Abstract
A mathematical model of the inner medullary collecting duct (IMCD) of the rat has been developed that is suitable for simulating luminal buffer titration and ammonia secretion by this nephron segment. Luminal proton secretion has been assigned to an HKATPase, which has been represented by adapting the kinetic model of the gastric enzyme by Brzezinski et al. (P. Brzezinski, B. G. Malmstrom, P. Lorentzon, and B. Wallmark. Biochim. Biophys. Acta 942: 215–219, 1988). In shifting to a 2 H^{+}:1 ATP stoichiometry, the model enzyme can acidify the tubule lumen ∼3 pH units below that of the cytosol, when luminal K^{+} is in abundance. Peritubular base exit is a combination of ammonia recycling and
HCO3−
flux (either via
Cl−/HCO3−
exchange or via a Cl^{−} channel). Ammonia recycling involves
NH4+
uptake on the NaKATPase followed by diffusive NH_{3} exit [S. M. Wall. Am. J. Physiol. 270 (Renal Physiol. 39): F432–F439, 1996]; model calculations suggest that this is the principal mode of base exit. By virtue of this mechanism, the model also suggests that realistic elevations in peritubular K^{+} concentration will compromise IMCD acid secretion. Although ammonia recycling is insensitive to carbonic anhydrase (CA) inhibition, the base exit linked to
HCO3−
flux provides a CAsensitive component to acid secretion. In model simulations, it is observed that increased luminal NaCl entry increases ammonia cycling but decreases peritubular
Cl−/HCO3−
exchange (due to increased cell Cl^{−}). This parallel system of peritubular base exit stabilizes acid secretion in the face of variable Na^{+} reabsorption.
 epithelial proton transport
 protonpotassiumadenosinetriphosphatase
 chloride/bicarbonate exchange
 ammonia transport
proton secretion along the inner medullary collecting duct (IMCD) results in reclamation of delivered
HCO3−
and titration of luminal buffers and occurs in association with ammonia secretion. When buffer presentation is not excessive, luminal pH can be depressed 3 units below that of plasma. Within the luminal membrane, HKATPase appears to be the principal proton transporter (11, 15, 29), although this view is not universal, and an important role for the HATPase has also been advanced (19). At the peritubular membrane, there are at least two possible mechanisms for base exit. One is via
HCO3−
flux, either through
Cl−/HCO3−
exchange (12, 22) or as
HCO3−
permeation of peritubular Cl^{−} channels (13). Support for this mechanism derives from microperfusion experiments in which luminal
HCO3−
reabsorption is substantially reduced by application of either acetazolamide or SITS (18, 25). The second mechanism involves ammonia recycling, and this has been established in isolated perfused tubule preparations. Particularly in the IMCD, in which peritubular concentration of
NH4+
is high, Wall (27) has recently identified a novel paradigm for base exit. In this scheme,
NH4+
enters on the NaKATPase in competition for K^{+} (28) (with perhaps additional entry via K^{+} channels), elevates cytosolic ammonia, and thus promotes diffusive exit of NH_{3}. A prediction from this mechanism is that base exit, and thus luminal acid secretion, should be influenced by the rate of Na^{+} reabsorption, as well as the peritubular concentration of K^{+}, and that this rate should be uninfluenced by carbonic anhydrase (CA) activity.
To represent these features of acid/base transport in an IMCD model, a robust representation of the luminal membrane HKATPase is required, which can yield flux estimates over the wide range of realistic luminal pH and K^{+} concentrations. Although there is no specific information available on the kinetics of the renal HKATPase, there has been considerable effort to define the reaction steps of the gastric enzyme, and these data have been incorporated into a kinetic model by Brzezinski et al. (3). In the work that follows, that model of the gastric ATPase has been adapted to serve as a representation of the renal HKATPase. The adaptation involves changing the stoichiometry from one H^{+} and one K^{+} per ATP hydrolyzed to two. This achieves a more realistic limiting luminal pH, which is also a function of luminal K^{+} concentration. With this HKATPase incorporated in the epithelial model of the IMCD, the calculations will illustrate the quantitative feasibility of Wall’s proposal (27) for the role of ammonia in acid secretion and will also demonstrate that a substantial component may still be sensitive to CA inhibition. This model will also illustrate the interaction of K^{+} with IMCD acid secretion: to inhibit from the peritubular side and to enhance from the luminal side. Finally, when the model is configured as a set of coalescing tubules, it permits a prediction of the luminal disequilibrium pH and examination of its determinants.
MODEL HKATPASE
Figure 1 displays a reaction scheme for the HKATPase. It is essentially that of Brzezinski et al. (3), with the only difference being their specification of one H^{+} and one K^{+} transported for each ATP hydrolyzed. Although it is secure that the number of H^{+} and K^{+} exchanged are equal, in studies of the gastric enzyme there is controversy regarding the ratio of H^{+} per ATP, with reported values of 1:1 (14, 17) or 2:1 (16, 20). Using a 1:1 stoichiometry, Brzezinski et al. (3) were able to select rate coefficients for the reactions which were compatible with published kinetic data on the gastric enzyme (30). Table 1 lists the reactions of their scheme, along with rate coefficients for each reaction. These coefficients in Table1 are identical to those developed by Brzezinski et al. (3), with the exception of the backward rate constant of reaction 6, which has taken a factor of 100 larger. For the coefficients in Table 1, the pK
_{a} for H^{+}binding in this step is 3.0, rather than 1.0 attributed by Brzezinski et al. The motivation for changing this parameter derives from examining steadystate simulations of the HKATPase transporting against a pH gradient (see below).
Fig. 1.
Scheme for HKATPase, adapted from that of Brzezinski et al. (3). The cytosolicfacing enzyme is denoted E_{1}, and the luminalfacing enzyme is E_{2}. A stoichiometry of 2 H^{+} and 2 K^{+} per ATP is illustrated.
Table 1.
Reactions of a gastric HKATPase
The mathematical model corresponding to the scheme for the gastric enzyme contains 10 unknown concentrations, c_{i}, representing the intermediate reaction products listed in Table2. There are also seven specified solutes, which include the internal and external concentrations of H^{+} and K^{+} and the cytosolic concentrations of ATP, ADP, and P_{i}. Corresponding to each of the 11 reactions is a net reaction velocity, v_{i}
, specified naturally as the difference between forward and backward rates. Because the reaction scheme is branched, there are nine conservation equations in the velocities0=vi−vi+1
Equation 1where i = 2, … , 7 and i = 10, and0=v1+v11−v2
0=v8−v9−v10
This form of the equations assumes a steady state for the reaction intermediates, and it is only these steadystate equations that are used in the model calculations of this report. The equation count is complete at 10, with specification of overall conservation of enzyme∑i=110 ci=1
Equation 2The system of 10 equations is linear in the concentrations, c_{i}, and is solved using a linear equation solver.
Table 2.
Variable concentrations of the models
In the work of Brzezinski et al. (3), they apply their model to two sets of steadystate experimental data in figures 4 and 5 of their study. When these steadystate calculations are repeated using the parameters of Table 1, there is no significant difference between the two sets of model predictions. The models do distinguish themselves, however, when one tries to assess the ability of the HKATPase to transport protons against a pH gradient. For the calculations of Fig. 2, the cytosolic pH is set to 7.4, and K^{+} concentrations for lumen and cytosol are 20 and 150 mM, respectively. Cell ATP is 2.0 mM, ADP is 0.04 mM, and P is 5.0 mM (26). Reaction rate is determined as luminal pH is varied and is plotted as fraction of the rate at luminal pH 7.4. Figure 2 shows that the HKATPase of Brzezinski et al. still transports at 50% of its maximal rate down at pH 0.4. For the HKATPase configured in this report, 50% transport occurs near pH 2 and virtually ceases near luminal pH 1. This pattern of transport would appear to be satisfactory for a gastric enzyme, but to keep collecting duct pH greater than 4.0, further modification of the kinetic model was necessary.
Table 3 displays reaction steps for an HKATPase with a 2:1 H^{+} to ATP stoichiometry and is one of several possibilities. In this case, it was assumed that unloading and loading of two identical sites would occur sequentially. Selection of kinetic coefficients could also have proceeded in a number of ways, but here they were derived from those of the unimolecular enzyme. Consider the reaction schemes for one and two binding sitesH+E KF⇄KBHE
2H+E kf⇄kb H+HEkf⇄kb H2E
where the rate constants for the unimolecular reaction are capitalized, and those for the bimolecular reaction are lower case. The corresponding reaction velocities, V and v, areV=KF[H][E]−KB[HE]
Equation 3
v=kf2[H]2kf[H]+kb[E]−kb2kF[H]+kb[H2E]
Equation 4If one requires (at pH 7.4) the forward velocities of the two schemes to be equal and the backward velocities of the two schemes are also equal, then settingκ2=kf[H]kb2=KF[H]KB
Equation 5one obtainskfKF=1+1κ
kbKB=1+κ
Equation 6Thus for the association reaction shown above, the bimolecular coefficients of Table 3 can be obtained from the unimolecular coefficients of Table 1. For dissociation, the analysis is similar. With this parameter selection process, the energy difference at equilibrium (at pH 7.4) between product and substrate is also preservedlog[H2E][E]=log[HE][E]=2log(κ)
Equation 7
Table 3.
Reactions of a renal HKATPase
Table 2 displays the 14 model variables used for the bimolecular version of the HKATPase. With model equations similar to those for the unimolecular version, the parameters of Table 3 were used to generate a prediction of HKATPase flux as a function of luminal pH. This curve is plotted in Fig. 2, (2H^{+}/ATP) and shows 50% transport around luminal pH 5, with cessation near pH 4. This transport profile is more like that expected for a renal enzyme, and the parameter set of Table 3 has been used in all the epithelial calculations of this and the preceding study (31). The flux rate of this HKATPase is obviously dependent upon the concentration of luminal potassium, and the ability of luminal K^{+} to modulate proton secretion has been examined in Fig.3. Here, the same calculation of Fig. 2 has been performed for fivefold variations in luminal K^{+}. All of the fluxes have been normalized to the maximum flux at lumen K^{+} of 125 mM. For these parameters, it appears that the maximum flux through the HKATPase at neutral lumen pH is relatively insensitive to luminal K^{+}. However, the pH for halfmaximal flux and pH for zero flux each increase by ∼0.7 unit [log_{10}(5)] for each 80% decrease in luminal K^{+} concentration.
Fig. 2.
HKATPase reaction rate as a function of luminal pH. Lumen and cell K^{+} are fixed at 20 and 150 mM; cell pH is 7.40; and lumen pH is varied from 0.3 to 7.40. For all calculations, cell ATP, ADP, and P_{i} are 2.0, 0.04, and 5.0 mM, respectively. The curve labeled “Brzezinski et al.” was obtained using their published parameters (3), the curve labeled 1H^{+}/ATP uses the modified parameters of Table 1, and the curve labeled 2H^{+}/ATP uses the parameters of Table 3.
Fig. 3.
HKATPase reaction rate as a function of luminal pH. For the parameters of Table 3 (2H^{+}/ATP) and for cytosolic conditions as for the calculations of Fig. 2, HKATPase reaction rate is plotted as a function of luminal pH. The four curves illustrate the effect of varying luminal K^{+} from 1 to 125 mM.
MODEL CALCULATIONS
In the calculations that follow, the mathematical model of the IMCD epithelium and the system of IMCD tubules is that of the companion study (31). Unless stated otherwise, the model parameters are the baseline parameters (Table 1 in Ref. 31), and the boundary conditions are those of the outerinner medullary junction (OIMJ) (Table 2 in Ref. 31). In Fig. 4, the epithelial model of IMCD has been used to predict the rate of luminal acidification via the HKATPase when ambient (luminal and peritubular)
NH4+
concentration is varied. In thetop panels of Fig. 4, the cytosolic conditions are displayed, and the bottom shows the rate of luminal membrane H^{+} secretion. In each panel of Fig.4, there are two curves, one computed for complete inhibition of carbonic anhydrase (−CA) and one in which cytosolic CA catalysis is present at baseline levels, i.e., 100fold acceleration of CO_{2} hydration (+CA). Regardless of the CA activity, increasing the
NH4+
concentration leads to cytosolic acidification and enhancement of H^{+}secretion. The fact that the two curves in the bottom of Fig. 4are nearly parallel may be interpreted to mean that the
NH4+
effect on acid secretion is relatively insensitive to CA activity. The distance between these curves is the CAsensitive component of acid secretion. In the presence of high ambient
NH4+,
the CAsensitive component is only 23% of the total H^{+} secretion, but when
NH4+
is nearly absent, H^{+}transport is reduced 78% by CA inhibition. With inhibition of CA, there is an alkaline disequilibrium pH of the cytosol, generated by H^{+} extrusion via the HKATPase and depletion of cytosolic H_{2}CO_{3}. Because CA inhibition impairs
HCO3−
generation without affecting
HCO3−
exit, there is a decrease in cell
HCO3−
despite the alkalosis.
Fig. 4.
Impact of ammonia on IMCD acid secretion. Using the IMCD epithelial model with baseline parameters and OIMJ boundary conditions, the ambient (luminal and peritubular) ammonia concentration is varied from 0.2 to 10 mM. Curves designated −CA are solutions in which the rates of hydration of CO_{2} (0.145 s^{−1}) and dehydration of H_{2}CO_{3} (49.6 s^{−1}) in the cell are those of free solution. Curves designated +CA have reaction rates 100fold higher. Bottom: rate of luminal proton secretion via the HKATPase. Top: cytosolic conditions.
The cytosolic acidification in the presence of high
NH4+
derives from metabolically driven
NH4+
uptake, secondary increase in cytosolic NH_{3} concentration, and (peritubular and luminal) diffusive exit of NH_{3}. According to Wall and Koger (28), much of this
NH4+
uptake is mediated by the NaKATPase, with competition between peritubular
NH4+
and K^{+}. As indicated by their results, the relative affinity of
NH4+
and K^{+} for the external binding site was taken to be 0.2 in these model calculations. This scheme predicts that increases in peritubular K^{+}concentration, by decreasing ammonia cycling, should increase cytosolic pH and decrease luminal H^{+} secretion. This prediction is examined in Fig. 5, which uses the epithelial model of IMCD with baseline parameters to see the impact of peritubular KCl addition or removal. The bottom of Fig. 5 shows that the rate of luminal membrane H^{+} secretion decreases monotonically with increasing peritubular K^{+}. Themiddle panels of Fig. 5 resolve the change in base exit into components referable to ammonia cycling or
HCO3−
exit. In the left middleof Fig. 5, total peritubular
NH4+
uptake is displayed as the sum of pump and channelmediated
NH4+
fluxes. The decrease in pump flux with increasing peritubular K^{+} is as expected from competition on the NaKATPase; the decrease in channel flux is smaller and derives from cytosolic depolarization. In the right middleof Fig. 5, the total
HCO3−
exit is displayed as the sum of channel flux and that of the
Cl/HCO3−
exchanger. In this model, increasing peritubular K^{+} increases cell Cl^{−}concentration (Fig. 5, top right ) as a result of the prominent activity of the KCl cotransporter (see companion study, Ref.31). With increased cell Cl^{−}, there is decreased
HCO3−
exit via the
Cl−/HCO3−
exchanger. Thus increases in peritubular K^{+} concentration depress both mechanisms for base exit by the IMCD cell, although the effect on ammonia cycling is most important. The change in transport rate of the HKATPase must derive from either changes in cytosolic K^{+}concentration or pH or both. As peritubular K^{+}concentration increases, there is a monotonic increase in cytosolic K^{+} concentration from 110 to 225 mM (not shown), which tends to diminish the transport rate of the HKATPase. For peritubular K^{+} concentrations above 7.5 mM, the decrease in base exit (ammonia cycling and
HCO3−
flux) results in progressive cytosolic alkalinization (Fig. 5, top left ), which also slows the transport rate of the HKATPase. When peritubular K^{+} decreases below 7.5 mM, the decrease in cell K^{+} enhances the rate of the HKATPase sufficiently to induce a small cytosolic alkalinization (overriding the facilitated base exit).
The rate of peritubular uptake of
NH4+
via the NaKATPase should also be influenced by the rate of IMCD Na^{+} transport. In Fig. 6, the epithelial model is utilized to simulate variation in luminal NaCl from 2 to 110 mM. Over this range of luminal Na^{+}, the peritubular pump rate for Na^{+} ranges from 0.9 to 6.1 nmol ⋅ s^{−1} ⋅ cm^{−2}. In these calculations, however, there is virtually no change in luminal membrane H^{+} secretion (Fig. 6, bottom) nor in cell pH (Fig. 6, top left ). The middle panels of Fig. 6show that the changes in Na^{+} transport lead to opposing effects on ammonia cycling and
HCO3−
exit. In the left middle of Fig. 6, one can see that the pump flux of
NH4+
has nearly halted at the lowest luminal NaCl concentration. There is a small increase in conductive
NH4+
flux, consequent to the decrease in cell
NH4+.
As luminal NaCl decreases, there is a decrease in cytosolic Cl^{−}(Fig. 6, top right ) due to decreased flux through the luminal membrane NaCl cotransporter (see companion study, Ref. 31). With the decrease in cell Cl^{−}, there is increased peritubular
HCO3−
exit via the
Cl−/HCO3−
exchanger (Fig.6, right middle). The decrease in cell Cl^{−} also acts to increase cell K^{+} (via the KCl cotransporter), which hyperpolarizes the cell, and also acts to increase conductive
NH4+
uptake. Thus, this model predicts that with these two mechanisms for base exit operating in parallel, the change in
NH4+
transport via the NaKATPase will be opposed by conductive
NH4+
uptake and
HCO3−
exit. It is suggested that this configuration could act to stabilize acid secretion over a wide range of IMCD Na^{+} transport.
Luminal proton secretion titrates residual luminal
HCO3−
and
HPO42−,
as well as secreted NH_{3}, and the tubule model can be used to predict the profile of acidification along the IMCD. The calculations of Fig.7 were done using the IMCD tubule model and display the changes in buffer flow from inlet to tip. For all of the buffers, most of the flow changes occur within the first 2 mm of tubule, by virtue of the exponential decline in surface area. In the Fig. 7, left, the baseline OIMJ inlet conditions were used. This yields virtually complete reabsorption of
HCO3−,
accounting for 2 nmol/s of IMCD acid secretion, as well as complete titration of
HPO42−,
which accounts for another 1.4 nmol/s. The increase in
NH4+,
1.8 nmol/s, represents a 50% increase in delivered load of
NH4+
(3.6 nmol/s). Thus the total net acid secretion by this IMCD is 5.2 nmol/s. These calculations were repeated when luminal
HCO3−
was increased to 10 mM (by replacement of 5 mM Cl^{−}), and the results are shown in Fig. 7, right. In this case, delivered load of
HCO3−
has doubled, and delivered load of
HPO42−
has increased 43%, but net acid excretion only increases 13%. Here, the increase in
HCO3−
reabsorption (total 3.3 nmol/s) is blunted by a decrease in
HPO42−
titration (total 1.2 nmol/s) and a decrease in NH_{3} trapping (net 1.4 nmol/s). Thus, total acid secretion is 5.9 nmol/s, and appears to be relatively insensitive to delivered load of buffer. The dependence of
HCO3−
excretion on delivery is viewed more systematically in the calculations of Fig. 8, in which the entering
HCO3−
is varied between 5 and 15 mM. Figure 8 shows only the endluminal conditions as a function of the inlet
HCO3−.
There is an apparent threshold corresponding to an entering concentration of ∼6 mM, or an entering
HCO3−
load of 2.4 nmol/s, above which
HCO3−
begins to be spilled in a progressively alkaline urine.
Figure 8 also shows that with the increase in endluminal
HCO3−,
the acid disequilibrium pH increases in magnitude. This quantity, pH_{deq}, is the difference between the luminal pH and that which would obtain under equilibrium hydration of dissolved CO_{2}
pHdeq=log10 CM(HCO3−)CM(H2CO3)−log10 CM(HCO3−)(kh/kd)CM(CO2)
Equation 8or−pHdeq=log10 kdCM(H2CO3)khCM(CO2)
Equation 9When substantial phosphate is present, it is difficult to resolve the disequilibrium pH as a simple function of basic transport parameters; however, an intuitive expression can be developed when phosphate is negligible. Using the notation of this model, recall that the luminal generation of a species,s
_{M}(i ), is the sum of the gradient in axial flow plus its reabsorptive flux. For the hydrated CO_{2}species, we have the conservation equation (Eq. 8
of the companion study, Ref. 31).sM(HCO3−)+sM(H2CO3)
Equation 10
=AM[khCM(CO2)−kdCM(H2CO3)]
(A
_{M} is tubule crosssectional area), and since H_{2}CO_{3} is present in micromolar quantities, the approximationsM(HCO3−)≈AM[khCM(CO2)−kdCM(H2CO3)]
Equation 11The equation for electroneutrality of proton buffering (Eq. 11
of the companion study, Ref. 31) may be writtensM(HCO3−)+sM(HPO42−)=sM(H+)+sM(NH4+)
Equation 12so that when phosphate is negligiblesM(HCO3−)≈sM(H+)+sM(NH4+)
Equation 13
=sM(H+)−sM(NH3)
where the last equality derives from conservation of total ammonia. Because of the tiny axial gradients of H^{+} and NH_{3}, only the reabsorptive fluxes contribute significantly to the generation term, so thatsM(HCO3−)=BM[JM(H+)−JM(NH3)]
Equation 14where B
_{M} (in cm) is tubule circumference, and J
_{M} is total reabsorptive flux (in mmol ⋅ s^{−1} ⋅ cm^{−2}). Substituting for
sM(HCO3−)
yieldskdCM(H2CO3)=khCM(CO2)
Equation 15
−(2/rM)[JM(H+)−JM(NH3)]
where the ratio of tubule circumferenceB
_{M}/A
_{M} = 2/r
_{M}. Thus in the case of negligible phosphate, the disequilibrium pH may be approximated−pHdeq=log10 1−JM(H+)−JM(NH3)(rM/2)khCM(CO2)
Equation 16This expression reveals that the magnitude of the disequilibrium pH increases according to the ratio of H^{+}secretion (less NH_{3} secretion) relative to the rate of hydration of CO_{2}. Notably, it is independent of the concentration of
HCO3−
(except as it influences H^{+} secretion), of H_{2}CO_{3}permeability, of luminal flow rate, and of the axial pH gradient. The accuracy of this approximation is displayed in Table4, for which the calculations of Fig. 8 are repeated, with the exception that the entering total phosphate has been decreased by an order of magnitude to 1.1 mM, so that the endluminal phosphate is only 5.4 mM.
Table 4.
Estimation of endIMCD disequilibrium pH
As indicated in Fig. 3, the rate of H^{+} secretion by the model HKATPase is dependent upon luminal K^{+} concentration over the full range of acid luminal pH. In all of the foregoing calculations, luminal K^{+} concentration was high. The entering K^{+} concentration was 50 mM, which, despite reabsorptive flux, increased further with luminal fluid removal (Fig. 3of companion study, Ref. 31). Below a certain value of entering concentration (in these calculations, ∼20 mM), the luminal K^{+} declines along the IMCD, and with sufficiently low entering K^{+}, this IMCD can achieve luminal concentrations of less than 1.0 mM. This is displayed in Fig.9, which shows luminal
HCO3−
and K^{+} as a function of distance for entering K^{+} concentrations of 7, 11, and 15 mM. (Entering Cl^{−} is adjusted with the K^{+}.) Clearly, as lumen K^{+} diminishes,
HCO3−
reabsorption is impaired. This is viewed more systematically in Fig. 10, for which entering K^{+} concentration is varied from 11 to 35 mM and for which the tip K^{+} concentration is predicted to vary from 1.2 to 52 mM. What is plotted in Fig. 10 is the endluminal pH as a function of the logarithm of the endluminal K^{+}concentration; the scales of both axes are identical, and the slope is close to unity over the whole range of K^{+} concentrations. This may be interpreted as a limiting chemical potential for the activity of this model HKATPase, either as 3 pH units or the energetic equivalent in H^{+} plus K^{+} gradients.
Fig. 10.
Correlation of endluminal K^{+} concentration with endluminal pH. The IMCD tubule model is used with baseline parameters. Boundary conditions are those of OIMJ, with the exception that entering KCl is varied from 11 to 35 mM in 2 mM steps. Lumen pH at the IMCD tip is shown as a function of the logarithm of tip K^{+}concentration. Indicated points correspond to calculated values.
The role of a vacuolar HATPase in IMCD function has not been established and has not been included in this model. Indeed, Wall et al. (29) found no bafilomycininhibitable proton secretion by the isolated perfused IMCD, and Bastani (2) could not detect this pump immunocytochemically in terminal IMCD. Nevertheless, Schwartz (19) has reported HATPase activity in IMCD cells in culture. Furthermore, in the perfused tubule (29) only about onehalf of luminal proton secretion could be inhibited by removal of luminal K^{+} or application of Sch28080, and the immunocytochemical investigation (2) also failed to detect HKATPase in terminal IMCD. Transport by the HATPases has been represented by an empiric expression devised by Strieter et al. (24), based on the data of Andersen et al. (1) for turtle bladderJ(H+)=J(H+)max⋅[1.0+exp[ξ⋅(μ¯MI(H+)−μ¯0)]]−1
Equation 17where J(H^{+})_{max} is the maximum proton flux, and
μ¯MI(H+)
is the electrochemical potential difference of H^{+} from the cytosol to the lumen; ξ defines the steepness of the function, and
μ¯0
defines the point of halfmaximal activity. The important finding of Andersen et al. (1) was that the proton flux depended upon both electrical and chemical components of the proton potential and that the flux went from maximal to zero over a range of the proton potential of 180 mV (or 3 pH units or 17.5 J/mmol). The data of Andersen et al. (Ref. 1, figure 9) are approximately represented by choosing ξ = 0.4 and
μ¯0
= −4.0 J/mmol. In the calculations of Fig. 11, this HATPase has been installed in the luminal cell membrane of the IMCD epithelium, replacing some of the HKATPase. The densities of the two proton pumps were chosen so that when lumen pH = 7.4, each contributed equally to proton secretion, and at the luminal pH of OIMJ (pH 6.6) the total proton secretion was equal to that of the model presented here (exclusively HKATPase). This apportionment was intended to represent the contingency that all of the noninhibitable proton secretion found by Wall et al. (29) was that of an HATPase. In Fig. 11, the cell pH and the luminal membrane proton secretion are plotted as luminal pH is varied. Although the two proton fluxes are equal at neutral pH, it is apparent that at the entering luminal pH of OIMJ, only 15% of the proton flux is attributable to the HATPase. It is clear that the electronegativity of the cell interior prevents the vacuolar HATPase from acidifying the lumen below pH 5.5. Even if present, the HATPase cannot contribute significantly when the IMCD lumen has been acidified.
Fig. 11.
Effect of luminal pH on IMCD proton secretion. The HATPase defined byEq. 17
has been incorporated into the model epithelium, replacing some of the luminal HKATPase. The constraints of this replacement are that the total proton secretion under OIMJ conditions remains unchanged and that the contribution of the two proton transporters be equal at lumen pH 7.4.
DISCUSSION
The key feature in the development of this model of the IMCD was fashioning a suitably realistic HKATPase to represent luminal proton secretion. In the absence of data on a renal enzyme, the gastric model of Brzezinski et al. (3) was adapted in two aspects. One modification was the change of one of their rate constants, to decrease the affinity for luminal H^{+} binding from pH 1 to pH 3. Without this change, the published parameters appear to allow the pump to transport vigorously against luminal pH well below 1.0, and there is no indication that Brzezinski et al. (3) explored this aspect of their model. This change does not appear to affect the concordance of the model with the experimental data that was considered. Although no attempt was made to exhaustively examine other parameter configurations, it was found that decreasing the affinity of luminal K^{+} binding also yielded a realistic limiting transport gradient, but also destroyed model agreement with the experimental determination of the ATP dependence of steadystate flux. The second modification was to change the stoichiometry from 1 H^{+} per ATP to 2. This was done by assuming sequential binding at two identical sites for H^{+} and K^{+}. Again, there was no specific experimental guidance for this adaptation, but it did afford a model enzyme whose limiting transport gradient was ∼3 pH units, rather than 6. With this model HKATPase, urine pH could be reduced to 4.0 when K^{+} was abundant, and urine K^{+} could be reduced to 1 mM when urine
HCO3−
was sufficient. Although low concentrations of urine K^{+} impair maximal acidification, the calculations (Fig. 10) suggest that this might not be apparent until urine K^{+} was less than 10 mM.
Complementary to the concern of H^{+} secretion against a gradient is the assignment of the rate of H^{+} transport when ion gradients are small (i.e., specifying the number of pumps). A relatively complete picture ofIMCD acidification in the rat was obtained via microcatheterization by Graber et al. (9). They found early
HCO3−
to be 5.2 mM, corresponding to a delivered load of 205 nmol/min, which was nearly completely reabsorbed (tip
HCO3−,
0.6 mM; tip
HCO3−
flow, 26 nmol/min). Delivery of
NH4+
to IMCD was 193 nmol/min, which increased to 462 nmol/min by papilla tip. Thus they estimated total acid secretion by IMCD to be 450 nmol/min. These results were felt to be compatible with the microcatheterization data of Sonnenberg et al. (21) who found a 60–70% increase in urinary
NH4+
flow from IMCD base to papilla tip, with a total excretion of ∼360 nmol/min. In the model IMCD epithelium, luminal acid secretion was approximately 1 nmol ⋅ s^{−1} ⋅ cm^{−2}, or 60 pmol ⋅ min^{−1} ⋅ mm^{−1}. This translated into a total IMCD net acid excretion of 312 nmol/min with 120 attributable to
HCO3−
reabsorption and 108 attributable to
NH4+
addition; total
NH4+
excretion was 324 nmol/min. Thus the selection of the model rate of luminal H^{+} secretion was conservative. The importance of this comparison is that the rates of IMCD proton secretion observed in the perfused tubule are quite a bit lower. Galla et al. (7) obtained a rate of 38 pmol ⋅ min^{−1} ⋅ mm^{−1}in the early segment of IMCD, which fell to 7 in the terminal segment. Of note, these perfusions were done in the absence of bath
NH4+.
Wall (27) perfused terminal IMCD and observed the impact of (6 mM) peritubular
NH4+
addition was to increase H^{+} secretion from 1.2 to 4.0 pmol ⋅ min^{−1} ⋅ mm^{−1}. Apparently massive increases in IMCD H^{+} secretion can be achieved in
HCO3−loaded
rats, as reported by Richardson and Kunau (18). Their micropuncture of papillary collecting duct of 145 to 210g rats, yielded estimates of IMCD
HCO3−
reabsorption of 1,000–3,000 nmol/min. In particular, they found that even with massively high delivered
HCO3−,
fractional
HCO3−
reabsorption remained about onethird, with no evidence of saturation. A similar fractional reabsorption was observed in the present model IMCD (Fig. 8).
The disequilibrium pH of the IMCD luminal fluid derives from delayed dehydration of H_{2}CO_{3}, and this provides proof of H^{+} secretion plus the absence of luminal CA. In the absence of phosphate buffer (but allowing ammonia), the analysis here estimated the disequilibrium pH as a logarithmic function of the ratio of net acid secretion to the hydration rate of CO_{2} (i.e., the equilibrium dehydration rate of H_{2}CO_{3}). The accuracy of this estimate was verified in the model IMCD. In the presence of phosphate buffer, an analytic estimate is more difficult to obtain, but comparison of Fig. 8 with Table 4 suggests that at comparable luminal pH, the presence of phosphate had relatively little impact on the disequilibrium pH. In the rat, the disequilibrium pH was found to be −0.26 in a microcatheterization study of normal rats in which papillary urine pH was 5.5 (10) and −0.40 in micropuncture of bicarbonateloaded rats (5). The calculations of this model appear to be in good agreement with these findings.
The possibility that peritubular
NH4+
uptake (with diffusive NH_{3} exit) might provide a proton source for luminal acidification was first articulated by Wall (27), based on studies in isolated perfused IMCD; in her view, the most important mechanism for
NH4+
uptake was the NaKATPase, although transport via a K^{+} channel was not excluded. Despite the relatively low affinity for
NH4+
on the sodium pump (28), the concentration of interstitial
NH4+
could be high [e.g., 9.2 and 22.5 mM in control and acidosis, by Stern et al. (23); 2.1 and 6.1 mM in control and acidosis, by Good et al. (8)]. The calculations of the model suggest that in the briskly transporting tubule in vivo, this scheme for ammonia recycling is quantitatively plausible. The model does suggest, however, that peritubular
NH4+
uptake via K^{+} channels is also important. This flux is driven by the peritubular electrical potential difference, and, with peritubular
NH4+
permeability 20% of peritubular K^{+} permeability, the channel flux of
NH4+
accounted for 38% of
NH4+
uptake under baseline conditions. It must be acknowledged that this model lacks representation of peritubular extrusion of
NH4+
via the KCl cotransporter, although given the high cell K^{+}concentration, this term would be expected to be small. Competition of
NH4+
for K^{+} on the NaKATPase also raises the possibility that peritubular K^{+}may blunt acid secretion. Vasa recta K^{+} concentrations have been found to be 6–54 mM, depending on the state of the animal (4). In the model calculations (Fig. 5), it came as no surprise that peritubular K^{+} concentrations in this range did substantially blunt pumpmediated
NH4+
uptake. What had not been anticipated was the model prediction that peritubular K^{+} would also decrease CAinhibitable base exit, by increasing cytosolic Cl^{−}. This aspect of model transport was derived from the importance of a peritubular KCl cotransporter, whose presence has been inferred (see companion study, Ref. 31). The impact of peritubular K^{+} to blunt IMCD acid secretion appears to be compatible with micropuncture observations obtained in rats with selective aldosterone deficiency (6). Finally, it should be noted that in this model, the parallel system of peritubular base exit (ammonia cycling and
Cl−/HCO3−
exchange) provided stability of acid secretion over a very broad range of Na^{+} transport. This was another unanticipated aspect of the model function but one which appears to be physiologically attractive.
The calculations of this study illustrate the requirement that a mathematical model of IMCD which aims to simulate H^{+}secretion by this segment must be comprehensive. In addition to a robust representation of luminal membrane proton transport, it must encompass the multiple pathways for peritubular base exit. In this, one is obliged to provide links to Na^{+} transport, to peritubular K^{+} concentration and peritubular membrane potential difference, and to variation in cytosolic Cl^{−}concentration. With respect to disorders of acid excretion, a number of specific transport defects have been identified, and additional ones have been suggested. It is clear from the calculations presented that in the presence of parallel transport systems, it is difficult to intuitively assess the impact of single transport defects. It is intended that the model presented here may serve as a useful tool for identifying the quantitative importance of transport abnormalities underlying observed acid/base disorders.
Acknowledgments
This investigation was supported by National Institute of Arthritis, Diabetes, and Digestive and Kidney Diseases Grant 1RO1DK29857.
Footnotes

Address for reprint requests: A. M. Weinstein, Dept. of Physiol. and Biophysics, Cornell University Medical College, 1300 York Ave., New York, NY 10021.
 Copyright © 1998 the American Physiological Society