## Abstract

In assessing disorders of potassium excretion, urine composition is used to calculate the transtubular gradient (TTKG), as an estimate of tubule fluid concentration, at a point when the fluid was last isotonic to plasma, namely, within the cortical collecting duct (CCD). A mathematical model of the CCD has been developed, consisting of principal cells and α- and β-intercalated cells, and which includes Na^{+}, K^{+}, Cl^{−}, HCO
, CO_{2}, H_{2}CO_{3}, phosphate, ammonia, and urea. Parameters have been selected to achieve fluxes and permeabilities compatible with data obtained from perfusion studies of rat CCD under the influence of both antidiuretic hormone and mineralocorticoid. Both epithelial (flat sheet) and tubule models have been configured, and model calculations have focused on the determinants of the TTKG. Using the epithelial model, luminal K^{+} concentrations can be computed at which K^{+}secretion ceases (0-flux equilibrium), and this luminal concentration derives from the magnitude of principal cell peritubular uptake of K^{+} via the Na-K-ATPase, relative to principal cell peritubular membrane K^{+} permeability. When the model is configured as a tubule and examined in the context of conditions in vivo, osmotic equilibration of luminal fluid produces a doubling of the initial K^{+} concentration, which, depending on delivered load, may be substantially greater than the zero-flux equilibrium value. Under such circumstances, the CCD will be a site for K^{+} reabsorption, although the relatively low permeability ensures that this reabsorptive flux is likely to be small. Osmotic equilibration may also raise luminal NH_{3} concentrations well above those in cortical blood. In this situation, diffusive reabsorption of NH_{3} provides a mechanism for base reclamation without the metabolic cost of active proton secretion.

- distal nephron
- aldosterone
- urine acidification
- ammonia transport

micropuncture study of potassium transport in the rat kidney has identified the accessible portion of the distal tubule as the principal site for potassium secretion (28, 29). Further along the nephron, there is little change in potassium flow, at least from a comparison of potassium delivery to the collecting duct with its appearance in the final urine. These data stand in contrast to a substantial body of subsequent investigation of the collecting duct in vitro. Examination of the cortical collecting tubule of the rabbit (35, 46, 52,54), and of the rat (40, 59), has demonstrated that this segment is a site of sodium reabsorption and potassium secretion, with transport activity enhanced by aldosterone and antidiuretic hormone (ADH). Dispersion in the data does exist with regard to the component of sodium reabsorption that is matched by potassium secretion, where a number of reports indicate that this fraction is about one-half (35, 40, 46), whereas others have observed that potassium secretion may be three-fourths (35, 52) or one-fourth (35, 54, 59) of the sodium flux. These observations have motivated extensive study of the principal cell of the cortical collecting duct (CCD), in which the luminal membrane Na^{+} and K^{+} channels, in series with peritubular membrane Na-K-ATPase, mediate the Na^{+} for K^{+}exchange by this segment. Indeed, clinical disorders of renal potassium excretion are implicitly referred back to the cortical collecting tubule with the calculation of the transtubular potassium gradient (13, 72).

A mathematical model of the CCD should be the appropriate instrument for extrapolating tubule function from perfusion conditions to those in vivo. Such a model must represent the principal cell and both intercalated cell types of this tubule segment, along with the solute species to allow simulation of Na^{+}, K^{+}, Cl^{−}, and acid-base transport. The only previous model of the cortical collecting duct was that of Strieter et al. (55,56), which represented the perfused tubule of the rabbit. That model was used to investigate the factors defining the limiting luminal Na^{+} concentration, at which Na^{+} reabsorption was brought to a halt. One conclusion of that work was that the inverse dependence of luminal membrane Na^{+} channel permeability on luminal Na^{+} concentration was key to representing the very low, limiting concentrations that had been observed, and that feature has been retained in the present model. In general, rat and rabbit cortical collecting tubules are similar with respect to the overall rates of transport and transepithelial electrical potentials. Notable differences include the fraction of intercalated cells that are α-cells [60% in rat (1, 60); 30% in rabbit (33)], and the chloride conductance of the peritubular membrane of the principal cell [greater in rabbit than rat (33,44)]. In the CCD model developed here, the α-cell has been updated to conform to that recently published in a model of the outer medullary collecting duct of the rat (70), which includes a kinetic representation of the peritubular Cl^{−}/HCO
exchanger and a luminal membrane H-K-ATPase.

The primary focus of the present work is the potassium gradient that can be established across the cortical collecting tubule. First, the model CCD is described and is compared with rat tubule transport rates and permeabilities. Modeled as an epithelium, this CCD is a flat sheet of cells between specified bathing solutions. This version of the model predicts K^{+} fluxes as a function of bath conditions and is incorporated into a program that computes the limiting luminal concentration, at which K^{+} flux ceases. In these calculations, the zero-flux K^{+} gradient is found to vary directly with the rate of Na^{+} transport, and, inversely, with the peritubular membrane K^{+} permeability. Modeled as a tubule, this CCD predicts the solute flows in vivo, given estimates of the entering conditions from micropuncture of late distal tubule. Using a high tubule water permeability typical for the effect of ADH, it is found that CCD osmolality equilibrates early, and CCD K^{+}concentration is near its limiting value for most of the tubule length. Indeed, under conditions of high K^{+} delivery, CCD luminal K^{+} concentration can rise well above its equilibrium value, and thus the CCD can become a site for K^{+} reabsorption.

## MODEL CCD

The model CCD is depicted in Fig. 1, in which the epithelium contains three cell types and a common intercellular space, all bounded by luminal and peritubular solutions. In the CCD tubule model, the epithelial compartments line the tubule lumen, and luminal concentrations vary axially as a consequence of transport. Within each compartment the concentration of species*i* is designated C_{α}(*i*), where α is lumen (M), interspace (E), principal cell (P), α-intercalated cell (A), β-intercalated cell (B), or peritubular solution (S). Within the epithelium, the flux of solute *i* across membrane αβ is denoted *J*
_{αβ}(*i*) (mmol · s^{−1} · cm^{2}), where αβ may refer to tight junction (ME), interspace basement membrane (ES), any of the luminal cell membranes (MP, MA, or MB), lateral cell membranes (PE, AE, or BE), or basal cell membranes (PS, AS, or BS). Along the tubule lumen, axial flows of solute are designated F_{M}(*i*) (mmol/s). The 12 model solutes are Na^{+}, K^{+}, Cl^{−}, HCO
, CO_{2}, H_{2}CO_{3}, HPO
, H_{2}PO
, NH_{3}, NH
, H^{+}, and urea, as well as an impermeant species within the cells, and possibly within the lumen. These comprise the minimal set of solutes that will permit representation of net acid excretion.

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 (68, 70). Within a cell (I; I = P, A, or B), the generation of volume,*s*
_{I}(v), or of solute *i*, [*s*
_{I}(*i*)], is equal to its net export plus its accumulation
Equation 1where V_{I} is the compartment volume (cm^{3}/cm^{2}). The interspace exchanges with all of the model compartments, so that mass generation is written
Equation 2
Within the tubule lumen, mass generation is appreciated as an increase in axial flux, as transport into the epithelium, or as local accumulation
Equation 3where *B*
_{M} is the tubule circumference, and*A*
_{M} is the tubule cross-sectional area. With this notation, the equations of mass conservation for volume and for the nonreacting species (Na^{+}, K^{+} , Cl^{−}, and urea) are written
Equation 4where α = P, A, B, E, or M. For the phosphate and for the ammonia buffer pairs, there is conservation of total buffer
Equation 5
Equation 6Although peritubular Pco
_{2} will be specified, the CO_{2} concentrations of the cells, interspace, and lumen are model variables. The relevant reactions are
where dissociation of H_{2}CO_{3} is rapid and assumed to be at equilibrium. Because HCO
and H_{2}CO_{3} are interconverted, mass conservation requires
Equation 7
for α = P, A, B, or E, whereas for the tubule lumen
Equation 8
In each compartment (α = P, A, B, E, or M), conservation of total CO_{2} is expressed
Equation 9Corresponding to conservation of protons is the equation for conservation of charge for all the buffer reactions
Equation 10where *z _{i}
* is the valence of species

*i*. In this model, conservation of charge for the buffer reactions (

*Eq. 10*) may be rewritten Equation 11The solute equations are completed with the chemical equilibria of the buffer pairs: HPO :H

_{2}PO , NH

_{3}:NH , and HCO :H

_{2}CO

_{3}. Corresponding to the electrical potentials, ψ

_{α}, for α = P, A, B, E, or M, is the equation for electroneutrality Equation 12With respect to water flows, volume conservation equations for lumen, interspace, and cells can be used to compute the five unknowns: luminal volume flow, lateral interspace hydrostatic pressure, and the three cell volumes. (Cell hydrostatic pressure is set equal to luminal pressure; total cell impermeant content is assumed fixed.) Across each cell membrane, the volume fluxes are proportional to the hydroosmotic 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}Equation 13where V

_{E0}and

*A*

_{ES0}are reference values for volume and outlet area, respectively, and ν

_{E}is a compliance. Along the lumen, hydrostatic pressure changes according to an equation for Poiseuille flow Equation 14Solute transport across the model membranes is either electrodiffusive (through a porous matrix or via a channel), coupled to the electrochemical potential gradients of other solutes (via a cotransporter or an antiporter), or coupled to metabolic energy (via an ATPase). This is expressed in the model by the flux equation Equation 15 In

*Eq. 15*, the first term is the Goldman relation for ionic fluxes, where

*h*

_{αβ}(

*i*) is a solute permeability, and C

_{α}(

*i*) and C

_{β}(

*i*) are the concentrations of

*i*in compartments α and β, respectively. Here Equation 16is a normalized electrical potential difference (PD), where

*z*

_{i}is the valence of

*i*,

*F*is the Faraday,

*RT*is the product of gas constant and temperature, and ψ

_{α}−ψ

_{β}is the PD between compartments α and β. In this model, all of the permeabilities,

*h*

_{αβ}(

*i*), are constant, with the exception of the Na

^{+}permeability of the luminal membrane of the principal cell. For this channel, an inverse relationship between Na

^{+}permeability and both luminal and cytosolic Na

^{+}concentrations has been described in several epithelia and represented by Civan and Bookman (10) as Equation 17 The second term of the solute flux (in

*Eq. 15*) specifies the coupled transport of species

*i*and

*j*according to linear nonequilibrium thermodynamics, where the electrochemical potential of

*j*in compartment α is Equation 18For 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

^{−}/HCO exchange across the peritubular membrane of the α-intercalated cell. Here, the kinetic model for AE1 (70) has been used, with a single transporter density parameter representing its activity.

In this model, there are three ATPases. Within the peritubular membrane (both lateral and basal membranes) of all three cell types (I = P, A, B), the Na-K-ATPase is represented by an expression
Equation 19
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^{+}
Equation 20The pump flux of K^{+} plus NH
reflects the 3:2 stoichiometry
Equation 21with the transport of either K^{+} or NH
determined by their relative affinities,*K*
_{K} and*K*
_{NH
}
Equation 22Analogous expressions are written for active transport at the basal cell membranes,*J*
(Na^{+}). Within the luminal membrane of the α-cell and the peritubular membrane of the β-cell, there is a proton ATPase. An empirical expression representing the H^{+}-ATPase was devised by Strieter et al. (55), approximating data of Andersen et al. (3) for turtle bladder
Equation 23
where *J*(H^{+})_{max} is the maximum proton flux,
_{MI} (H^{+}) is the electrochemical PD of H^{+} from the cytosol to the lumen, ξ_{MI} defines the steepness of the function, and
_{0} defines the point of half-maximal activity. The important finding of Andersen et al. (3) was that the proton flux depended on 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). Within the luminal membrane of the α-intercalated cell, there is an H-K-ATPase, which has been given a full kinetic representation (69).

## MODEL PARAMETERS

The parameters displayed in Table1 were selected so that the model tubule would correspond to the CCD of the rat. Where rat data were not available, rabbit measurements were considered for guidance. The distal nephron of the rat accessible to micropuncture includes a distal convoluted tubule (DCT), connecting segment, and initial collecting duct. The CCD includes this initial collecting duct and the cortical collecting tubule within the medullary ray (27). In the rat, the cortical collecting tubule is short, ∼1.5 mm, (37), so that a total CCD length of 2.0 mm has been chosen to represent the nephron segment between the last accessible micropuncture and the medullary collecting duct. In the tubule calculations, it will be assumed that all of the coalescing of nephrons in the arcade is complete, so that the model tubule segment is unbranched. In the rabbit, measurements of inner and outer tubule diameters are 25 and 35 μm, giving luminal and epithelial volumes of 490 and 960 pl/mm, respectively, comparable to those reported for the rat (36). In the rat CCD, intercalated cells comprise ∼40% of epithelial volume, with 21.4 and 15.7% α- and β-cells, respectively (23). In the rabbit, lateral interspace volume is ∼11% of epithelial volume (71). For a total epithelial volume of ∼5 × 10^{−4}cm^{3}/cm^{2}, this yields a principal cell volume of 3 × 10^{−4} cm^{3}/cm^{2} and α- and β-intercalated cell volumes of 1.2 and 0.8 × 10^{−4} cm^{3}/cm^{2}, respectively. The cellular compartments, along with the important transport pathways, are displayed in Fig. 2.

In rat CCD, the principal cell luminal and peritubular membrane areas have been reported to be ∼4,000 and 25,000 cm^{2}/cm^{3} cell volume (50, 61), which translates into 1.2 and 7.5 cm^{2}/cm^{2}epithelial area, respectively. The important water channels of this epithelium are restricted to the principal cells (34), in which the luminal membrane, containing aquaporin-2, is rate limiting for water flow. For most of the calculations, ADH is assumed to be present and luminal water permeability has been selected to achieve an epithelial water permeability (*P*
_{f}) of ∼0.1 cm/s (9, 37). The peritubular membrane unit water conductance was taken to be two-thirds that of the luminal membrane, so that with its sixfold area amplification, the peritubular*P*
_{f} was four times that of the luminal membrane. Electrophysiology of the principal cell indicates that the luminal conductance is enhanced by both aldosterone and ADH. With both hormones present, the fractional apical resistance may be 0.7–0.8, indicating a luminal conductance ∼30% of that of the peritubular membrane (42, 44). Under these circumstances, an absolute luminal conductance has been reported to be 16 mS/cm^{2}(41). Most of this luminal membrane conductance is due to potassium, with a transference number of 0.73 (44). For the model parameters (Table 1), the maximal Na^{+}permeability was set equal to the K^{+} permeability, and the value shown (∼22% of the K^{+} permeability) derives from the inhibitory effect of ambient Na^{+} (*Eq. 17
*). This value of Na^{+} permeability was found to yield transepithelial Na^{+} fluxes in the reported range when both aldosterone and ADH are present. Luminal membrane NH
permeability was set at 20% of that of K^{+}, comparable to NH
permeation of other K^{+} channels. Luminal membrane Cl^{−} permeability was set (arbitrarily) at 10% of that of K^{+}, and HCO
permeability to 20% of that of Cl^{−}, rendering conductive anion fluxes negligible. A luminal membrane NaCl cotransporter has been included, on the basis of a single report that 50% of Na^{+} reabsorption by rat CCD is thiazide sensitive (57). In view of the fact that this finding has not received confirmation (6, 43), the baseline flux through this pathway has been set low (3.6% of Na^{+} flux), but the impact of greater cotransport is explored in the model calculations. In the rat, the peritubular membrane is nearly selective for K^{+} (42, 44). In this model, the K^{+} permeability is such that the peritubular conductance is 2.5 times that of the luminal membrane. The peritubular NH
permeability is again 20% of that for K^{+}; Cl^{−} permeability is 5% that of K^{+}; and HCO
permeability is 20% that of Cl^{−}. Within the peritubular membrane of principal cells of rabbit CCD, there are also electroneutral cotransporters, Na^{+}/H^{+} (63, 65), and Cl^{−}/HCO
(63, 66). The density of these transporters was adjusted to achieve realistic cell pH and Cl^{−} [e.g., 13 mmol/l for cell Cl^{−}(5); principal cell pH 7.3–7.4 (48)]. Because of the lack of direct information, the unit membrane NH_{3} and urea permeabilities were taken to be equal and adjusted to achieve agreement with overall epithelial permeabilities for these solutes.

Membrane areas of intercalated cells of the CCD have been determined and indicate 4,800 and 14,400 cm^{2}/cm^{3} cell volume for the luminal and peritubular membranes of α-cells, and 2,150 and 21,000 cm^{2}/cm^{3} for β-cell membranes (60), respectively. When corrected for the volume density of α- and β-cells (1.2 and 0.8 × 10^{−4}cm^{3}/cm^{2}, respectively), these membrane areas are 0.6 and 1.7 cm^{2}/cm^{2} for the α-cell and 0.2 and 1.7 cm^{2}/cm^{2} epithelial area for the β-cell, respectively. The important α-cell transport pathways are shown in Fig. 2 and include two luminal membrane proton ATPases in series with peritubular Cl ^{−}/HCO
(AE1) and a Cl^{−} channel. The parameters are essentially those selected for the α-cell of the outer medullary collecting duct (70), with the exception that the density of the H-K-ATPase and peritubular K^{+} channel have been decreased by 60%, whereas the density of the H^{+}-ATPase has been increased by 33%. The H-K-ATPase is present in intercalated cells of rat and rabbit CCD (47, 49), although under control conditions its proton secretory rate appears to be less than that of the H^{+}-ATPase (31). The decreased value taken for peritubular K^{+} conductance remains compatible with the electrophysiology of the α-cell (33). The important β-cell transport pathways are a luminal membrane Cl^{−}/HCO
exchanger in series with peritubular H^{+}-ATPase and Cl^{−} channel. Under control conditions, no H-K-ATPase activity has been identified in this cell, although it may become important in the correction of metabolic alkalosis (16) or under conditions of low sodium intake (47). The luminal membrane anion exchanger is different from AE1 (12), and, to respect this difference, the nonequilibrium thermodynamic formulation has been used. The luminal membrane has no significant electrical conductance, and, as in the α-cell, the peritubular membrane is dominated by the chloride conductance (33). Peritubular Na^{+}/H^{+} and Cl^{−}/HCO
exchangers have been included in view of their presence in β-cells of rabbit CCD (65, 67,74). The β-cell unit membrane permeabilities for nonelectrolytes have been assumed to be identical to those chosen for the α-cell. In the rat, cytoplasm of intercalated cells stains intensely for carbonic anhydrase (26), although the membrane-bound isoform (CA-IV) is absent (7). Thus the rate coefficients for CO_{2} hydration and dehydration (*Eq. 7
*) have been taken as those of the uncatalyzed reaction within the tubule lumen and lateral intercellular space and for full catalysis (10,000-fold greater) within α- and β-cells. In view of some staining within principal cells (26), the coefficients were taken to be 10-fold greater than the uncatalyzed rate.

Values for the tight junctional conductance of the rat CCD have been found to be 11–13 mS/cm^{2}, perhaps two- to threefold greater than that for the rabbit tubule (41, 44). As in rabbit, the Cl^{−} permeability appears to be slightly larger than that for Na^{+} (44). In the model of rabbit CCD (55), it had been noted that the low value of Na^{+} permeability was essential to achieving tubule fluid Na^{+} concentrations as low as those observed. In preliminary calculations for this model of the rat tubule, it was observed that if tight junctional conductance were twice that of rabbit CCD, then paracellular backflux would be unacceptably large: overall epithelial Na^{+} secretion for a “late distal” luminal fluid composition (35 mM NaCl concentration). Indeed, even with junctional conductance comparable to rabbit CCD, 5 mS/cm^{2}, the paracellular backflux of Na^{+} is still two-thirds of the reabsorptive Na^{+} flux across the principal cell (Fig. 2). That lower conductance has been selected for this model. The tight junctional Cl^{−}-to-Na^{+} permeability ratio has been set at 1.2, consistent with observation in rats (44), and perhaps somewhat lower than in rabbits (64). Junctional K^{+} and NH
permeabilities were assumed to be equal to that of Cl^{−} and that of HCO
to be 25% of the value for Cl^{−}. The junctional urea permeability was set equal to half the measured epithelial urea permeability (25). In models of renal tubule segments, the basement membrane is a permeability barrier to the lateral interspace and allows for the possibility that the interspace can act as an unstirred layer, with solute concentrations distinct from those of the peritubular bath. In this model, the overall conductance of the basement membrane is ∼1,000 mS/cm^{2}, with relative solute permeabilities comparable to their mobility in solution.

## MODEL CALCULATIONS

Suitability of the parameter choices is assessed, in part, by examining predicted fluxes and permeabilities. Table2 contains the solutions of the model equations for the open-circuited epithelium, when lumen and bath solutions are equal, comparable to solutions used in perfusion studies. The computed compartment volumes (principal:α:β:interspace) are 59, 22, 14, and 5%, respectively, of a total epithelial volume of 6 × 10^{−4}cm^{3}/cm^{2}. The electrical PD of tubule lumen (−18.4 mV) and of the peritubular membrane of the principal cell (−79.2 mV) are similar to those found in tubules under the influence of both aldosterone and ADH (42, 43). Within the principal cell, the Cl^{−} concentration is low (12.7 mM) but still above its equilibrium concentration of 5.8 mM. This is a consequence primarily of the luminal NaCl cotransporter, although the peritubular Cl^{−}/HCO
exchanger contributes ∼23% of the entering Cl^{−}. This cytosolic Cl^{−}concentration is within the range of determinations using Cl^{−}-sensitive microelectrodes [Cl^{−} activity ∼9 mmol/l (43)]. When the transepithelial solute fluxes are scaled to a tubule of 25 μm diameter, principal cell Na^{+} reabsorption is 91.6 pmol · mm^{−1} · min^{−1} with a paracellular backflux of 23.0, giving a net reabsorptive Na^{+} of 68.6 pmol · mm^{−1} · min^{−1}. Close to half of this is balanced by principal cell K^{+} secretion of 30.0 and close to half by Cl^{−} reabsorption of 30.5 pmol · mm^{−1} · min^{−1}. The Cl^{−} flux is primarily paracellular (22.2), with smaller components across the β-cell (5.2) and principal cell (3.1). The remainder of the Na^{+} flux is balanced by an equivalent reabsorptive “HCO
” flux (pmol · mm^{−1} · min^{−1}) of 8.2, comprised β-cell HCO
secretion (5.2) in parallel with α-cell H^{+} secretion (13.4). These values for transepithelial ionic fluxes are within ranges that would be appropriate for tubules under the influence of both aldosterone and ADH. It may also be noted that with ambient total ammonia concentrations of 1.0 mM, the model predicts net reabsorption of ammonia, despite the lumen negative PD. This is due to the acid disequilibrium within the lateral interspace (due to peritubular Na^{+}/H^{+}), with diffusion trapping of NH_{3}.

Table 3 displays the results of simulating idealized epithelial permeability determinations. For these calculations, the model represents a short-circuited epithelium in vitro bathed by the equal luminal and peritubular solutions in Table 2, plus an additional luminal impermeant at a concentration 0.1 mM. Calculations were performed in which each luminal solute concentration in turn was lowered and then raised by 0.1 mM. The change in solute flux relative to the change in concentration is listed in Table 3 as the permeability, *H*
_{M}(*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 shown in *column 2* in Table 3(mS/cm^{2}). The total conductance is ∼8 mS/cm^{2}; by design, it is 30–50% of the measured conductance in rat tubules (42, 44), more like that in rabbit tubules. For comparison with the model tubule, permeability measurements in rat CCD have been made for urea, 0.4 × 10^{−5} cm/s (25), for NH
, 2.6 × 10^{−5} cm/s (14), and for NH_{3}, 0.024 cm/s (14). In Fig.3, the model of the voltage-clamped epithelium is used to examine the effect of transepithelial electrical PD on ion flux. Each panel corresponds to a different solute species, Na^{+}, K^{+}, Cl^{−}, and “HCO
,” where “HCO
” is the sum of HCO
reabsorption and H^{+}secretion. In each panel, both transjunctional and total fluxes are displayed. It is apparent that throughout an 80-mV variation in transepithelial PD, nearly all of the Cl^{−} flux is transjunctional, nearly all of the K^{+} flux is transcellular, and the “HCO
” flux is small. The Na^{+} flux remains reabsorptive down to −60 mV, due to a substantial transcellular component that is relatively insensitive to transepithelial PD. Although the paracellular K^{+}permeability is slightly greater than that for Na^{+} (Table1), the small magnitude of the junctional K^{+} flux is due to the small magnitude of the ambient K^{+} concentration. The principal cell K^{+} permeability (luminal and peritubular membranes in series) is only about threefold greater than that of the tight junction; the much greater sensitivity of transcellular K^{+} flux to PD is due to the high intracellular K^{+} concentrations maintained by active peritubular K^{+} uptake.

Variations in sodium transport by the model CCD are examined in Figs.4-6. In Fig. 4, epithelial PD and ion fluxes are calculated over a range of variation in luminal Na^{+} concentrations. The perfusion and bathing solutions are as in Table 2, with the exception that the luminal HCO
concentration has been decreased to 5 mM (replaced by Cl^{−}). Figure 4, *left*, corresponds to experiments in which Na^{+} is replaced by an impermeant cation, and Cl^{−} is constant (∼135.5 mM), whereas NaCl variation (equal changes in Na^{+} and Cl^{−}, with isosmotic replacement by an impermeant) is shown on the*right*. The curves on the *left* are similar to those calculated by Strieter et al. (55; see Fig. 8) in simulating experiments by Stokes (52). As in the previous model, net reabsorptive Na^{+} flux continues down to luminal Na^{+} concentrations below 10 mM, and K^{+}secretion varies over the whole range of Na^{+}concentrations. With NaCl variation (*right*), the major differences in epithelial transport are the smaller excursion in luminal PD and the nearly constant rate of K^{+} secretion at all luminal Na^{+} concentrations >30 mM. Principal cell function during this NaCl variation is examined in more detail in Fig.5. Here, the change in luminal membrane permeability (relative to the fixed luminal K^{+} permeability) as luminal Na^{+}is varied is shown (*top left*). Thus, even though the luminal membrane Na^{+} potential increases progressively with increasing luminal Na^{+} concentration (*middle left*), the decrease in Na^{+} permeability produces a luminal membrane electrical potential that is relatively constant at higher luminal Na^{+} (*top right*). This means a relatively constant luminal membrane K^{+} potential (*middle right*) and thus a stable K^{+} flux (*bottom right*). Variation in CCD Na^{+}reabsorption can be examined over an even broader range by varying the density of luminal membrane Na^{+} channels, and, in the calculations of Fig. 6, this permeability has been varied from 3 to 300% of control. As is shown on the *left*, the Na^{+} permeability has been varied in isolation, whereas on the *right*, there is also proportional variation in the density of the peritubular Na-K-ATPase. In each set of calculations, enhanced luminal Na^{+} entry hyperpolarizes the epithelium and increases K^{+} secretion, Cl^{−} reabsorption, and even “HCO
” reabsorption. It is apparent, however, that coordination of peritubular exit with luminal entry amplifies the luminal signal. This is true even for the reabsorptive Cl^{−} flux, which is primarily paracellular.

Figure 7 summarizes the impact of changes in Na^{+} transport on the other fluxes. Throughout the figure, the predicted K^{+} and Cl^{−} transport are plotted as a function of the rate of Na^{+} reabsorption. In the replotting of the data from Fig. 6 (*top*), it is apparent that both K^{+} and Cl^{−} fluxes are nearly linear functions of Na^{+} flux and nearly through the origin. This implies that the relative fraction of Na^{+} reabsorption balanced by K^{+} and by Cl^{−} fluxes is nearly constant over the full range of transport [consistent with observations of Stokes (52)]. In contrast, for the simulations of luminal NaCl variation (Fig. 4), the changes in Na^{+} flux are nearly completely balanced by changes in Cl^{−} flux (*bottom right*). This curve does not go through the origin, so that for small Na^{+}fluxes there is essentially KCl secretion. [In the model, this K^{+} secretion derives from the diffusion potential set up by the NaCl gradient. Although direct KCl coupling within the luminal membrane of CCD principal cells has been suspected from studies of perfused rabbit tubules (73), it is not a feature of this model.] Figure 7 (*bottom left*) corresponds to simulations in which the luminal membrane NaCl cotransport coefficient is increased over a factor of 20, with a parallel increase in peritubular Cl^{−} permeability. In these calculations, there is no change in K^{+} flux with the variation in Na^{+}reabsorption. Here, even though the increased transcellular Na^{+} flux produces a proportional peritubular K^{+}uptake (Na-K-ATPase), the increase in Cl^{−} permeability depolarizes the peritubular membrane and enhances the return of K^{+} back across this membrane (not shown).

Beyond the effect of Na^{+} reabsorption, K^{+}secretion can be modulated by principal cell membrane K^{+}permeabilities, as well as luminal fluid K^{+} concentration. Figure 8 displays epithelial model predictions for PD and solute fluxes using the high-Na^{+}, low-K^{+} (5 mM) perfusion solution (Table 2) in the open-circuited epithelium. In Fig. 8 (*left*) luminal membrane K^{+} permeability is varied from 3 to 300% of baseline. As the luminal K^{+} permeability increases, K^{+}secretion is enhanced and the epithelium depolarizes, thus decreasing paracellular Cl^{−} reabsorption and paracellular Na^{+} backleak. Furthermore, with increasing luminal K^{+} permeability, the luminal membrane hyperpolarizes (not shown), thus enhancing transcellular Na^{+} reabsorption and peritubular K^{+} uptake, ultimately augmenting K^{+}secretion. In Fig. 8 (*right*) the effect of modulating peritubular K^{+} permeability from 3 to 300% of control is shown. As expected, increasing peritubular K^{+} permeability decreases K^{+} secretion and enhances Cl^{−}reabsorption. However, the striking feature of these calculations is the lack of effect on Na^{+} transport. In contrast to luminal K^{+} permeability variation, increasing peritubular K^{+} permeability hyperpolarizes both the tight junction and luminal cell membranes. As a consequence, both paracellular backflux and luminal reabsorptive Na^{+} flux are enhanced and cancel. It may also be noted that the overall impact on K^{+} flux is smaller with peritubular variation than with luminal variation of K^{+} permeabilities.

In Fig. 9, luminal KCl concentration is varied from 1.0 to 39 mM. In the panels on the*left*, the luminal perfusate is the high-Na^{+}solution in Table 2. Epithelial hyperpolarization with increasing luminal K^{+} (*top*) and a depolarization of the luminal membrane of the principal cell (*middle*) are shown. The curve labeled “K Potential” (*middle*) is the potential from lumen to cell, so that the crossing point from negative to positive potential corresponds to the transition from principal cell K^{+} secretion to reabsorption. The K^{+} fluxes are shown (*bottom*), where it is evident that most of the variation in epithelial K^{+} flux is due to changes in principal cell flux. With respect to flux across the luminal cell membrane, the transition from secretion to reabsorption occurs at a luminal K^{+} concentration of ∼25 mM; for the epithelium, this transition point is ∼23 mM. In the panels on the*right* in Fig. 9, the perfusion solution is a low-Na^{+}, low-Cl^{−} (35 mM) solution characteristic of early CCD conditions (see below). The potentials are not very different, and the ability of the tubule to secrete K^{+} is not affected.

The maximal luminal K^{+} concentration that can be sustained by epithelial K^{+} secretion, or zero-flux K^{+}concentration, can be estimated analytically with reference to the scheme depicted in Fig. 2. Considering only K^{+} and combining lateral and basal cell membranes into a peritubular membrane, the fluxes may be estimated from the K^{+}potentials by
Equation 24
where *J*
_{PS} is total peritubular K^{+} flux, and μ_{M} and μ_{P} are computed relative to peritubular conditions
The coefficients *L*
_{αβ}(mmol^{2} · J^{−1} · s^{−1}) are related to membrane K^{+} permeabilities,*h*
_{αβ} (cm^{3}/s), and conductances,*g*
_{αβ} (mS/cm^{2})
Equation 25where
_{αβ} is a (logarithmic) mean membrane concentration
Because cellular K^{+} balance requires*J*
_{MP} = *J*
_{PS}
Equation 26luminal membrane K^{+} secretion is zero when
Equation 27and, ignoring K^{+} transport by the α-intercalated cell, overall epithelial K^{+} flux is zero when
Equation 28For a relatively nonconductive tight junction, *Eq. 28
*is approximated by *Eq. 27
*, which indicates that the limiting K^{+} concentration is determined largely by the peritubular membrane K^{+} permeability relative to the peritubular uptake of K^{+} via Na-K-ATPase. It should be influenced little by luminal membrane K^{+} permeability, because when luminal membrane K^{+} flux is zero, the permeability is irrelevant. (With reference to *Eq. 27
*, the impact of luminal membrane K^{+} conductance is greatest when K^{+} secretion is substantial and is an important determinant of both the magnitude of early tubular K^{+} secretion and of overall epithelial K^{+} conductance.) Table 4 uses the parameters in Table 1 and the K^{+} concentrations in Table 2 to estimate the limiting K^{+} concentration (*Eq. 28
*). For luminal electrical PD of −30 mV, this concentration is 23 mM, comparable to the value obtained from Fig. 9. To compute the limiting K^{+} concentrations directly, the CCD epithelial model has been configured as a subroutine in a program that uses Newton iteration to identify the luminal K^{+}concentration (via KCl adjustments) at which secretion is zero, and the results are displayed in Fig. 10. The*top* two panels compute limiting K^{+}concentrations over the range of variation in luminal Na^{+}permeability considered in Figs. 6 and 7; the *middle* panels show the limiting concentrations computed for variation in membrane K^{+} permeability considered in Fig. 8; the *bottom left* panel contains limiting K^{+} concentrations when coupled NaCl transport is varied 20-fold as in Fig. 7; and the*bottom right* panel examines the effect of varying luminal NaCl concentration, as in Figs. 4 and 7. Consistent with *Eq.27
*, increasing Na^{+} flux two- to threefold via increasing Na^{+} channel permeability increases the limiting K^{+} gradient two- to threefold. Also consistent with*Eq. 27
* is the insensitivity of the limiting K^{+}gradient to luminal K^{+} permeability and a hyperbolic sensitivity of the limiting gradient to the peritubular permeability. Figure 10 (*bottom*) demonstrates that it is possible to modulate Na^{+} transport without any effect on the limiting K^{+} gradient: in the case of coupled NaCl transport, there is parallel peritubular Cl^{−} channel activation, and in the case of luminal Na^{+} there is modulation of luminal membrane Na^{+} conductance. In both cases, changes in membrane potentials act to leave the limiting K^{+} gradient relatively unaffected.

When the CCD epithelium is configured as a tubule, it permits predictions about its function in vivo. Figure 2
*A* displays estimates of conditions and fluxes in the early collecting duct. Conditions are taken to resemble those in late distal tubule: Na^{+} = 35.0, K^{+} = 12.0, Cl^{−} = 36.2 (5, 11), and HCO
= 7.0 mM (8, 22). For an axial volume flow of 7.5 nl · min^{−1} · tubule^{−1} (54 μl/min for all 7,200 tubules) and a single-kidney glomerular filtration rate of 500 μl/min, these concentrations correspond to ∼2.5, 30, 3, and 3.5% of filtered loads of Na^{+}, K^{+}, Cl^{−}, HCO
, respectively. The total phosphate, 3.9 mM, is a lower concentration than reported (11), but for a plasma concentration of 2.5 mM (22) it represents a delivery of 16% of filtered load and, from that perspective, is consistent with observations (11). The urea concentration, 30 mM, corresponds to ∼60% of filtered load and, although a low concentration, is compatible with determinations of urea flow at this point (30). The ammonia concentration, 2 mM, sits between measured values for late distal tubule fluid in the rat (21,38). With these values as input, the concentrations and flows have been determined for a 2.0-mm tubule segment, whose peritubular solution is shown in Table 2. Figure 11shows the tubule fluid osmolality and solute concentrations (*left*) and axial volume and solute flows (*right*). It is apparent that over the first 20% of the tubule length, about half of the tubule water is reabsorbed in the transition to isotonicity, and consequently luminal solute concentrations double. Moreover, along the length of the tubule, ∼20% of the delivered NaCl is reabsorbed, resulting in additional water loss and still further increases in luminal K^{+} concentration. Although K^{+} is initially secreted (−15.2 pmol · mm^{−1} · min^{−1}, Fig.2
*A*), luminal K^{+} concentration rises above its limiting value, so that by the tubule exit, K^{+} flux is reabsorptive (6.1 pmol · mm^{−1} · min^{−1}). The end-tubule concentrations and fluxes have been indicated in Fig.2
*B*. By this point in the tubule, luminal Na^{+} and K^{+} have increased to 77 and 31 mM, respectively, and luminal PD has hyperpolarized slightly to −29 mV. Overall CCD K^{+} flux is small, with reabsorption ∼6% of delivered load. Osmotic concentration of luminal solutes also enhances urea and ammonia reabsorption (9 and 38% of delivered load, respectively). The whole-tubule economy of solute fluxes has been summarized in Table5.

The observations in Fig. 11 are that luminal K^{+}concentration can be driven above its limiting value, by either pure water abstraction or isosmotic NaCl reabsorption. Either of these features can be enhanced by changing luminal conditions. In Fig.12, entering flow and concentrations are identical to those in Fig. 11, with the exception that entering KCl concentration has been doubled to 24 mM. Now, osmotic abstraction of water rapidly increases luminal K^{+} concentration to 45 mM, nearly twice the limiting K^{+} gradient, and under these circumstances there is an increase in both absolute and fractional K^{+} reabsorption (∼25% of delivered load). What should also be noted is that although the luminal K^{+} concentration is much higher than can be sustained by K^{+} secretion, the relaxation of luminal K^{+} toward the limiting value is slow, due to the relatively low K^{+} permeability of this segment. The end-luminal K^{+} concentration is still more than eightfold greater than the peritubular concentration. The impact of isosmotic NaCl reabsorption is emphasized in Fig.13, wherein the initial conditions are those in Fig. 11, with the exception that the entering flow is 20% of baseline. With such a sluggish flow, NaCl reabsorption is ∼80% of delivered load and occurs more rapidly than K^{+}reabsorption. This produces a sustained increase in luminal K^{+} concentration, to 33 mM by tubule end.

It should not be concluded from the foregoing figures that this model CCD cannot be a site of net K^{+} secretion. In Fig.14, the luminal membrane Na^{+} permeability and the density of peritubular Na-K-ATPase have both been increased by 50% (as in Fig. 6, *right*). In the CCD epithelial model, using perfusion bath conditions, this parameter change yields Na^{+} reabsorption and K^{+}secretion of 94.1 and −43.8 pmol · mm^{−1} · min^{−1}, respectively, comparable to the maximal fluxes observed in (ADH- and DOC-treated) rat tubules in vitro (40). Figure 14 contains the model prediction of such a tubule in vivo, using the perfusion conditions of Fig. 11. In this case, CCD Na^{+} reabsorption increases from 19 to 32%, the tubule hyperpolarizes to −35 mV, and the K^{+} reabsorption at baseline now becomes a net K^{+} secretion equal to 15% of delivered load (Table 5). With these parameters, the limiting K^{+} concentration is ∼35 mM, and the final tubular K^{+} concentration is nearly 40 mM. Considerations of osmotic water flux suggest that, by blunting the ADH effect, luminal hypotonicity might also render the CCD a site for K^{+} secretion. In the calculations of Fig.15, the luminal*P*
_{f} is set at 10% of its baseline value, and luminal osmolality reaches only 240 mosmol/kgH_{2}O. Along most of the tubule length, the luminal K^{+} concentration remains less than its limiting value, and, until the very end of the tubule, the K^{+} flux remains secretory. Thus, instead of the 6% overall K^{+} reabsorption at baseline, low tubule water permeability results in a 14% increase in K^{+} delivery to the medullary collecting duct. Furthermore, what had been 38% NH
reabsorption by the antidiuretic CCD becomes a net secretion of 3% (Table 5). Finally, as had been discerned experimentally, the greatest K^{+} secretion can be achieved when luminal Cl^{−} is replaced by an impermeant anion. This is illustrated in Fig. 16, in which 30 mM of an impermeant have been substituted for luminal Cl^{−}. In this tubule, distal delivery of K^{+} is now 28% greater than the entering load. Compared with control, both luminal hyperpolarization and decreased volume reabsorption combine to render conditions favorable for sustained K^{+} secretion.

## DISCUSSION

In the assessment of disorders of potassium excretion in humans, it has become customary to calculate the transtubular K^{+}gradient (TTKG)
Equation 29
*Equation 29
* is intended to estimate the tubule fluid K^{+} concentration at a point at which it was last isotonic to plasma, namely, the CCD. The formula assumes that there is negligible transport of solute in the medullary collecting duct, so that K^{+} concentration changes from cortex to final urine derive only from water abstraction in the medullary collecting duct. (This is an important limitation to the ultimate significance of the TTKG. If, for example, half of the sodium and urea delivered to the medullary collecting duct were to be reabsorbed there, then the computed TTKG would overestimate the CCD K^{+} concentration by approximately a factor of 2.) Nevertheless, in the antidiuretic human kidney, the TTKG does increase with K^{+} loading or administration of mineralocorticoid (13). It is tempting, therefore, to try to interpret the TTKG with reference to ion transport by CCD and its modulation by mineralocorticoid. In the present work, a model of the CCD was developed with parameters assigned to simulate observed fluxes and permeabilities of rat tubules in vitro under standard perfusion conditions. In these experiments, CCD K^{+}secretion is responsive to both ADH and mineralocorticoid, and these observations focus attention on the limiting luminal K^{+}concentration that can be established by tubular secretion. The principal finding of this modeling effort is that, in the antidiuretic CCD in vivo, water abstraction may be a critical factor in determining the TTKG, in the sense that the TTKG may be well above the limiting K^{+} gradient.

This conclusion regarding the magnitude of the limiting K^{+}gradient is a quantitative one and contingent on the accuracy of the model parameters, specifically the transport rates and certain key permeabilities. With respect to overall transport rates, Table6 provides a summary of observations of CCD fluxes from rabbit and rat under similar perfusion conditions (luminal Na,^{+} 145 mM; K^{+} , 5 mM) with or without additional ADH and mineralocorticoid. The baseline parameter set selected for the model calculations yielded net Na^{+} reabsorption and K^{+} secretion of 68.6 and −29.8 pmol · mm^{−1} · min^{−1}, respectively, and a transepithelial PD of −18.3 mV, consistent with perfusions in which both hormone effects are present. The measured Na^{+} flux largely determines the choice of luminal membrane Na^{+} permeability of the principal cell. Given this permeability, measurements of the K^{+} transference number for the luminal membrane (44) provide an estimate of luminal K^{+} permeability. Finally, the measurement of the fractional apical resistance (together with the observation that, in the rat, the peritubular membrane is nearly entirely K^{+}selective) determines peritubular K^{+} permeability. From these considerations, it appears that the overall permeability of the transcellular K^{+} pathway is likely to be severalfold greater than that across the tight junction. Because of the high cytosolic K^{+} concentration, the difference in K^{+} conductance between paracellular and transcellular pathways is higher still in this model, ∼10-fold greater. It must be acknowledged that in the above argument, experimental determination of tight junctional K^{+} permeability is not available, so that the measurement of tight junctional conductance was used as a surrogate (with the assumption that K^{+} permeability is 20% greater than that of Na^{+}).

From the parameters selected here, the prediction for the limiting K^{+} concentration was 23 mM (Fig. 10). This may be compared with micropuncture determinations of late distal fluid-to-plasma K^{+} concentration ratio of 3.8 in rats on a low-Na^{+} diet and treated with DOCA (29). In a microperfusion study, the limiting K^{+} concentration of rat DCT has been estimated to be 13–15 mM by extrapolating between secretory (10 mM) and reabsorptive (25 mM) luminal K^{+}concentrations (18). Perhaps a more direct determination of the limiting K^{+} concentration is the “stationary K^{+} concentration” observed in a split drop, 30.6 mM under control conditions (2). There do not appear to be measurements of limiting K^{+} gradients in perfused rat CCD, although, in a study of rabbit CCD, with prolonged contact time the tubules could raise luminal K^{+} concentration to >100 mM (19). An approximate analysis of the limiting luminal K^{+} potential indicated that it is roughly the ratio of principal cell peritubular uptake of K^{+} relative to the peritubular membrane K^{+} conductance and is influenced little by luminal membrane K^{+} conductance. Although it was possible to achieve high limiting K^{+} concentrations by decreasing peritubular membrane K^{+} permeability, electrophysiology of rabbit CCD has indicated that peritubular K^{+} conductance actually increases under the influence of mineralocorticoid (39). Thus the effect of aldosterone to enhance the limiting K^{+} concentration is likely to be due to enhanced peritubular K^{+} uptake plus luminal hyperpolarization, rather than decreasing K^{+} backleak.

Once DCT fluid reaches the water-permeable segment of the nephron, osmotic equilibration produces a near doubling of solute concentrations. If tubule fluid K^{+} is less than its limiting concentration, it can increase rapidly toward the limiting gradient (Fig. 11), and if tubule fluid K^{+} is at its limiting concentration at the start of CCD, it can increase well above the limiting gradient (Fig. 12). In the first case, K^{+}secretion by CCD is shut off, and in the second case K^{+}transport by CCD becomes reabsorptive. This osmotic equilibration is contingent on ADH-mediated hypotonic reabsorption. In the absence of ADH, the persistence of luminal hypotonicity maintains a favorable secretory gradient for K^{+} along the length of CCD (see Fig.15). These calculations highlight the necessity for ADH-modulated K^{+} transport to maintain K^{+} balance during transitions between antidiuresis and diuresis. They also underscore the energetic advantage of this organization of transport along the nephron, wherein the bulk of K^{+} secretion is completed before water abstraction (28, 29).

After osmotic equilibration with hypotonic reabsorption, there may be further isotonic reabsorption of luminal fluid due to NaCl transport by CCD. In this case, the limiting K^{+} concentration is no longer the zero-flux concentration used to obtain *Eqs. 27
* and *
28
* but rather some higher concentration that will support reabsorptive K^{+} flux at a rate comparable to the volume reabsorption, *J*
_{v}. If one ignores paracellular and intercalated cell fluxes, then the limiting condition satisfies
Equation 30where *L*
_{MS} is the series equivalent coefficient specified in *Eq. 26
*. Setting C
to be the limiting concentration that satisfies the zero-flux in*Eq. 27
* and denoting
one obtains
Equation 31
in which it has been assumed that luminal PD is little changed by the isotonic NaCl reabsorption. Defining an equivalent K^{+}permeability by
*Eq. 31
* may be rewritten
Equation 32where the approximation in *Eq. 31
* requires that*J*
_{v}/*h*
_{MS} be small. For the baseline model parameters, Table 4 shows the equivalent permeability,*h*
_{MS}, and in view of the end-tubule volume reabsorption, *J*
_{v} = 6 × 10^{−6} cm^{3}/cm^{2}, the reabsorptive volume flow is sufficiently small to apply *Eq. 31
*.

The estimated increment in limiting K^{+} concentration (Table4) is 15–30% above the zero-flux limiting concentration. From these considerations, the increase in the limiting K^{+}gradient due to isotonic NaCl transport should be a small contribution to the TTKG. This is not to say that the overall K^{+}reabsorption by CCD must be small. With reference to Fig. 13, in which the impact of slow axial flow is considered, when there was reabsorption of 75% of the delivered Na^{+} there was reabsorption of 60% of delivered K^{+} and an approach of luminal K^{+} concentration to the estimated limiting gradient. In one set of observations in normal rats, the TTKG was found to be 10, and the estimated CCD K^{+} concentration was back-calculated to be 40 mM (Table 8 in Ref. 72). This most closely corresponds to the situation in Fig. 12, in which the luminal K^{+} concentration delivered to CCD was 24 mM, which approximately doubled during osmotic equilibration. In this simulation, CCD K^{+} reabsorption was ∼20% of delivered load.

A critical issue for model development was estimating the relative contributions of K^{+} secretion and Cl^{−}reabsorption to maintaining overall electroneutral solute transport by CCD. The fraction of Na^{+} reabsorption that was balanced by K^{+} secretion was selected to be ∼50%, according to the observations of Schafer and Troutman (40). What had not been anticipated in construction of the model was that, once this ratio had been selected for the baseline case, it persisted over the whole range of Na^{+} reabsorption rates engendered by varying luminal Na^{+} permeability (Fig. 7). In contrast, increasing delivered NaCl load resulted in increased NaCl reabsorption with little change in K^{+} flux. Most of the Cl^{−}reabsorption by this model tubule was paracellular. Transport assigned to the β-cell was assumed to be small (17%). Supporting this assumption is the observation that when luminal Cl^{−} was removed from perfused (control) rat CCD, there was no significant change in HCO
reabsorption (17). A small fraction of Cl^{−} reabsorption (10%) was assigned to coupled transport across the principal cell via an NaCl cotransporter (57). It was found that even a small peritubular Cl^{−} conductance was sufficient to reduce cytosolic Cl^{−} to levels compatible with experimental observation (43). As long as changes in the rate of luminal membrane NaCl cotransport and peritubular membrane Cl^{−} permeability varied in parallel, there would be little discernible change in cytosolic Cl^{−} concentrations. When Na^{+} fluxes were varied by increasing luminal NaCl cotransport, there was no change in K^{+} secretion, despite the increased flux through the Na-K-ATPase. (The peritubular Cl^{−} current depolarized the epithelium, so that despite increased peritubular uptake of K^{+}, secretion was not enhanced.) Application of bradykinin, which can alter CCD Cl^{−} secretion without changing transepithelial PD, is difficult to understand in the absence of a transcellular electroneutral mechanism for Cl^{−}reabsorption (58). In simulations not shown, the impact of decreasing α-cell H^{+}-ATPase density on CCD K^{+}secretion was negligible, so that these calculations do not offer insight into the K^{+} wasting of distal renal tubule acidosis.

Issues of CCD acid-base transport have not been systematically explored in this initial presentation of the model. In the normal rat, the CCD is a net proton-secreting segment, but the rate of luminal acidification appears to be lower than either DCT or medullary collecting duct. Length-specific proton secretory rate in late DCT is perhaps comparable to that of CCD, whereas early DCT may be threefold greater (62). In the calculations of this model, osmotic water loss doubles luminal HCO
concentration in the early portion of the tubule, and there is a relatively stable concentration along the remainder of the segment. In the reference condition, ∼16% of the delivered HCO
was reabsorbed, but slow flow could enhance this substantially. A subtle, but perhaps important observation is that in antidiuresis there is also an abrupt increase in luminal ammonia concentrations, with brisk reabsorption of significant amounts of base (NH_{3}). Quantitatively, this base reabsorption is a significant fraction (2/3) of the net HCO
reabsorption. When this base exit from the lumen was eliminated in diuresis (Fig. 15 and Table 5), CCD HCO
reabsorption also virtually ceased. This mechanism for HCO
reclamation seems not to have been recognized previously. However, it should not be specific to some special feature of this model. Rather, substantial NH_{3}reabsorption by the CCD follows directly from the delivery of ammonia at a concentration an order of magnitude higher than in cortical blood, further concentration of ammonia and HCO
by water abstraction, and a high CCD permeability to NH_{3}. Although overall collecting duct transport of ammonia has been found to be small (21), prior models of the medullary collecting duct have predicted significant ammonia addition (69, 70). Thus sequential ammonia reabsorption and secretion along the collecting duct would not be incompatible with the overall micropuncture accounting.

In sum, with the emphasis of the present work on K^{+}excretion, CCD model parameters have been selected to represent the rat kidney under the influence of both ADH and aldosterone. The model was used to identify determinants of the transtubular K^{+}gradient. From the perspective of an epithelial model, the zero-flux equilibrium is the natural calculation, and in this the luminal concentration derives from the magnitude of principal cell peritubular K^{+} uptake relative to peritubular membrane K^{+}permeability. When the model is configured as a tubule and there is isosmotic NaCl reabsorption, the condition of luminal K^{+}equilibrium requires that overall K^{+} flux be reabsorptive and keep pace with the reabsorptive volume flux. This provides a small increment to the zero-flux equilibrium concentration. The most striking effect, however, in the simulation of conditions in vivo comes with osmotic equilibration of luminal fluid. With this, there is a doubling of the initial K^{+} concentration, which, depending on delivered load, may be substantially greater than the equilibrium value. This implies that the CCD could be a site for K^{+}reabsorption, although the relatively low permeability ensures that luminal K^{+} concentration declines slowly. Thus assessment of the TTKG from final urine composition may yield a value well above that which can be rationalized from transport studies in isolated perfused tubules.

## Acknowledgments

This investigation was supported by Public Health Service Grant 1-R01-DK-29857 from the National Institute of Diabetes and Digestive and Kidney Diseases.

## Footnotes

Address for reprint requests and other correspondence: A. M. Weinstein, Dept. of Physiology and Biophysics, Weill Medical College of Cornell University, New York, NY 10021.

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. Section 1734 solely to indicate this fact.

- Copyright © 2001 the American Physiological Society