INTRODUCTION

TOP ABSTRACT INTRODUCTION MODEL CCD MODEL PARAMETERS MODEL CALCULATIONS DISCUSSION REFERENCES

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 -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 -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

TOP ABSTRACT INTRODUCTION MODEL CCD MODEL PARAMETERS MODEL CALCULATIONS DISCUSSION REFERENCES

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(i), where  is lumen (M), interspace (E), principal cell (P), -intercalated cell (A), -intercalated cell (B), 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 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, CO₂, H₂CO₃, HPO, H₂PO, NH₃, 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.

View larger version (16K): [in this window] [in a new window]   Fig. 1.   Schematic representation of cortical collecting duct (CCD) epithelium, consisting of principal and intercalated cells and lateral intercellular space (LIS). The model tubule lumen is lined by this epithelium. Intraepithelial fluxes are designated J(i), where the subscript refers to luminal cell membranes (MP, MA, MB), lateral cell membranes (PE, AE, BE), basal cell membranes (PS, AS, BS), tight junction (ME), 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 (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

(1)

where V_I is the compartment volume (cm³/cm²). The interspace exchanges with all of the model compartments, so that mass generation is written

(2)

Within the tubule lumen, mass generation is appreciated as an increase in axial flux, as transport into the epithelium, or as local accumulation

(3)

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 volume and for the nonreacting species (Na⁺, K⁺ , Cl, and urea) are written

(4)

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

(5)

(6)

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. Because HCO and H₂CO₃ are interconverted, mass conservation requires

(7)

for  = P, A, B, or E, whereas for the tubule lumen

(8)

In each compartment ( = P, A, B, E, or M), conservation of total CO₂ is expressed

(9)

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

(10)

where z_i is the valence of species i. In this model, conservation of charge for the buffer reactions (Eq. 10) may be rewritten

(11)

The solute equations are completed with the chemical equilibria of the buffer pairs: HPO:H₂PO, NH₃:NH, and HCO:H₂CO₃. Corresponding to the electrical potentials, , for  = P, A, B, E, or M, is the equation for electroneutrality

(12)

With 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

(13)

where V_E0 and A_ES0 are reference values for volume and outlet area, respectively, and _E is a compliance. Along the lumen, hydrostatic pressure changes according to an equation for Poiseuille flow

(14)

Solute 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

(15)

In Eq. 15, 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

(16)

is 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 is the PD between compartments  and . In this model, all of the permeabilities, h(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

(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  is

(18)

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/HCO exchange across the peritubular membrane of the -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

(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⁺

(20)

The pump flux of K⁺ plus NH reflects the 3:2 stoichiometry

(21)

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

(22)

Analogous expressions are written for active transport at the basal cell membranes, J(Na⁺). Within the luminal membrane of the -cell and the peritubular membrane of the -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

(23)

where J(H⁺)_max is the maximum proton flux, _MI (H⁺) is the electrochemical PD 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. (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 -intercalated cell, there is an H-K-ATPase, which has been given a full kinetic representation (69).

	MODEL PARAMETERS

TOP ABSTRACT INTRODUCTION MODEL CCD MODEL PARAMETERS MODEL CALCULATIONS DISCUSSION REFERENCES

The parameters displayed in Table 1 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% - and -cells, respectively (23). In the rabbit, lateral interspace volume is ~11% of epithelial volume (71). For a total epithelial volume of ~5 × 10⁴ cm³/cm², this yields a principal cell volume of 3 × 10⁴ cm³/cm² and - and -intercalated cell volumes of 1.2 and 0.8 × 10⁴ cm³/cm², respectively. The cellular compartments, along with the important transport pathways, are displayed in Fig. 2.

View larger version (25K): [in this window] [in a new window]   Fig. 2.   A: CCD transport pathways along with the model concentrations (mM) and fluxes (pmol · mm1 · min1) computed at the initial portion of the tubule. B: CCD transport pathways along with the model concentrations (mM) and fluxes (pmol · mm1 · min1) computed at the distal portion of the tubule (x = 2 mm, corticomedullary junction).

In rat CCD, the principal cell luminal and peritubular membrane areas have been reported to be ~4,000 and 25,000 cm²/cm³ cell volume (50, 61), which translates into 1.2 and 7.5 cm²/cm² 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/cm² (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 NH₃ 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 cm²/cm³ cell volume for the luminal and peritubular membranes of -cells, and 2,150 and 21,000 cm²/cm³ for -cell membranes (60), respectively. When corrected for the volume density of - and -cells (1.2 and 0.8 × 10⁴ cm³/cm², respectively), these membrane areas are 0.6 and 1.7 cm²/cm² for the -cell and 0.2 and 1.7 cm²/cm² epithelial area for the -cell, respectively. The important -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 -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 -cell (33). The important -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 -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 -cells of rabbit CCD (65, 67, 74). The -cell unit membrane permeabilities for nonelectrolytes have been assumed to be identical to those chosen for the -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 CO₂ 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 - and -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/cm², 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/cm², 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/cm², with relative solute permeabilities comparable to their mobility in solution.

	MODEL CALCULATIONS

TOP ABSTRACT INTRODUCTION MODEL CCD MODEL PARAMETERS MODEL CALCULATIONS DISCUSSION REFERENCES

Suitability of the parameter choices is assessed, in part, by examining predicted fluxes and permeabilities. Table 2 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:::interspace) are 59, 22, 14, and 5%, respectively, of a total epithelial volume of 6 × 10⁴ cm³/cm². 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 · mm¹ · min¹ with a paracellular backflux of 23.0, giving a net reabsorptive Na⁺ of 68.6 pmol · mm¹ · min¹. 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 · mm¹ · min¹. The Cl flux is primarily paracellular (22.2), with smaller components across the -cell (5.2) and principal cell (3.1). The remainder of the Na⁺ flux is balanced by an equivalent reabsorptive "HCO" flux (pmol · mm¹ · min¹) of 8.2, comprised -cell HCO secretion (5.2) in parallel with -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 NH₃.

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/cm²). The total conductance is ~8 mS/cm²; 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 × 10⁵ cm/s (25), for NH, 2.6 × 10⁵ cm/s (14), and for NH₃, 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⁺ (Table 1), 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.

View larger version (26K): [in this window] [in a new window]   Fig. 3.   Impact of transepithelial potential difference (PD) on CCD ion fluxes. Calculations use the epithelial model with equal luminal and peritubular solutions (Table 2). "HCO"refers to the sum of proton secretion plus HCO reabsorption.

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.

View larger version (32K): [in this window] [in a new window]   Fig. 4.   Effect of luminal Na variation on CCD function. Calculations use the open-circuited epithelial model. Left: luminal Na decreased by substitution with an impermeant cation. Right: varied luminal NaCl and addition of a neutral impermeant for osmotic balance.

View larger version (32K): [in this window] [in a new window]   Fig. 5.   Principal cell luminal membrane during luminal NaCl variation. Calculations are those in Fig. 4 (right) using the open-circuited epithelial model with luminal NaCl variation. hMP(Na) and hMP(K), luminal membrane ionic permeabilities, with hMP(Na) being a function of luminal Na+ concentration (Eq. 17). Middle: electrochemical potentials of Na+ and K+, respectively, across the luminal membrane of the principal cell.

View larger version (33K): [in this window] [in a new window]   Fig. 6.   Variation of principal cell luminal membrane Na+ permeability, hMP(Na). Left: variation of principal cell luminal membrane Na+ in isolation. Right: variation of peritubular Na-K-ATPase density in parallel with hMP(Na). The open-circuited epithelial model is used with identical (high-Na+) luminal and peritubular solutions of Table 2.

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).

View larger version (32K): [in this window] [in a new window]   Fig. 7.   Epithelial Cl and K+ fluxes as a function of Na+ flux. Top: replotting of results from the calculations in Fig. 6, in which principal cell luminal membrane Na+ permeability is varied alone (left), or in parallel with peritubular Na-K-ATPase density (right). Bottom left: luminal NaCl cotransport coefficient of the principal cell increased over a 20-fold range in parallel with the peritubular membrane Cl permeability. Bottom right: replotting of the calculations in Fig. 4, in which luminal NaCl is varied.

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.

View larger version (31K): [in this window] [in a new window]   Fig. 8.   Variation of principal cell membrane K+ permeabilities. Left: variation of luminal membrane permeability. Right: variation of peritubular K+ permeability. The open-circuited epithelial model is used, with identical (high-Na+) luminal and peritubular solutions of Table 2.

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.

View larger version (39K): [in this window] [in a new window]   Fig. 9.   Variation of luminal K+ concentration by addition of KCl. Baseline perfusion and bath using high-Na+ solutions in Table 2 (left) and luminal perfusate in vivo condition in Fig. 2 (right). Middle: electrical PD and K+ potential across the principal cell luminal membrane, with the cytosol as reference (positive potentials support reabsorptive fluxes). Bottom: transepithelial K+ fluxes resolved into cellular and paracellular components.

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 ^<