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 -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, K_Cl and K_HCO3. 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

(1)

Corresponding to the two unknowns, x' and x", are the model equations for conservation of total carrier, x_T

(2)

and for zero net flux of carrier

(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

(4)

(5)

This linear system is solved for x' and x"

(6)

where
+(1+"+")(P'_b'+P'_c')

View larger version (12K): [in this window] [in a new window]   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

(7)

(8)

and the net efflux for bicarbonate which must be equal and opposite to that for chloride

(9)

It should be observed that for the net flux to equal zero when bathing media are equal (b' = b" and c' = c")

(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

(11)

so that

(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, J_bm^o, J_bmⁱ, and J_bm^s) and apparent affinities (K_bm^o, K_bmⁱ, and K_bm^s). With reference to Eq. 12, the model defines these measured quantities

(13)

Equation 13 indicates that the three self-exchange experiments depend upon only three composite parameters, namely, the geometric mean of the translocation constants
and the ratios of the affinities to the translocation constants K'_b/x_TP'_b and K_b/x_TP_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

(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 K_bm^s/J_bm^s and K_bm^o/J_bm^o, the ratio K_bmⁱ/J_bmⁱ 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, K_I, of 172 mmol/l. In the context of this model, this inhibition is represented as an effect on the transporter abundance, x_T, relative to a maximal abundance, x_Tm

(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 K_bm^s. 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.

View this table: [in this window] [in a new window]   Table 1.   Cl/HCO3 exchanger model development

View larger version (37K): [in this window] [in a new window]   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.

View larger version (35K): [in this window] [in a new window]   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 C(i), where  is lumen (M), interspace (E), cell (I), or peritubular solution (S). Within the epithelium the flux of solute i across membrane is denoted J(i) (mmol · s¹ · cm²), where 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 F_M(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.

View larger version (16K): [in this window] [in a new window]   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 J(i), where the subscript 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 (s(i)) is equal to its net export plus its accumulation

(17)

(18)

where V 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.

(19)

where B_M is the tubule circumference, and A_M 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

(20)

where  = E, I, or M. For the phosphate and for the ammonia buffer pairs, there is conservation of total buffer

(21)

(22)

Although peritubular PCO₂ will be specified, the CO₂ concentrations of the cells, interspace, and lumen are model variables. The relevant reactions are
where dissociation of H₂CO₃ is rapid, and assumed to be at equilibrium. Since HCO₃ and H₂CO₃ are interconverted, mass conservation requires

(23)

for  = I or E, whereas for the tubule lumen

(24)

In each compartment ( = I, E, or M), conservation of total CO₂ is expressed as

(25)

Corresponding to conservation of protons is the equation for conservation of charge for all the buffer reactions

(26)

where z_i is the valence of species i. In this model, conservation of charge for the buffer reactions takes the form

(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, , for  = E, I, or M, is the equation for electroneutrality

(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, C_I(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, V_E, and its basement membrane area, A_ES, are functions of interspace hydrostatic pressure, P_E

(29)

where V_E0 and A_ES0 are reference values for volume and outlet area, respectively, and _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

(30)

In Eq. 30, the first term is the Goldman relation for ionic fluxes, where h(i) is a solute permeability, and C(i) and C(i) are the concentrations of i in compartments  and , respectively. Here

(31)

is a normalized electrical potential difference, where z_i is the valence of i, and    is the potential difference between compartments  and . 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  is

(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

(33)

where J(H⁺)_max is the maximum proton flux, and _MI(H⁺) is the electrochemical potential difference of H⁺ from the cytosol to the lumen; _MI defines the steepness of the function, and 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  = 0.4 and 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.

View larger version (26K): [in this window] [in a new window]   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

in which the half-maximal Na⁺ concentration, K_Na, increases linearly with internal K⁺, and the half-maximal K⁺ concentration, K_K, increases linearly with external Na⁺ (19). The pump flux of K⁺ plus NH₄⁺ reflects the 3:2 stoichiometry

(35)

with the transport of either K⁺ or NH₄⁺ determined by their relative affinities, K_K and K_NH⁺4

(36)

Analogous expressions are written for active transport at the basal cell membrane, J_IS^act.

	MODEL PARAMETERS

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

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 · mm¹ · min¹) 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 (V_I) of about 0.3 × 10³ 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).

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.

View larger version (21K): [in this window] [in a new window]   Fig. 6.   OMCD cellular transport pathways, along with model cell fluxes (pmol · s1 · cm2) 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 that of chloride, bicarbonate permeability as 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, P_Na:P_K:P_Cl = 3.9:5.9:4.8 × 10⁶ cm/s (27, 38, 58). In the rat, OMCD NH₄⁺ permeability is 1.3 × 10⁵ 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 × 10⁵) 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

TOP 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.