Binding of a Single Rb+ Increases Na+/K+-ATPase, Activating Dephosphorylation without Stoichiometric Occlusion*

We used partially purified Na+/K+-ATPase from pig kidney to study dephosphorylation, occlusion, and ATPase activity in the same enzyme preparation and in media of identical composition containing 10 μm ATP and different concentrations of Rb+, used as a K+ congener. The experiments were performed using a rapid-mixing apparatus with a time resolution of 3.5 ms. The main findings were as follows. (i) At sufficiently low Rb+ concentration the initial rate of dephosphorylation was higher than that of occlusion, (ii) as [Rb+] tended to zero the slope of the time course of occlusion but not that of the time course of dephosphorylation approached zero and, (iii) as Rb+ concentration increased, ATPase activity first increased and, after passing through a maximum, tended to a value that was lower than that observed in media without Rb+. None of these results is compatible with the currently held idea that binding of a single Rb+ to the E2P conformer of the ATPase does not modify the rate of dephosphorylation and strongly suggest that a single Rb+ does promote dephosphorylation through a mechanism that is not stoichiometrically coupled to Rb+ occlusion. If this mechanism is included in the currently accepted scheme for ATP hydrolysis by the Na+/K+-ATPase, a reasonable prediction of the experimental results is obtained.

We used partially purified Na ؉ /K ؉ -ATPase from pig kidney to study dephosphorylation, occlusion, and ATPase activity in the same enzyme preparation and in media of identical composition containing 10 M ATP and different concentrations of Rb ؉ , used as a K ؉ congener. The experiments were performed using a rapidmixing apparatus with a time resolution of 3.5 ms. The main findings were as follows. (i) At sufficiently low Rb ؉ concentration the initial rate of dephosphorylation was higher than that of occlusion, (ii) as [Rb ؉ ] tended to zero the slope of the time course of occlusion but not that of the time course of dephosphorylation approached zero and, (iii) as Rb ؉ concentration increased, ATPase activity first increased and, after passing through a maximum, tended to a value that was lower than that observed in media without Rb ؉ . None of these results is compatible with the currently held idea that binding of a single Rb ؉ to the E 2 P conformer of the ATPase does not modify the rate of dephosphorylation and strongly suggest that a single Rb ؉ does promote dephosphorylation through a mechanism that is not stoichiometrically coupled to Rb ؉ occlusion. If this mechanism is included in the currently accepted scheme for ATP hydrolysis by the Na ؉ /K ؉ -ATPase, a reasonable prediction of the experimental results is obtained.
During its physiological operation, the plasma membrane Na ϩ /K ϩ -ATPase couples the hydrolysis of ATP to the transport of three intracellular sodium ions in exchange for two extracellular potassium ions. According to the currently accepted scheme of ATP hydrolysis, this takes place in a series of steps that include at least ( Fig. 1) (i) the (Na ϩ ϩ Mg 2ϩ )-dependent phosphorylation by ATP of the E 1 conformer of the enzyme to form the E 1 P phosphoenzyme (phosphorylation requires the binding of three Na ϩ ions that are taken up from the cytosol and trapped in E 1 P), (ii) the E 1 P 3 E 2 P transition of the phosphoenzyme (this is accompanied by the release of Na ϩ to the extracellular medium), (iii) the activation by extracellular K ϩ of the dephosphorylation of E 2 P (it is generally accepted that this requires the binding of two K ϩ ions that are taken up from the extracellular medium and trapped in E 2 ), and finally (iv), the return of E 2 to E 1 with the release K ϩ into the cytosol. The binding of ATP to a noncatalytic site in E 2 whose K m (Ϸ200 M) is at least 10 3 times higher than the K m of the active site of the ATPase (Ϸ0.2 M) leads to a 100-fold increase in the rate of the E 2 (K 2 ) 3 E 1 ϩ 2K ϩ reaction. Hence, as the concentration of ATP tends to zero the release of K ϩ becomes the slowest step of the Na ϩ /K ϩ -ATPase cycle (see Refs. 1 and 2). The Na ϩ and K ϩ taken up by the enzyme in steps (i) and (iii) are in a state in which they exchange slowly with the incubation media. This is usually called "occlusion" and is considered to be the expression of an intermediate step in the movement of Na ϩ and K ϩ across the ATPase from one surface of the membrane to the other. No direct structural information with enough resolution is yet available to identify the occlusion domains in the enzyme.
Phosphorylation is absolutely dependent on Na ϩ , whereas for activation of dephosphorylation K ϩ can be replaced with varying degrees of effectiveness by Rb ϩ , Cs ϩ , Li ϩ , NH 4 ϩ , or Tl ϩ . At least Rb ϩ (3) or Tl ϩ (4) also replaces K ϩ for occlusion. Dephosphorylation persists in the absence of K ϩ , albeit at a rate that is 100ϫ slower than in the presence of K ϩ . This gives rise to an ATPase activity, called the Na ϩ -ATPase, which requires Na ϩ and Mg 2ϩ and is coupled to the transport of Na ϩ . Na ϩ -ATPase is switched off as K ϩ or its congeners drive dephosphorylation toward the pathway that leads to occlusion. The model in Fig. 1 has three branching points at the E 2 P states that lead to three pathways. The velocity of dephosphorylation in the absence of products (v dephos ) will be the sum of the contributions of the three pathways, i.e. v dephos ϭ k 40 ͓E 2 P͔ ϩ k 41 ͓E 2 PK͔ ϩ k 42 ͓E 2 PK 2 ͔ (Eq. 1) The relative abundance of each state of E 2 P will depend on the concentration of K ϩ . Assuming rapid equilibrium for the addition of K ϩ , this dependence will be where K K1 and K K2 are equilibrium dissociation constants and E 2 P 0 ϭ [E 2 P] ϩ [E 2 PK] ϩ [E 2 PK 2 ]. E 2 P 0 will be equal to the concentration of E 2 P in the absence of K ϩ when all the enzyme is catalyzing Na ϩ -ATPase activity. Equations 2-4 illustrate the more general fact that, as [K ϩ ] goes from zero to infinity, the following will occur. (i) [E 2 P] will decrease continuously toward zero (Equation 2), and the same will happen with Na ϩ -ATPase activity. (ii) [E 2 PK] will start at zero, pass through a maximum, and then tend to zero (Equation 3). If we accept the general view that k 40 ϭ k 41 , then this process, which is usually not explicitly included in the reaction schemes of the Na ϩ /K ϩ -ATPase, will have no expression on v dephos . (iii) [E 2 PK 2 ] will start at zero and tend to saturation at a value equal to the total amount of E 2 P along an S-shaped curve with zero initial slope (Equation 4).
The existence at non-saturating [K ϩ ] of pathways not requiring occlusion means that the rate of occlusion will be lower than the ATPase activity, a difference that will be cancelled at saturating [K ϩ ] when all the reaction flow passes through the E 2 (K 2 ) 3 E 1 ϩ 2K ϩ step (cf. Equations 1 and 4).
We have shown (5) that if we assume that k 41 ϭ k 40 , then the pathways not involving occlusion are too slow to explain why deocclusion rate is slower than ATPase activity at nonsaturating [K ϩ ]. We also showed that the most economical way to solve this discrepancy is to abandon the restriction that k 41 ϭ k 40 and to posit that k 41 Ͼ k 40 ; that is, that the binding of a single K ϩ to E 2 P increases the rate of dephosphorylation. Moreover, we proposed that dephosphorylation of E 2 PK does not involve occlusion of the ion. The experiments in this paper are an attempt to obtain direct evidence for this hypothesis. For this, we looked at the effects of Rb ϩ as congener of K ϩ on the initial rates of occlusion and of dephosphorylation and on the ATPase activity in enzymes incubated at the same temperature and in media of identical composition.
The experiments were designed to test the following predictions. (i) If occlusion were strictly dependent on the binding of two Rb ϩ per E 2 P, then the function relating the rate of occlusion to Rb ϩ concentration should be a sigmoid curve of zero initial slope (see Equation 4). (ii) If acceleration of dephosphorylation took place after the binding of a single Rb ϩ to E 2 P, then dephosphorylation rate should increase with [Rb ϩ ] along a function of positive initial slope. (iii) If (i) and (ii) were fulfilled, then at sufficiently low [Rb ϩ ] the velocity of dephosphorylation should be higher than that of occlusion. (iv) At micromolar [ATP], ATPase activity versus [Rb ϩ ] should first increase above the value of the Na ϩ -ATPase as a consequence of activation by a single Rb ϩ and, after passing through a maximum, tend to a lower value as [Rb ϩ ] reaches saturation. This is so because it is known that under these conditions the E 2 (K 2 ) 3 E 1 ϩ 2K ϩ step is slower than any other.
We further analyzed the consequences of our hypothesis by simulating the behavior of the model in Fig. 1 to which we added the pathway enclosed within the shaded area. E x was included in the reaction scheme to evaluate whether the dephosphorylation of E 2 PK leads to occlusion. To perform this we fitted analytical solutions of the model to the experimental data of the effects of [Rb ϩ ] on the steady-state levels of ATPase activity, EP, and occluded Rb ϩ , measured in media of identical composition and temperature as those used to measure the rates of dephosphorylation and occlusion. The values of the kinetic parameters thus obtained were used to predict the effects of Rb ϩ on the above mentioned rates.
Results reported here constitute the first detailed study of the time course of Rb ϩ occlusion through the physiological route. This allowed us also for the first time to correlate the kinetics of occlusion with that of dephosphorylation. A preliminary report of some of the results presented in this work has been published (6).

EXPERIMENTAL PROCEDURES
Na ϩ /K ϩ -ATPase was partially purified from pig kidney outer medulla according to Jensen et al. (7) and kindly provided by the Department of Biophysics, University of Århus, Denmark. All the results shown in this paper were obtained from experiments performed at 25°C with enzyme suspended in reaction media containing 150 mM NaCl, 10 M ATP, 0.7 mM MgCl 2 , 0.2 mM EDTA, and 25 mM imidazole-HCl (pH 7.4 at 25°C) and the concentrations of Rb ϩ indicated in the figures.
The time courses of dephosphorylation and of occlusion were measured in experiments carried out using a rapid-mixing apparatus, SFM4/Q, developed by Bio-Logic (France). The experiments were started adding the desired concentrations of Rb ϩ to the incubation medium in which the enzyme (final concentration 40 -50 g protein/ ml) performed steady-state Na ϩ -ATPase activity. The time course of dephosphorylation was measured by monitoring the transition of the 32 P-labeled phosphoenzyme to a new steady state after the addition of enough of RbCl as to attain the concentrations indicated in the legends to the figures. Reactions were stopped by chemical quenching with trichloroacetic acid (final concentration 10 g/100 ml) as described by Schwarzbaum et al. (8). The time course of occlusion was measured using [ 86 ]Rb ϩ . Because acid denaturation by chemical quenching releases occluded Rb ϩ , occlusion was measured after stopping the reaction by rapid cooling-and-washing following the procedure developed in our laboratory (9). This procedure, which preserves the structural integrity of the enzyme, is able to stop deocclusion reactions that proceed with rate constants of up to 25 s Ϫ1 without significant loss of Rb ϩ (9).
The initial rates of Rb ϩ -dependent dephosphorylation and of Rb ϩ occlusion were calculated from (i) the solution at t ϭ 0 of the first derivative of the functions fitted to the complete time courses (Fig. 2), (ii) the slope of the straight lines fitted to data of occluded Rb ϩ and of phosphoenzyme concentrations measured at the time of addition of Rb ϩ (t ϭ 0) and after a 93-ms-long incubation. This alternative ("singletime measurements"), which was particularly convenient to increase the precision of the comparison between the initial rates of occlusion and dephosphorylation, allowed a higher density of data at the lower Rb ϩ concentrations (Fig. 3, B-E), when the rates were small enough to fulfill initial rate conditions after 93 ms.
The steady-state concentration of occluded Rb ϩ was measured by Occluded K ϩ is shown between parentheses. The scheme does not show the addition and release of Na ϩ , K ϩ , and Mg 2ϩ . The lowercase ks are the rate constants of the elementary steps indicated by neighboring arrows. K ATP is the equilibrium constant for the dissociation of ATP from E 2 (K 2 )ATP. K K1 and K K2 have the same meaning for the dissociation of K ϩ from E 2 PK and E 2 PK 2 , respectively. Analyze the consequences of binding of a single K ϩ to E 2 P was made explicit in the pathway enclosed within the shaded area. E x was included to test the possibility of occlusion after dephosphorylation of E 2 PK. The steps governed by k 50 and k 5A were considered irreversible because of the high concentration of Na ϩ (150 mM) used in the experiments. Phosphorylation and dephosphorylation are irreversible because of the lack of products.
mixing in the rapid-mixing apparatus equal volumes of a suspension of Na ϩ /K ϩ -ATPase (80 -100 g of protein/ml) in reaction media with the same media containing 20 M ATP and different concentrations of 86 Rb ϩ . Reaction mixtures were incubated for 3.5 s before being injected into the cooling-and-washing chamber. Control experiments using aging times from 3 to 10 s showed no variation in the amounts of occluded rubidium, indicating that 3.5 s was enough to reach steady state.
The steady-state concentration of phosphoenzyme was measured by the same procedure as that used for occlusion except that [␥-32 P]ATP was used. After 5.6 s, the reaction was quenched by mixing 3 volumes of the reaction mixture with 2 volumes of an ice-cold solution of 25 g/100 ml trichloroacetic acid and 50 mM H 3 PO 3 .
ATPase activity was determined according to Schwarzbaum et al. (8), measuring the amount of [ 32 P]P i released from [␥-32 P]ATP. The reaction medium was identical as that used for the occlusion and dephosphorylation experiments. Enzyme concentration was 10 g of protein/ ml. Incubation times varied between 5 and 40 s to avoid the hydrolysis of more than 10% of the ATP, thus ensuring initial rate conditions. Blanks were measured in the media where all Na ϩ was replaced by K ϩ .
Theoretical equations were adjusted to the results by weighted nonlinear regression based on the Gauss-Newton algorithm using commercial programs (Excel, Sigma-Plot, and Mathematica for Windows). Weighting factors were calculated as the reciprocal of the variance of experimental data.
[␥-32 P]ATP was either purchased as such or synthesized using the procedure of Glynn and Chappel (10), except that no unlabeled orthophosphate (P i ) was added. Carrier-free [ 32 P]P i , [ 86 Rb]RbCl, and [␥-32 P]ATP were from PerkinElmer Life Sciences. ATP, enzymes, and reagents for the synthesis of [␥-32 P]ATP were from Sigma. All other reagents were of analytical grade.

Comparison between the Initial Rates of Dephosphorylation and
Occlusion-We measured the time courses of dephosphorylation and of Rb ϩ occlusion in parallel experiments using the same enzyme preparation in media of the same composition and temperature. All time courses were determined after the addition of Rb ϩ to enzymes that perform steady-state Na ϩ -ATPase activity. Results are shown in Fig. 2,  A (dephosphorylation) and B (Rb ϩ occlusion).
We calculated the initial velocities of Rb ϩ -dependent dephosphorylation (v dephos,Rb where ␣ is a coefficient whose value will be either 1 or 0 depending on whether or not E 2 PRb leads to the occlusion of a single Rb ϩ . Note that, under the conditions of our experiments, the initial rate of Rb ϩ -depend-   Fig. 1 corresponding, respectively, to conditions (i), (ii), and (iii) and the values of the rate constants given in Table 3. ent dephosphorylation will be the difference between the rates of breakdown and of formation of EP. Because before the addition of Rb ϩ the enzyme performs Na ϩ -ATPase activity in steady state, the rate of dephosphorylation will be exactly balanced by that of phosphorylation. For this reason the term Ϫk 40 E 2 P 0 appears in Equation 5.
If each state of E 2 P is expressed as in Equations 2-4, then at very low [Rb ϩ ], Equations 5 and 6 can, respectively, be approximated by v dephos,Rb and v occ 0 Ϸ E 2 P 0ͫ ␣k 41 which show that both v occ 0 and v dephos,Rb 0 include a linear and a parabolic term in [Rb ϩ ]. In the case of v dephos,Rb 0 the initial slope will be positive (as in the results in Fig. 3B) only if k 41 Ͼ k 40 ; that is, if there is activation by a single Rb ϩ . In the case of v occ 0 , a positive initial slope indicates ␣ Ͼ 0. If one Rb ϩ were occluded in E x , ␣ would be equal to 1, and the initial slope of v occ 0 would be slightly larger than that of v dephos,Rb 0 (cf. linear terms in Equations 7 and 8). This is in stark contrast with our results in which the initial slope of v occ 0 is about one-third that of v dephos,Rb 0 , indicating the absence of stoichiometric occlusion.
Effects of Rb ϩ Concentration on ATPase Activity, Phosphoenzyme, and Occluded Rb ϩ in Steady State-In the experiment whose results are shown in Fig. 4, ATPase activity was measured in media containing 10 M ATP and from zero to 1000 M Rb ϩ . It can be seen that the response to [Rb ϩ ] is biphasic, with an initial activation that shows no signs of sigmoidicity and a maximum at around 40 M Rb ϩ (see the inset) followed by a decrease to an asymptotic value that is lower than that observed at zero [Rb ϩ ]. We have shown previously (11) that inhibition by Rb ϩ such as that seen in the second phase in Fig. 4 is not caused by displacement of Na ϩ from the sites from which it activates the ATPase.
The results in Fig. 4 are similar to those obtained by other authors (12,13). They also agree with the observation by Vilsen (14) who found an analogous response to K ϩ using 3 mM ATP in a mutant of rat kidney Na ϩ /K ϩ -ATPase with very low affinity for ATP.
The most likely cause of the activation by Rb ϩ is that k 41 Ͼ k 40 . The lack of sigmoidicity in the activation phase supports the idea that activation reflects the binding of only one Rb ϩ to the phosphoenzyme. Notice that if k 41 ϭ k 40 , ATPase activity would continuously decrease with [Rb ϩ ]. This is so because the only effect of Rb ϩ would be to shift the rate-limiting step from that governed by k 40 (2.4 s Ϫ1 ) to that governed by the E 2 to E 1 transition, whose value at 10 M ATP is 1.3 s Ϫ1 (5). This mechanism would account for the inhibition by Rb ϩ shown in Fig. 4 and was first proposed by Post et al. (12) in their pioneering work on occlusion.
The steady-state levels of EP and of Rb occ (Figs. 5 and 6, respectively) are sigmoid functions of Rb ϩ concentration. Comparison of these   Table 2). The short dashed, dotted, and long dashed lines are, respectively, simulations of the scheme in Fig.  1 for the conditions (i), (ii), and (iii) and the values of the rate constants given in Table 3.   Table 2). The short dashed, dotted, and long dashed lines are, respectively, simulations of the scheme in Fig. 1 for the conditions (i), (ii), and (iii) and the values of the rate constants given in Table 3. The inset is the detail of the plots up to 100 M [Rb ϩ ].
results with those in Fig. 3 shows that the apparent affinity for Rb ϩ is much higher for the steady-state results (K 0.5 Ϸ 100 -200 M) than for the effects of Rb ϩ on the initial velocities. The reason of this discrepancy is that initial velocity only measures the rate of Rb ϩ binding, whereas the steady-state measurements include the effect of the formation and breakdown of occluded Rb ϩ . Because the rate of deocclusion is very low at 10 M ATP, the enzyme accumulates in its occluded state, which acts as a sink.
In Fig. 7 we plotted the sum of the concentrations of EP and of occluded enzyme ( 1 ⁄ 2 Rb occ ) as a function of [Rb ϩ ]. It can be seen that the sum reaches a minimum at about 200 M Rb ϩ . This result is consistent with the idea that at intermediate Rb ϩ concentrations there is an accumulation of enzyme states that are neither phosphorylated nor occluded. These states might correspond to those, like E x , that participate in the activation of dephosphorylation by a single Rb ϩ .
The experimental data of the steady-state values of ATPase activity (Fig. 4), phosphorylation (Fig. 5), and occlusion (Fig. 6) were fitted to the following equation, where Y 0 and Y 2 are the values of the steady-state property being measured in the absence and presence of saturating concentrations of Rb ϩ , respectively. Y 1 is the contribution of Y at intermediate Rb ϩ concentrations, and K 1 and K 2 are apparent constants for the dissociation of Rb ϩ from the enzyme. As can be seen by the continuous lines that fit the experimental points, Equation 9 gave an excellent description of the three steady-state results. Inspection of the best-fitting values of the parameters in Table 1 shows the following. (i) For Rb ϩ occlusion, Y 0 and Y 1 tended to values that were not significantly different from zero. This is consistent with the lack of states with only one Rb ϩ occluded per enzyme. (ii) Y 1 Ͼ Y 0 Ͼ Y 2 for ATPase activity determines the existence of the maximum, and that the activity at non-limiting Rb ϩ concentration is less than the Na ϩ -ATPase activity (Fig. 4). (iii) In the case of EP levels, Y 0 Ͼ Y 1 Ͼ Y 2 , which fits with the continuously decreasing curve of EP versus [Rb ϩ ]. (iv) Y 2 for Rb ϩ occlusion is twice the value of Y 0 for EP, as it is to be expected if all the enzyme went from EP to E 2 (Rb 2 ) as [Rb ϩ ] went from zero to infinity.
(v) The ratio Y 2 /2 for Rb occ is equal to the value of E 2 (Rb 2 ) at saturating [Rb ϩ ]. Under this condition Y 2 /2 times the apparent rate constant for the E 2 (Rb 2 ) 3 E 1 ϩ 2Rb ϩ step at 10 M ATP (1.3 s Ϫ1 ) approaches closely the value of Y 2 for the ATPase activity. This fits the idea that at saturating [Rb ϩ ] all the reaction flow takes place through this step. (vi) The values of K 1 and K 2 for each steady-state property tested are sufficiently close as to make it likely that the three properties obey the same mechanism. In fact, if the average value of K 1 and K 2 for the three properties are used for fitting Y 0 , Y 1 , and Y 2 , the resulting curves describe very well the experimental data (results not shown).
The Value of k 41 -In contrast with k 40 or k 42 , there is no experimental condition that allows estimating k 41 separately from k 40 or k 42 (cf. Equation 3 with Equations 2 and 4). In view of this we used an indirect method to estimate k 41 based on assuming rapid-equilibrium binding of Rb ϩ to E 2 P, in which case the initial rate of Rb ϩ -dependent dephosphorylation will be v dephos,Rb When [Rb ϩ ] Ͻ Ͻ K Rb2 , Equation 10 will approach v dephos,Rb (Eq. 12) Equations 11 and 12 show that, knowing the values of E 2 P 0 and of k 40 , the values of k 41 , K Rb1 , and ␤ can be estimated by fitting Equation 11 to the experimental results of v dephos,Rb 0 . When this is done using the experiment in Fig. 3, the best-fitting value of k 41 is 24 Ϯ 16 s Ϫ1 or 18 Ϯ 11 s Ϫ1 depending on the value of E 2 P 0 used (see Table 2). These values lay between the usually accepted values for k 40 (2.5 s Ϫ1 ) and k 42 (250 -500 s Ϫ1 ; for references see Table 3).
Comparison between the Experimental Results and the Simulations of the Scheme in Fig. 1-The experimental results presented insofar are consistent with our hypothesis that the binding of a single Rb ϩ to E 2 P activates dephosphorylation without leading to the stoichiometric  Table 3.   Fig. 1 at ͓Rb ϩ ͔ ϭ 0 using the rate constants in Table 3. b EP measured in the absence of Rb ϩ .
occlusion of Rb ϩ . To see if the explicit inclusion of our hypothesis into the Post-Albers model (Fig. 1) is able to predict the experimental results in this paper, we fitted the analytical solution of the scheme in Fig. 1 to the data in Figs. 4 -6 considering three alternative conditions, (i) k 41 ϭ k 40 and no occlusion of Rb ϩ in E x takes place (␣ ϭ 0), (ii) k 41 independent of k 40, and E x occluding one Rb ϩ (␣ ϭ 1), and (iii) k 41 independent of k 40, and E x not occluding Rb ϩ (␣ ϭ 0).
In all cases the fitting was performed using the whole set of steadystate experimental values in Figs. 4 -6. To reduce the number of independent parameters to be fitted we kept some of them constant as indicated in Table 3. This is justified because there is reliable experimental information on the values of these parameters from both our work and the work of others (see the legend of Table 3). The set of parameters obtained by this procedure were used to plot theoretical values of the initial rates of dephosphorylation and occlusion and to compare them with the experimental results (Fig. 3, C-E). It can be seen that condition (i) is unable to fit the results. A better fit is obtained making k 41 independent of k 40 but considering that E x occludes one Rb ϩ (condition (ii)), and the best fit is achieved assuming no occlusion in E x and keeping k 41 independent of k 40 (condition (iii)). Condition (iii) also gives the best description of the steady-state values in Figs. 4 -6. The inability of condition (i) to fit the results of ATPase activity is particularly noticeable (Fig. 4).
It must be noted, however, that even condition (iii) does not fully account for the experimental results. This is especially so for the initial rates of occlusion and dephosphorylation in which, although it is the only one that predicts a difference between both rates, condition (iii) underestimates the slope of the rate of occlusion at very low [Rb ϩ ].
It could be argued that, for our experimental results, the slope of v occ 0 ϭ f ([Rb ϩ ]) when [Rb ϩ ]3 0 is not zero because one Rb ϩ is occluded in E x (␣ ϭ 1) but that a faction of the occluded Rb ϩ is washed out during its measurement. If this were so, the rate of deocclusion would be the rate of the E x 3 E 1 reaction (k 6 ). The best value of this constant (11 s Ϫ1 , see Table 3) falls far below the values of deocclusion rate that can be stopped by our quenching procedure without significant loss of Rb ϩ (see "Experimental Procedures" and Rossi et al. (9)). Therefore, an artifact due to the washing of occluded Rb ϩ during its measurement does not seem to be the cause of our results. A more likely explanation may be that Rb ϩ occlusion does take place but in only one of the microscopic states of E 2 PRb. An example of these states would be species of E 2 P that bind one Rb ϩ and one Na ϩ , as has been suggested by Vilsen (14).
It is not known if the ATPase activated by a single Rb ϩ is able to drive active transport. This cannot be discarded in view of the observation by Beaugé et al. (18) who showed that a response similar as that observed in Fig. 4 is elicited by extracellular K ϩ . The data presented here were collected in a non-compartmentalized system that does not allow deciding if transport occurs. Even if a compartmentalized system were available, to measure the stoichiometry of net K ϩ pumping with 10 M ATP (and, hence, with a very low pump-to-leak ratio) is a very difficult task. For this reason, all the reliable measurements of stoichiometry of transport in the Na ϩ pump have been performed with optimal concentrations of ATP and Na ϩ and K ϩ (19). In conclusion, results in this paper indicate the presence of an until now unknown mode of behavior of the Na ϩ / K ϩ -ATPase, whose further study may help to understand the mechanism of active transport of Na ϩ and K ϩ .

TABLE 3
Values of the kinetic parameters used for simulations of the scheme in Fig. 1

for different conditions
We assumed that the binding of the two Rb ϩ to E 2 P and that binding of ATP to E 2 (Rb 2 ) takes place in rapid equilibrium. The values of the total enzyme concentration (E T ) and of k 40 were obtained by fitting the solutions of the model to the results of EP and ATPase activity at ͓Rb ϩ ͔ ϭ 0 and fixed thereafter. Only K Rb1 , k 6 , and (except in condition k 41 ϭ k 40 , ␣ ϭ 0) k 41 were adjusted; the rest of the values were kept constant as given in the