A mathematical model of the outer medullary collecting duct of the rat

Alan M. Weinstein

Department of Physiology and Biophysics, Weill Medical College of Cornell University, New York, New York 10021


    ABSTRACT
TOP
ABSTRACT
INTRODUCTION
MODEL AE1
MODEL OMCD
MODEL PARAMETERS
MODEL CALCULATIONS
DISCUSSION
REFERENCES

A mathematical model of the outer medullary collecting duct (OMCD) has been developed, consisting of alpha -intercalated cells and a paracellular pathway, and which includes Na+, K+, Cl-, HCO3-, CO2, H2CO3, phosphate, ammonia, and urea. Proton secretion across the luminal cell membrane is mediated by both H+-ATPase and H-K-ATPase, with fluxes through the H-K-ATPase given by a previously developed kinetic model (Weinstein AM. Am J Physiol Renal Physiol 274: F856-F867, 1998). The flux across each ATPase is substantial, and variation in abundance of either pump can be used to control OMCD proton secretion. In comparison with the H+-ATPase, flux through the H-K-ATPase is relatively insensitive to changes in lumen pH, so as luminal acidification proceeds, proton secretion shifts toward this pathway. Peritubular HCO3- exit is via a conductive pathway and via the Cl-/HCO3- exchanger, AE1. To represent AE1, a kinetic model has been developed based on transport studies obtained at 38°C in red blood cells. (Gasbjerg PK, Knauf PA, and Brahm J. J Gen Physiol 108: 565-575, 1996; Knauf PA, Gasbjerg PK, and Brahm J. J Gen Physiol 108: 577-589, 1996). Model calculations indicate that if all of the chloride entry via AE1 recycles across a peritubular chloride channel and if this channel is anything other than highly selective for chloride, then it should conduct a substantial fraction of the bicarbonate exit. Since both luminal membrane proton pumps are sensitive to small changes in cytosolic pH, variation in density of either AE1 or peritubular anion conductance can modulate OMCD proton secretory rate. With respect to the OMCD in situ, available buffer is predicted to be abundant, including delivered HCO3- and HPO42-, as well as peritubular NH3. Thus, buffer availability is unlikely to exert a regulatory role in total proton secretion by this tubule segment.

proton-potassium-activated adenosinetriphosphatase; AE1; urine acidification; ammonia transport


    INTRODUCTION
TOP
ABSTRACT
INTRODUCTION
MODEL AE1
MODEL OMCD
MODEL PARAMETERS
MODEL CALCULATIONS
DISCUSSION
REFERENCES

INACCESSIBLE TO MICROPUNCTURE, transport by the outer medullary collecting duct (OMCD) has been inferred from in vitro studies of rabbit and rat tubules. Such experiments have established that the OMCD is a proton-secreting nephron segment with a lumen-positive electrical potential (5, 43, 59). For most of the tubule (inner stripe), there is no discernible active sodium transport (58), although at least 60% of the OMCD cells resemble principal cells from the cortical collecting duct (26, 51, 57). Acid secretion occurs via electrogenic H+-ATPase and electroneutral H-K- ATPase (3, 22, 64), with all of the intercalated cells of this segment (and none of the principal cells) displaying luminal membrane staining for the H+-ATPase (1) and H-K-ATPase (6). The luminal membrane of the intercalated cells has virtually no electrical conductance, whereas that of the peritubular cell membrane is dominated by a chloride pathway (33, 46). Proton secretion is contingent upon the presence of peritubular chloride (60), presumably the result of peritubular HCO3- exit in exchange for Cl-. The anion exchanger specific to the erythrocyte, AE1, has been identified as that of the peritubular cell membrane of OMCD intercalated cells (54, 66). Indeed, mutations of AE1 have recently been associated with a clinical defect in urinary acidification (62).

A mathematical model of the OMCD provides a means for considering the cellular interaction of the membrane components of acid secretion. It also provides a means of extrapolating from in vitro observations to the likely conditions in vivo. The transport characteristics of the H+-ATPase have been known for some time (2) and have been used in a mathematical model of the cortical collecting tubule (61). More recently, construction of a model of the inner medullary collecting duct (IMCD) (71) required revision of a full kinetic model of the H-K-ATPase (10). In the present work, transport properties of AE1 in erythrocytes at 38°C (20, 31) have been used to fashion a kinetic model of this peritubular anion exchanger. These components, along with a peritubular anion channel, provide the critical elements for simulation of the alpha -intercalated cell, or equivalently, the OMCD. In what follows, each of these four transporters appears to be quantitatively important in intercalated cell proton secretion and thus could be a suitable candidate for regulation of OMCD acidification. For the tubule in vivo, the model provides a means of resolving luminal proton secretion into its three components: titration of luminal HCO3-, titration of secreted NH3, and titration of luminal HPO42-. Calculations suggest that for OMCD in vivo, changes in buffer availability may shift the luminal composition but are not likely to have a substantial effect on net acid excretion by this segment.


    MODEL AE1
TOP
ABSTRACT
INTRODUCTION
MODEL AE1
MODEL OMCD
MODEL PARAMETERS
MODEL CALCULATIONS
DISCUSSION
REFERENCES

Figure 1 depicts a scheme for a carrier, X, which may be oriented toward the external (X') or internal (X") membrane face, where either HCO3- or Cl- may be bound. In this scheme, it is assumed that anion exchange proceeds via sequential translocation, the so-called "ping-pong" mechanism (12, 25). It is also assumed that anion binding is rapid relative to translocation, so that the concentration of bound carrier at each face is determined from equilibrium binding constants, KCl and KHCO3. With respect to chloride, there is NMR evidence supporting this assumption (13). The carrier is not assumed to be symmetric, so that 1) distinct binding constants are specified for each membrane face, and 2) the translocation constant for outside to inside flux (P') will not necessarily be equal to that for inside to outside flux (P"). Denote b', c', b", and c" as the concentrations of bicarbonate and chloride within each bath, and bx', cx', and x' and bx", cx", and x" are the concentrations of bound and free carrier on each membrane face. Then, the equilibrium condition implies that the ratios of bound to free carrier may be represented
<AR><R><C>&bgr;′=<FR><NU>bx′</NU><DE>x′</DE></FR>=<FR><NU>b′</NU><DE>K′<SUB>b</SUB></DE></FR><IT>, &bgr;″=</IT><FR><NU><IT>bx″</IT></NU><DE><IT>x″</IT></DE></FR><IT>=</IT><FR><NU><IT>b″</IT></NU><DE><IT>K″</IT><SUB>b</SUB></DE></FR><IT>,</IT></C></R><R><C><IT>
&ggr;′=</IT><FR><NU><IT>cx′</IT></NU><DE><IT>x′</IT></DE></FR><IT>=</IT><FR><NU><IT>c′</IT></NU><DE><IT>K′</IT><SUB>c</SUB></DE></FR><IT>, </IT>and<IT> &ggr;″=</IT><FR><NU><IT>cx″</IT></NU><DE><IT>x″</IT></DE></FR><IT>=</IT><FR><NU><IT>c″</IT></NU><DE><IT>K″</IT><SUB>c</SUB></DE></FR></C></R></AR> (1)
Corresponding to the two unknowns, x' and x", are the model equations for conservation of total carrier, xT
x′+bx′+cx′+x″+bx″+cx″=x<SUB>T</SUB> (2)
and for zero net flux of carrier
P′<SUB>b</SUB><IT>bx′+P′</IT><SUB>c</SUB><IT>cx′=P″</IT><SUB>b</SUB><IT>bx″+P″</IT><SUB>c</SUB><IT>cx″</IT> (3)
In Eq. 3, the left-hand and right-hand sides represent the unidirectional inward and outward fluxes of the carrier. There is no flux of unloaded carrier, corresponding to strict 1:1 stoichiometry for a two-ion system. Using the equilibrium conditions of Eq. 1, Eqs. 2 and 3 may be rewritten
x′(1+&bgr;′+&ggr;′)+x″(1+&bgr;″+&ggr;″)=x<SUB>T</SUB> (4)

x′(P′<SUB>b</SUB><IT>&bgr;′+P′</IT><SUB>c</SUB><IT>&ggr;′</IT>)<IT>−x″</IT>(<IT>P″</IT><SUB>b</SUB><IT>&bgr;″+P″</IT><SUB>c</SUB><IT>&ggr;″</IT>)<IT>=0</IT> (5)
This linear system is solved for x' and x"
x′=<FR><NU>x<SUB>T</SUB>(<IT>P″</IT><SUB>b</SUB><IT>&bgr;″+P″</IT><SUB>c</SUB><IT>&ggr;″</IT>)</NU><DE><IT>&Sgr;</IT></DE></FR><IT>  x″=</IT><FR><NU><IT>x</IT><SUB>T</SUB>(<IT>P′</IT><SUB>b</SUB><IT>&bgr;′+P′</IT><SUB>c</SUB><IT>&ggr;′</IT>)</NU><DE><IT>&Sgr;</IT></DE></FR> (6)
where
&Sgr;=(1+&bgr;′+&ggr;′)(P″<SUB>b</SUB><IT>&bgr;″+P″</IT><SUB>c</SUB><IT>&ggr;″</IT>)
+(1+beta "+gamma ")(P'bbeta '+P'cgamma ')


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Fig. 1.   Kinetic scheme for AE1. Carrier X may be oriented toward the external (X') or internal (X") membrane faces, where either HCO3- or Cl- are bound. Anion binding is rapid relative to translocation, so that the concentration of bound carrier at each face is determined from equilibrium binding constants, Kc and Kb. The carrier is not assumed to be symmetric, so that the translocation constants for outside to inside flux (P') will not necessarily be equal to those for inside to outside flux (P"). There is no slippage of empty carrier.

Thus one obtains expressions for the unidirectional influx and efflux of bicarbonate and chloride
J′<SUB>b</SUB><IT>=P′</IT><SUB>b</SUB><IT>bx′=P′</IT><SUB>b</SUB><IT>&bgr;′x′=</IT><FR><NU><IT>x</IT><SUB>T</SUB></NU><DE><IT>&Sgr;</IT></DE></FR><IT> P′</IT><SUB>b</SUB><IT>&bgr;′</IT>(<IT>P″</IT><SUB>b</SUB><IT>&bgr;″+P″</IT><SUB>c</SUB><IT>&ggr;″</IT>)

J″<SUB>b</SUB><IT>=P″</IT><SUB>b</SUB><IT>bx″=P″</IT><SUB>b</SUB><IT>&bgr;″x″=</IT><FR><NU><IT>x</IT><SUB>T</SUB></NU><DE><IT>&Sgr;</IT></DE></FR><IT> P″</IT><SUB>b</SUB><IT>&bgr;″</IT>(<IT>P′</IT><SUB>b</SUB><IT>&bgr;′+P′</IT><SUB>c</SUB><IT>&ggr;′</IT>) (7)

J′<SUB>c</SUB><IT>=P′</IT><SUB>c</SUB><IT>cx′=P′</IT><SUB>c</SUB><IT>&ggr;′x′=</IT><FR><NU><IT>x</IT><SUB>T</SUB></NU><DE><IT>&Sgr;</IT></DE></FR><IT> P′</IT><SUB>c</SUB><IT>&ggr;′</IT>(<IT>P″</IT><SUB>c</SUB><IT>&ggr;″+P″</IT><SUB>b</SUB><IT>&bgr;″</IT>) (8)

J″<SUB>c</SUB><IT>=P″</IT><SUB>c</SUB><IT>cx″=P″</IT><SUB>c</SUB><IT>&ggr;″x″=</IT><FR><NU><IT>x</IT><SUB>T</SUB></NU><DE><IT>&Sgr;</IT></DE></FR><IT> P″</IT><SUB>c</SUB><IT>&ggr;″</IT>(<IT>P′</IT><SUB>c</SUB><IT>&ggr;′+P′</IT><SUB>b</SUB><IT>&bgr;′</IT>)
and the net efflux for bicarbonate which must be equal and opposite to that for chloride
J<SUP>n</SUP><SUB>b</SUB><IT>=J″</IT><SUB>b</SUB><IT>−J′</IT><SUB>b</SUB><IT>=</IT><FR><NU><IT>x</IT><SUB>T</SUB></NU><DE><IT>&Sgr;</IT></DE></FR> [<IT>P″</IT><SUB>b</SUB><IT>&bgr;″P′</IT><SUB>c</SUB><IT>&ggr;′−P′</IT><SUB>b</SUB><IT>&bgr;′P″</IT><SUB>c</SUB><IT>&ggr;″</IT>] (9)
It should be observed that for the net flux to equal zero when bathing media are equal (b' = b" and c' = c")
<FR><NU>P″<SUB>c</SUB></NU><DE><IT>K″</IT><SUB>c</SUB></DE></FR> <FR><NU><IT>K′</IT><SUB>c</SUB></NU><DE><IT>P′</IT><SUB>c</SUB></DE></FR><IT>=</IT><FR><NU><IT>P″</IT><SUB>b</SUB></NU><DE><IT>K″</IT><SUB>b</SUB></DE></FR> <FR><NU><IT>K′</IT><SUB>b</SUB></NU><DE><IT>P′</IT><SUB>b</SUB></DE></FR> (10)
The two sides of Eq. 10 have been recognized by Fröhlich and Gunn (16) as the ratio of the concentrations of unbound carrier, outward:inward facing, when the bathing media are either all chloride or all bicarbonate. This ratio has been denoted the "asymmetry factor," A, and by virtue of Eq. 10, must be independent of the identity of the ambient anion.

Recently, Brahm and coworkers (20, 31) investigated the kinetics of AE1 at 38°C in erythrocytes in a system that could be used to examine either bicarbonate or chloride self-exchange. When bicarbonate self-exchange is under consideration, the ambient chloride concentrations are zero, and unidirectional fluxes of bicarbonate must be equal. This restricts the representation of the experiment to the top half of Fig. 1, and thus only four of the eight model parameters are relevant. According to Eq. 7 the unidirectional efflux of bicarbonate must be
J″<SUB>b</SUB><IT>=x</IT><SUB>T</SUB> <FR><NU><IT>P″</IT><SUB>b</SUB><IT>&bgr;″P′</IT><SUB>b</SUB><IT>&bgr;′</IT></NU><DE>(<IT>1+&bgr;′</IT>)(<IT>P″</IT><SUB>b</SUB><IT>&bgr;″</IT>)<IT>+</IT>(<IT>1+&bgr;″</IT>)(<IT>P′</IT><SUB>b</SUB><IT>&bgr;′</IT>)</DE></FR> (11)
so that
<FR><NU>1</NU><DE>J″<SUB>b</SUB></DE></FR><IT>=</IT><FR><NU><IT>1</IT></NU><DE><IT>x</IT><SUB>T</SUB></DE></FR> <FENCE><FR><NU><IT>1</IT></NU><DE><IT>P′</IT><SUB>b</SUB></DE></FR><IT>+</IT><FR><NU><IT>1</IT></NU><DE><IT>P″</IT><SUB>b</SUB></DE></FR><IT>+</IT><FR><NU><IT>K′</IT><SUB>b</SUB></NU><DE><IT>P′</IT><SUB>b</SUB><IT>b′</IT></DE></FR><IT>+</IT><FR><NU><IT>K″</IT><SUB>b</SUB></NU><DE><IT>P″</IT><SUB>b</SUB><IT>b″</IT></DE></FR></FENCE> (12)
For the bicarbonate studies, three protocols were used: changing extracellular bicarbonate only (with cytosolic bicarbonate fixed), changing cytosolic bicarbonate only, and changing both symmetrically. Corresponding to each of these experiments are maximal self-exchange rates (in their notation, Jbmo, Jbmi, and Jbms) and apparent affinities (Kbmo, Kbmi, and Kbms). With reference to Eq. 12, the model defines these measured quantities
<FR><NU>1</NU><DE>J<SUP>o</SUP><SUB>bm</SUB></DE></FR><IT>=</IT><FR><NU><IT>1</IT></NU><DE><IT>x</IT><SUB>T</SUB></DE></FR> <FENCE><FR><NU><IT>1</IT></NU><DE><IT>P′</IT><SUB>b</SUB></DE></FR><IT>+</IT><FR><NU><IT>1</IT></NU><DE><IT>P″</IT><SUB>b</SUB></DE></FR><IT>+</IT><FR><NU><IT>K″</IT><SUB>b</SUB></NU><DE><IT>P″</IT><SUB>b</SUB><IT>b″</IT></DE></FR></FENCE>  <FR><NU><IT>K</IT><SUP>o</SUP><SUB>bm</SUB></NU><DE><IT>J</IT><SUP>o</SUP><SUB>bm</SUB></DE></FR><IT>=</IT><FR><NU><IT>K′</IT><SUB>b</SUB></NU><DE><IT>x</IT><SUB>T</SUB><IT>P′</IT><SUB>b</SUB></DE></FR>

<FR><NU>1</NU><DE>J<SUP>i</SUP><SUB>bm</SUB></DE></FR><IT>=</IT><FR><NU><IT>1</IT></NU><DE><IT>x</IT><SUB>T</SUB></DE></FR> <FENCE><FR><NU><IT>1</IT></NU><DE><IT>P′</IT><SUB>b</SUB></DE></FR><IT>+</IT><FR><NU><IT>1</IT></NU><DE><IT>P″</IT><SUB>b</SUB></DE></FR><IT>+</IT><FR><NU><IT>K′</IT><SUB>b</SUB></NU><DE><IT>P′</IT><SUB>b</SUB><IT>b′</IT></DE></FR></FENCE>  <FR><NU><IT>K</IT><SUP>i</SUP><SUB>bm</SUB></NU><DE><IT>J</IT><SUP>i</SUP><SUB>bm</SUB></DE></FR><IT>=</IT><FR><NU><IT>K″</IT><SUB>b</SUB></NU><DE><IT>x</IT><SUB>T</SUB><IT>P″</IT><SUB>b</SUB></DE></FR> (13)

<FR><NU>1</NU><DE>J<SUP>s</SUP><SUB>bm</SUB></DE></FR><IT>=</IT><FR><NU><IT>1</IT></NU><DE><IT>x</IT><SUB>T</SUB></DE></FR> <FENCE><FR><NU><IT>1</IT></NU><DE><IT>P′</IT><SUB>b</SUB></DE></FR><IT>+</IT><FR><NU><IT>1</IT></NU><DE><IT>P″</IT><SUB>b</SUB></DE></FR></FENCE>  <FR><NU><IT>K</IT><SUP>s</SUP><SUB>bm</SUB></NU><DE><IT>J</IT><SUP>s</SUP><SUB>bm</SUB></DE></FR><IT>=</IT><FR><NU><IT>1</IT></NU><DE><IT>x</IT><SUB>T</SUB></DE></FR> <FENCE><FR><NU><IT>K′</IT><SUB>b</SUB></NU><DE><IT>P′</IT><SUB>b</SUB></DE></FR><IT>+</IT><FR><NU><IT>K″</IT><SUB>b</SUB></NU><DE><IT>P″</IT><SUB>b</SUB></DE></FR></FENCE>
Equation 13 indicates that the three self-exchange experiments depend upon only three composite parameters, namely, the geometric mean of the translocation constants
<FR><NU>1</NU><DE>x<SUB>T</SUB><IT>P′</IT><SUB>b</SUB></DE></FR><IT>+</IT><FR><NU><IT>1</IT></NU><DE><IT>x</IT><SUB>T</SUB><IT>P″</IT><SUB>b</SUB></DE></FR>
and the ratios of the affinities to the translocation constants K'b/xTP'b and K''b/xTP''b. Although the experimental studies of Brahm and colleagues (31) can provide six observations, if the model is applicable, then three dependence relations among these observations should be satisfied. Even when only two of the experiments are performed, variation of the external anion and symmetric variation of the anions (31), the model predicts
<FR><NU>1</NU><DE>J<SUP>o</SUP><SUB>bm</SUB></DE></FR><IT>=</IT><FR><NU><IT>1</IT></NU><DE><IT>J</IT><SUP>s</SUP><SUB>bm</SUB></DE></FR><IT>+</IT><FR><NU><IT>1</IT></NU><DE><IT>b″</IT></DE></FR> <FENCE><FR><NU><IT>K</IT><SUP>s</SUP><SUB>bm</SUB></NU><DE><IT>J</IT><SUP>s</SUP><SUB>bm</SUB></DE></FR><IT>−</IT><FR><NU><IT>K</IT><SUP>o</SUP><SUB>bm</SUB></NU><DE><IT>J</IT><SUP>o</SUP><SUB>bm</SUB></DE></FR></FENCE> (14)
where b" is the constant internal bicarbonate concentration used in the study. Furthermore, this analysis indicates that these three experiments cannot suffice to solve for all four model parameters, P'b, P''b, K'b, and K''b. Indeed, within the constraints of the experimental data, one is free to select the ratio K'b/K''b arbitrarily, and then using the three composite parameters, the four model parameters are determined. Finally, Eq. 10 provides a relationship between the bicarbonate parameters and the chloride parameters. Although the two sets of parameters were obtained from independent self-exchange experiments, the absence of metabolic coupling requires equality of the asymmetry factors
(15)
In Table 1, AE1 self-exchange parameters have been abstracted from the work of Brahm and colleagues (20, 31). For both bicarbonate and chloride studies, data from variation of external anion concentration and symmetric variation of anion concentration have been used. For consistency with the bicarbonate study, the chloride data obtained from red cell ghosts was selected. From the ratios Kbms/Jbms and Kbmo/Jbmo, the ratio Kbmi/Jbmi was obtained as a difference (Eq. 13) and was obtained similarly for chloride. It is immediately apparent that the data selected do not satisfy the equilibrium Eq. 15. This discrepancy was noted by the authors, who preferred to attribute it to experimental error, rather than inapplicability of the ping-pong scheme (31). Indeed, Eq. 15 can be satisfied by choosing different values for the affinities, all still within the published standard errors. These modified values appear in the second column of each section of Table 1, and the computation showing satisfaction of Eq. 15 is indicated there. With respect to the consistency of the model data with the scheme of Fig. 1 (i.e., satisfaction of Eq. 14), both the original and modified values for both anions give decent agreement, and this computation is also included. As indicated above, a kinetic model consistent with these experimental data could be built with any value for the ratio of affinities for either anion. In the case of chloride, the study of Liu et al. (42) suggests that the ratio of internal and external affinities is close to 1.0. Assuming a similar ratio for bicarbonate, values for the translocation constants and affinities are indicated in Table 1. Finally, the study of Gasbjerg et al. (20) indicated that internal bicarbonate appeared to inactivate the anion exchanger in a noncompetitive way, with a half-maximal inhibitory concentration, KI, of 172 mmol/l. In the context of this model, this inhibition is represented as an effect on the transporter abundance, xT, relative to a maximal abundance, xTm
<FR><NU>x<SUB>T</SUB></NU><DE><IT>x</IT><SUB>Tm</SUB></DE></FR><IT>=</IT><FR><NU><IT>1</IT></NU><DE><IT>1+b″/K</IT><SUB>I</SUB></DE></FR> (16)
In Fig. 2, the kinetic model of AE1 is used to simulate several self-exchange experiments from which the input parameters were generated. The four top panels of Fig. 2 illustrate HCO3- self-exchange, for which ambient Cl- = 0, and the calculations correspond to the experiments displayed in figure 5 of Gasbjerg et al. (20). In Fig. 2, A and B, external HCO3- is varied while the internal concentration is fixed, either at 50 or 165 mmol/l. The slightly smaller values for the efflux rates obtained from the model (more apparent in Fig. 2B), derive from the higher value taken for Kbms. In Fig. 2, D and E, internal HCO3- is varied, either alone or symmetrically. The appearance of a maximal efflux rate in a neighborhood of 100 mmol/l HCO3- is a consequence of the internal site for noncompetitive inhibition of the exchanger. The two bottom panels of Figure 2 illustrate Cl- self-exchange, in which ambient HCO3- = 0, and the calculations correspond to experiments in red cell ghosts shown in figures 2 and 4 of Knauf et al. (31). In Fig. 2C external Cl- is varied, with internal Cl- = 175 mmol/l, and in Fig. 2F, the concentrations on both sides of the membrane are varied symmetrically. For these simulations, there is little discrepancy with the experimental data.

                              
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Table 1.   Cl-/HCO3- exchanger model development



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Fig. 2.   Model simulations of self-exchange of HCO3- or Cl- via AE1. In A and B, model Eq. 7 (using parameters from Table 1) is evaluated over a range of external (CE) HCO3-, while the internal concentration (CI) is fixed, either at 50 or 165 mM; ambient Cl- is absent. In D and E, model Eq. 7 is solved while internal HCO3- is varied, either alone or symmetrically. In C, model Eq. 8 is evaluated over a range of external Cl-, with internal Cl- = 175 mM; ambient HCO3- is absent. In F, the Cl- concentrations on both sides of the membrane are varied symmetrically.

Figure 3 displays calculations illustrating Cl-/HCO3- flux by the model AE1 operating as an exchanger in the neighborhood of a reference condition: internal HCO3- and Cl- concentrations of 26 and 29 mmol/l, and external concentrations of 26 and 114 mmol/l, respectively. This reference is approximately that of the model tubule developed below. Each panel of Fig. 3 illustrates the variation of a single internal (A and B) or external (C and D) anion concentration (solid curves). The most obvious feature of Fig. 3 is the greater sensitivity of model fluxes with variation of cytosolic concentrations (compared with variation of external concentrations), with the greatest sensitivity to changes in internal HCO3-. The numbers C(dJ/dC) are the derivatives of the fluxes with respect to the fractional change in ion concentration, taken at the reference, and are essentially derivatives with respect to chemical potential. For each panel of Fig. 3 and each value of the logarithmic derivative, a dotted curve is drawn to approximate the exchange rate as a linear function of the logarithm of the abscissa.


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Fig. 3.   Cl-/HCO3- flux by the model AE1 operating as an exchanger (Eq. 9) in the neighborhood of a reference condition: internal HCO3- and Cl- concentrations 26 and 29 mM, and external concentrations 26 and 114 mM, respectively. In A and B, internal HCO3- and internal Cl- are varied; in C and D, the external anion concentrations are the independent variables. The dashed curves are best-fit single exponentials through the reference condition.


    MODEL OMCD
TOP
ABSTRACT
INTRODUCTION
MODEL AE1
MODEL OMCD
MODEL PARAMETERS
MODEL CALCULATIONS
DISCUSSION
REFERENCES

The model outer medullary collecting duct formulated here will be essentially that found in the inner stripe, in which transport activity appears to be that of the intercalated cells. With this simplification, all transport pathways will be ascribed to either a transcellular pathway across intercalated cells or to a paracellular pathway. The model will be formulated both as an OMCD epithelium, with specified luminal and peritubular conditions, or as a tubule, in which luminal concentrations vary axially. Figure 4 displays both models, in which cellular and intercellular compartments line the tubule lumen. Within each compartment the concentration of species i is designated Calpha (i), where alpha  is lumen (M), interspace (E), cell (I), or peritubular solution (S). Within the epithelium the flux of solute i across membrane alpha beta is denoted Jalpha beta (i) (mmol · s-1 · cm-2), where alpha beta may refer to luminal cell membrane (MI), tight junction (ME), lateral cell membrane (IE), basal cell membrane (IS), or interspace basement membrane (ES). Along the tubule lumen, axial flows of solute are designated FM(i) (mmol/s). The 12 model solutes are Na+, K+, Cl-, HCO3-, CO2, H2CO3, HPO42-, H2PO4-, NH3, NH4+, H+, and urea, as well as an impermeant species within the cells and possibly within the lumen. These are the minimal set of solutes that will permit representation of net acid excretion.


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Fig. 4.   Schematic representation of outer medullary collecting duct (OMCD) epithelium, consisting of intercalated cell and lateral intercellular space (LIS), and tubule model, whose lumen is lined by this epithelium. Intraepithelial fluxes are designated Jalpha beta (i), where the subscript alpha beta refer to luminal cell membrane (MI), tight junction (ME), lateral cell membrane (IE), basal cell membrane (IS), or interspace basement membrane (ES). Along the tubule lumen, axial flows are designated FM(i).

To formulate the equations of mass conservation with multiple reacting solutes, consider first an expression for the generation of each species within each model compartment. Within a cell or interspace, the generation of i (salpha (i)) is equal to its net export plus its accumulation
s<SUB>I</SUB>(<IT>i</IT>)<IT>=J</IT><SUB>IE</SUB>(<IT>i</IT>)<IT>+J</IT><SUB>IS</SUB>(<IT>i</IT>)<IT>−J</IT><SUB>MI</SUB>(<IT>i</IT>)<IT>+</IT><FR><NU>d</NU><DE>d<IT>t</IT></DE></FR> [V<SUB>I</SUB>C<SUB>I</SUB>(<IT>i</IT>)] (17)

s<SUB>E</SUB>(<IT>i</IT>)<IT>=J</IT><SUB>ES</SUB>(<IT>i</IT>)<IT>−J</IT><SUB>ME</SUB>(<IT>i</IT>)<IT>−J</IT><SUB>IE</SUB>(<IT>i</IT>)<IT>+</IT><FR><NU>d</NU><DE>d<IT>t</IT></DE></FR> [V<SUB>E</SUB>C<SUB>E</SUB>(<IT>i</IT>)] (18)
where Valpha is the compartment volume (in cm3/cm2). Within the tubule lumen, solute generation is appreciated as an increase in axial flux, as transport into the epithelium, or as local accumulation.
s<SUB>M</SUB>(<IT>i</IT>)<IT>=</IT><FR><NU><IT>∂</IT>F<SUB>M</SUB>(<IT>i</IT>)</NU><DE><IT>∂x</IT></DE></FR><IT>+B</IT><SUB>M</SUB>[<IT>J</IT><SUB>ME</SUB>(<IT>i</IT>)<IT>+J</IT><SUB>MI</SUB>(<IT>i</IT>)]

+<FR><NU>∂</NU><DE>∂t</DE></FR> [A<SUB>M</SUB>C<SUB>M</SUB>(<IT>i</IT>)] (19)
where BM is the tubule circumference, and AM is the tubule cross-sectional area. With this notation, the equations of mass conservation for the nonreacting species (Na+, K+, Cl-, and urea) are written
s<SUB>&agr;</SUB>(i)=0 (20)
where alpha  = E, I, or M. For the phosphate and for the ammonia buffer pairs, there is conservation of total buffer
s<SUB>&agr;</SUB>(HPO<SUP>2−</SUP><SUB>4</SUB>)<IT>+s<SUB>&agr;</SUB></IT>(H<SUB>2</SUB>PO<SUP>−</SUP><SUB>4</SUB>)<IT>=0</IT> (21)

s<SUB>&agr;</SUB>(NH<SUB><IT>3</IT></SUB>)<IT>+s<SUB>&agr;</SUB></IT>(NH<SUP>+</SUP><SUB>4</SUB>)<IT>=0</IT> (22)
Although peritubular PCO2 will be specified, the CO2 concentrations of the cells, interspace, and lumen are model variables. The relevant reactions are
H<SUP><IT>+</IT></SUP><IT>+</IT>HCO<SUP>−</SUP><SUB>3</SUB><IT> ⇄ </IT>H<SUB><IT>2</IT></SUB>CO<SUB><IT>3</IT></SUB> <LIM><OP><ARROW>⇄</ARROW></OP><LL><IT>k</IT><SUB>h</SUB></LL><UL><IT>k</IT><SUB>d</SUB></UL></LIM> H<SUB><IT>2</IT></SUB>O<IT>+</IT>CO<SUB><IT>2</IT></SUB>
where dissociation of H2CO3 is rapid, and assumed to be at equilibrium. Since HCO3- and H2CO3 are interconverted, mass conservation requires
s<SUB>&agr;</SUB>(HCO<SUP>−</SUP><SUB>3</SUB>)<IT>+s<SUB>&agr;</SUB></IT>(H<SUB><IT>2</IT></SUB>CO<SUB><IT>3</IT></SUB>)

= V<SUB><IT>&agr;</IT></SUB>[<IT>k</IT><SUB>h</SUB>C<SUB><IT>&agr;</IT></SUB>(CO<SUB><IT>2</IT></SUB>)<IT>−k</IT><SUB>d</SUB>C<SUB><IT>&agr;</IT></SUB>(H<SUB><IT>2</IT></SUB>CO<SUB><IT>3</IT></SUB>)] (23)
for alpha  = I or E, whereas for the tubule lumen
s<SUB>M</SUB>(HCO<SUP>−</SUP><SUB>3</SUB>)<IT>+s</IT><SUB>M</SUB>(H<SUB><IT>2</IT></SUB>CO<SUB><IT>3</IT></SUB>)

=A<SUB>M</SUB>[<IT>k</IT><SUB>h</SUB>C<SUB>M</SUB>(CO<SUB><IT>2</IT></SUB>)<IT>−k</IT><SUB>d</SUB>C<SUB>M</SUB>(H<SUB><IT>2</IT></SUB>CO<SUB><IT>3</IT></SUB>)] (24)
In each compartment (alpha  = I, E, or M), conservation of total CO2 is expressed as
s<SUB>&agr;</SUB>(HCO<SUP>−</SUP><SUB>3</SUB>)<IT>+s<SUB>&agr;</SUB></IT>(H<SUB><IT>2</IT></SUB>CO<SUB><IT>3</IT></SUB>)<IT>+s<SUB>&agr;</SUB></IT>(CO<SUB><IT>2</IT></SUB>)<IT>=0</IT> (25)
Corresponding to conservation of protons is the equation for conservation of charge for all the buffer reactions
<LIM><OP>∑</OP><LL>i</LL></LIM> z<SUB>i</SUB>s<SUB>&agr;</SUB>(i)=0 (26)
where zi is the valence of species i. In this model, conservation of charge for the buffer reactions takes the form
s<SUB>&agr;</SUB>(H<SUP>+</SUP>)<IT>+s<SUB>&agr;</SUB></IT>(NH<SUP>+</SUP><SUB>4</SUB>)<IT>−s<SUB>&agr;</SUB></IT>(HCO<SUP>−</SUP><SUB>3</SUB>)<IT>−s<SUB>&agr;</SUB></IT>(HPO<SUP>2−</SUP><SUB>4</SUB>)<IT>=0</IT> (27)
The solute equations are completed with the chemical equilibria of the buffer pairs: HPO42-:H2PO4-, NH3:NH4+, and HCO3-:H2CO3. Corresponding to the electrical potentials, psi alpha , for alpha  = E, I, or M, is the equation for electroneutrality
<LIM><OP>∑</OP><LL>i</LL></LIM> z<SUB>i</SUB>C<SUB><IT>&agr;</IT></SUB>(<IT>i</IT>)<IT>=0</IT> (28)
With respect to water flows, volume conservation equations for lumen, interspace, and cell can be used to compute the three unknowns: luminal volume flow, lateral interspace hydrostatic pressure, and cell volume. (Cell hydrostatic pressure is set equal to luminal pressure; total cell impermeant content is assumed fixed.) This approach has been adopted for the epithelial model with fixed peritubular conditions but is not satisfactory for the tubule for which the large variations in peritubular osmolality would impact unrealistically on cytosolic electrolytes. As utilized previously in modeling the IMCD (70), the approach to the tubule model has been to restrict simulations to steady-state problems and to assume that cell volume homeostasis has been achieved by adjustment of an impermeant osmolyte, b. Thus with cell volume specified and fixed, CI(b) is the model variable used to satisfy the equations for fluid balance across the luminal and peritubular cell membranes. Across each cell membrane, the volume fluxes are proportional to the hydrosmotic driving forces. With respect to the lateral interspace, its volume, VE, and its basement membrane area, AES, are functions of interspace hydrostatic pressure, PE
<FR><NU>V<SUB>E</SUB></NU><DE>V<SUB>E<IT>0</IT></SUB></DE></FR><IT>=</IT><FR><NU><IT>A</IT><SUB>ES</SUB></NU><DE><IT>A</IT><SUB>ES<IT>0</IT></SUB></DE></FR><IT>=1.0+&ngr;</IT><SUB>E</SUB>(P<SUB>E</SUB><IT>−</IT>P<SUB>S</SUB>) (29)
where VE0 and AES0 are reference values for volume and outlet area, respectively, and nu E is a compliance.

Solute transport is either electrodiffusive (e.g., via a channel), coupled to the electrochemical potential gradients of other solutes (e.g., via a cotransporter or an antiporter), or coupled to metabolic energy (via an ATPase). This is expressed in the model by the flux equation
J<SUB>&agr;&bgr;</SUB>(i)=h<SUB>&agr;&bgr;</SUB>(i)&zgr;<SUB>&agr;&bgr;</SUB>(i)<FENCE><FR><NU>C<SUB><IT>&agr;</IT></SUB>(<IT>i</IT>)<IT>−</IT>C<SUB><IT>&bgr;</IT></SUB>(<IT>i</IT>)<IT>e</IT><SUP>−&zgr;<SUB><IT>&agr;&bgr;</IT></SUB>(<IT>i</IT>)</SUP></NU><DE><IT>1−e</IT><SUP>−<IT>&zgr;<SUB>&agr;&bgr;</SUB></IT>(<IT>i</IT>)</SUP></DE></FR></FENCE>

+<LIM><OP>∑</OP><LL>j</LL></LIM> L<SUB>&agr;&bgr;</SUB>(i, j)[<A><AC>&mgr;</AC><AC>&cjs1171;</AC></A><SUB>&agr;</SUB>(j)−<A><AC>&mgr;</AC><AC>&cjs1171;</AC></A><SUB>&bgr;</SUB>(j)]+J<SUP>act</SUP><SUB>&agr;&bgr;</SUB>(<IT>i</IT>) (30)
In Eq. 30, the first term is the Goldman relation for ionic fluxes, where halpha beta (i) is a solute permeability, and Calpha (i) and Cbeta (i) are the concentrations of i in compartments alpha  and beta , respectively. Here
&zgr;<SUB>&agr;&bgr;</SUB>(i)=<FR><NU>z<SUB>i</SUB>F</NU><DE>RT</DE></FR> (&psgr;<SUB>&agr;</SUB>−&psgr;<SUB>&bgr;</SUB>) (31)
is a normalized electrical potential difference, where zi is the valence of i, and psi alpha  - psi beta is the potential difference between compartments alpha  and beta . The second term of the solute flux equation specifies the coupled transport of species i and j according to linear nonequilibrium thermodynamics, where the electrochemical potential of j in compartment alpha  is
<A><AC>&mgr;</AC><AC>&cjs1171;</AC></A><SUB>&agr;</SUB>(j)=RT ln[C<SUB><IT>&agr;</IT></SUB>(<IT>j</IT>)]<IT>+z<SUB>j</SUB>F&psgr;<SUB>&agr;</SUB></IT> (32)
For each of these transporters, the assumption of fixed stoichiometry for the coupled fluxes allows the activity of each transporter to be specified by a single coefficient. The exception to this representation of coupled fluxes is that of Cl-/HCO3- exchange across the peritubular membrane, referable to AE1. Here the kinetic model developed above has been used, so that a single transporter density parameter suffices to represent its activity.

In this model, there are two proton ATPases within the luminal cell membrane. The H-K-ATPase is identical to that which has been developed for the model of the IMCD (71), with only the transporter density adjusted to suit the change in context. Also in earlier work, an empiric expression representing the H+- ATPase was devised by Strieter et al. (61), approximating data of Andersen et al. (2) for turtle bladder
J(H<SUP>+</SUP>)<IT>=J</IT>(H<SUP>+</SUP>)<SUB>max</SUB>

×[1.0+exp[<IT>&xgr;·</IT>(<IT><A><AC>&mgr;</AC><AC>&cjs1171;</AC></A></IT><SUB>MI</SUB>(H<SUP>+</SUP>)<IT>−<A><AC>&mgr;</AC><AC>&cjs1171;</AC></A><SUB>0</SUB></IT>)]]<SUP>−1</SUP> (33)
where J(H+)max is the maximum proton flux, and <A><AC>&mgr;</AC><AC>¯</AC></A>MI(H+) is the electrochemical potential difference of H+ from the cytosol to the lumen; xi MI defines the steepness of the function, and <A><AC>&mgr;</AC><AC>¯</AC></A>0 defines the point of half-maximal activity. The important finding of Andersen et al. (2) 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. (figure 9 in Ref. 2) are approximately represented by choosing xi  = 0.4 and <A><AC>&mgr;</AC><AC>¯</AC></A>0 -4.0 J/mmol. Figure 5 illustrates the response of each of these proton pumps to changes in luminal and cytosolic conditions in a neighborhood of a reference condition: lumen and cell pH, 7.34, lumen and cell K+, 45 and 130 mmol/l, and transmembrane potential difference (PD), 42 mV. The pump densities were taken so that at the reference point, the contributions of each transporter were equal. In Fig. 5, left, luminal pH is varied while cytosolic conditions are fixed. Transport by the H+-ATPase increases with luminal alkalinization and decreases nearly 90% with acidification of the lumen by 1 pH unit. In this model H-K-ATPase, transport is predicted to be quite insensitive to luminal pH near the reference, only declining after a 2 pH unit reduction. In Fig. 5, right, cytosolic pH has been varied, and it is apparent that both ATPases are relatively sensitive to small changes in cell pH.


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Fig. 5.   Proton fluxes as a function of luminal and cytosolic conditions in the neighborhood of a reference condition: lumen and cell pH, 7.34; lumen and cell K+, 45 and 130 mM, respectively; and transmembrane potential difference (PD), 42 mV. Pump densities were chosen so that at the reference point, fluxes through each transporter were equal. Left: luminal pH is varied while cytosolic conditions are fixed. Right: cytosolic pH is the independent variable.

Within the peritubular membrane, the Na-K- ATPase is represented by the expression
J<SUP>act</SUP><SUB>IE</SUB>(Na<SUP>+</SUP>)<IT>=</IT>[<IT>J</IT><SUP>act</SUP><SUB>IE</SUB>(Na<SUP>+</SUP>)]<SUB>max</SUB>

×<FENCE><FR><NU>C<SUB>I</SUB>(Na<SUP>+</SUP>)</NU><DE>C<SUB>I</SUB>(Na<SUP>+</SUP>)<IT>+K</IT><SUB>Na</SUB></DE></FR></FENCE><SUP><IT>3</IT></SUP><FENCE><FR><NU>C<SUB>E</SUB>(K<SUP>+</SUP>)</NU><DE>C<SUB>E</SUB>(K<SUP>+</SUP>)<IT>+K</IT><SUB>K</SUB></DE></FR></FENCE><SUP><IT>2</IT></SUP>
in which the half-maximal Na+ concentration, KNa, increases linearly with internal K+, and the half-maximal K+ concentration, KK, increases linearly with external Na+ (19). The pump flux of K+ plus NH4+ reflects the 3:2 stoichiometry
J<SUP>act</SUP><SUB>IE</SUB>(K<SUP>+</SUP>)<IT>+J</IT><SUP>act</SUP><SUB>IE</SUB>(NH<SUP>+</SUP><SUB>4</SUB>)<IT>=</IT>−(<IT>2/3</IT>)<IT>J</IT><SUP>act</SUP><SUB>IE</SUB>(Na<SUP>+</SUP>) (35)
with the transport of either K+ or NH4+ determined by their relative affinities, KK and KNH+4
<FR><NU>J<SUP>act</SUP><SUB>IE</SUB>(NH<SUP>+</SUP><SUB>4</SUB>)</NU><DE><IT>J</IT><SUP>act</SUP><SUB>IE</SUB>(K<SUP>+</SUP>)</DE></FR><IT>=</IT><FR><NU>C<SUB>E</SUB>(NH<SUP>+</SUP><SUB>4</SUB>)</NU><DE><IT>K</IT><SUB>NH<SUP><IT>+</IT></SUP><SUB><IT>4</IT></SUB></SUB></DE></FR><IT>·</IT><FR><NU><IT>K</IT><SUB>K</SUB></NU><DE>C<SUB>E</SUB>(K<SUP>+</SUP>)</DE></FR> (36)
Analogous expressions are written for active transport at the basal cell membrane, JISact.


    MODEL PARAMETERS
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ABSTRACT
INTRODUCTION
MODEL AE1
MODEL OMCD
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The parameters displayed in Table 2 were selected so that the model tubule might correspond most closely to the OMCD of the rat. Where rat data were not available, rabbit measurements were used for guidance. With respect to acidification, there seems to be little to distinguish the outer and inner stripes of the rat OMCD: reported proton secretory rates in vitro (in pmol · mm-1 · min-1) for the outer stripe [10.2 (5), 22.1 (18), and 37.6 (22)] and for the inner stripe [24.4 (15) and 13.1 (47)] are similar and cover a broad range; the fractional content of intercalated cells appears to be about 35% for both segments (26, 51, 57); and there is no evidence in the rat for the presence of membrane-bound carbonic anhydrase (CA-IV), either from histochemical (9) or functional studies (15). [This is in contrast to the rabbit, for which membrane-bound CA appears to be present in the inner stripe but not the outer stripe (55)]. Thus, in view of the relatively short length of the outer stripe of rat OMCD (0.5 mm), compared with the inner stripe (1.5 mm) (32), the whole tubule has been approximated as a uniform 2-mm segment with a 30-µm inner diameter. For a tubule thickness of 9 µm, the 35% intercalated cell fraction corresponds to an intercalated cell volume (VI) of about 0.3 × 10-3 cm3/cm2 of epithelium. Estimates of intercalated cell surface area suggest a ratio of peritubular to luminal membrane of between 3 and 4 to 1 and an absolute luminal membrane area of about 2 cm2/cm2 epithelium (45, 51, 57). The volume of the lateral intercellular space was taken to be about 10% of the epithelial volume (with a relatively small compliance), a value comparable to that observed in cortical collecting duct (72).

                              
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Table 2.   OMCD parameters

Figure 6 depicts several of the important cellular transport pathways. Both H-K-ATPase and H+- ATPase are contained within the luminal membrane of OMCD (73), and the rates of proton secretion by H-K-ATPase relative to H+-ATPase have been identified in both rat [2.5 (22)] and rabbit [0.7 (4), 0.8 (64), 1.0 (68), and 2.0 (3)]. To select pump densities for the model OMCD, proton transport via the two ATPases was set approximately equal at neutral luminal pH, and relative activity of the pumps as a function of luminal pH is explored in the model calculations. There is no evidence for any other coupled transport pathway within the luminal membrane, and in the rabbit OMCD, electrophysiological study indicates no significant luminal membrane conductance (33, 34, 46). Furthermore, there are no detectable aquaporin AQP-2 water channels in the luminal cell membrane of intercalated cells in the rat (17, 48). Accordingly, the total luminal membrane water permeability was set at 1% of the peritubular membrane water permeability. In view of the intense staining for carbonic anhydrase within OMCD cells (44), the rate constants for full catalysis (10,000-fold increase) were assumed for the cytosolic compartment.


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Fig. 6.   OMCD cellular transport pathways, along with model cell fluxes (pmol · s-1 · cm-2) computed for luminal and peritubular conditions representing corticomedullary junction (Table 3).

The peritubular membrane of rat OMCD contains Na-K-ATPase (52), and its density was selected to obtain a suitably low cytosolic Na+ concentration. In addition to the Cl-/HCO3- exchanger, a peritubular Na+/H+ exchanger is present in rabbit OMCD (8). It has been demonstrated in intercalated cells (in addition to principal cells), where it is capable of proton extrusion rates comparable to that of the proton pumps (41, 69). Its density coefficient was selected to yield fluxes comparable to those of the H+-ATPase. The model peritubular membrane also contains a coupled phosphate transporter, which maintains a small entry flux. Although the total conductance of the peritubular membrane in rabbit OMCD intercalated is unknown, it has been established that the principal ion permeability is that for chloride (33, 34), while that for potassium is much smaller (46). Whereas a variety of chloride channels show substantial bicarbonate conductance (39, 50), bicarbonate conductance of the intercalated cell peritubular membrane has not been demonstrated. Koeppen (33) did find significant steady-state membrane depolarization with reduction in peritubular HCO3-, but the time course was slow, and no rapid depolarization was evident. The peritubular chloride permeability for the model cell was estimated from the constraints of cell PD (-30 to -40 mV), a suitable cell chloride concentration, and the need to recycle all of the Cl- uptake through AE1 back out through this channel. Potassium permeability was taken as <FR><NU>1</NU><DE>10</DE></FR> that of chloride, bicarbonate permeability as <FR><NU>1</NU><DE>8</DE></FR> that of chloride, and NH4+ permeability as 1/4 that of potassium.

Overall epithelial electrical conductance (in mS/cm2) has been measured in OMCD only for rabbit and was found to be slightly higher in outer stripe [3.7 (34) and 5.7 (36)] than in inner stripe [1.9 (33), 2.2 (36), and 3.4 (46)]. These conductances are compatible with estimates of ion permeability, PNa:PK:PCl = 3.9:5.9:4.8 × 10-6 cm/s (27, 38, 58). In the rat, OMCD NH4+ permeability is 1.3 × 10-5 cm/s (14). Presumably, these epithelial ion permeabilities reflect the properties of the OMCD tight junctions. For the selection of model tight junction solute permeabilities, it has been assumed that OMCD K+ permeability is approximately that of NH4+ and that the relative ion permeabilities in rat are comparable to those of rabbit. This yields an overall epithelial conductance for rat OMCD about twice that of rabbit. The interspace basement membrane conductance was assumed to be about two orders of magnitude greater than that of the tight junction, and solute permeabilities were proportional to diffusivity in free solution.

Membrane permeabilities have also been assigned for the non-ionic species: water, urea, NH3, CO2, and H2CO3. In the rabbit, antidiuretic hormone (ADH)-stimulated OMCD water permeability has been reported as 0.046 cm/s, an increase about 30-fold above the unstimulated permeability (29). Although a value for rat OMCD is not available, water permeabilities for the two species are comparable in cortical collecting tubule. For the model calculations, a water permeability about half-maximal was assumed and referred entirely to the "paracellular" pathway (which includes the principal cell-lateral interspace route). All membrane and tight junction reflection coefficients are assumed to be 1.0, while those for interspace basement membrane are 0.0. The overall urea permeability has been measured for rat OMCD (3.5 × 10-5) and is about 10-fold greater than that for rabbit (23). In the absence of information about the transepithelial route for urea permeation, for this model, 45% of the epithelial permeability has been ascribed to the intercalated cell (with uniform unit membrane urea permeability), and the remainder paracellular. With respect to NH3, the rat OMCD permeability, 0.012 cm/s (14), is sufficiently high to reflect diffusion limitation across the cellular layer, rather than membrane limitation. Within the scope of this model, it suffices to ascribe this permeability to cell membranes, with uniform unit membrane permeability. This avoids creating a paracellular NH3 pathway wherein one presumes the (unrealistic) routing of the bulk of the NH3 flux through the lateral interspace. Similar concerns apply to CO2, so that CO2 permeabilities have been assumed equal to those of NH3. H2CO3 has been assumed to permeate at 1% the rate of CO2.


    MODEL CALCULATIONS
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ABSTRACT
INTRODUCTION
MODEL AE1
MODEL OMCD
MODEL PARAMETERS
MODEL CALCULATIONS
DISCUSSION
REFERENCES

Table 3 and Fig. 6 display the solution of the equations for the epithelial model of OMCD with lumen and bath conditions suggestive of the corticomedullary junction. Overall, the lumen is isotonic to blood, with a urea concentration comparable to that from a cortical nephron (23). The concentrations of Na+, K+, and Cl- are within the range reported for the last accessible micropuncture site (7). The higher values used here reflect the transition to isotonicity (via water abstraction) within the cortex of the antidiuretic kidney. From another perspective, in later calculations the total volume flow into the model OMCD will be assumed to be 7.2% of glomerular filtration rate (GFR), or 36 µl/min. With this assumption, the concentrations chosen correspond to Na+ and K+ delivery to this segment of 3.6% and 65% of filtered loads. The HCO3- concentration, 10 meq/l, corresponds to a delivery of 2.9% of filtered load, which may be compared with 6.4% delivery found at the last micropuncture site (11). The NH4+ concentration, 2 meq/l, also yields a delivered load close to that reported for the rat (53). The luminal total phosphate concentration corresponds to approximately 85% fractional reabsorption in proximal nephron.

                              
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Table 3.   Model solution for corticomedullary junction

The computed concentrations of Na+, K+, and Cl- within the model cell (Table 3) are within the range of values obtained by microprobe for intercalated cells of the cortical collecting duct (7, 21). The peritubular membrane PD, -35.6 mV, is comparable to values obtained in intercalated cells of the OMCD of the rabbit (30, 36). The open-circuit PD of the model, +1.2 mV, is low, but this PD also reflects the negative Na+ diffusion potential (which dominates the positive K+ diffusion potential). In a calculation in which the luminal ion composition is identical to that of the bath, the open-circuit PD is +4.8 mV. These values are consistent with the PD determinations in rat OMCD [+2.5 mV (22) and +1.05 mV (15)] as well as some measurements in rabbit (4), albeit lower than other observations in rabbit OMCD [+16.5 mV (36) and +10.6 mV (59)]. The total proton secretory rate, 594 pmol · s-1 · cm-2 (34 pmol · min-1 · mm-1), increases when the composition of the luminal perfusion solution is identical to that of the peritubular bath (characteristic of in vitro tubule studies), 722 pmol · s-1 · cm-2 (41 pmol · min-1 · mm-1). These values are high even for the rat [24.4 pmol · min-1 · mm-1 (15)], but as will be indicated below, this rate of proton secretion is only just capable of titrating the base delivery to the OMCD.

In view of the large transepithelial solute concentration gradients across OMCD, the overall epithelial solute permeabilities have a substantial impact on luminal solute flows. Table 4 displays the results of simulating idealized epithelial permeability determinations. For these calculations, a short-circuited tubule epithelium in vitro was represented, bathed by equal luminal and peritubular solutions of composition (mmol/l) 140 Na+, 10 K+, 119 Cl-, 25 HCO3-, 1.5 CO2, 3.9 total phosphate, 5.0 urea, 1.0 NH4+, and 0.1 impermeant. A series of calculations were performed in which each luminal solute concentration in turn was lowered and then raised by 0.1 mmol/l. The change in solute flux relative to the change in concentration is listed in Table 4 as the permeability, HM(i) (in cm/s), and is the average of the two determinations. Alternatively, epithelial ion permeability was determined by imposing a transepithelial voltage (positive and negative 0.1 mV). The change in ion flux relative to voltage, when multiplied by z(i)F is the partial conductance, GM, shown on the right (in mS/cm2). The permeabilities displayed are, by design, comparable to those cited in the previous section. In particular, these permeabilities predict a substantial secretory flux of sodium, as well as significant reabsorption of potassium and urea by OMCD.

                              
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Table 4.   Model epithelial permeabilities

Table 5 and Figure 7 display the predictions of the model OMCD configured as a 2-mm tubule within the renal medulla. Initial conditions are those already specified at the corticomedullary junction, with an inlet volume flow of 5 nl/min for each tubule, or 36 µl/min for the 7,200 OMCD of the rat. The peritubular composition at the endpoint includes a doubling of Na+ and K+ concentrations, along with an increase of NH4+ to 3 mmol/l and an increase of urea to 20 mmol/l. Peritubular composition at intermediate points are determined by linear interpolation. The differential equations of mass conservation were cast as a centered difference scheme and integrated with a mesh of 40 points; accuracy was confirmed by refining this spacing. Table 5 indicates that osmotic equilibration was nearly complete with reabsorption of about half the entering volume flow. There is secretion of Na+ equal to 25% of the delivered load, with reabsorption of approximately 40% of entering potassium and 60% of entering urea. [Fortuitously, the absolute rates of Na+ absorption and K+ secretion are nearly equal, and thus resemble Stokes' (58) observation of OMCD Na+ and K+ fluxes, but bath and perfusion conditions here are quite different from those experiments.] Proton secretion along the OMCD results in reabsorption of 50% of delivered HCO3- (24 pmol/min), more than doubling of luminal NH4+ (13 pmol/min), and titration of luminal HPO42- (8 pmol/min). The axial profiles of pH, HCO3-, and NH4+ are shown in Fig. 7. Within the first 0.6 mm of OMCD, a disequilibrium pH of about 0.4 units is developed, and NH3 influx into this portion of the tubule is sluggish. In the latter portion of OMCD, where the acid lumen is fully developed, NH3 influx is linear, reflecting the increase in peritubular concentration. Luminal acidification results in a small decrease in the rate of proton secretion, largely due to its impact on transport by the H+-ATPase.

                              
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Table 5.   OMCD solute reabsorption



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Fig. 7.   Predicted acidification along the OMCD configured as a 2-mm tubule within the renal medulla. Initial conditions are those of the corticomedullary junction, with an inlet volume flow of 5 nl/min (Table 5). Left: calculations using baseline parameters, in which hydration of CO2 within the lumen is uncatalyzed. Right: there is a 10,000-fold increase in this reaction rate. For all panels, the abcissa is position along the tubule. Top: lumen pH. Middle: luminal concentrations of HCO3- and NH4+. Bottom: total proton secretion and that referable to the H+-ATPase. CA, carbonic anhydrase.

The case in which luminal CA is present is considered in Fig. 7, right. Proton secretion now consumes 75% of delivered HCO3- (38 pmol/min), but makes a smaller contribution to increase luminal NH4+ (8 pmol/min) and titrates luminal HPO42- (6 pmol/min). Thus the rate of net acid secretion in the presence of luminal CA (52 pmol/min) exceeds that in the absence of CA (45 pmol/min), but the differences are not predicted to be striking. As indicated in the previous section, the rate coefficient for the hydration of CO2 in the cytosol has been set at 10,000 times the uncatalyzed value. For the rate of proton secretion by OMCD, calculations with this model indicate that any value for the hydration rate coefficient greater than 1% of the selected value (i.e., anything greater than 100 times the uncatalyzed value) yields virtually identical cytosolic pH, and thus no impact on luminal proton secretion. Within the lateral interspace, no catalysis of CO2 hydration has been assumed. With reference to Table 3, an acid disequilibrium pH of 0.14 units develops, and this is due to the activity of the Na+/H+ exchanger within the peritubular membrane. Model calculations in which full CA catalysis is extended to the lateral interspace have no significant impact on luminal acid secretion but do increase NH3 secretion by removing an intraepithelial acid compartment. The NH3 fluxes of Table 3 describe a situation in which acidification of the interspace shunts NH3 taken up across the basal cell membrane and returns NH4+ back to the peritubular blood. This effect appears to be largely due to the small NH3 fluxes under the assumed corticomedullary junction conditions, but its impact diminishes rapidly in importance as peritubular NH4+ concentrations increase along the medulla.

In view of the uncertainty of the outer medullary interstitial NH4+ concentration profile, tubule transport is examined over a a range of values for peritubular NH4+. This is done in the calculations of Fig. 8, which utilize the tubule model with the same initial conditions as in Fig. 7. In these calculations, peritubular NH4+ remains at 1 mmol/l at x = 0, while the concentration at x = 2 mm is varied from 1 to 9 mmol/l. For each simulation, interstitial NH4+ concentration along the tubule is the linear interpolation between start and endpoints. Figure 8, top, displays the end-luminal HCO3- and NH4+ concentrations; the middle shows end-luminal pH; and the bottom contains the three components of acid excretion by a single OMCD segment: HCO3- reabsorption, NH4+ addition, and HPO42- titration. It is clear that with the increase in peritubular NH4+, there is progressive buffer addition to the tubule lumen and progressive tubule fluid alkalinization. At all but the lowest peritubular NH4+ concentrations, there is an increase in end-luminal HCO3- concentration above that at the inlet (10 mmol/l). The curve labeled "total proton secretion" is the sum of the three buffer changes, and is relatively insensitive to changes in peritubular NH4+. Thus, availability of ammonia buffer can shift the luminal composition substantially but appears to have little impact on total proton secretion by OMCD. There is also uncertainty regarding the outer medullary interstitial urea profile, and Table 6 displays the results of exploring OMCD urea reabsorption over a range of conditions. As with the NH4+ calculations, the initial conditions remain those of Fig. 7, while the peritubular urea concentration at x = 2 mm is varied from 20 to 100 mmol/l; interstitial urea concentrations are the linear interpolation. It is clear that at all concentrations, the model predicts substantial outer medullary urea reabsorption.


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Fig. 8.   Acid/base transport by OMCD: effect of peritubular NH4+. The model OMCD is solved as terminal (x = 2 mm) peritubular NH4Cl is varied from 1 to 9 mmol/l, and this concentration is taken as the abcissa. Other conditions for lumen and peritubular bath are as in Fig. 7. Top: end-luminal concentrations of HCO3- and NH4+. Middle: pH. Bottom: resolves total proton secretion into its component buffer titrations: changes in axial flow of HCO3-, HPO42-, and NH4+ along the tubule.


                              
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Table 6.   OMCD urea reabsorption

Although pH dependence of the important OMCD transporters has been illustrated above (Figs. 3 and 5), in the calculations that follow, the epithelial model has been used to examine interactions of these components, with emphasis on possible regulatory sites. The impact of luminal HCO3- concentration on OMCD proton secretion is displayed in Fig. 9. The bath conditions are those of the corticomedullary junction (Table 3), with the exception of luminal HCO3--for-Cl- substitution. Figure 9 includes peritubular PD, cytosolic pH, and the components of luminal proton secretion and peritubular base exit (lateral plus basal terms). By design, the contributions of H+-ATPase and H-K- ATPase are nearly equal under baseline conditions (HCO3- = 10 mmol/l). The curve labeled "base exit" is equal in magnitude to the "total" proton secretion, but it is clearly less than the sum of HCO3- fluxes via AE1 and the conductive pathway. This is due to the presence of peritubular proton reabsorption by the Na+/H+ exchanger. Over the range of luminal HCO3- concentrations (1.0 to 50 mmol/l), there is a relatively steep dependence of flux through the H+-ATPase. With luminal acidification, H+-ATPase flux is nearly shut off, resulting in cell acidification and peritubular depolarization. Since the H-K-ATPase is stimulated by the decrease in cytosolic pH and is relatively insensitive to lumen pH, the decrease in lumen HCO3- actually results in a small increase in flux through this transporter. At the peritubular membrane, HCO3- fluxes through the two exit pathways move (fortuitously) in parallel, although the driving forces for each pathway are distinct. In Figs. 10-13, the density of each of the four important components has been varied over approximately 100-fold range, from 0.03 to 3.0 times the baseline value, and a transport tableau similar to that of Fig. 9 has been generated. In all of these calculations, the bath conditions are those of the corticomedullary junction. In Fig. 10, the density of the luminal membrane H+-ATPase has been varied. The curves are virtually identical to those of Fig. 9, confirming the impact of luminal pH as solely through this transporter. In Fig. 11, the density of the H-K-ATPase has been varied, and the effect is more complex. Although total proton secretion is nearly identical, in the absence of the electrical impact of the H+-ATPase, there is peritubular membrane hyperpolarization with the decrease in H-K-ATPase activity. This is due to the decrease in cytosolic Cl- concentration coincident with the decrease in cell HCO3-. The hyperpolarization exacerbates the cytosolic acidosis associated with the decrease in luminal proton secretion. In short, modulating the H+-ATPase results in less derangement of cell composition than modulating the H-K-ATPase. Figure 12 contains results of calculations in which the AE1 density is varied. As expected, decreasing AE1 alkalinizes the cell and shifts HCO3- exit to the conductive pathway. Since both of the luminal proton transporters are sensitive to cytosolic pH, changes in AE1 activity modulate total proton secretion. Coincident with the decrease in AE1, there is a decrease in cytosolic Cl-, and as in Fig. 11, peritubular hyperpolarization. Finally, Fig. 13 examines the effect of modulating the conductive pathway. Since there is no evidence for separate HCO3- and Cl- channels, the peritubular anion permeability for each species has been varied proportionally. As with changes in AE1, decreases in HCO3- conductance alkalinize the cell and decrease proton secretion. The peritubular membrane hyperpolarizes because of the decrease in Cl- conductance (despite the increase in cell Cl-), due to the relatively greater contribution of K+ to total membrane conductance. Ultimately, increases in cell Cl- start to diminish the AE1 flux.


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Fig. 9.   alpha -Intercalated cell function: variation of luminal HCO3-. Calculations use the OMCD epithelial model. Bath conditions are those of the corticomedullary junction (Table 3), with the exception of luminal HCO3--for-Cl- substitution. Peritubular PD, cytosolic pH, and the components of luminal proton secretion and peritubular base exit (lateral plus basal terms) are shown.



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Fig. 10.   alpha -Intercalated cell function: variation of luminal H+-ATPase. Calculations use the OMCD epithelial model with corticomedullary junction bath conditions. Luminal membrane H+-ATPase density, relative to control, is plotted on the abcissa, and the tableau of dependent cellular variables is as in Fig. 9.



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Fig. 11.   alpha -Intercalated cell function: variation of luminal H-K-ATPase. Calculations use the OMCD epithelial model with corticomedullary junction bath conditions. Luminal membrane H-K-ATPase density, relative to control, is plotted on the abcissa, and the tableau of dependent cellular variables is as in Fig. 9.



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Fig. 12.   alpha -Intercalated cell function: variation of peritubular AE1. Calculations use the OMCD epithelial model with corticomedullary junction bath conditions. Peritubular membrane AE1 density, relative to control, is plotted on the abcissa, and the tableau of dependent cellular variables is as in Fig. 9.



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Fig. 13.   alpha -Intercalated cell function: variation of peritubular bicarbonate conductance, g(HCO3-), and chloride conductance, g(Cl-). Calculations use the OMCD epithelial model with corticomedullary junction bath conditions. Peritubular membrane Cl- and HCO3- permeabilities are varied proportionally, and their value, relative to control, is plotted on the abcissa. The tableau of dependent cellular variables is as in Fig. 9.

This epithelial model affords the opportunity to consider the impact of modulating proton secretion on the volume of the intercalated cells. For the calculations of Figs. 10-13, the predicted cell volumes (relative to baseline) are displayed in Fig. 14. It is clear that changing the rate of transport by changing the H-K-ATPase density buys a new set of problems. Doubling this transporter density (about a 20% increase in total proton transport) results in a near doubling of cell volume. In contrast, changing the flux through the H+-ATPase leaves cell volume virtually unchanged. Indeed, the hyperpolarization associated with the increase in flux through this transporter produces a slight decrease in cell volume. Changes in AE1 density produce relatively modest parallel changes in cell volume. Modulating peritubular anion permeability shows less of an effect on cell volume when this pathway is increased, due to the concomitant Cl- exit. However, conductances below about 30% of baseline start to yield steep increases in cell volume. In proximal tubule, it has been shown that changes in flux through the Na-K-ATPase modulate the (dominant) peritubular K+ conductance via changes in cytosolic ATP concentration and the effect of ATP to shut the K+ channel (63). In Fig. 14, top left, the effect of coordinate regulation of H-K-ATPase and peritubular K+ permeability are considered. For these calculations, the three-fold increase in H-K-ATPase density is accompanied by a two-fold increase in peritubular K+ permeability, and for each 17% decrease in H-K-ATPase density there is a concomitant decrease of about 10% in the K+ permeability. The result is nearly perfect cell volume homeostasis on the side of increasing H-K-ATPase. This modulated K+ permeability also proves to be homeostatic with respect to cytosolic pH, as seen by comparing the acid/base tableau of Fig. 15 with that of Fig. 11. It also shifts the H-K-ATPase increase from depolarizing to hyperpolarizing and thus enhances total base exit.


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Fig. 14.   alpha -Intercalated cell volume: modulation of luminal and peritubular membrane transporters. For the calculations of Figs. 10-13, the predicted cell volumes (relative to baseline) are displayed in the respective panel. The additional curve in the H-K-ATPase panel [modulated g(K+)] is the result of a calculation in which peritubular K+ permeability is varied in proportion to the change in H-K-ATPase density (Fig. 15).



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Fig. 15.   alpha -Intercalated cell function: variation of luminal H-K-ATPase and peritubular K+ permeability. Calculations use the OMCD epithelial model with corticomedullary junction bath conditions. Luminal membrane H-K-ATPase density (abcissa) is varied as in Fig. 11, and the variation in peritubular membrane K+ permeability is two-thirds that of the pump density. The tableau of dependent cellular variables is as in Fig. 9.

To this point, all of the calculations have examined steady-state behavior of the OMCD. The model has been formulated, however, with inclusion of time-dependent accumulation terms in the mass balance Eqs. 17-19, and the computer code represents these terms in a centered difference scheme. This permits calculation of OMCD transients, and in particular, simulation of experiments in which cell pH is measured following acid loading. Figure 16 illustrates a protocol similar to that used by Kuwahara et al. (40), in which the contributions of the H+-ATPase and H-K-ATPase toward pH recovery are compared. In these calculations, the initial condition is a steady state in which ambient ammonia has been raised to 20 mM, luminal K+ has been reduced to near zero, and peritubular Na+/H+ exchange is blocked. Specifically, the peritubular solution is that of Table 3, with the exception of 19 mM NH4+ for Na+ substitution; the luminal solution is identical to the bath, with the exception of 4.95 mM Na+ for K+ substitution; and the Na+/H+ coefficient has been reduced to 0.1% of its value in Table 2. Since two of three of its acid extruders are nonfunctional, the OMCD pH is low (6.8). At t = 0, the ambient ammonia is restored to 1 mM, and the cell acidifies further, to nearly pH 6.0. At this point, the H+-ATPase is solely responsible for the observed recovery of approximately 0.01 pH unit per second. At t = 90, luminal K+ is restored, luminal H-K-ATPase is activated, and the recovery rate increases fivefold. The greater contribution of the H-K-ATPase could have been predicted from its greater sensitivity to cytosolic acidification (Fig. 3). Furthermore, in these calculations, the H+-ATPase is additionally hindered by cellular hyperpolarization to nearly -70 mV (due to low cell Cl-, consequent to low cell HCO3-). It is clear from these considerations that a number of factors (including the cell pH at which one chooses to restore luminal K+) can impact on the apparent contribution of the two proton pumps.


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Fig. 16.   Time course of cellular pH recovery from an acid load. The initial condition is a steady-state solution in which ambient ammonia has been raised to 20 mM, luminal K+ is near zero, and peritubular Na+/H+ exchange is blocked. At t = 0, ambient ammonia is restored to 1 mM. The cell rapidly acidifies and then, by t = 10, starts its pH recovery due to action of the H+-ATPase. The value of dpH/dt was computed as the slope at t = 30. At t = 90, luminal K+ is restored, luminal H-K-ATPase is activated, and the recovery rate increases fivefold (the slope at t = 93).


    DISCUSSION
TOP
ABSTRACT
INTRODUCTION
MODEL AE1
MODEL OMCD
MODEL PARAMETERS
MODEL CALCULATIONS
DISCUSSION
REFERENCES

This model, the first to represent OMCD, is, in essence, a model of the proton-secreting alpha -intercalated cell, in parallel with a lateral interspace. Although OMCD contains both principal cells and intercalated cells, no active transport activity has been ascribed to the principal cell. It must be acknowledged that although these two cell types are distinguished morphologically and immunocytochemically, only a single cell type has been recognized electrophysiologically in the inner stripe of OMCD (33, 46). For the purposes of this model, their data has been assumed applicable to the intercalated cell. The principal cell is most certainly the water pathway and should also serve as a route for CO2, H2CO3, NH3, and urea. To represent the epithelial permeabilities for these species, a full principal cell model was not considered necessary: water and urea permeabilities could be lumped within the paracellular pathway, and CO2, H2CO3, and NH3 fluxes could be attributed to a route through the intercalated cell without significant distortion of cytosolic solute concentrations. The four crucial membrane transporters for this model are the two luminal proton ATPases, H+-ATPase and H-K-ATPase, and the two peritubular base exit pathways, the Cl-/HCO3- exchanger and a Cl- channel that admits HCO3-. Proton transport by the H+-ATPase is a function of transmembrane proton-motive force, as described by Andersen et al. (2), and was cast into functional form by Strieter et al. (61) for a model of the alpha -intercalated cell of cortical collecting tubule. The H-K-ATPase of this model is that which had been developed for the IMCD (71) and had been an adaptation of the gastric H-K-ATPase by Brzezinski et al. (10). The representation of the Cl-/HCO3- exchanger of OMCD is new.

The peritubular membrane Cl-/HCO3- exchanger of the alpha -intercalated cell has been identified with the erythroid band 3 anion exchanger, AE1 (1, 54). Although this transporter has been the object of extensive experimental investigation and theoretical analysis (16, 49), it appears that no computable kinetic model for this transporter has, as yet, been formulated. This refers specifically to assigning parameter values within the scheme of binding constants and translocation rates depicted in Fig. 1. The present model was enabled by the recent experimental observations of Brahm and colleagues (20, 31), who systematically delineated the 38°C kinetics of the upper limb (HCO3- self-exchange) and the lower limb (Cl- self-exchange) of the ping-pong mechanism. Their work provided estimates of internal and external anion affinities along with maximal fluxes. Even with these extensive data, the model AE1 remains incompletely determined. Each limb of the scheme is defined by four parameters: two affinities plus forward and backward translocation rates. The experimental observations were sufficient to determine the model coefficients up to a single free parameter in each limb. For the parameter selection here (Table 1), the ratio of internal and external binding affinities was set at 1. This was based on information obtained from other chloride studies (42) and is presumed to be true for bicarbonate as well. The resulting model is thus completely compatible with all of the experimental observations (Fig. 2), but a family of such models could have been generated, by varying the bicarbonate affinity ratio. Parenthetically, in calculations not shown, exchange of internal Cl- for external HCO3- was remarkably insensitive to this ratio, and no useful experiments to determine this parameter could be devised.

In this model OMCD, peritubular base exit proceeds via both AE1 and via a conductive pathway, nearly equally under baseline conditions. The magnitude of the conductive chloride pathway is bounded from below by measurements of intercalated cell chloride concentration and peritubular PD and by the requirement that at least 50% of luminal proton secretion be matched by peritubular chloride flux (recycled from entry via AE1). The observation made in developing this model is that even if HCO3- permeability of the chloride channel is one-eighth of that for chloride, the driving forces are such that about one-half of the generated HCO3- should be reabsorbed via this pathway. Where specific measurements of single channel HCO3--to-Cl- permeability ratios have been obtained, a figure of 1:8 appears to be a conservative underestimate (39, 50). Furthermore, in cultured OMCD cells, a HCO3- conductance has been identified (37). Thus in this model, even when peritubular AE1 activity is reduced to near zero, model proton secretion decreases by only a one-third (Fig. 12). This model prediction, however, is at odds with the experimental observation that OMCD proton secretion is eliminated by removal of peritubular chloride or by application of a stilbene inhibitor of AE1 (60). It is also at odds with another study in which OMCD cell pH was monitored and removal of ambient chloride reduced peritubular HCO3- permeability by 90% (28). A number of explanations could be invoked to rationalize this important discrepancy. It is possible that the OMCD peritubular chloride channel is much more selective than others in favor of chloride. Alternatively, with peritubular chloride removal, either cell shrinkage or alkalinization might inhibit anion channel activity. It is also possible that much of the peritubular exit of chloride occurs in an electroneutral manner (e.g., KCl cotransport) so that the conductive pathway is much smaller in magnitude than estimated here. Resolution of this issue could be achieved with direct determination of whole cell conductance with and without the presence of peritubular chloride, and in the presence of AE1 inhibition. Indeed, Koeppen (35) speculated that the slow depolarization of SITS-inhibited OMCD cells might be due to loss of cell chloride via a KCl cotransporter.

Another aspect of these model calculations has been the different character of luminal membrane proton transport exhibited by the two proton ATPases. The H+-ATPase appears to be more sensitive to luminal pH so that flux through this transporter is nearly shut off when the lumen falls below pH 6.5 (Figs. 5 and 9). The model H-K-ATPase functions virtually undiminished until lumen pH falls below 5.0. Both pumps are sensitive to relatively small changes in cytosolic pH. Model calculations suggest that the H+-ATPase transport density could be modulated over a broad range without serious derangement of cell pH or cell volume (Figs. 10 and 14). In this model, the hyperpolarization associated with increased H+-ATPase activity acts directly to enhance peritubular Cl- and HCO3- exit and indirectly to enhance Cl-/HCO3- exchange. In contrast, modulating the H-K-ATPase density led to substantial swings in cell volume, becoming quite marked when the transport activity was increased. The difficulties associated with changes in H-K-ATPase activity derive from the relatively small peritubular K+ permeability of this model cell. However, this permeability is congruent with the experimental finding of a small, if not vanishing, K+ conductance (33, 46). One possible resolution to this difficulty was illustrated in model calculations (Figs. 14 and 15), namely, parallel activation of peritubular K+ permeability with changes in H-K-ATPase activity. Although this has precedent in proximal tubule pump-leak coupling via ATP, there appear to be important differences in OMCD. In OMCD, the H-K-ATPase is spatially removed from the K+ channel, and if ATP were the mediator, then one might expect confusing signals with changes in H+-ATPase activity. Alternatively, the derangements in cell volume with changing H-K-ATPase activity might be substantially mitigated if there were an important electroneutral K+ exit pathway (i.e., KCl cotransport).

Several calculations examined the impact of CA activity on OMCD transport. It was found that cytosolic CA was critical to maintain proton secretion at normal rates. At reaction rates less than 100-fold greater than the uncatalyzed rate of CO2 hydration, the cell alkalinized and both luminal proton pumps showed diminished transport. Catalysis was assumed to be absent in the lateral intercellular space, and by virtue of the peritubular Na+/H+ exchanger, the interspace became a region with an acid disequilibrium pH. In comparison with a model in which CO2 hydration in the interspace was fully catalyzed, the absence of CA here had no appreciable effect on OMCD proton secretion. This acidic intraepithelial compartment could, however, shunt cytosolic NH3 back to the peritubular bath. This effect was substantial only at low peritubular NH3 concentrations. In the rat, luminal CA is absent (9, 15), and the uncatalyzed rate of CO2 hydration was used in the model lumen. This is in contrast to the inner stripe OMCD of the rabbit, where CA is present and where inhibition abolishes the bulk [62% (55)], if not all (65) of bicarbonate reabsorption. The data of Tsuruoka and Schwartz (65) are particularly strong on this point, showing nearly full inhibition of HCO3- reabsorption with either benzolamide or a lumen-restricted CA inhibitor and restoring reabsorption by adding CA to the luminal perfusate. The present model OMCD cannot reproduce those results (Fig. 7). This is because the predicted luminal acid disequilibrium (0.4 pH unit) has no effect on H-K-ATPase and approximately a 55% effect on transport by the H+-ATPase. Luminal H-K-ATPase is found in rabbit OMCD (4, 64). It is possible that the reported effect of CA inhibition derives from proton pump kinetics different from those assumed here, or perhaps from a direct effect of the inhibitor on the pumps.

In the model calculations of this report, buffer was abundant, including delivered HCO3- and HPO42-, as well as peritubular NH3. As indicated by Flessner et al. (15), the OMCD is sufficiently permeable to NH3 that high rates of proton secretion can be sustained by virtue of NH3 availability alone. Peritubular NH4+ concentrations within outer medulla have not been determined, but measurements within papilla include 2.1 mmol/l (24) and 9.2 mmol/l (56), suggesting ambient concentrations well above those of systemic plasma. In this model, increasing peritubular NH4+ simply shifted the buffer composition of the end-OMCD urine but had little impact on the rate of total proton secretion (Fig. 8). This is in contrast to inner medullary acidification where NH4+ can serve as a proton donor, and increases in peritubular NH4+ enhance luminal proton secretion. In inner medulla, a significant fraction of proton secretion is dependent on NH4+ availability and thus independent of cytosolic CA (67, 71). Overall, total proton secretion by this model OMCD was approximately 50 pmol/min, or for a single kidney with 7,200 OMCD, 6.3 nmol/s. This may be compared with the estimate for total inner medullary proton secretion by the rat kidney of 5.2 nmol/s (71). The HCO3- delivery to IMCD from the model outer medulla is 3.1 nmol/s, with a volume delivery rate of 0.27 µl/s (Table 5). This base delivery is within the capability of the model IMCD to acidify the urine. When the end-luminal solution from this OMCD (Table 5) is deployed as initial conditions for the IMCD model using the same papillary interstitial values (Table 2 in Ref. 70), luminal acidification proceeds to pH 5.7 with a HCO3- concentration 1.3 mmol/l.

In summary, the use of mass balance equations that can accommodate several buffer systems has permitted formulation of a model of the acid-secreting intercalated cell of the rat outer medulla and thus a representation of OMCD. For this model, the luminal membrane proton pumps have been adapted from the IMCD, and a kinetic model for the peritubular Cl-/HCO3- exchanger is newly devised. Although this model AE1 is fully compatible with erythrocyte data at 38°C, it is not unique, and the means to create a family of compatible anion exchangers has been indicated. With respect to the intercalated cell, there are two model observations that suggest further investigation. 1) If all of the chloride entry via AE1 recycles across a peritubular chloride channel, and if this channel is anything other than highly selective (HCO3-:Cl- permeability ratio <1:8), then it should conduct a substantial fraction of the bicarbonate exit. 2) If all of the peritubular K+ exit is conductive, then variation in luminal membrane H-K-ATPase activity is predicted to result in significant derangement of cell volume. Both of these model conclusions could be invalidated if peritubular KCl cotransport were present. With respect to the OMCD in situ, available buffer appears to be present well in excess and is unlikely to exert a regulatory role in total proton secretion by this tubule segment.


    ACKNOWLEDGEMENTS

I thank Dr. Philip A. Knauf for several very helpful discussions regarding AE1 function and for critical reading of portions of this manuscript.


    FOOTNOTES

This investigation was supported by National Institute of Arthritis, Diabetes, and Digestive and Kidney Disease Grant 1-R01-DK-29857.

Address for reprint requests and other correspondence: A. M. Weinstein, Department of Physiology and Biophysics, Weill Medical College of Cornell University, 1300 York Avenue, New York, NY 10021 (E-mail: alan{at}nephron.med.cornell.edu).

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. §1734 solely to indicate this fact.

Received 27 October 1999; accepted in final form 27 January 2000.


    REFERENCES
TOP
ABSTRACT
INTRODUCTION
MODEL AE1
MODEL OMCD
MODEL PARAMETERS
MODEL CALCULATIONS
DISCUSSION
REFERENCES

1.   Alper, SL, Natale J, Gluck S, Lodish HF, and Brown D. Subtypes of intercalated cells in rat kidney collecting duct defined by antibodies against erythroid band 3 and renal vacuolar H+-ATPase. Proc Natl Acad Sci USA 86: 5429-5433, 1989[Abstract].

2.   Andersen, OS, Silveira JEN, and Steinmetz PR. Intrinsic characteristics of the proton pump in the luminal membrane of a tight urinary epithelium. J Gen Physiol 86: 215-234, 1985[Abstract].

3.   Armitage, FE, and Wingo CS. Luminal acidification in K-replete OMCDi: contributions of H-K-ATPase and bafilomycin-A1-sensitive H-ATPase. Am J Physiol Renal Fluid Electrolyte Physiol 267: F450-F458, 1994[Abstract/Free Full Text].

4.   Armitage, FE, and Wingo CS. Luminal acidification in K-replete OMCDi: inhibition of bicarbonate absorption by K removal and luminal Ba. Am J Physiol Renal Fluid Electrolyte Physiol 269: F116-F124, 1995[Abstract/Free Full Text].

5.   Atkins, JL, and Burg MB. Bicarbonate transport by isolated perfused rat collecting ducts. Am J Physiol Renal Fluid Electrolyte Physiol 249: F485-F489, 1985[Abstract/Free Full Text].

6.   Bastani, B. Colocalization of H-ATPase and H, K-ATPase immunoreactivity in the rat kidney. J Am Soc Nephrol 5: 1476-1482, 1995[Abstract].

7.   Beck, F-X, Dorge A, Rick R, Schramm M, and Thurau K. The distribution of potassium, sodium and chloride across the apical membrane of renal tubular cells: effect of acute metabolic alkalosis. Pfluegers Arch 411: 259-267, 1988[ISI][Medline].

8.   Breyer, MD, and Jacobson HR. Regulation of rabbit medullary collecting duct cell pH by basolateral Na/H and Cl/base exchange. J Clin Invest 84: 996-1004, 1989[ISI][Medline].

9.   Brown, D, Zhu XL, and Sly WS. Localization of membrane-associated carbonic anhydrase type IV in kidney epithelial cells. Proc Natl Acad Sci USA 87: 7457-7461, 1990[Abstract].

10.   Brzezinski, P, Malmstrom BG, Lorentzon P, and Wallmark B. The catalytic mechanism of gastric H+/K+-ATPase: simulations of pre-steady-state and steady-state kinetic results. Biochim Biophys Acta 942: 215-219, 1988[ISI][Medline].

11.   Buerkert, J, Martin D, and Trigg D. Segmental analysis of the renal tubule in buffer production and net acid formation. Am J Physiol Renal Fluid Electrolyte Physiol 244: F442-F454, 1983[ISI][Medline].

12.   Falke, JJ, and Chan SI. Evidence that anion transport by band 3 proceeds via a ping-pong mechanism involving a single transport site. A 35Cl NMR study. J Biol Chem 260: 9537-9544, 1985[Abstract/Free Full Text].

13.   Falke, JJ, Kanes KJ, and Chan SI. The kinetic equation for the chloride transport cycle of band 3. A 35Cl and 37Cl NMR study. J Biol Chem 260: 9545-9551, 1985[Abstract/Free Full Text].

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