>+</UP></SUP>[E<UP>T</UP>]+k<SUP><UP>−</UP></SUP>[E<UP>T</UP>′] (Eq. 10)
Equation 10 may be written as follows,
<FR><NU><UP>d</UP>([E<UP>T</UP>]+[E<UP>TT</UP>])</NU><DE><UP>d</UP>t</DE></FR>+<FENCE><FR><NU>k<SUP><UP>+</UP></SUP></NU><DE>1+<FR><NU>[<UP>T</UP>]</NU><DE>K</DE></FR></DE></FR>+k<SUP><UP>−</UP></SUP></FENCE>([E<UP>T</UP>]+[E<UP>TT</UP>])−k<SUP><UP>−</UP></SUP>[E]<SUB><UP>tot</UP></SUB>=0 (Eq. 11)
where [E]tot is the total enzyme concentration. [ET] + [ETT] is an exponential function of time, as follows,
y<SUB>t</SUB>−y<SUB><UP>eq</UP></SUB>=[y<SUB>0</SUB>−y<SUB><UP>eq</UP></SUB>]e<SUP><UP>−</UP>k<SUB>app</SUB>t</SUP> (Eq. 12)
with
k<SUB>app</SUB>=<FR><NU>k<SUP><UP>+</UP></SUP></NU><DE>1+<FR><NU>[<UP>T</UP>]</NU><DE>K</DE></FR></DE></FR>+k<SUP><UP>−</UP></SUP> (Eq. 13)
y0 and yeq are the initial and equilibrium values of the sum of [ET] + [ETT], respectively. ETT is here assumed to be the only active form. Since [ETT] is a constant fraction of [ET] + [ETT], it obeys the same exponential law. The rate of ATP hydrolysis is proportional to [ETT] at any time. It can then be written as follows.
[E<UP>TT</UP>]<SUB>t</SUB>−[E<UP>TT</UP>]<SUB><UP>eq</UP></SUB>=([E<UP>TT</UP>]<SUB>0</SUB>−[E<UP>TT</UP>]<SUB><UP>eq</UP></SUB>)e<SUP><UP>−</UP>k<SUB>app</SUB>t</SUP> (Eq. 14)
with
[E<UP>TT</UP>]<SUB><UP>eq</UP></SUB>=<FR><NU>[E]<SUB><UP>tot</UP></SUB></NU><DE>1+<FENCE>1+<FR><NU>k<SUP><UP>+</UP></SUP></NU><DE>k<SUP><UP>−</UP></SUP></DE></FR></FENCE><FR><NU>K</NU><DE>[<UP>T</UP>]</DE></FR></DE></FR> (Eq. 15)
Equation 14 shows that the time course of the activity depends on the initial concentration of ETT, which is itself determined by the time of ATP preincubation.

Two extreme situations can be distinguished in Scheme 1.

Situation 1-- If TTX is added at reactivating concentration a long time after ATP, all forms are present upon TTX addition. The activity, proportional to [ETT]0, is initially low and then increases to reach its equilibrium value.

Situation 2-- Without incubation with ATP, ET and ETT are the only forms initially present. The activity starts at its maximal level and decreases to its equilibrium value.

Equation 13, which gives kapp as a function of [TTX] (Scheme 1c) applies to both situations.

The final multistate model (Scheme 2) will be developed from Scheme 1b, which is a simplified form of Scheme 1a in which the equilibrium between the ET and ETT states is instantaneously reached.

Multistate Kinetics of Reactivation at High Concentrations of TTX (Scheme 2)

In Scheme 2, the enzyme may bear zero, one, or two TTX molecules. The left three ATP-free states will not be considered here. When ATP is added, the substates initially present are forms A (active, no TTX bound), I1 (inhibited, one TTX molecule bound), and O (overstimulated, two TTX molecules bound). States I1 and O are in rapid equilibrium.

Situation 1-- The incubation of the inhibited enzyme with ATP for a long time directly leads to the formation of states I2 and D (ET and ET' in Scheme 1). The addition of reactivating TTX leads to a situation quite identical to that described in Scheme 1, with the same prediction, an activity initially low that increases to reach its equilibrium value, with a rate constant, kapp, defined in Equation 13.

Situation 2-- When ATP is added after reactivating TTX or, at the same time, the enzyme is initially under states I1 and O (state A can be neglected), the system then evolves toward the formation of states S, I2, and D. The quasi-equilibrium between states I1 and O is described by the dissociation constant K0 as follows.
K<SUB>0</SUB>=<FR><NU>[<UP>I</UP><SUB>1</SUB>][<UP>T</UP>]</NU><DE>[<UP>O</UP>]</DE></FR> (Eq. 16)
If [X] = [I1] + [O] and if the irreversible conversion of states I1 and O, respectively, into states I2 and S occurs with the same rate constant ka, the evolution of [X] is given by the equation,
<FR><NU><UP>d</UP>[<UP>X</UP>]</NU><DE><UP>d</UP>t</DE></FR>=<UP>−</UP>k<SUB>a</SUB>[<UP>X</UP>] (Eq. 17)
where
[<UP>X</UP>]<SUB>t</SUB>=[<UP>X</UP>]<SUB>0</SUB>e<SUP><UP>−</UP>k<SUB>a</SUB>t</SUP> (Eq. 18)
Likewise, if [Y] = [I2] + [S], where the following is true,
[<UP>I</UP><SUB>2</SUB>]=<FR><NU>[<UP>Y</UP>]</NU><DE>1+<FR><NU>[<UP>T</UP>]</NU><DE>K</DE></FR></DE></FR> (Eq. 19)
the evolution of [Y] is given as follows.
<FR><NU><UP>d</UP>[<UP>Y</UP>]</NU><DE><UP>d</UP>t</DE></FR>=<UP>−</UP>k<SUP><UP>+</UP></SUP>[<UP>I</UP><SUB>2</SUB>]+k<SUP><UP>−</UP></SUP>[<UP>D</UP>]+k<SUB>a</SUB>[<UP>X</UP>] (Eq. 20)
Using Equation 19, Equation 20 becomes the following,
<FR><NU><UP>d</UP>[<UP>Y</UP>]</NU><DE><UP>d</UP>t</DE></FR>=<UP>−</UP><FR><NU>k<SUP><UP>+</UP></SUP></NU><DE>1+<FR><NU>[<UP>T</UP>]</NU><DE>K</DE></FR></DE></FR> [<UP>Y</UP>]+k<SUP><UP>−</UP></SUP>[<UP>D</UP>]+k<SUB>a</SUB>[<UP>X</UP>] (Eq. 21)
where K, k+, and k- have the same definitions as in Scheme 1. Since the following is true,
[E]<SUB><UP>tot</UP></SUB>=[<UP>X</UP>]+[<UP>Y</UP>]+[<UP>D</UP>] (Eq. 22)
Equation 21 becomes the following.
<FR><NU><UP>d</UP>[<UP>Y</UP>]</NU><DE><UP>d</UP>t</DE></FR>+<FR><NU>k<SUP><UP>+</UP></SUP></NU><DE>1+<FR><NU>[<UP>T</UP>]</NU><DE>K</DE></FR></DE></FR> [<UP>Y</UP>]=k<SUP><UP>−</UP></SUP>([E]<SUB><UP>tot</UP></SUB>−[<UP>Y</UP>]−[<UP>X</UP>])+k<SUB>a</SUB>[<UP>X</UP>] (Eq. 23)
The initial conditions are [X]0 = [E]tot and [Y]0 = 0. Thus, Equation 23 becomes the following.
<FR><NU><UP>d</UP>[<UP>Y</UP>]</NU><DE><UP>d</UP>t</DE></FR>+<FENCE><FR><NU>k<SUP><UP>+</UP></SUP></NU><DE>1+<FR><NU>[<UP>T</UP>]</NU><DE>K</DE></FR></DE></FR>+k<SUP><UP>−</UP></SUP></FENCE>[<UP>Y</UP>]=k<SUP><UP>−</UP></SUP>[E]<SUB><UP>tot</UP></SUB>+(k<SUB>a</SUB>−k<SUP><UP>−</UP></SUP>)[E]<SUB><UP>tot</UP></SUB>e<SUP><UP>−</UP>k<SUB>a</SUB>t</SUP> (Eq. 24)
The solution of this equation is as follows,
[<UP>Y</UP>]<SUB>t</SUB>=[<UP>Y</UP>]<SUB>1</SUB>e<SUP><UP>−</UP>k<SUB>1</SUB>t</SUP>+[<UP>Y</UP>]<SUB>2</SUB>e<SUP><UP>−</UP>k<SUB>2</SUB>t</SUP>+[<UP>Y</UP>]<SUB>3</SUB> (Eq. 25)
with
k<SUB>1</SUB>−k<SUP><UP>−</UP></SUP>+<FR><NU>k<SUP><UP>+</UP></SUP></NU><DE>1+<FR><NU>[<UP>T</UP>]</NU><DE>K</DE></FR></DE></FR> (Eq. 26)
and
k<SUB>2</SUB>=k<SUB>a</SUB> (Eq. 27)
[<UP>Y</UP>]<SUB>3</SUB>=<FR><NU>[E]<SUB><UP>tot</UP></SUB></NU><DE>1+<FR><NU>k<SUP><UP>+</UP></SUP></NU><DE>k<SUP><UP>−</UP></SUP><FENCE>1+<FR><NU>[<UP>T</UP>]</NU><DE>K</DE></FR></FENCE></DE></FR></DE></FR> (Eq. 28)
[<UP>Y</UP>]<SUB>2</SUB>=<FR><NU>[E]<SUB><UP>tot</UP></SUB></NU><DE><FR><NU>k<SUP><UP>+</UP></SUP></NU><DE>(k<SUB>a</SUB> k<SUP><UP>−</UP></SUP>)<FENCE>1+<FR><NU>[<UP>T</UP>]</NU><DE>K</DE></FR></FENCE></DE></FR>−1</DE></FR> (Eq. 29)
and
[<UP>Y</UP>]<SUB>1</SUB>=<UP>−</UP>[<UP>Y</UP>]<SUB>3</SUB>−[<UP>Y</UP>]<SUB>2</SUB> (Eq. 30)
If VOver is the activity of state O and VStim is the activity of state S, the global ATPase activity V as a function of time is Vt = VOver[O] + VStim[S], which gives the equation,
V<SUB>t</SUB>=<FR><NU>V<SUB><UP>Over</UP></SUB>[<UP>X</UP>]<SUB>t</SUB></NU><DE>1+<FR><NU>K<SUB>0</SUB></NU><DE>[<UP>T</UP>]</DE></FR></DE></FR>+<FR><NU>V<SUB><UP>Stim</UP></SUB>[<UP>Y</UP>]<SUB>t</SUB></NU><DE>1+<FR><NU>K</NU><DE>[<UP>T</UP>]</DE></FR></DE></FR> (Eq. 31)
or
V<SUB>t</SUB>=<FENCE><FR><NU>V<SUB><UP>Over</UP></SUB>[E]<SUB><UP>tot</UP></SUB></NU><DE>1+<FR><NU>K<SUB>0</SUB></NU><DE>[<UP>T</UP>]</DE></FR></DE></FR>+<FR><NU>V<SUB><UP>Stim</UP></SUB>[<UP>Y</UP>]<SUB>2</SUB></NU><DE>1+<FR><NU>K</NU><DE>[<UP>T</UP>]</DE></FR></DE></FR></FENCE>e<SUP><UP>−</UP>k<SUB>2</SUB>t</SUP>+<FR><NU>V<SUB><UP>Stim</UP></SUB>[<UP>Y</UP>]<SUB>1</SUB></NU><DE>1+<FR><NU>K</NU><DE>[<UP>T</UP>]</DE></FR></DE></FR> e<SUP><UP>−</UP>k<SUB>1</SUB>t</SUP>+<FR><NU>V<SUB><UP>Stim</UP></SUB>[<UP>Y</UP>]<SUB>3</SUB></NU><DE>1+<FR><NU>K</NU><DE>[<UP>T</UP>]</DE></FR></DE></FR> (Eq. 32)
The model is described by the sum of two exponential decays characterized by two distinct rate constants: k1, which is [TTX]-dependent (Equation 24), and k2, which is not (Equation 25).

In Situation 2, we took into account the kinetic data collected after 1 min in order to focus on the slow component of the decay (Fig. 6b). The rate constant that characterized this decay proved to be [TTX]-independent (Fig. 6c) and was thus identified as ka.
    REFERENCES
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Abstract
Introduction
Procedures
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References

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