The Occlusion of Rb(+) in the Na(+)/K(+)-ATPase. I. The identity of occluded states formed by the physiological or the direct routes: occlusion/deocclusion kinetics through the direct route.

Occlusion of K(+) or its congeners in the Na(+)/K(+)-ATPase occurs after K(+)-dependent dephosphorylation (physiological route) or in media lacking ATP and Na(+) (direct route). The effects of P(i) or ATP on the kinetics of deocclusion of the K(+)-congener Rb(+) formed by each of the above mentioned routes was independent of the route of occlusion, which suggests that both routes lead to the same enzyme intermediate. The time course of occlusion via the direct route can be described by the sum of two exponential functions plus a small component of very high velocity. At equilibrium, occluded Rb(+) is a hyperbolic function of free [Rb(+)] suggesting that the direct route results in enzyme states holding either one or two occluded Rb(+). Release of occluded Rb(+) follows the sum of two decreasing exponential functions of time, corresponding to two phases with similar sizes. These phases are not caused by independent physical compartments. The rate constant of one of the phases is reduced up to 30 times by free Rb(+). When Rb(+) is the only pump ligand, the kinetics of occlusion and deocclusion through the direct route are consistent with an ordered-sequential process with additional independent step(s) interposed between the uptake or the release of each occluded Rb(+).

Occlusion of K ؉ or its congeners in the Na ؉ /K ؉ -ATPase occurs after K ؉ -dependent dephosphorylation (physiological route) or in media lacking ATP and Na ؉ (direct route). The effects of P i or ATP on the kinetics of deocclusion of the K ؉ -congener Rb ؉ formed by each of the above mentioned routes was independent of the route of occlusion, which suggests that both routes lead to the same enzyme intermediate. The time course of occlusion via the direct route can be described by the sum of two exponential functions plus a small component of very high velocity. At equilibrium, occluded Rb ؉ is a hyperbolic function of free [Rb ؉ ] suggesting that the direct route results in enzyme states holding either one or two occluded Rb ؉ . Release of occluded Rb ؉ follows the sum of two decreasing exponential functions of time, corresponding to two phases with similar sizes. These phases are not caused by independent physical compartments. The rate constant of one of the phases is reduced up to 30 times by free Rb ؉ . When Rb ؉ is the only pump ligand, the kinetics of occlusion and deocclusion through the direct route are consistent with an orderedsequential process with additional independent step(s) interposed between the uptake or the release of each occluded Rb ؉ .
The coupling of the hydrolysis of ATP to the active transport of Na ϩ and K ϩ in the Na ϩ /K ϩ -ATPase (EC 3.6.2.37) takes place through several elementary steps including: (i) the Na ϩ -dependent phosphorylation of the ATPase by ATP, (ii) the K ϩactivated hydrolysis of the phosphoenzyme thus formed (1), (iii) conformational changes of both the phospho and the dephosphoenzymes, and (iv) the occlusions of Na ϩ and K ϩ . When occluded, the access of Na ϩ or K ϩ to the bulk of the solvent is strongly restricted, probably because they are moving through the ATPase as part of their active transport. K ϩ can be replaced by Rb ϩ , Cs ϩ , Tl ϩ , or NH 4 ϩ in all the reactions in which it participates.
Occlusion of K ϩ was first proposed on the basis of indirect evidences by Post and co-workers (1) and then confirmed by other researchers who showed that the K ϩ -congeners Rb ϩ (2)(3)(4) or Tl ϩ (5) became associated to the Na ϩ /K ϩ -ATPase in such a way that they are only slowly removed by cation exchange resins (2,6) or by extensive washings (3)(4)(5).
A minimal sequence of steps for the formation and release of occluded K ϩ or its congeners and its coupling to cation transport (7) based on the currently accepted reaction scheme of the Na ϩ /K ϩ -ATPase (for references, see Ref. 4) shown in Fig. 1, would be as follows.
The macroscopic distinction between intra-and extracellular location of ligands is lost in fragmented membrane preparations. Following Glynn and Karlish (7), we call E 1 the conformer of the pump with high affinity for ATP and for intracellular Na ϩ , which catalyzes the reversible transfer of the terminal phosphate of ATP to the enzyme with formation of E 1 P. E 2 is the conformer of the pump with high affinity for extracellular K ϩ and low affinity for ATP. E 2 catalyzes the reversible transfer of orthophosphate between E 2 P and water. E 1 and E 2 also differ in their spectroscopic properties (8) and in their reactivity to proteolytic enzymes (9). Extracellular K ϩ binds to the E 2 P (Reaction a in Scheme 1) remaining exchangeable with the medium until P i is released leaving K ϩ occluded in E 2 . In agreement with other authors (see Ref. 7) we call this the physiological route of occlusion because it not only requires the physiological operation of the pump but seems to be a necessary step of this operation (4). Occluded K ϩ is released into the intracellular medium (Reactions c-e in Scheme 1). This step is accelerated about 200 times after intracellular ATP binds to E 2 (K ϩ n ) occluded (Reactions d-e in Scheme 1). This effect does not involve the hydrolysis of ATP and is exerted at a site whose affinity is much lower than that of the active site of the ATPase (3,4,8). Occlusion of K ϩ can also be attained by the reversal of Reaction c in Scheme 1. As * This work was supported by grants from the Consejo Nacional de Investigaciones Científicas y Técnicas, Agencia Nacional de Promoción Científica y Tecnológica, and Universidad de Buenos Aires, Argentina. The costs of publication of this article were defrayed in part by the payment of page charges. This article must therefore be hereby marked "advertisement" in accordance with 18  in previous papers (see Ref. 7) we call this the direct route because it does not involve other intermediates than those formed between K ϩ and the enzyme. Occlusion through the direct route leads to the equilibrium distribution between free and occluded K ϩ , while the physiological route leads to a steady state, whose duration will depend on the supply of ATP and on the accumulation of P i and ADP (7).
Under physiological conditions, a small fraction of the pump units exchange intra-for extracellular K ϩ (K ϩ /K ϩ exchange (10)). This requires P i and needs but does not consume ATP and does not lead to net transport. It probably expresses the reversible shuttling of the ATPase between the states shown in Reactions a-e of Scheme 1. Hence, direct as well as physiological occlusion routes would be used during the normal operation of the pump. Scheme 1 and Fig. 1 also show that occlusion can follow the binding of K ϩ to E 1 ATP. This will not be considered here (but see Ref. 11).
During the physiological operation of the pump, the value of the coefficient "n" in Scheme 1 and Fig. 1 seems to be two (4,5) since occlusion only takes place when two K ϩ (or Rb ϩ ) ions are bound. This is consistent with the observation that 2 K ϩ ions are transported in each Na ϩ /K ϩ -ATPase cycle (12).
There is no a priori reason for positing that the two occlusion routes lead to the same enzyme states as it is assumed in Scheme 1 and Fig. 1. On the basis that in general, only identical intermediates having identical distribution will show the same kinetic behavior, we studied the kinetics of deocclusion in media containing different concentrations of P i or ATP. Studies of this kind have been performed by Forbush (3) but because of his setting he could not discard the possibility that "different but related occluded states could interconvert." This proviso does not apply to our experimental procedure (13). Moreover, the results presented here extend the conditions employed both by Forbush and by ourselves (14), to a much wider range of ATP or P i concentrations. The second part of this paper is an analysis of the equilibrium distribution between free and occluded Rb ϩ and of the kinetics of formation and breakdown of enzyme states holding occluded Rb ϩ using the direct route.

EXPERIMENTAL PROCEDURES
Enzyme-Na ϩ /K ϩ -ATPase was partially purified from pig kidney (15). The specific activity at the time of preparation ranged from 19 to 28 mol of P i min Ϫ1 (mg protein) Ϫ1 measured at 37°C in media with 150 mM NaCl, 20 mM KCl, 3 mM ATP, 4 mM MgCl 2 , and 25 mM imidazole-HCl, pH 7.4. The variability in specific activity was reflected in the maximal amount of occluded rubidium obtained (Rb occ, max ) but in all cases the molar ratio Rb occ, max /(ADP-binding sites) was not significantly different from 2.
Reagents and Reaction Conditions-[ 86 Rb]RbCl and [ 32 P]orthophosphoric acid were from PerkinElmer Life Science. [␥-32 P]ATP was synthesized using the procedure of Glynn and Chappell (16), except that no unlabeled orthophosphate (P i ) was added. All other reagents were of analytical grade. Incubations were performed at 25°C in media containing 25 mM imidazole-HCl (pH 7.4 at 25°C) and 0.25 mM EDTA. The concentrations of other components varied according to the experiments and are indicated under "Results." Before starting a reaction, the components to be mixed were diluted in the reaction media. Control experiments (not shown) indicated that, under the conditions used, the enzyme retained its activity for up to 240 min long incubation.
Measurement of Rubidium Occlusion-Following Rossi et al. (13), quenching of occlusion reactions was attained by means of the quick drop in temperature, in ligand concentrations and in free [ 86 Rb]Rb ϩ . Occluded Rb ϩ was considered equal to that retained by the enzyme after washing with at least 300 ml of an ice-cold washing solution containing 30 mM KCl and 20 mM imidazole-HCl (pH 7.4 at 0°C) flowing at a rate of 40 ml/s. The procedure uses a rapid-mixing apparatus (RMA) 1 (SFM4 from Bio-Logic, France) connected to a quenching and washing chamber. Depending on the incubation time, the occlusion reactions were performed either in a test tube or in the RMA, but in all cases the reaction was stopped injecting the enzyme suspension from the RMA into the quenching and washing chamber at a flow rate of 1-5 ml/s (13).
Equilibrium Rb ϩ Occlusion through the Direct Route-70 to 150 g/ml of Na ϩ /K ϩ -ATPase were incubated during 15 to 120 min in media with [ 86 Rb]RbCl. Blanks were estimated from the amount of [ 86 Rb]Rb ϩ retained by the filters when the enzyme was omitted. Their values were similar to those obtained with heat-inactivated enzyme (30 min at 65°C) or with native enzyme in media with 40 mM Na ϩ . Blank values, which were usually much lower than 10% of the amount of occluded [ 86 Rb]Rb ϩ , were independent of the mass of enzyme and linearly related to the Rb ϩ concentration (13).
Steady-state Rb ϩ Occlusion through the Physiological Route-95 to 110 g/ml Na ϩ /K ϩ -ATPase was incubated with [ 86 Rb]RbCl in media containing NaCl, MgCl 2 , and micromolar ATP. Incubation lasted for 3 s to ensure steady-state conditions. Blanks were estimated from samples lacking ATP.
The Time Course of Formation of Occluded Rubidium through the Direct Route-Enzyme suspension (222 g of protein/ml) in 25 mM imidazole-HCl (pH 7.4 at 25°C) and 0.25 mM EDTA, was mixed in the RMA with an equal volume of the same medium having different concentrations of [ 86 Rb]RbCl and incubated for different lengths of time. Then, 0.57 ml of the incubation mixture was squirted into the quenching and washing chamber.
The Time Course of Rb ϩ Deocclusion-This was done looking at the decrease of occluded [ 86 Rb]Rb ϩ after isotopic dilution of the [ 86 Rb]Rb ϩ . When the effects of ATP or P i were compared, 1 volume of the incubation suspension containing the occluded species was mixed with 1 volume of a solution having sufficient unlabeled Rb ϩ as to give a 100-or 200-fold decrease in the specific activity of [ 86 Rb]Rb ϩ and to set [Rb ϩ ] at 10 mM. When the kinetics of deocclusion was studied using Rb ϩ as the only pump ligand, isotopic dilution was attained adding enough of a solution of identical composition as to cause a 20-fold decrease in the specific activity of [ 86 Rb]Rb ϩ .
ATPase Activity in the Presence of Inorganic Orthophosphate-This was measured as the amount of [ 32 P]P i released from [␥-32 P]ATP, according to the method described by Schwarzbaum et al. (17) slightly modified to quantitatively extract the P i present in the media. Incubation time was short enough as to prevent the hydrolysis of more than 10% of the ATP present and to ensure initial rate conditions. Enzyme concentration was 8 g of protein/ml and blanks consistent in an assay were all the Na ϩ in the medium was replaced by K ϩ .
Data Analysis and Development of Theoretical Models-Equations were adjusted to the results by nonlinear regression using commercial programs (Excel 7.0 for Windows TM and Sigma-Plot TM 2.0 for Windows TM ). The goodness of fit of a given equation to the experimental results was evaluated by the AIC criterion (18) defined as AIC ϭ N ln(SS) ϩ 2 P, with N ϭ number of data, P ϭ number of parameters, and SS ϭ sum of weighted square residual errors. Unitary weights were considered in 1 The abbreviation used is: RMA, rapid mixing apparatus.
FIG. 1. A kinetic scheme for ATP hydrolysis by the Na ؉ /K ؉ -ATPase based on the Albers-Post model. E 1 is the conformer of the enzyme that binds ATP to the active site to form the enzyme-substrate complex, E 1 ATP. E 1 P and E 2 P are two conformational states of a phosphoenzyme formed by the covalent binding of the terminal phosphate of ATP to the enzyme. Occluded K ϩ is held in enzyme forms (or form groups) E 2 (K n ) and E 2 (K n )ATP, where ATP is noncovalently bound. Arrows in dotted lines indicate the physiological route for occlusion, while arrows in dashed lines indicate the direct route of occlusion. all cases. It is obvious that AIC values may be positive or negative. The best equation was considered that which gave the lower value of AIC.
To test kinetic models we developed a procedure (19) for its use in Mathematica TM for Windows TM (version. 4.1). This includes the following steps. (i) The set of differential equations that describe the model together with the corresponding conservation equations is worked out; (ii) initial values are assigned heuristically to the rate constants and to the total enzyme concentration (E T ), then the numerical solutions of the set of differential equations is obtained; (iii) the solutions are compared with the experimental values and the rate constants and E T are corrected using a procedure based on the Gauss-Newton algorithm (20); (iv) the corrected values are used to obtain a new set of numerical solutions; and (v) steps iii and iv are repeated until the standard deviation of the residual errors reaches a minimal constant value.

Effects of the Route of Occlusion on the Kinetics of Deocclusion
We measured the amount of occluded [ 86 Rb]Rb ϩ (Rb occ ) remaining at different times after isotopic dilution in media with different concentrations of either ATP or P i . A good description of the time course of deocclusion was obtained with the sum of two exponential functions of time plus a time-independent term (Equation 1 below). Regression analysis to fit exponentials can yield strong statistical correlation between rate coefficients (k values) and amplitudes (A values). This leads to high standard errors of the parameters, which may affect the evaluation of the significance of differences between two sets of values. To circumvent this, we also compared the data by means of a graphical procedure. This was based on the fact that, if the kinetics of deocclusion were the same for the enzyme states formed via the two routes of occlusion then the time courses of Rb ϩ loss from both kind of intermediates should differ only in a constant factor contained in the initial amount of occluded Rb ϩ . The effect of this factor can be canceled out by dividing each data value by the initial amount of occluded Rb ϩ (Rb occ,0 ). This procedure is simpler and relies on less assumptions than that we had used before, which took into account Rb occ when time tends to infinity (14). We calculated Rb occ,0 extrapolating the function that we had used during regression. Rb occ,0 obtained by the direct route was larger than that obtained by the physiological route (cf. panels A and B) because during steady-state hydrolysis of ATP at limiting [Rb ϩ ], enzyme states not containing occluded Rb ϩ , such as phosphoenzymes, will represent a significant fraction of the total amount of enzyme. It is clear that ATP induced a large increase in the rate of loss of occluded [ 86 Rb]Rb ϩ . As already mentioned, a good description of the whole set of results was obtained using the sum of two exponential functions of time plus a time-independent term corresponding to the amount of Rb ϩ that remains occluded when the time after dilution tends to infinity, i.e.

Effect of ATP on the Loss of Occluded [ 86 Rb]Rb ϩ
The best fitting values of the parameters of Equation 1 are given in Table I and were used to calculate the continuous lines in Fig. 2. For comparative purposes, Table I also includes the values of the parameters obtained by fitting to the same data a single exponential function of time plus a constant term; these were used to draw the dotted lines in Fig. 2 (panels A and B). As judged from the values of sum of squares (SS) and AIC in Table  I it is apparent that Equation 1 gives a better description of the results. As mentioned above we also performed a graphical comparison of the two set of data. Plots of the time course of deocclusion normalized for Rb occ,0 in media with 10 or 100 M ATP are given in panels C and D, respectively. It can be seen that the experimental points for the two occlusion routes are almost superimposable. It is noteworthy that the value of A ϱ cannot be fully explained by the new isotopic equilibrium reached after dilution of [ 86 Rb]Rb ϩ . If this were so residual Rb occ (A ϱ ) should be 100 or 200 times less than Rb occ,0 (i.e., A 1 ϩ A 2 ϩ A ϱ ) whereas as shown in Table I, the actual values of A ϱ were significantly higher. We also measured the release of occluded Rb ϩ into media containing from 0 to 2500 M ATP. A single exponential function of time plus a constant term was adjusted to the data for each ATP concentration since this gave sufficient quantitative information for performing paired comparisons of deocclusion rates. The best fitting values of the rate coefficients are plotted as a function of [ATP] in Fig. 3. The continuous curves in the figure show that the effect of ATP on them can be adequately described by a hyperbolic function.
The results in Fig. 3 make it clear that the parameters of Equation 2 are the same regardless of the route followed to occlude Rb ϩ .

Effect of P i on the Loss of Occluded [ 86 Rb]Rb ϩ
We looked at the time course of loss of occluded [ 86 Rb]Rb ϩ formed either through the direct or the physiological routes in media containing from 0 to 8 mM P i . All media also contained 2.5 M ATP to ensure equal exposure to the nucleotide in all samples.  Table I. The data were analyzed both by regression and by the graphical method. Fig. 4 shows that P i induced a large increase in the rate of deocclusion and that at all P i concentrations the shape of the time courses is the same regardless of the route of occlusion (panels C-G). The results were adequately fitted by Equation 1. Since regression showed that A 1 and A 2 had values that were not significantly different, a more economical fit was obtained by setting A 1 ϭ A 2 ϭ A in Equation 1. Results in Fig.  4 also show that Rb occ,0 (see also Fig. 5, panel C) was smaller for the species occluded by the direct route. This is fully explained by the decrease induced by Mg 2ϩ in the equilibrium level of Rb occ (11).
The parameters of Equation 1 that gave best fit to the results in Fig. 4 were used to draw the continuous lines in this figure and are plotted in Fig. 5 as a function of [P i ]. As shown in panels A and B, k 1 and k 2 increase with [P i ] along rectangular hyperbolas of the form (cf. Equation 2).
The best fitting values of k 0 , k ϱ and K 0.5 of Equation 3 given in Table II show that the K 0.5 for P i is the same for k 1 and k 2 and that k ϱ for k 2 has comparable values as that for ATP-stimulated deocclusion (cf . Table II and legend to Fig. 3). All the parameters of P i -stimulated deocclusion were practically independent of the route of occlusion. Therefore, as in the case of ATP, the kinetics of the deocclusion accelerated by P i was the same regardless of the route followed to reach occlusion. According to Scheme 1, the effect of P i on deocclusion is caused by the stimulation by P i of the reversal of Reaction b. This view is supported by the observations by Forbush (21) that Rb ϩ (K ϩ ) is released into the suspending medium of right-side out membrane vesicles loaded with P i , following the incorporation of Mg 2ϩ into the intravesicular medium.
To check this we looked at the effect of P i on the ATPase activity. Inhibition of the ATPase by P i is a necessary, but not sufficient, condition for attributing the activation of deocclusion by P i to the reversal of Reaction b in Scheme 1. Fig. 6 shows the results of our measurements of steady-state ATP hydrolysis as a function of [P i ] in media of identical composition and temperature as those used in the deocclusion experiments. P i acted as a partial inhibitor of the ATPase reducing its activity to about half when [P i ] tended to infinity. Inhibition took place along a hyperbola that was half-maximal at 0.86 Ϯ 0.12 mM P i (continuous line in Fig. 6). The figure also shows that the effect of P i is enhanced by increasing [Mg 2ϩ ].
If reversal of Reaction b in Scheme 1 were the cause of acceleration of deocclusion by P i then it should be accompanied by the phosphorylation of the enzyme. Our attempts to measure EP formation from P i at the same conditions as those used in the experiments shown in Fig. 6 yielded phosphoenzyme levels, which were a small fraction of the blanks and that therefore gave too much scatter as to allow reliable conclusions (experiments not shown). Low levels of EP under analogous conditions as those used by us has been reported by others (22) and could be explained if E 2 P were formed from E 2 (Rb 2 ) with an overall rate constant (20 s Ϫ1 , see values of k 2 in Fig. 5, panel C, and in Table II), which is much slower than the rate constant for breakdown of E 2 P at nonlimiting concentration of Rb ϩ (at least 230 s Ϫ1 (4)).

Equilibrium and Kinetic Properties of the Direct Occlusion Route
The Time Course of Equilibration between Free and Occluded Rb ϩ -We measured occlusion and deocclusion of Rb ϩ via the  direct route with Rb ϩ as the only pump ligand. Na ϩ /K ϩ -ATPase in media containing Rb ϩ concentrations going from 3 to 228 M was incubated at 25°C for periods ranging from 0.037 to 180 s and then Rb occ was measured.
The results are given in Fig. 7  Best fit to each of the curves in Fig. 7, panels A and B, was attained by the following function of time.
The continuous lines in Fig. 7 Table II. ing hyperbolic functions of the concentration of Rb ϩ of the shape, where A i and A i,max are either A 1 and A 1,max or A 2 and A 2,max . The values of K 0.5 and of A i,max of the two exponential terms were not significantly different from each other and when a single coefficient A was used to adjust both exponential terms in Equation 4, the best fitting values obtained for this parameter were not different from those obtained from the independent adjustment of A 1 and A 2 . Panel B in Fig. 8 shows that Equation 4 adequately described the total amount of occluded Rb ϩ in equilibrium with free Rb ϩ (Rb occ,ϱ ), calculated as A 0 ϩ A 1 ϩ A 2 (Equation 4). The best fitting value of K 0.5 was nonsignificantly different from those which adjusted A 1 and A 2 . Panel B also shows that A 0 followed a saturable function of [Rb ϩ ]. In view of its small relative size (it did not exceed 15% of Rb occ,ϱ ) no attempt was made to fit an equation to these data.
Panels C and D show plots of the rate coefficients k 1 and k 2 of Equation 1. It is apparent that k 1 was about 10 times larger than k 2 and increased in an approximately linear fashion with [Rb ϩ ] (panel C), whereas k 2 remained practically unaffected by changes in the concentration of the cation (panel D). The linear response of k 1 to [Rb ϩ ] was confirmed in independent experiments covering Rb ϩ concentrations not used in the experiment in Fig. 7.
The Shape of the Rb occ Versus [Rb ϩ ] Curve-The hyperbolic response of Rb occ,ϱ to [Rb ϩ ] (panel B in Fig. 8) is difficult to harmonize with the evidence that indicates that occlusion only takes place when two Rb ϩ are trapped per ATPase molecule since this would yield sigmoidal and not hyperbolic curves (for review, see Refs. 4 and 6, but also see Refs. [23][24][25]. In view of this, we investigated more thoroughly the equilibrium distribution between occluded and free Rb ϩ in media containing Rb ϩ in concentrations spread along a 0.05 to 472 M range and placing sufficient points at very low concentrations of Rb ϩ , where sigmoidicity would be more manifest. Results in Fig. 9, show that under these conditions the Rb occ versus [Rb ϩ ] curve was still hyperbolic, even at the lowest [Rb ϩ ] tested (inset).
The Release of Occluded Rb ϩ -In a preliminary experiment, we found that the 20-fold dilution of the enzyme associated to the isotopic dilution to measure the release of occluded [ 86 Rb]Rb ϩ had no effect on the equilibrium distribution between free and occluded Rb ϩ . Therefore, the procedure allowed us to determine the loss of occluded [ 86 Rb]Rb ϩ under conditions in which the equilibrium between free and occluded Rb ϩ is preserved.
We studied the kinetics of the loss of occluded Rb ϩ using ATPase preparations equilibrated during 15 min in media containing from 2 to 500 M [ 86 Rb]Rb ϩ . As shown in Fig. 10, like the experiments in which ATP or P i were present, the loss followed a double exponential function as Equation 1. The results also show that the release of Rb ϩ was markedly slowed down as the concentration of Rb ϩ in the media was increased. This effect of Rb ϩ is shown more clearly in the inset to panel A of Fig. 10 in which the results of the experiments with the lowest and highest [Rb ϩ ] tested (2 and 500 M) were scaled dividing Rb occ at any incubation time by Rb occ,0 . The plot strongly suggests that only one of two exponential components of the time course is involved in the inhibition by Rb ϩ . This was confirmed plotting the best fitting values of the rate coefficients k 1 and k 2 of each of the curves in Fig. 10 against the concentration of Rb ϩ . It is clear (Fig. 11) that as [Rb ϩ ] increased, one of the constants (arbitrarily designated as k 2 ) dropped to a very low value while the other remained unaffected by Rb ϩ . Inhibition of the loss of Rb ϩ was not mimicked by either 1 mM Na ϩ or 0.7 mM free Mg 2ϩ (not shown). In media with no Rb ϩ , k 1 becomes sufficiently near to k 2 as to make the time course of loss of Rb ϩ describable by a single exponential function.
Regression analysis showed that the inhibitory effect of Rb ϩ on k 2 was exerted along a decreasing hyperbolic function of [Rb ϩ ], i.e. as follows.
The continuous line that fits the experimental points in panel B of Fig. 11 is a plot of this equation for the best fitting values of the parameters (see legend to Fig. 11). The effect of Rb ϩ on k 2 is exerted with a K I of about 7.5 M, which is similar to that obtained for the effect of Rb ϩ on the equilibrium level of Rb occ (cf. Fig. 9). Although k 2ϱ is only about 3% of k 20 , our results strongly suggest that k 2ϱ is significantly different from zero since its value is about 10 times larger than that of its standard error and remained different from zero when [Rb ϩ ] was raised to 10 mM (experiment not shown).

FIG. 6. Na ؉ /K ؉ -ATPase activity as a function of [P i ], under similar conditions as those used in the deocclusion experiments.
Measurements were performed in media of the same composition and temperature as those used to obtain the results shown in Fig.  4  In the experiment shown in Fig. 12 the incubation leading to occlusion was carried out in media containing 200 M Rb ϩ during only 0.34 s and the time course of deocclusion was measured and compared with that of enzyme in which occlusion was allowed to reach equilibrium. According to the results in Figs. 7 Fig. 12 shows that for the enzyme incubated during 0.34 s, Rb ϩ loss again followed a biphasic time course, with phases of similar amplitudes. After scaling (panel B), this time course and that for the enzyme that had reached equilibrium between free and occluded Rb ϩ were superimposable, indicating that they both follow similar kinetics.

DISCUSSION
The main conclusion of the first part of this paper is that, within a wide range of ATP and P i concentrations, the same kinetics of Rb ϩ loss is observed for occluded states formed via the direct or the physiological routes. The most economical explanation for these results is that both occlusion routes lead to the same enzyme state(s). If different enzyme states were formed, these should be distributed in equilibrium already at the start of the time course measurements.
Although we used ATP and P i -Mg only as tools to probe the behavior of the occluded intermediates, as a collateral result two properties of their action were apparent and merit some consideration. These were the inhibition by P i and the complex shape of the deocclusion curves.
The Effect of P i on Deocclusion-On their face value, our observation of partial inhibition of the ATPase by P i is incompatible with reactions involving the reversible release of P i as it is the case of that in Fig. 1. Partial inhibition could happen if a second ligand were needed for P i to act and were present at limiting concentrations. Mg 2ϩ is the most likely candidate for this since it is known that full inhibition of the ATPase activity by P i requires high [Mg 2ϩ ] (26,27) and results in this paper  Fig. 7. Panel A is a plot of A 1 (q) and A 2 (Ⅺ). Panel B is a plot of A 0 (f) and Rb occ , when t 3 ϱ (Rb occ,ϱ (OE)), that was calculated as A 0 ϩ A 1 ϩ A 2 . Panels C and D are plots of the rate coefficients that govern the fast (k 1 ) and slow (k 2 ) components, respectively. Vertical bars are Ϯ1 S.E. The continuous lines in A are plots of Equation 5 for the best fitting values (Ϯ1 S.E.) of its parameters that were K 0.5 (M) ϭ 5 Ϯ 1 (for A 1 ), and 3.5 Ϯ 0.6 (for A 2 ) and A imax (nanomole of Rb ϩ (mg protein) Ϫ1 ) 1.8 Ϯ 0.1 (for A 1 ) and 1.58 Ϯ 0.06 (for A 2 ). The continuous line in B is a plot of Equation 5 for the best fitting values of its parameters that were K 0.5 ϭ 5.7 Ϯ 0.5 M and Rb occ,max ϭ 3.85 Ϯ 0.09 nmol of Rb ϩ (mg protein) Ϫ1 .

FIG. 8. The dependence with [Rb ؉ ] of the parameters in Equation 4 that gave best fit to the experimental data in
FIG. 9. The equilibrium distribution between free and occluded Rb ؉ in media containing from 0.05 to 472 M Rb ؉ . Rb occ was measured after a 15-min long incubation, instead of extrapolating the time courses of formation of Rb occ as in the experiment in Fig. 7. The continuous line is the plot of a rectangular hyperbola for the best fitting values of the parameters which were (Ϯ1 S.E.) Rb occ,max ϭ 3.68 Ϯ 0.07 nmol of Rb ϩ (mg protein) Ϫ1 and K 0.5 ϭ 5.7 Ϯ 0.4 M. The inset is the initial part of the curve, corresponding to the initial 1% of the Rb ϩ concentration. (Fig. 6) show that doubling [Mg 2ϩ ] increased the inhibitory effect of P i . An attractive explanation of the effect of Mg 2ϩ is the proposal by Sachs (27) that inhibition by P i requires the formation of the Mg 2ϩ -P i complex. In this case, partial inhibition by P i would be present when [Mg 2ϩ ] were insufficient to attain a suitable concentration of the Mg 2ϩ -P i complex. This would not happen if P i and Mg 2ϩ acted separately where at sufficiently high [P i ], the rate of P i release from the enzyme and the overall activity of the ATPase will tend to zero at any [Mg 2ϩ ].
The Complex Kinetics of the Release of Rb ϩ -In all conditions tested the kinetics of Rb ϩ release was describable by the sum of two exponential functions of time, plus a time-independent term (A ϱ in Equation 1) (see also Refs. 3, 23, and 28). As mentioned under "Results," and in contrast with what happen when Rb ϩ is the only pump ligand, A ϱ is higher than that predicted by the isotopic equilibrium reached after dilution of [ 86 Rb]Rb ϩ . Therefore, it is likely that additional and much slower deocclusion phases, not matched by exponential functions of time in Equation 1, are present forcing the regression procedure to increase the value of A ϱ .
The interpretation of the multiple phases in the deocclusion curves is likely to depend on the deocclusion conditions. The kinetics of this process in the case of P i is very similar to that when Rb ϩ is the only pump ligand. It does not seem to be far fetched to explain the behavior in the presence of P i by adapting to it the treatment developed below for deocclusion in this condition (see also Refs. 21,28,and 29).
ATP poses a different problem in the analysis of deocclusion kinetics. The rate constant for the dissociation of ATP from the occluded state is much larger than the constant of deocclusion of Rb ϩ (4) indicating that ATP binds in rapid equilibrium to this state. Under these conditions, ATP would promote the simultaneous release of two occluded Rb ϩ (3), yielding a single exponential for the deocclusion curve. This contrasts with our results (see also Ref. 3), which show that two phases are always present and that the rate coefficient of the slower phase is considerably lower than the value that would make it kinetically compatible with the turnover of the pump. It seems therefore inescapable to conclude that, in media with ATP, the slow phases represent deocclusion pathways unrelated to the physiological operation of the pump. Likely candidates for these pathways are the release of K ϩ from extracellular instead than from intracellular sites, as suggested for the route of spontaneous deocclusion by Forbush (Ref. 28, and see also Ref. 7), and/or the presence of heterogeneous or nonfunctional enzyme units in our preparations.
The Quantitative Behavior of Equilibrium Occlusion When Rb ϩ Is the Only Pump Ligand: The Equilibrium Constant for Direct Occlusion-It seems reasonable to posit that occlusion comprises at least two steps: (i) the binding of Rb ϩ on sites from for the values of the parameters that gave best fit to the experimental results (see Fig. 11). In these curves, the values of A ϱ corresponded very well to those expected from calculations based on the isotopic dilution. which it is exchangeable with free Rb ϩ , and (ii) the occlusion of the bound Rb ϩ . This view is supported by the observation (inset to Fig. 7, panel B) that initial velocity of occlusion follows hyperbolic kinetics as a function of the concentration of free Rb ϩ . In what follows, we shall call "bound" the Rb ϩ that is not occluded, to distinguish it from the occluded Rb ϩ (which is also bound). Assuming, for the sake of simplicity, that only one Rb ϩ binds and becomes occluded per enzyme, a two-step occlusion reaction can be written as follows.
Scheme 2 defines two equilibrium constants, K deocc and K diss .
The amount of occluded Rb ϩ in equilibrium with free Rb ϩ will follow a rectangular hyperbola.
shows that K app and ERb occ,max will depend on K deocc and/or K diss as follows, ERb occ,max ϭ E T 1 ϩ K deocc (Eq. 10) so that, as K deocc goes from infinity to zero, the fraction of enzyme holding occluded Rb ϩ will go from zero to one and K app will go from K diss to zero. The properties of Scheme 2, which can be extended to the binding and occlusion of more that one Rb ϩ and to processes having more than two steps, show that K app ϭ K diss only when K deocc tends to infinity, i.e. when there is no occlusion. In any other condition K app Ͻ K diss . For the same reasons, at saturating concentrations of free Rb ϩ , ERb occ will approach E T only if K deocc tends to zero. Therefore the observation that occluded Rb ϩ when [Rb ϩ ] tends to infinity is very close to twice E T (see "Experimental Procedures" and Refs. 4 and 5) strongly suggests that K deocc is low enough as to displace almost completely the equilibrium toward occlusion. In our experiments K app was about 5 M, whereas the K 0.5 for the effect of Rb ϩ on the initial rate of Rb ϩ uptake (inset to Fig. 7, panel B), which should estimate (probably as a lower limit) K diss was around 85 M. On the basis of Scheme 2, K deocc would be about 0.06. If these numbers were correct, Equation 10 would indicate that ERb occ,max is actually close to 2 ϫ E T .
The Shape of the Rb ϩ Occlusion Curve and the Apparent Affinity for Rb ϩ Occlusion-The experimental data indicating that occlusion only takes place when 2 mol of Rb ϩ are bound per mole of enzyme (see for instance, Refs. 4 -6), imply that the curve relating the equilibrium between free and occluded Rb ϩ should be sigmoidal instead of hyperbolic as in the experiments presented in this paper (see also, Refs. [23][24][25]. It is possible to force models requiring more than one Rb ϩ for occlusion, to approach a hyperbolic response by assigning to one of the Rb ϩ -binding and occluding sites a sufficiently high affinity as to be occupied (but not in the occluded state) by Rb ϩ at all concentrations tested (23). In this case the Rb occ versus [Rb ϩ ] function would depend on the titration with Rb ϩ of only the site with lower affinity in the enzyme. Although this hypothesis has the appeal of not modifying the stoichiometry of occlusion, at this stage of our knowledge it is no more than an ad hoc way to conciliate findings that seem to be contradictory. It seems therefore more reasonable to consider that the hyperbolic response observed in our experiments indicates that during direct occlusion, and when Rb ϩ is the only pump ligand, forms holding either one or two occluded Rb ϩ participate in the equilibrium between free and occluded Rb ϩ . Independent support for this hypothesis is provided by the biphasic time courses of formation and release of occluded Rb ϩ .
Another striking feature of the Rb occ ϭ ƒ([Rb ϩ ]) curve is its very high affinity for the cation. In fact using the nomenclature of Scheme 2, K app was about 5 M and K diss was around 85 M. It is difficult to conceive that with these values for affinity, occluded Rb ϩ (or K ϩ ) will be released efficiently into the cytosol whose concentration of K ϩ is about 150 mM. Similar considerations apply to the inhibition of deocclusion by free Rb ϩ which is also exerted with an affinity (K I about 7.5 M) which would make it difficult the rate of release of Rb ϩ into the cytosol. In this respect it is noteworthy that Forbush (21,28) studying the release of occluded Rb ϩ in media with P i and Mg 2ϩ found a high affinity inhibition by Rb ϩ and concluded that the release was through extracellular sites. He also suggested (28) that extracellular sites mediated the release of occluded Rb ϩ in the absence of other pump ligands. Fig. 12 showed that the time courses of Rb ϩ release from enzyme that had not reached occlusion equilibrium exhibited slow and fast components with the same size as those of enzymes in which Rb occ had reached equilibrium with free Rb ϩ . This rules out the possibility that the biphasic response is caused by the presence of two independent compartments since a single exponential curve should be apparent if only one of the two compartments were significantly occupied by the cation. The inadequacy of this hypothesis is also apparent in its inability to explain why the loss of Rb ϩ from one of the compartments is almost fully arrested by sufficiently high concentrations of Rb ϩ in the incubation media.

The Kinetics of Occlusion and Deocclusion Make Untenable Models Having Two Independent Rb ϩ -occluding Sites per ATPase Molecule-Our experiments in
The Leaky Single File Model for Rb ϩ Occlusion May Explain the Results-In equilibrium conditions, an alternative to the independent site model is to postulate that only one Rb ϩbinding site is accessible, through which two Rb ϩ transit sequentially into the occluded state. A similar mechanism was postulated by Glynn et al. (29) and Forbush (21,28) to explain Rb ϩ deocclusion in the presence of Mg-P i . The simplest reaction for this process is as follows.
In this scheme, the value of the rate constants that govern the equilibrium between Rb ϩ -bound states that are occluded or not occluded are supposed to be strongly poised toward the occluded states, so that intermediates with bound Rb ϩ were omitted. Moreover, the entry and occlusion of the second Rb ϩ requires a change in the position of the Rb ϩ that is already occluded (which in Scheme 3 is symbolized by its displacement to the left-hand side of E), followed by the occlusion of the second Rb ϩ that occupies the position left free. The sequential occlusion of the two Rb ϩ in Scheme 3 can be envisioned as taking place within a pocket which imposes a single-file mechanism for the displacement of Rb ϩ . The Rb ϩ occluded at the left-hand side of E would occupy a deeper position in the pocket than that at the right-hand side of E (Scheme 3). If this is taken together with the operation of a single-file mechanism it is apparent that the Rb ϩ occluded in the deeper position will only be released after the more superficially occluded Rb ϩ . Therefore, if k T and k ϪT are not fast enough, the occlusion and release of half of the occluded Rb ϩ will be slower than that of the other half. In the case of occlusion, the rate of entrance of the second Rb ϩ would be limited by the rate of "jumping" of the first, more superficially placed Rb ϩ to the deeper position in the pocket. This would be seen as an increase in Rb occ along two phases, an earlier one whose velocity will increase with [Rb ϩ ] followed by a slower one whose velocity will be independent of [Rb ϩ ]. In the case of release, the loss of the Rb ϩ placed in the deeper position of the pocket could be limited by a small value of k ϪT and/or by the lack of an empty place where to "jump" before leaving the enzyme. In Rb ϩ -free media and depending on whether k ϪT is limiting or not, this could be seen as a biexponential or monoexponential release, respectively. Nevertheless, as [Rb ϩ ] increases, the curve will become biexponential, the rate of release of one of the phases decreasing toward zero as the concentration of Rb ϩ in the media tends to infinity. Also in Scheme 3, it is not necessary to reach equilibrium between free and occluded Rb ϩ for Rb occ to appear as distributed between two compartments of similar capacities.
We call the model based on Scheme 3 the leaky single file model (see Fig. 13). The term Single File was selected for the reasons explained above. The term Leaky was incorporated because there is no complete blockage of the exit of the deeply occluded Rb ϩ , as it is shown by the small residual loss of Rb ϩ from the slow compartment even when free [Rb ϩ ] tends to infinity. This leak is not predicted by Scheme 3, but is represented in Fig. 13 by the off rate constants of the steps in dashed lines. A similar "leak" was found by Forbush (28) when study-ing the blocking effect of high concentrations of K ϩ and its congeners in media containing Mg-P i .
There is no structural information to support the idea of a "pocket" in the enzyme. We have posited such a mechanism to facilitate the visualization of the occlusion kinetics but any structure which imposes the restrictions of Scheme 3 will yield the same kinetics as the leaky single file model.
To analyze to what extent the leaky single file model is able to account quantitatively for the results presented in this paper we simulated the behavior of the reaction scheme in Fig. 13. This scheme includes Scheme 3 as the states connected by means of continuous lines but additional transitions were incorporated, not only to explain the leak at [Rb ϩ ] tending to infinity but also the existence of a rapid initial occlusion (the term A 0 in Equation 4), which was explored including the steps connected by dotted lines and incorporating the conformers E 1 and E 2 of the ATPase. Since both the Scheme 3 and the scheme in Fig. 13 consider Rb ϩ binding and occlusion as a single-step process, in both the initial rate of occlusion will be a linear function of the concentration of Rb ϩ . Since (inset to Fig. 7) for the range of [Rb ϩ ] used in our experiments, the deviation of the initial slope from linearity is small, to avoid further complications of the scheme in Fig. 13 we did not include the additional steps necessary to account for the saturable increase in the initial rate of occlusion with [Rb ϩ ] (see inset to panel B in Fig.  7 and Refs. 21 and 30).
To simulate the behavior of the model in Fig. 13 we applied the method described under "Experimental Procedures," including the following restrictions. (a) The rate constants of occlusion and deocclusion of Rb ϩ for a given site were considered independent of the occupancy of the other site, i.e. as follows.
The equivalence between the different pathways connecting the same initial and final states places the following additional restrictions on the values of the rate constants.
(c) When Rb ϩ occlusion was simulated, a time-independent parameter, which does not belong to the scheme in Fig. 13, was added to the equations to account for the eventual presence of  Fig. 13 fitted to the data shown in Figures 7, 9, and 10 The procedure used is described in "Data Analysis and Development of Theoretical Models," under "Experimental Procedures." The value of total enzyme (E T Ϯ 1 S.E.) in nmol (mg protein) Ϫ1 obtained from the fitting was 1.86 Ϯ 0.02 for the data in Figs. 7 and 9, or 2.7 Ϯ 0.02 for the data in Fig. 10. This was due to the variability in specific activity between the two different lots of enzyme used in the experiments, as described under "Experimental Procedures."

Rate constants
Values  13. A scheme of the leaky single file model for occlusion through the direct route. The Rb ϩ shown in parentheses is occluded Rb ϩ ; the occluded Rb ϩ shown at the left of E 2 is located in the deeper part of the pocket. The shaded enzyme states are those which hold occluded Rb ϩ . The arrows drawn with a continuous lines represent the pathways through which most of the reactions take place and that corresponds to Scheme 3 under "Discussion." The arrows drawn with dashed lines represent the very slow pathways which are included to explain the small loss of occluded Rb ϩ that persists when [Rb ϩ ] in the media tends to infinity. E 1 binds, but does not occlude Rb ϩ ; the dotted lines that connect E 1 with E 2 and with the two states of E 1 were incorporated into the model to see to what extent the pathways they represent are able to explain the initial rapid occlusion of Rb ϩ (A 0 in Equation 4). a term such as that symbolized by "A 0 " in Equation 4 of "Results." (d) In agreement with the usually held views on the properties of the E 1 and E 2 conformers of the Na ϩ /K ϩ -ATPase (Ref. 7, and see also comments to Scheme 1 and legend to Fig.  1), we assumed that only E 2 is capable to occlude Rb ϩ . Table III shows the best fitting values for each of the constants of the scheme in Fig. 13. The following comments seem to be pertinent: (i) k 1 , k Ϫ1 , k 2 , k Ϫ2 , k T , and k ϪT are at least 1 order of magnitude larger than the rate constants governing the transitions between the other states of E 2 . Therefore, the main pathway for occlusion and deocclusion is identical to that of Scheme 3. (ii) The transfer of occluded Rb ϩ at very low rates governed by the coefficients of k Ϫp1 and k Ϫp2 (cf. these with k Ϫ1 and k Ϫ2 ) from its deep position to the media would explain why the loss of occluded Rb ϩ from the slow compartment is not zero at [Rb ϩ ] tending to infinity. (iii) The very large values of the constants of the reactions that lead to occlusion after the binding of Rb ϩ to E 1 are not enough to explain the time-independent component of the occlusion curves.

Comparison between the Predicted and Observed Kinetics for
Occlusion and Release of Rb ϩ -The best-fitting values of the rate constants and numerical solution of the differential equations describing the scheme in Fig. 13  We also looked if the scheme in Fig. 13 was able to predict the distribution between the fast and slow components of the release of occluded Rb ϩ when the incubation leading to occlusion was short enough as to leave almost empty the slow compartment (see Fig. 12). To do this we used the best fitting values in Table III to generate discrete points of Rb occ corresponding to the same incubation times and Rb ϩ concentrations as those used in Fig. 12. Fig. 15 shows that the discrete values generated by the simulation are adequately fitted by Equation 1 (continuous lines). The legend to Fig. 15 shows that the best fitting value of A 2 was about 40% of A 1 ϩ A 2 , which is near to what it was observed in the experimental values shown in Fig. 12.
Final Remarks-Results in this paper show that, in the absence of other ligands, Rb ϩ occlusion and deocclusion via the direct route follow biphasic time courses and that this behavior is not due to the presence of independent compartments. This phenomenon can be quantitatively explained positing the existence of enzyme states with one and two occluded Rb ϩ formed through an ordered-sequential addition and release of the cation. The biphasic deocclusion kinetics and the micromolar apparent dissociation constant for Rb ϩ for blocking the release of half of the occluded Rb ϩ are in agreement with results presented here and by other authors for P i -stimulated deocclusion (28,29). This may indicate that the direct route of occlusion takes place through external, rather than internal sites of the Na ϩ /K ϩ -ATPase (see also Ref. 28), a view that facilitates to FIG. 14. Adjustment of the leaky single file model (continuous lines) to the experimental data in Figs. 7, 9, and 10. Continuous lines are numerical solutions of the differential equations corresponding to Scheme in Fig. 13 using the values of rate constants shown in Table III. Panels A and B show the curves fitted to the experimental data of occlusion (Fig. 7) and panels C and D show the curves fitted to the experimental data of release ( Fig. 10) of Rb ϩ . Panels A and C show the whole range of incubation times, whereas panels B and D are enlargements of the initial part of the time courses. Panel E shows the curves fitted to Rb occ versus [Rb ϩ ] (see Fig. 9) for the whole concentration range, and panel F is an enlargement of the first portion of the curve. Fig. 12 with the leaky single file model. Data were simulated with the model in Fig. 13 using the values of rate constants shown in Table III account for the micromolar apparent dissociation constant observed for the Rb occ versus [Rb ϩ ] curve in our equilibrium experiments. The participation of external sites in Rb ϩ deocclusion in the absence of other ligands is not considered by Scheme 1 and Fig. 1 and if it existed, it would require additional steps apart from those postulated in these schemes.