INTRODUCTION

Top Abstract Introduction Discussion References

IN THE RAT, the inner medullary collecting duct (IMCD) receives about 5% of the filtered fluid, containing roughly 3.0-4.5% of filtered Na⁺ and 20% of filtered K⁺ (3, 43). Final regulation of urinary osmolality, Na⁺, K⁺, and pH occurs as some 7,200 tubules merge 5 or 6 times to form 100-200 papillary collecting ducts (24, 32). In vivo studies in the rat have indicated that IMCD Na⁺ transport rates can be comparable to those of the proximal convoluted tubule (2, 7, 52), although rates in dissected tubules may be substantially lower than those in vivo (34). Although the luminal cell membrane contains a cation channel (29) and a component of Na⁺ reabsorption is amiloride inhibitable (21, 35, 62), the transepithelial potential difference (PD) of this segment is relatively small (16, 19, 34, 37, 47), and thiazides can inhibit a significant fraction of Na⁺ transport by the IMCD (35, 61). Under normal circumstances, the IMCD appears to be a site for K⁺ reabsorption (5, 7), whereas in times of K⁺ loading, IMCD K⁺ secretion can be detected (3, 7). There is luminal proton secretion along the IMCD (16, 53), which appears referable in large measure to an H-K-ATPase (14, 23, 31). Furthermore, the rate of acid secretion is modulated by NH₃ availability to mediate peritubular base exit (54).

From these considerations, any model of IMCD which seeks to represent Na⁺ transport must also provide a reasonable picture of K⁺ and acid/base fluxes. Although most of the important features of these cells have been represented in prior models, several aspects of IMCD have not been modeled previously in epithelial simulations. These include the finite rate of hydration of CO₂ within the cell and extracellular compartments, the luminal membrane H-K-ATPase, the large variation in peritubular solute composition over the length of the duct, and the progressive coalescing of the tubules to become the papillary collecting ducts. In this report, the basic structure of the model will be presented, with emphasis on the pathways for Na⁺ and K⁺ transport. In the companion study (59a), features of acid/base transport will be examined. In this work, the epithelial model will also be cast as a tubule model to ensure that the computed epithelial fluxes result in realistic axial solute gradients. The calculations to be presented suggest that the magnitude of the transcellular Na⁺ and K⁺ fluxes are such that both luminal NaCl cotransport and peritubular KCl cotransport must be important pathways in this epithelium.

	MODEL STRUCTURE

The 12 model solutes are Na⁺, K⁺, Cl, HCO₃, CO₂, H₂CO₃, HPO²₄, H₂PO₄, NH₃, NH⁺₄, H⁺, and urea, as well as an impermeant species within the cells and possibly within the lumen. These are the minimal set of solutes which will permit representation of net acid excretion. Compared with the previous cortical collecting duct (CCD) model of Strieter et al. (51), the additional solutes are H₂CO₃, the two ammonia species, and urea. As in previous work, the model will be composed of compliant cellular and intercellular compartments lining the tubule lumen (Fig. 1). Within each compartment, the concentration of species i is designated C(i ), where  is lumen (M), interspace (E), cell (I), or peritubular solution (S). Within the epithelium the flux of solute i across membrane is denoted J(i ) (mmol · s¹ · cm²), where may refer to luminal cell membrane (MI), tight junction (ME), lateral cell membrane (IE), basal cell membrane (IS), or interspace basement membrane (ES). Along the tubule lumen, axial flows of solute are designated F_M(i ) (in mmol/s). To formulate the equations of mass conservation with multiple reacting solutes, it has been convenient to identify the generation of each species as an intermediate variable. Within a cell or interspace, the generation of i (s(i )) is equal to its net export plus its accumulation

(1)

(2)

where V is the compartment volume (in cm³/cm²). Within the tubule lumen, solute generation is appreciated as an increase in axial flux, as transport into the epithelium, or as local accumulation

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

(4)

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

(5)

(6)

View larger version (14K): [in this window] [in a new window]   Fig. 1.   Schematic representation of the inner medullary collecting duct (IMCD) epithelium as cellular and paracellular (LIS) compartments. Epithelial model defines the transmural fluxes for the IMCD tubule. Tree at bottom represents the coalescing of 7,200 initial IMCD tubules to 112 terminal IMCD tubules. Variables and subscript notations are described in MODEL STRUCTURE.

One important feature of this model will be the handling of CO₂ and the disequilibrium pH. A medullary interstitial PCO₂ profile will be specified, but the CO₂ concentrations of the cells, interspace, and lumen are model variables. The relevant reactions are where dissociation of H₂CO₃ is rapid and assumed to be at equilibrium. Since HCO₃ and H₂CO₃ are interconverted, mass conservation requires

(7)

for  = I or E, whereas for the tubule lumen

(8)

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

(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 takes the form

(11)

The solute equations are completed with the chemical equilibria of the following buffer pairs: HPO²₄:H₂PO₄, NH₃:NH⁺₄, and HCO₃:H₂CO₃. Corresponding to the electrical potentials, , for  = E, I, or M, is the equation for electroneutrality

(12)

With respect to volume flow, the approach taken heretofore has been to utilize the volume conservation equations for lumen, interspace, and cell to compute the three unknowns: luminal volume flow, lateral interspace hydrostatic pressure, and cell volume. (Since cell hydrostatic pressure was set equal to luminal pressure, and total cell impermeant content was also fixed, changes in cell volume adjusted the concentration of the impermeant to achieve cell water balance.) In a model of IMCD epithelium, this approach is unsatisfactory, since the large variations in peritubular osmolality would impact unrealistically on cytosolic electrolytes. In vivo, chronic changes in interstitial osmolality leave cytosolic electrolyte composition relatively unperturbed because of compensating changes in organic osmolytes (42). Indeed, changes in cytosolic osmolyte concentrations can occur relatively rapidly in response to ambient conditions (41). Ideally, one would specify the concentrations of organic osmolytes as additional model variables with their own kinetics. However, what has been done here is to restrict the simulations to steady-state problems and assume that cell volume homeostasis has been achieved by adjustment of an impermeant osmolyte, b. Thus with cell volume specified and fixed, C_I(b) is the model variable used to satisfy the equations for fluid balance across the luminal and peritubular cell membranes. Across cell each membrane, the volume fluxes are proportional to the hydrosmotic driving forces. With respect to the lateral interspace, its volume, V_E, and its basement membrane area, A_ES, are functions of interspace hydrostatic pressure, P_E

(13)

where V_E0 and A_ES0 are reference values for volume and outlet area, respectively, and _E is a compliance.

Solute transport is either electrodiffusive (e.g., via a channel), coupled to the electrochemical potential gradients of other solutes (e.g., via a cotransporter or an antiporter), or coupled to metabolic energy (via an ATPase). This is expressed in the model by the flux equation

(14)

In Eq. 14, 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

(15)

is a normalized electrical PD, where z_i is the valence of i, and    is the PD between compartments  and . The second term of the solute flux equation specifies the coupled transport of species i and j according to linear nonequilibrium thermodynamics, where the electrochemical potential of j in compartment  is

(16)

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.

In this model, the Na-K-ATPase within the peritubular membrane is represented by the expression (12)

(17)

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⁺ (51). The pump flux of K⁺ plus NH⁺₄ reflects the 3:2 stoichiometry

(18)

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

(19)

Analogous expressions are written for active transport at the basal membrane, J^act_IS. In collecting duct, the ratio K_NH⁺4:K_K is approximately 0.2 (56). Across the luminal membrane there is an H-K-ATPase, and in view of the large range of luminal solute concentrations encountered in vivo, the representation of this pump must be suitably robust. This pump has been given a kinetic description and is presented in detail in the companion study (59a).

In this model, both the cross-sectional area and the epithelial transport area decrease as one proceeds from the outer-inner medullary junction (OIMJ) toward the papillary tip. Starting with 7,200 tubules at the base of IMCD, a series of 6 mergings of pairs of ducts reduces the final number of papillary collecting ducts by 1/64 to 113 (24). With each merging, the lateral surface for transport, B_M, and the axial cross section for flow, A_M, are halved. In the computer code, this is accomplished using a continuous formulation as a function of distance, x, along the IMCD of total length L

(20)

In these tubule calculations, the luminal hydrostatic pressure has been specified and assumed to be constant along the length of the tubule. This avoids the computation of a pressure drop along a system of coalescing, distensible tubules.

	MODEL PARAMETERS

In the selection of model parameters, effort was taken to achieve compatibility with observations in rat IMCD, where both in vivo and in vitro data are available. A simplification was made in representing the IMCD cell as a single entity, ignoring differences from base to papillary tip. Model parameters are indicated in Table 1. The luminal diameter, set to 30 µm, is comparable to the observation of Sands et al. (39), and cell volume, 8 × 10⁴ cm³/cm², is in the range reported by Flamion and Spring (9). Morphological data of Rastegar et al. (33) indicate apical-to-basolateral membrane area ratio of 9.5 and basolateral-to-tubular basement membrane area of 5.6. For a tubule with inner and outer radii of 15 and 23 µm, respectively, these measurements are compatible with an unamplified luminal membrane, whose area is 0.6 cm²/cm² relative to the tubule basement membrane. Total tubule length is 5 mm (61). The volume of the lateral intercellular space was taken to be ~10% of the epithelial volume (with a relatively small compliance), a value comparable to that observed in CCD (60). Carbonic anhydrase (CA) activity diminishes along the IMCD, from substantial staining at the outer third to negligible at the papillary tip (30). This is, presumably, the cytosolic enzyme CA-II, since the membrane bound carbonic anhydrase, CA-IV, does not appear to be present in IMCD (4, 55). Accordingly, within the lumen and lateral interspace, the rate constants for hydration and dehydration are the uncatalyzed values (13); within the cells, the coefficients have been assumed to be 100-fold larger (i.e., 1% of complete catalysis). Permeabilities of the tight junction were taken to yield compatibility with the overall electrical resistance of the epithelium and a relative anion selectivity (39). The interspace basement membrane conductance was assumed to be about two orders of magnitude greater than that of the tight junction, and solute permeabilities were proportional to diffusivity in free solution.

In Fig. 2, the important cellular transport pathways have been indicated. On the peritubular membrane is an Na-K-ATPase, with competition between NH⁺₄ and K⁺ for cellular uptake. The peritubular membrane of these cells contains Na⁺/H⁺ exchange (18, 57) and Cl/HCO₃ exchange (27, 48). A peritubular cotransporter for Na⁺-K⁺-2Cl has also been identified (17, 36). Also shown within the peritubular membrane is a cotransporter for HPO²₄, (in parallel with a small peritubular phosphate leak) so that the cells will have a nonzero concentration of phosphate, serving as a second internal buffer; there is, however, no vectorial transport of phosphate. The peritubular KCl cotransporter has not been identified experimentally but is a prediction of these model calculations and will be considered below. The conductance of the peritubular membrane is predominantly that for K⁺ (47). With such K⁺ channels, an NH⁺₄ permeability may also be associated which may be less than (20) or comparable (6) to that for K⁺. In this case, the NH⁺₄ permeability was taken to be 20% that for K⁺. The Cl permeability was assumed small, but not insignificant, namely 3% of the K⁺ permeability, and the HCO₃ permeability was one-half that for Cl. At the luminal membrane, the only significant conductance is that of a nonselective cation channel, with equal permeabilities to Na⁺ and K⁺ (29). The importance of this channel for luminal Na⁺ entry, relative to NaCl cotransport, is considered in the model calculations below. Proton secretion at the luminal membrane derives at least in part (14), and perhaps exclusively (23, 31, 58), from an H-K-ATPase. In the calculations of this study, this is the only luminal proton pump.

View larger version (21K): [in this window] [in a new window]   Fig. 2.   Important transport pathways of the IMCD cell. Model cell fluxes (pmol · s1 · cm2) are those obtained for the open-circuited epithelium, under conditions resembling those at the outer-inner medullary junction.

Although not pictured in Fig. 2, membrane permeabilities have been assigned for CO₂, H₂CO₃, NH₃, urea, and water. The unit membrane permeabilities for CO₂ and H₂CO₃ are those used for proximal tubule cell membrane (26) and have been assumed to apply to both luminal and peritubular membranes. The NH₃ permeability is derived from the whole epithelial determination of Flessner et al. (10), by assuming the unit membrane permeabilities of luminal and peritubular membranes are comparable. (Since the peritubular membrane area is 10-fold greater than the luminal, it is the luminal membrane permeability that is measured in the transepithelial determination.) Sands et al. (38) have found an IMCD urea permeability 1.7 × 10⁴ cm/s and 6.9 × 10⁴ cm/s in the absence and presence of ADH. Accordingly, the tight junction urea permeability was taken as 30% of the unstimulated epithelial permeability, and the unit membrane urea permeabilities of luminal and peritubular surfaces were assumed to be comparable (giving the peritubular membrane a 10-fold greater urea permeability). In this way, the unit membrane peritubular urea permeability (9 × 10⁵ cm/s) is comparable to that determined by Star (49). The stimulated luminal membrane urea permeability was taken as 20-fold greater than its unstimulated value, and all calculations were performed with the assumption that antidiuretic hormone (ADH) is present. ADH also produces a threefold increase in the IMCD water permeability (P_f) from 7 × 10³ to 19 × 10³ cm/s (38). As with urea, the tight junction water permeability was taken as 30% of the unstimulated epithelial value, and the unit water permeabilities of luminal and peritubular membranes were assumed comparable. A stimulated luminal membrane water permeability fivefold greater than the unstimulated value yielded a realistic overall epithelial water permeability.

	MODEL CALCULATIONS

Table 2 contains the results of a solution of the model equations for an open-circuited epithelium between bathing media in which the solute concentrations are suggestive of the OIMJ. The high luminal K⁺ (7) and low luminal HCO₃ (8, 15) have been observed in vivo. The luminal NH⁺₄ concentration is similar to that found from microcatheterization (15, 46). The peritubular concentrations of K⁺ (7) and NH⁺₄ (50) are both high, whereas the HCO₃ concentration is not very different from that of the systemic circulation (8). The computed cellular electrolyte concentrations are similar to those obtained by Sone et al. (42) using the electron microprobe. In particular, the cell Na⁺ is about twice as high, and the cell Cl is about five times higher than that found by the same workers in the CCD (1). In vitro, dissected rat IMCD cells have an acid pH (54), and this is also achieved with this set of model parameters. The solute concentrations within the lateral interspace largely reflect those of the peritubular medium, because of the relatively small diffusion barrier of the basement membrane. Here, one model prediction that emerges is a significant disequilibrium alkalosis within the interspace. This is due to the peritubular HCO₃ extrusion, the lack of membrane-bound CA, and the micromolar concentrations of H₂CO₃. The open-circuit PD is 13 mV, a value close to that first observed (37) but higher than subsequent determinations.

Figure 2 displays several of the important fluxes when the bathing media concentrations are those assigned to the OIMJ. Luminal membrane Na⁺ flux is divided between channel-mediated entry (26%) and NaCl cotransport; peritubular Na⁺ extrusion is via the Na-K-ATPase, with small uptake fluxes through Na⁺/H⁺ exchange, phosphate entry, and Na-K-2Cl cotransport. With reference to Table 2, the electrochemical gradient favoring blood-to-lumen Na⁺ flux produces a transjunctional backflux ~39% of the luminal membrane reabsorption. Under the conditions shown, luminal membrane K⁺ flux is also reabsorptive, both via the nonselective cation channel (29%) and the H-K-ATPase. At the peritubular membrane, there is also cellular uptake of K⁺ via the Na-K-ATPase and, to a minor degree, with Na-K-2Cl cotransport. Peritubular exit of K⁺ via channel-mediated flux is relatively small (12%) in comparison to KCl cotransport. The high luminal K⁺ concentration yields a reabsorptive paracellular flux, which is ~11% of the total transepithelial flux. With respect to Cl, reabsorption at the luminal membrane is cotransport with Na⁺ and at the peritubular membrane with K⁺; paracellular Cl flux is secretory but small. Plausibility of these overall epithelial Na⁺, K⁺, and Cl fluxes requires scrutiny of the performance of this IMCD when it is configured as system of tubules.

Using the geometry outlined above, the 7,200 tubules of a single kidney were perfused at 24 µl/min, or ~5% of single-kidney glomerular filtration rate (GFR) (43). The perfusing solution composition was that of the OIMJ (Table 2), so that Na⁺ delivery was 4% and K⁺ delivery 50% of estimated filtered loads. The peritubular solute concentrations at the terminal IMCD were taken to be identical to those at the OIMJ, with the exceptions of urea (which increased from 200 to 500 mM) and K⁺ (which increased from 10 to 20 mM), so that peritubular osmolality increased from 800 to 1,120 mosM. Along the IMCD length, peritubular concentrations were assumed to vary linearly. Model equations were solved using a first-order implicit differencing scheme, since the centered scheme used previously for proximal tubule calculations would not converge. For the baseline parameters, 800 spatial steps were required to reach the point of no further visual change in the solution curves. Figure 3 displays computed solute profiles and flows within the IMCD. About 71% of the entering Na⁺ and 65% of entering K⁺ has been reabsorbed, but with progressive water reabsorption there is a 24% increase in end-tubule Na⁺ and a 54% increase in end-tubule K⁺ concentrations. Proton secretion via the H-K-ATPase has reduced the luminal HCO₃ concentration from 5 mM to near zero, with pH falling nearly 3 units to 4.0. The NH⁺₄ concentration, initially 10 mM and increasing to 58 mM, reflects ammonia addition along the tubule equal to 47% of entering flow; the magnitude of this secretion is similar to that reported from microcatheterization (46).

View larger version (26K): [in this window] [in a new window]   Fig. 3.   Luminal profiles and flows predicted by the IMCD tubule model. Tubule cross section and surface area decreases exponentially from the outer-inner medullary junction (OIMJ) to the papillary tip, whereas area-specific transport rates remain uniform. Interstitial profile at x = 0 is that for the OIMJ, whereas at x = 0.5 cm there has been an increase of urea from 200 to 500 mM and an increase of K+ from 10 to 20 mM; for intermediate points, the interstitial concentrations are linear interpolations. OIMJ entering flow is 24 µl/min. Top: solute concentration profiles. Bottom: axial flows of solute and volume relative to the initial rate.

In view of the large transepithelial solute concentration gradients, accuracy of the fluxes is contingent on accuracy of the overall epithelial solute permeabilities. Table 3 displays the results of simulating idealized epithelial permeability determinations. For these calculations, a short-circuited tubule epithelium in vitro was represented, bathed by equal luminal and peritubular solutions of the following composition (mM): 140 Na⁺, 10 K⁺, 119 Cl, 25 HCO₃, 1.5 CO₂, 3.9 total phosphate, 5.0 urea, 1.0 NH⁺₄, and 0.1 impermeant. A series of calculations were performed in which the luminal solute concentrations were lowered and then raised by 0.1 mM. After subtracting off the small correction due to changing volume flux, 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 (in mS/cm²) and when multiplied by z(i )FC(i )/RT is the ionic permeability (in cm/s), and these are also included in Table 3. It is clear that the important conductances are those for Na⁺ and Cl, and to a minor extent K⁺ and HCO₃, and that together these confer a total epithelial conductance of 14.6 mS/cm² or 68  · cm². This may be compared with experimental conductance determinations of 13.7 (47) and 25.4 mS/cm² (39). The ion permeabilities for Na⁺ and Cl derived from the conductances are 1.15 and 1.85 × 10⁵ cm/s and are similar to those obtained by Sands et al. (39). The permeabilities derived from imposition of ion gradients are greater (2.32 and 2.29 × 10⁵ cm/s) and reflect the presence of electroneutral, coupled solute pathways. The model K⁺ permeability is 3.1 × 10⁵ cm/s; for comparison, a value for rat IMCD K⁺ permeability of 4.0 × 10⁵ has been obtained by Rocha and Kudo (34), using isotopic bath-to-lumen K⁺ secretion rates. The model permeabilities for NH⁺₄, 0.98 × 10⁵, and NH₃, 0.20 × 10² cm/s, are comparable to those obtained by Flessner et al. (10) for rat IMCD in vitro. The water permeability, P_f = 0.014 cm/s, and the urea permeability, 3.36 × 10⁴ cm/s, are similar to values reported for ADH-treated rat IMCD (34, 38).

Luminal membrane NaCl cotransport is a prominent feature of this model IMCD cell; the constraint on the magnitude of this component is examined in Fig. 4. In these calculations, luminal membrane NaCl cotransport is varied by varying the coefficients for NaCl cotransport and peritubular KCl cotransport proportionally. At the same time, the luminal membrane cation conductance (equal for Na⁺ and K⁺) is adjusted so that transcellular Na⁺ flux remains constant at 5.6 neq · s¹ · cm². Thus the fraction of luminal flux via cotransport was changed without changing the total luminal membrane Na⁺ flux. This was achieved by incorporating the IMCD model as a subroutine in a Newton iteration using the luminal cation conductance as the single independent variable. The boundary conditions for these calculations are those for the OIMJ, as in Table 2. The bottom of Fig. 4 displays the luminal membrane cation permeability determined from the Newton iteration. The middle of Fig. 4 is the fractional apical resistance, defined as the electrical resistance of the luminal membrane divided by the sum of luminal and peritubular resistances. Transepithelial and peritubular electrical PD values are plotted in the top of Fig. 4; the distance between these two curves is the luminal membrane PD. The arrows on each abcissa of Fig. 4 indicate the fractional luminal Na⁺ flux using the baseline parameters of Table 1, so that for this case the fractional apical resistance is 85%. The electrophysiological study of rat IMCD by Stanton (47) found a fractional apical resistance of 94 ± 5% (SD) along with a transepithelial PD of 3 mV. This observation suggests that the NaCl cotransporter cannot be any less important than its representation by the baseline model parameters.

View larger version (24K): [in this window] [in a new window]   Fig. 4.   Impact of electroneutral vs. conductive luminal Na+ entry on epithelial electrical parameters. Using OIMJ boundary conditions, the coefficient for Na-Cl cotransport is varied, and there is adjustment of the luminal cation (Na+ and K+) permeability so as to maintain transcellular Na+ flux fixed at 9.6 nmol · s1 · cm2. Results are plotted as a function of the fraction of luminal Na+ flux via the cotransporter, and arrow indicates baseline parameters. Bottom: luminal membrane cation permeability, which yields the desired transcellular Na+ flux. Middle: luminal resistance divided by the sum of luminal and peritubular resistances.

For the same calculations of Fig. 4, Fig. 5 displays the model K⁺ fluxes. The curves in the top of Fig. 5 indicate that the paracellular K⁺ is relatively minor in comparison to the total. The bottom of Fig. 5 contains the total luminal membrane K⁺, along with its two components, that through the cation channel and via the H-K-ATPase. It is apparent that as the channel permeability starts to increase (moving from right to left along the abcissa), the reabsorptive K⁺ flux through the channel also increases. However, with increasing channel permeability there is progressive luminal membrane depolarization (Fig. 4). Ultimately, the channel K⁺ flux declines and becomes secretory to the extent that it nullifies the reabsorptive pump flux, and net luminal membrane K⁺ flux is negligible.

View larger version (22K): [in this window] [in a new window]   Fig. 5.   Impact of electroneutral vs. conductive luminal Na+ entry on potassium fluxes. For the same calculations as Fig. 4, top contains the paracellular and total transepithelial K+ fluxes. Bottom: contains the transcellular K+ fluxes, broken down between the cation channel and the H-K-ATPase. Arrow indicates baseline parameters.

The impact of luminal Na⁺ concentration on the IMCD Na⁺ and K⁺ fluxes is examined in Fig. 6. In these calculations, two sets of model parameters are considered: calculations in Fig. 6, left, are obtained using the baseline parameters, in which about three-fourths of the luminal membrane Na⁺ flux is via cotransport; and calculations in Fig. 6, right, are obtained using parameters in which about three-fourths of the luminal membrane Na⁺ flux is conductive. This parameter set was derived from the calculations of Fig. 4, so that total luminal cell membrane Na⁺ fluxes are equal under OIMJ bathing conditions. The bathing conditions for these calculations are those of the OIMJ, except for the luminal NaCl concentration, which is shown on the abcissa of Fig. 6. For both Na⁺ and K⁺, and with either parameter set, the junctional fluxes are small. In the top of Fig. 6, it may be observed that for both sets of model parameters, the luminal membrane Na⁺ fluxes are equal at high luminal Na⁺ (by design), and both go to zero at a luminal Na⁺ concentration of 2 mM. However, in the midrange of luminal Na⁺ concentrations, the cotransport parameters provide a luminal membrane flux that is less sensitive to luminal Na⁺ concentration. As a consequence, the luminal Na⁺ concentration at which total epithelial Na⁺ flux falls to zero is near 20 mM when cotransport dominates but over 40 mM when the channel dominates. With respect to K⁺ fluxes, the salient difference between the two models is the relative independence of Na⁺ and K⁺ fluxes, when NaCl cotransport is most important.

View larger version (33K): [in this window] [in a new window]   Fig. 6.   Sensitivity of Na+ and K+ fluxes to luminal NaCl concentration. Calculations utilize the epithelial model, and bath conditions are those of the OIMJ, except for luminal NaCl concentration on the abcissa. Left: obtained using baseline parameters, in which 3/4 of luminal membrane Na+ flux is via cotransport. Right: obtained using parameters in which 3/4 of luminal membrane Na+ flux is conductive. Under OIMJ bathing conditions (110 mM NaCl), luminal cell membrane Na+ fluxes are equal for both parameter sets.

The ability of the model IMCD to reabsorb Na⁺ against an adverse electrochemical gradient is further examined in Fig. 7. For these calculations, all of the parameter sets generated from the calculations of Fig. 4 are utilized, and the abcissa identifies each set by the fractional luminal NaCl cotransport under OIMJ bathing conditions. For each parameter set, luminal NaCl concentration is varied to bring some Na⁺ flux to zero, and that concentration is plotted as the ordinate. This was achieved by incorporating the IMCD model as a subroutine in a Newton iteration using the luminal Na⁺ concentration as the single independent variable. The bottom curve in Fig. 7 demonstrates that this model cell can reabsorb Na⁺ down to a luminal concentration of ~2 mM and that this equilibrium value is independent of the mechanism of luminal Na⁺ uptake. The top curve of Fig. 7 signifies the value of luminal NaCl when the cellular reabsorptive flux is just equal to the paracellular backflux; this is the curve most relevant to predicting the equilibrium Na⁺ concentration of the luminal fluid in vivo. It is apparent that conductive uptake is less effective than the coupled NaCl cotransporter in clearing the lumen of Na⁺. In part, this is due to the more negative luminal PD, when luminal Na⁺ uptake is conductive. This is shown in the calculations of the middle curve of Fig. 7, in which transepithelial Na⁺ is brought to zero while the epithelium is short circuited; the small slope that remains for this middle curve reflects the differences in shape of the curves of luminal membrane Na⁺ flux as a function of luminal Na⁺ concentration, displayed in Fig. 6. At its best, this model epithelium can transport down to ~17 mM, although the rats examined by Diezi et al. (7) achieved end-papillary sodium concentrations of 11.4 mM and final urine concentrations of perhaps one-half that value.

View larger version (24K): [in this window] [in a new window]   Fig. 7.   Impact of electroneutral vs. conductive luminal Na+ entry on luminal Na+ equilibrium concentrations. Calculations of Fig. 4 generated parameter sets yielding identical transcellular Na+ flux under OIMJ boundary conditions but with varying fractional flux via the NaCl cotransporter. Using each of these parameter sets in the epithelial model, luminal NaCl concentration is decreased until a Na+ flux has been brought to zero. In the top curve is the total transepithelial Na+ flux under open-circuit conditions; in the middle curve is the total transepithelial Na+ flux under short-circuit conditions; and in the bottom curve is the transcellular Na+ flux. Arrow indicates baseline parameters.

Another prominent feature of this model is the peritubular KCl cotransporter, which has yet to be identified in this epithelium experimentally. In the model, if one decreases the coefficient for peritubular KCl cotransport, then cell K⁺ and Cl concentrations increase, and cell NaCl entry is blunted. The functional impact of the cotransporter can be replaced by proportional increases in both peritubular K⁺ and Cl conductances. The Cl pathway is required because of the substantial luminal Cl entry, and the K⁺ pathway is required by virtue of peritubular K⁺ uptake via the Na-K-ATPase. In the calculations of Fig. 8, as one moves along the abcissa from right to left, the peritubular KCl coefficient is decreased and the two ion conductances are proportionally increased in a Newton iteration that seeks to maintain transcellular Na⁺ flux constant at 5.6 nmol · s¹ · cm². The bounding solutions are those of the OIMJ. The four top panels of Fig. 8 show that even though only luminal Na⁺ flux was controlled, the variations in luminal Cl flux, cytosolic K⁺ and Cl, and peritubular PD are all minor. Thus, with respect to net fluxes and cell composition, the parallel conductance pathways could substitute for the cotransporter. In the bottom of Fig. 8, however, are the peritubular conductances for each parameter set. Although the peritubular conductance for IMCD has not been determined, one might consider comparison with values obtained for proximal tubule of rat, 0.011 (11), and rabbit, 0.026 mho/cm² (28). This consideration has motivated the choice of the baseline parameter at the rightmost point of these curves. Parenthetically, the fraction of peritubular KCl cotransport is greater than unity because the denominator of this fraction is transcellular Cl flux, while the numerator also includes reabsorption of the backflux through the peritubular Na-K-2Cl cotransporter.

View larger version (28K): [in this window] [in a new window]   Fig. 8.   Replacement of peritubular KCl cotransport with K+ and Cl conductances to maintain constant Na+ flux. Epithelial model is used with OIMJ boundary conditions. In these calculations, the coefficient for KCl cotransport is varied, and there is proportionate adjustment of both peritubular K+ and Cl conductances so as to maintain transcellular Na+ flux constant at 9.6 nmol · s1 · cm2. Right-most point of each curve corresponds to the baseline parameter set.

Rocha and Kudo (36) have observed in perfused IMCD that atrial natriuretic factor (ANF) eliminates net reabsorption of Na⁺ and Cl, both by enhanced secretory flux and decreased reabsorptive flux. Their isotopic flux determinations indicated ANF increases the bath-to-lumen fluxes of Na⁺, Cl, and K⁺, and this increase is blocked by furosemide. The impact of an isolated activation of peritubular Na-K-2Cl cotransport is examined in Fig. 9. The model equations are solved over a range of luminal NaCl concentrations and compare the baseline parameter set (curves labeled "low flux") with a parameter set in which the peritubular Na-K-2Cl coefficient has been increased 10-fold ("high flux"). The top of Fig. 9 displays the flux of Na⁺ through the cotransporter; under OIMJ conditions, the high flux is approximately sevenfold greater than control. With activation of the cotransporter, cell Cl concentrations are ~15 mM higher, under all luminal NaCl concentrations (Fig. 9, bottom). Nevertheless, the middle of Fig. 9 shows that the impact of isolated Na-K-2Cl cotransport on net transepithelial Na⁺ flux is small. These calculations suggest that ANF action requires impact on an additional Na⁺ transport step, most likely luminal entry.

View larger version (25K): [in this window] [in a new window]   Fig. 9.   Impact of peritubular Na-K-2Cl cotransport on IMCD function. As in Fig. 6, the epithelial model is solved over a range of luminal NaCl concentrations. Curves obtained using the baseline parameters are designated "low flux." Solutions for a parameter set in which the coefficient for the peritubular Na-K-2Cl cotransporter is increased 10-fold are designated "high flux."

	DISCUSSION

Top Abstract Introduction Discussion References

This is the first mathematical model of the IMCD to be developed that includes a simulation of the IMCD epithelial cell. It was intended that the scope of this model should include acid secretion by this nephron segment, and this has mandated inclusion of ammonia and phosphate buffers and has enriched the assortment of transport components. Several aspects of this model are novel, and have not been utilized previously in other epithelial models. Specifically, these include the finite rate of hydration of CO₂ and the activity of a luminal cell membrane H-K-ATPase. Whereas previous nephron segment models have consisted of a single cylinder surrounded by a uniform bath, the IMCD model required a system of coalescing tubules traversing an interstitium whose composition varies as a function of distance. In this regard, two simplifications were made in the formulation of this model. The first is that cell volume is constant, and balance of luminal and peritubular water fluxes is secured by adjusting the concentration of an impermeant intracellular solute. The importance of this simplification is avoidance of grossly exaggerated swings in the concentration of intracellular electrolytes with changes in peritubular or luminal osmolality. For the moment, it avoids the issue of identifying volume-mediated solute transport steps, as well as identifying the kinetics of organic osmolytes. The trade-off with this simplification is that only steady-state model calculations are meaningful. The second simplification is the assumption of uniform hydrostatic pressure along the IMCD. With six generations of tubule merging, there is a 64-fold reduction in tubule number, despite only a 25% decline in the volume flow from base to tip in antidiuresis. Thus the tubule fluid velocity is expected to increase some 16-fold from base to tip and should be comparable to a proximal tubule flow rate. Under diuretic conditions, the tip flow rate may further increase 10-fold. For a rigid cylinder under Poiseuille flow, the predicted pressure drop at the higher flows would be unrealistic, but rather than consider a system of distensible tubules, the issue has been suppressed for the purpose of this model.

After model geometry, the most important issue to secure is the rate of Na⁺ transport. A number of investigators have provided information on this issue, and their results have been summarized in Table 4. Perhaps the most direct information comes from Ullrich and Papavassiliou (52), who microperfused medullary capillaries of the rat with a Ringer solution and assessed IMCD Na⁺ transport rate by shrinking drop observation. In their preparation, paracellular backleak should have been negligible, so that their reported flux rate, 4.0 nmol · s¹ · cm², may be identified with transcellular Na⁺ reabsorption. Micropuncture studies in young rats (5, 7) have reported their results as the fraction of filtered Na⁺ reabsorbed per length of tubule (0.6% and 1.5%, respectively). To translate these into absolute flux rates per unit length requires an assumption of the number of papillary collecting ducts at the level of micropuncture (for 1,000 IMCD, 7 or 17 nmol · s¹ · cm², respectively). Microcatheterization has been undertaken in older rats (2, 43), and these investigations report Na⁺ reabsorption over the whole IMCD as a fraction of filtered load (1.75% and 3.0%, respectively). To obtain absolute flux rates, one needs an estimate of the total branching length of the IMCD (using 14,400 mm, 4.6 and 7.9 nmol · s¹ · cm², respectively). Thus, the data obtained in vivo are in general agreement and yield an IMCD Na⁺ transport rate comparable to that of proximal tubule. Transport rates obtained from in vitro IMCD preparations are substantially lower. In the isolated perfused IMCD of the rat, Rocha and Kudo found Na⁺ reabsorption to be ~1.0 nmol · s¹ · cm² (34, 35). Suspension of rabbit IMCD has been found to utilize oxygen at 0.65 µmol O₂ · min¹ · g wet wt¹ (62). Even if all of this were for Na⁺ transport, it accounts for only 0.3 nmol · s¹ · cm² (6 ATP per O₂ and 3 Na⁺ per ATP). More directly, in this same preparation, ouabain-sensitive K⁺ uptake was found to be ~25% of tissue K⁺ per minute (25), or ~0.7 nmol · s¹ · cm². In a cell culture of rat IMCD, Wall and Koger (56) observed a maximal Na-K-ATPase activity of 200 nmol · mg protein¹ · min¹, or 1.5 nmol · s¹ · cm². In the model calculations presented here, the Na⁺ flux rate was chosen to be consistent with in vivo data: IMCD reabsorption of about three-fourths of delivered Na⁺ load, or ~3% of estimated filtered Na⁺ load.

With respect to model development, the importance of the absolute Na⁺ transport rate derives from the constraint it imposes on the magnitude of rheogenic transport. The only reported cellular electrophysiology on IMCD is that of Stanton (47), who found a small transepithelial electrical PD and a high fractional apical resistance. Admittedly, these findings could, in part, be an artifact of the in vitro preparation and its low reabsorptive Na⁺ flux. In the calculations of this model, only when channel-mediated luminal membrane Na⁺ reabsorption was less than 25% of the total was the fractional apical resistance greater than 85%. Although one early determination of transepithelial PD was 11 mV (37), subsequent investigations have shown it close to zero (16, 19). Indeed, the transepithelial PD of this model, 12 mV, is high and is attributable largely to the component of rheogenic Na⁺ transport. This suggests that the bulk of Na reabsorption is electrically silent, as via a thiazide-sensitive NaCl cotransporter, which has been identified both in vivo (61) and in vitro (35). Unfortunately, there are no studies available that can confirm the presence of such a cotransporter within the IMCD, either by in situ hybridization or by immunohistochemistry. Although amiloride can increase luminal membrane electrical resistance (47) and decrease perfused tubule Na⁺ reabsorption (35), along with tubule oxygen consumption (62), there is no sure way to use these observations to guarantee the importance of this pathway in vivo. Of note, the model calculations argue only that electrogenic Na⁺ transport is a small fraction of the total; they do not preclude Na⁺-urea cotransport (22) as a significant component within the electrogenic fraction. It is of interest that the model cell could reabsorb Na⁺ down to 2 mM of luminal Na⁺, regardless of rheogenic or electroneutral luminal uptake. However, when the cellular flux is channel mediated, there is a higher transepithelial PD, greater tight junctional backflux, and luminal Na⁺ equilibrium concentration can become unrealistically high. The discrepancy between the luminal Na⁺ concentration that zeroes cell membrane Na⁺ flux and that which zeroes epithelial Na⁺ flux reflects, in part, the Na⁺ permeability of the tight junction. Although the model permeabilities have tried to remain faithful to reported estimates, it is likely that a tighter tight junction will be necessary to simulate the low urine Na⁺ concentrations of which the Na⁺-deprived rat is capable (7).

Consequent to the high reabsorptive Na⁺ flux by IMCD is substantial peritubular K⁺ uptake via the Na-K-ATPase and, of necessity, a peritubular exit step for K⁺. Were it known, the overall electrical conductance of the peritubular membrane would serve as a constraint on acceptable values for the peritubular K⁺ permeability. In the absence of this information, if the peritubular resistance of proximal tubule substitutes as a constraint on this parameter, then most of the peritubular K⁺ exit must be electroneutral, as for example, via a KCl cotransport. A similar problem was confronted in the development of the model of proximal convoluted tubule (59). In that setting, peritubular KCl cotransport was proposed as a resolution to a modeling exigency; the presence of this transporter in proximal tubule was subsequently identified by Sasaki et al. (40). Overall, in control conditions, the IMCD demonstrates little K⁺ transport (5, 7, 43); in vitro (34), or under Na⁺ or K⁺ depletion (7), K⁺ transport is reabsorptive; whereas in volume expansion (44, 45), or under K⁺ loading, secretory K⁺ flux has been observed. In this model, K⁺ flux was reabsorptive due to the inclusion of a briskly transporting luminal membrane H-K-ATPase. The flux through this transporter was deliberately highlighted in order to demonstrate its capacity to achieve an acid luminal pH. Given the generous delivery of buffer in these model calculations, the degree of K⁺ reabsorption was obligatory. It should be noted (Fig. 5) that the K⁺ flux through the luminal membrane cation channel is predicted to be small; with the baseline parameters, it is 25% of the total, but with higher fractional luminal membrane NaCl cotransport, this fraction declines toward zero. Thus, without specific activation of H-K-ATPase, the model would be compatible with substantially smaller K⁺ reabsorption. Furthermore, these calculations suggest that when the luminal membrane cation channel is activated, the channel K⁺ flux can become secretory.

The picture of the IMCD which emerges from this modeling effort is that of a briskly transporting epithelium, in some ways comparable to proximal tubule. The coalescing of tubules within the inner medulla diminishes total tubular surface and, most importantly, prevents axial flow from becoming sluggish. Thus, to modify urinary solute excretion, transport rates must be substantial. Assignment of a definitive set of model parameters is close to impossible, because of the profound impact of regulation on this segment. Indeed, Na⁺ transport may go from strongly reabsorptive to secretory. However, it has been useful to attempt to generate a "baseline" set of parameters with which to try to simulate in vivo transport under antidiuretic conditions. The calculations presented here have suggested that, like the proximal tubule, electroneutral cotransport mechanisms are likely to be most important in the transcellular ion fluxes of IMCD. Specifically, under normal circumstances, luminal membrane NaCl cotransport is likely responsible for at least 75% of Na⁺ reabsorption, and KCl cotransport appears to be the dominant pathway for K⁺ flux across the peritubular membrane. In contrast to proximal tubule, flux across the tight junction appears to be small, and a modulating role for the lateral interspace is not apparent. In the companion study (59a), acid secretion and ammonia transport by this model epithelium will be considered in more detail.