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【关键词】 Chang
Department of Physiology and Biophysics Weill Medical College of Cornell University, New York, New York
ABSTRACT
A model of rat early distal convoluted tubule (DCT) is developed in conjunction with a kinetic representation of the thiazide-sensitive NaCl cotransporter (TSC). Realistic constraints on cell membrane electrical conductance require that most of the peritubular Cl reabsorption proceeds via a KCl cotransporter,along with most of the K+ recycled from the Na-K-ATPase. The model tubule reproduces the saturable Cl reabsorption of DCT but not the micropuncture finding of linear Na+ flux in response to load, more likely a feature of late DCT (CNT). As in proximal tubule, early DCT HCO3 reabsorption is mediated by a luminal Na+/H+ exchanger (NHE), but in contrast to proximal tubule, the DCT exchanger is operating closer to equilibrium. In the model DCT, two consequences of the lesser driving force for NHE exchange are an acidic cytosol and wider swings in NHE flux with perturbations of luminal composition. Variations in luminal NaCl provide a challenge to cell volume, which can be blunted by volume dependence of the KCl cotransporter. Cell swelling can also be induced by increases in peritubular K+ concentration. In this case, volume-dependent inhibition of TSC could provide volume homeostasis that also enhances distal Na+ delivery, and ultimately enhances renal K+ excretion. In the model DCT, proton secretion is blunted by peritubular HCO3, so that there is little contribution by this segment to the maintenance of metabolic alkalosis. During alkalosis, the model predicts that increasing luminal NaCl concentration enhances NHE flux, so that these calculations provide no support for a role of early DCT in recovery from Cl depletion alkalosis.
thiazide-sensitive cotransporter; sodium reabsorption; cell volume homeostasis; metabolic alkalosis
THE EARLY DISTAL CONVOLUTED TUBULE (DCT) IS AN IMPORTANT site for renal Na+ and HCO3 reabsorption (53). With volume flow rate perhaps 20% of glomerular filtration rate and luminal Na+ concentration slightly under half that of plasma, one has an estimate that Na+ delivery is just under 10% of filtered load. Distal micropuncture has demonstrated that in normal circumstances about three-fourths of this Na+ load is reabsorbed prior to the collecting duct, divided more or less equally between early DCT and connecting tubule (11, 17, 22, 33, 46, 58). This requires reabsorption of 34% of filtered Na+ in a 1-mm tubule segment, yielding a Na+ transport rate about half that of proximal tubule. Reabsorption is, however, load dependent, and can vary severalfold. This Na+ flux is mostly NaCl, via the thiazide-sensitive cotransporter (TSC), but it is also via luminal Na+/H+ exchange (NHE). The magnitude of the HCO3 load to DCT is roughly 6% of filtered load, so that for all 36,000 nephrons, it is approximately 1.7 μmol/min, and still perhaps two- or threefold greater than net acid excretion for the rat (5, 43, 78). If half of this delivered load is reabsorbed by early DCT, the rate of HCO3 reabsorption is about a third that of proximal tubule. Within the DCT cell, the possibility that TSC or NHE fluxes may perturb cytosolic Na+ concentration, and that each cytosolic anion (Cl or HCO3) may perturb the concentration of the other, provides for mutual interaction between the two luminal Na+ entry pathways. Additional complexity is added when peritubular anion exit pathways are considered, and these interactions are susceptible to examination in DCT models.
The first mathematical model of DCT was the work of Chang and Fujita (7, 9). Although their first version (7) did not represent acid/base transport, their second version (9) was a comprehensive model that included both early DCT and CNT segments. A key feature of these models was a kinetic representation of the TSC based on data from the cloned flounder transporter (8). One of the strengths of their modeling effort was the care taken by Chang and Fujita to ensure fidelity of the model solute fluxes with experimental data. Since that work, however, additional data have become available specifically for the kinetics of rat TSC. It is also possible to improve the DCT cell representation by inclusion of the recently documented peritubular KCl cotransporter and thus remediate an unrealistically low peritubular electrical resistance. In the present model, cell volume is allowed to vary with changes in solute content. This provides a simulation that can be used to consider volume challenges to DCT that come with changes in luminal solute delivery or with peritubular composition. It also provides a means of representing volume-dependent compensatory mechanisms that can be examined for their homeostatic effect and their impact on transport. In the present work, the focus is exclusively on the early DCT cell, with its electroneutral Na+ entry pathways (TSC and NHE), in series with electroneutral peritubular anion exit pathways (KCl cotransport and Cl/HCO3 exchange). In the companion manuscript (76a), this early DCT segment will be placed in series with CNT to encompass transport by the "micropuncture accessible" DCT.
A KINETIC MODEL OF THE THIAZIDE-SENSITIVE NaCl COTRANSPORTER
Figure 1 is a scheme for the NaCl cotransporter, similar to that considered by Chang and Fujita (8). The empty carrier is denoted by X' or X'' according to its availability on the external (luminal) or internal (cytosolic) face of the cell membrane, and ion binding is not assumed to be ordered. The scheme differs from theirs by omitting metolazone, and by the assumption of equilibrium solute binding. In the model of Chang and Fujita, binding and unbinding of solutes were represented explicitly with forward and backward rate coefficients. However, they observed that when solute concentrations were in the physiological range, binding was rapid relative to translocation, so that the assumption of equilibrium solute binding produced only trivial differences in predicted transmembrane NaCl flux. Following Chang and Fujita, it is also assumed that binding affinities are symmetric across the membrane. Denote by x, xn, xc, and xnc the concentrations of X, Na-X, Cl-X, and NaCl-X on each membrane face, with the appropriate (') or ('') designation. Similarly, n' and c' (n'' and c'') denote the luminal (and cytosolic) concentrations of Na+ and Cl.
Set Kn and Kc as the equilibrium binding concentration (mM) for Na+ and Cl to the unloaded carrier. Then, one can define the normalized concentrations
(1)
(2)
If Knc is the equilibrium binding concentration for Cl to the Na+-loaded carrier, and Kcn that for Na+ to the Cl-loaded carrier, then the density of fully loaded carrier, xnc is expressed
(3)
Recognizing the thermodynamic constraint, Kn?Knc = Kc?Kcn, either expression could serve as the denominator for xnc.
The coefficients depend only on the known solute concentrations and affinities and allow the species of bound carrier to be expressed in terms of free carrier on either side of the membrane.Thus the conservation of total carrier, xT, is represented
(4)
In the scheme of Fig. 1, only an empty or fully loaded carrier can traverse the membrane. Translocation of an empty carrier is represented by rate coefficients P'0 and P0' for influx and efflux and by P'nc and Pnc' for a loaded carrier. By virtue of model symmetry, P'0?Pnc' = P0'?P'nc. The second model equation states that at stationary state, there is no net flux of carrier (unloaded plus loaded):
(5)
When these two equations are solved for x' and x'', one obtains
(6)
in which
(7)
Thus the unidirectional Na+ influx, J'Na, and efflux, JNa', are
(8)
(9)
so that net NaCl flux into the cell is
(10)
These flux equations are the basis for fitting the model TSC to experimental data. The number of free parameters is reduced by half with the assumption of affinity symmetry (with respect to inside and outside binding sites), and by half again with the assumption of equilibrium binding. Finally, the thermodynamic constraint on the coefficients around a cycle brings the number of free parameters to five: Kn, Kc, Knc, P'0, and P'nc. One of the translocation rates, P0'', is set arbitrarily and corresponds to assignment of the TSC density. (Equivalently, this density disappears when the experimental data are normalized to a "control" flux.) To obtain TSC model parameters, Chang and Fujita (8) relied on data from Tran et al. (59), who examined metolazone binding to renal cortical vesicles as a function of ambient Na+ and Cl. They also used flux data from Fig. 2 in the study of Gamba et al. (24), in which the TSC was expressed in oocytes and Na+ uptake was displayed as a function of ambient Na+ and Cl concentrations. Since that model was published, there has been a more comprehensive study of the kinetics of Na+ flux into oocytes expressing the TSC. In the work of Monroy et al. (49), the authors present two series of uptake experiments: in the first, external Na+ is fixed at one of several values, and for each value, an apparent Km for external Cl is determined; in the second series, external Cl is fixed, and for each value, an apparent Km for external Na+ is determined.
To analyze the experiments of Monroy et al., one can start with the equation for unidirectional Na+ influx and compute the maximal Na+ influx as ' is allowed to grow large, J'Na, max
(11)
Then, the external Na+ concentration at which the flux J'Na is half-maximal is obtained by solving for ' in the linear equation
(12)
namely,
(13)
In this equation, the Km for Na+ is a function of the external Cl concentration, equivalently, '. A similar equation (with ' and ' interchanged) is obtained for the Cl concentration, which drives half-maximal Na+ influx for a fixed external Na+ concentration. For this case, "maximal" is understood to mean the limit as external Cl concentration is increased, while leaving Na+ constant.
Table 1 displays four sets of kinetic coefficients for the model TSC. The first column contains those obtained from Chang and Fujita (8). The second column coefficients were obtained from a fit of the five independent kinetic coefficients to the data in Fig. 3 of Monroy et al. (49). In those experiments, early unidirectional Na+ uptake into oocytes was measured as a function of external Na+ concentration (with Cl fixed at 40 mM), or as a function of external Cl concentration (with Na+ fixed at 40 mM). To do this fit, the model requires specification of the internal ion concentrations, which were not reported. However, the authors indicate that for 24 h before the uptake studies the oocytes were kept in medium that contained 96 mM Na+, but was Cl free. Accordingly, for the calculations here, the internal Na+ and Cl concentrations were assumed to be 10 and 2 mM, respectively. It should be noted that in the calculations of Chang and Fujita (8), the internal concentrations were assumed to be 10 and 40 mM, but in the data used for that work, a Cl-free incubation was not part of the protocol. In calculations not shown, these kinetic parameters were not found sensitive to the internal ion concentrations. To do the fit, a sum-squared error function between the model predictions and the data was created, and the minimization routine VE08AD (Harwell Subroutine Library) was applied. Figure 2 shows simulations of the experiments in Fig. 3 from Monroy et al. (49) (along with their data points) for each of the sets of coefficients in Table 1. In the top panels, the parameters of Chang and Fujita (8) are used, and these predict an uptake that is less sensitive to external solutes than is observed. In this regard, it should be acknowledged that the experimental protocol of Gamba et al. (24) used to obtain these coefficients was different from that of Monroy et al. (49); in particular, flux studies by Gamba et al. (24) were done at 22°, whereas Monroy et al. (49) worked at 30°.
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Alternatively, one can use the theoretical expression for the value of the apparent Km of ion uptake (Eq. 13) to fit the model to data. This allows all of the studies performed by Monroy et al. (49), and summarized in their Tables 1 and 2, to register in the parameter fit. In their Table 1, the apparent Km for Cl uptake was determined for five external Na+ concentrations (2, 5, 10, 20, and 40 mM); in their Table 2, the apparent Km for Na+ uptake was determined for six values of external Cl (2, 4, 8, 10, 20, and 40 mM). When this is done (again using VE08AD as the minimization routine), the results of Fig. 3 are obtained. The bars in the top panels show the Km data, in which one ion is fixed in the external bath and the other varied. The bars in the bottom panels are the model predictions for the same experiments, using the parameter set that minimizes the square of the difference from the 11 experimental Km. The coefficients obtained from this fit are summarized in column 4 of Table 1. The experiments in Fig. 3 in Monroy et al. (49) generated the data displayed as the rightmost column of each of the panels. When the optimal coefficients are used to simulate the experiments in Fig. 3 of Monroy et al., the bottom panel of Fig. 2 is obtained. Consistent with the lower predicted Km, the predicted ion sensitivities are greater than observed in the single study. Finally, it is possible to fit the five model parameters to the two Km determinations from Fig. 3 of Monroy et al., ignoring the individual data points. Another set of optimal coefficients is obtained from this procedure, and these are shown in column 3 of Table 1. They are clearly quite different from those of the point fit, with a 52-fold decrease in Cl affinity of the empty carrier, and a 27-fold increase in Cl affinity of the Na+-bound carrier. Nevertheless, when these optimal coefficients are used to simulate the experiment, the panels of the third row of Fig. 2 are obtained. The predicted curves are nearly identical to those of the second row of panels and provide an alternative set of coefficients compatible with the data. In the DCT model developed below, the coefficients of Table 1, column 4, which are based upon the maximum amount of experimental data, are used to provide a kinetic model of the TSC.
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MODEL FORMULATION
The model DCT will follow the scheme that has been used previously for a tubule with a single cell type (76). In brief, the model will be formulated both as a DCT epithelium, with specified luminal and peritubular conditions, or as a tubule, in which luminal concentrations vary axially. Figure 4 displays both models, in which cellular and lateral intercellular (LIS) compartments line the tubule lumen. Within each compartment the concentration of species i is designated C(i), where is lumen (M), interspace (E), cell (I) or peritubular solution (S). Within the epithelium, the flux of solute i across membrane is denoted J(i) (mmol?s1?cm2), where may refer to luminal cell membrane (MI), tight junction (ME), lateral cell membrane (IE), basal cell membrane (IS), or interspace basement membrane (ES). Along the tubule lumen, axial flows of solute are designated FM(i) (mmol/s). The 12 model solutes are Na+, K+, Cl, HCO3, CO2, H2CO3, HPO42, H2PO4, NH3, NH4+, H+, and urea, as well two impemeant species within the cells, a nonreactive anion and cytosolic buffer. These are 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. Within a cell or interspace, the generation of i [s(i)] is equal to its net export plus its accumulation
(14)
(15)
where V is the compartment volume (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.
(16)
where BM is the tubule circumference, and AM is the tubule cross-sectional area. With this notation, the equations of mass conservation for the nonreacting species (Na+, K+, Cl, and urea) are written
(17)
where = E, I, or M. For the phosphate and for the ammonia buffer pairs, there is conservation of total buffer
(18)
(19)
Although peritubular PCO2 will be specified, the CO2 concentrations of the cells, interspace, and lumen are model variables. The relevant reactions are
(20)
where dissociation of H2CO3 is rapid and assumed to be at equilibrium. Because HCO3 and H2CO3 are interconverted, mass conservation requires
(21)
for = I or E, whereas for the tubule lumen
(22)
In each compartment ( = I, E, or M), conservation of total CO2 is expressed
(23)
Corresponding to conservation of protons is the equation for conservation of charge for all the buffer reactions
(24)
where zi is the valence of species i. In this model, conservation of charge for the buffer reactions (Eq. 24) takes the form
(25)
The solute equations are completed with the chemical equilibria of the buffer pairs HPO42:H2PO4, NH3:NH4+, and HCO3:H2CO3. Corresponding to the electrical potentials, for = E, I, or M, is the equation for electroneutrality
(26)
where for the cellular compartment ( = I), the sum includes the contribution of the impermeant anion plus the unprotonated impermeant buffer.
With respect to water flows, volume conservation equations for lumen, interspace, and cell can be used to compute the three unknowns: luminal volume flow, lateral interspace hydrostatic pressure, and cell volume. (Cell hydrostatic pressure is set equal to luminal pressure; total cell impermeant content is assumed fixed.) Across each cell membrane, the volume fluxes are proportional to the hydrosmotic driving forces. With respect to the lateral interspace, its volume, VE, and its basement membrane area, AES, are functions of interspace hydrostatic pressure, pE:
(27)
where VE0 and AES0 are reference values for volume and outlet area, 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
(28)
In this equation, 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 . Here,
(29)
is a normalized electrical potential difference (PD), where zi 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
(30)
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 exceptions to this representation of coupled fluxes are the two sodium transporters of the luminal membrane for which kinetic models are available, namely, the TSC, developed above, and an Na+/H+ exchanger (73). For each of these transporters, a single density parameter suffices to represent its activity. In this model, there is a single transport ATPase, the peritubular Na-K-ATPase, represented by the expression
(31)
in which the half-maximal Na+ concentration, KNa, increases linearly with internal K+, and the half-maximal K+ concentration, KK, increases linearly with external Na+ (25). The pump flux of K+ plus NH4+ reflects the 3:2 stoichiometry
(32)
with the transport of either K+ or NH4+ determined by their relative affinities, KK and KNH4+
(33)
MODEL PARAMETERS
The parameters for the model epithelium were selected so that the tubule might correspond to the early DCT of the rat. The data available for comparison come almost exclusively from flux determinations and electrophysiology performed in the setting of micropuncture and microperfusion experiments. To illustrate the variability within this database, a number of measurements for DCT have been summarized in a table as an APPENDIX. Some variability in transmural fluxes derives from differences in DCT solute delivery (axial flow rate and concentration), because delivery modulates reabsorption and secretion by this segment. As a consequence, comparisons of model with data mandate fidelity of the simulation to the experimental conditions. Additionally, some of the variability of experimental flux determinations derives from technique, specifically with respect to Na+, fluxes measured during microperfusion appear to be substantially less than those obtained in micropuncture studies. Nevertheless, a small number of workers have been able to microperfuse both early and late segments of DCT in isolation, and these data permit one to estimate relative contributions of DCT and CNT to the overall solute flux. In particular, a relatively uniform rate of Na+ reabsorption along DCT, observed in early micropuncture (46), was confirmed in segmental microperfusion (10, 19). Alternatively, although DCT K+ secretion was initially plotted as a linear function of tubule distance (45), nearly all occurs beyond the early segment (57, 61). With respect to HCO3 reabsorption, there is considerable variability in the fluxes, but microperfusion data suggest that the early DCT is an important, if not the dominant site for H+ secretion.
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Appendix. DCT Fluxes
Figure 5 displays the important transporters of the model DCT cell, and Table 2 contains the assigned permeability coefficients expressed per unit area of epithelium. To rationalize them to unit membrane permeabilities, one divides by the area of luminal membrane (4.7 cm2/cm2 epithelium), of lateral cell membrane (64.3 cm2/cm2 epithelium), or of basal cell membrane (4.7 cm2/cm2 epithelium), chosen in accordance with the measurements of Pfaller (50). In all cases, the unit permeabilities of lateral and basal membranes were assumed equal. The model TSC developed above is situated within the luminal membrane, and in parallel with this is an Na+/H+ exchanger. The Na+/H+ antiporter has been demonstrated functionally (21, 69) and identified as NHE2 (6). There are insufficient data to develop a kinetic representation of NHE2, so for this model the kinetics of NHE3 have been used (73). Of note, although an H+-ATPase had been incorporated into a prior DCT model (9), it does not appear here because this pump has not been detected in early DCT using either transport inhibitors (2, 21, 70) or antibody staining (4, 51). The electrical properties of the luminal membrane have been examined only in the rabbit, in which the conductance was principally for K+ with a smaller Na+ component (80), although the absolute magnitudes of these conductances are not known. In the rat, the transition between DCT and CNT is less distinct than in the rabbit, with the identification of sodium channels (ENaC) within DCT (39). With the model parameters in Table 2, the luminal membrane K+ and Na+ conductances are 3.5 and 0.4 mS/cm2, yielding an overall electrical resistance 250 ?cm2. It is assumed that the NH4+-to-K+ permeability ratio for the luminal K+ channel is 20%. Finally, a KCl cotransport pathway within the luminal cell membrane has been postulated on the basis of transport studies demonstrating an electroneutral dependence of K+ secretion on luminal Cl concentration (18, 61).
Within the peritubular cell membrane, the Na-K-ATPase mediates exit of Na+ in exchange for K+ plus NH4+ with stoichiometry 3-Na+:2-(K+ plus NH4+). The relative affinities of K+ and NH4+, for the Na-K-ATPase, are comparable in proximal tubule (34), but in inner medullary collecting duct the K+ affinity is greater by a factor of 4 (67). This ratio is not known for DCT, so in view of its cortical location, and to minimize DCT ammonia reabsorption, equal affinities were assumed. In the model, the major pathway for K+ and Cl exit across the peritubular membrane is the K-Cl cotransporter. In rabbit DCT, KCC4 is present within the peritubular membrane (64), whereas human DCT contains RNA for KCC1 (38). The magnitude of the permeability of this exit pathway will be considered in the calculations below, but its critical role in this model stands in contrast to the exclusive reliance on parallel K+ and Cl conductances in the peritubular membrane by Chang and Fujita (7, 9). Both K+ and Cl conductances have been identified electrophysiologically in the DCT peritubular membrane of the rabbit, with that for K+ dominant (63, 80). These ionic permeabilities are also included in this model (in a ratio of 3:1, K+-to-Cl), but it will be shown that realistic constraints on the overall peritubular conductance limit the traffic through these two pathways to a small component of the peritubular flux. For the K+ permeability in Table 2, the K+ conductance of the peritubular (combined lateral and basal) membrane is 12.5 mS/cm2; the Cl conductance is 5.4 mS/cm2. The overall peritubular conductance is 19.4 mS/cm2, corresponding to a resistance, 52 ?cm2, about fivefold more conductive than the luminal cell membrane. It is assumed that the NH4+-to-K+ permeability ratio for the peritubular K+ channel is 20%. It is also assumed that HCO3 permeates the Cl channel, with a HCO3-to-Cl permeability ratio of 1:2. Nevertheless, to accommodate the necessary HCO3 exit pathway is an electroneutral Cl/HCO3 flux, the major peritubular HCO3 exchanger. In this regard, AE2 has been identified within the peritubular membrane of rat DCT (1). The Na+/H+ exchanger NHE1 has been localized to the peritubular membrane of DCT (3) and is represented here using linear nonequilibrium thermodynamic formalism. In the model, its presence helps to stabilize cell pH along the tubule length.
Membrane permeabilities have also been assigned for the nonionic species: water, CO2, H2CO3, urea, and NH3. In the rat DCT, the volume fluxes measured by Costanzo (10) suggest that the early segment has about half the water permeability of the late segment. If one assumes luminal diameters of 15 and 18 μm, respectively (20), the observed fluxes translate into overall epithelial water permeabilities of 1.1 x 104 and 2.1 x 104 ml?s1?cm2?osmol1, or water permeabilities equal to 0.0059 and 0.012 cm/s for early and late DCT. In the model, the peritubular membrane water permeability was taken to be that used for the inner medullary collecting duct (IMCD) cell (74), so that with a luminal membrane water permeability 15% of peritubular (per unit membrane area), the overall epithelial water permeability was reproduced. In the absence of specific data for CO2, the unit permeabilities for luminal and peritubular membranes were taken equal, and set to the value used for both membranes in IMCD. A similar choice was made for the H2CO3 permeability. In the case of urea, again absent data, the unit permeabilities for luminal and peritubular membranes were also set equal to the unit permeability used previously for proximal tubule (2.0 x 106 cm/s) (71). With this choice (plus the small tight junctional urea permeability), one recovers the measured value for rat DCT urea permeability, 1.0 x 105 cm/s (30). For this DCT model, the unit membrane permeabilities to NH3 were also set equal to the value taken for proximal tubule membranes (2.0 x 103 cm/s) (72). It was found that if the value for NH3 permeability were set much higher, then the model DCT would show unrealistically high rates of ammonia reabsorption.
There are few epithelial permeability measurements to guide the selection of tight junctional parameters of DCT. The overall electrical conductance increases as one proceeds from early to late DCT, with measured values of 2.6 going to 4.0 mS/cm2 (44), from 6 to 25 mS/cm2 (12), and from 6.8 to 16.1 mS/cm2 (11). When these data are combined with an estimate of the Na+-to-Cl conductance ratio, 1.75 (early) and 2.67 (late), one has meaningful constraints on the two important tight junction permeabilities (44). In Table 2, the Na+ and Cl permeability coefficients are 8.0 x 105 and 5.0 x 105 cm/s, respectively. When luminal Na+ is 65 mM and luminal Cl is 56 mM, these permeability coefficients translate into conductances of 3.0 and 1.5 mS/cm2, and this accounts for 90% of the tight junction conductance.In this model, both tight junctional and transcellular pathways contribute to the overall K+ permeability. Good and Wright (28) found DCT K+ flux to be a linear function of luminal K+ concentration (their Fig. 7), and the slope yields an estimate of 9.3 x 105 cm/s for the K+ permeability of a tubule of 2.2-mm length and 15-μm diameter. In the model tight junction, the permeabilities of Na+, K+, and NH4+ were set equal. With this assignment, and the cell membrane K+ permeabilities estimated from the electrical considerations above, the overall DCT epithelial K+ permeability matches that derived from the data of Good and Wright. For the tight junction permeabilities of HCO3, CO2, and H2CO3, values equal to that for Cl were used, and for phosphate species, 20% of the Cl permeability. Tight junctional urea permeability was set at 25% of Na+ permeability, relatively small compared with its transcellular permeability.
All membrane and tight junction reflection coefficients are assumed to be 1.0, whereas those for interspace basement membrane are 0.0. 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. The values shown are those used previously for the interspace basement membrane of IMCD (74). Staining for carbonic anhydrase is positive within the DCT cell, especially in the basal region (40). Accordingly, rate constants for hydration of CO2 were assumed to be 1,000-fold greater than those for the uncatalyzed reaction in both cytosol and lateral intercellular space. (Full catalysis would be 10,000-fold greater, but increasing the model rate constant had little effect on the predicted concentrations and fluxes.) Within the tubule lumen, a collapse of the acid lumen by infusion of carbonic anhydrase (0.83 pH unit) was first noted by Rector et al. (52), and an acid disequilibrium pH of 0.37 unit was found under control conditions by Malnic et al. (43). Alternatively, DuBose et al. (16) only observed a disequilibrium pH under conditions of high HCO3 delivery (0.49 unit). In the model, luminal catalysis was taken to be 10-fold greater than the uncatalyzed coefficients. This was slow enough to still yield a disequilibrium pH (0.4 units; see below), but it was found that if the uncatalyzed rate coefficients were used, then an unrealistically low luminal pH was obtained (close to 6.0), which shut off proton secretion by the luminal Na+/H+ within the model DCT.
MODEL CALCULATIONS
Table 3 contains the open-circuit solution of the DCT epithelial model when luminal conditions are those expected near the tubule inlet (see APPENDIX). Luminal Na+, K+, Cl, and HCO3 were taken to be 65, 2.0, 56, and 8 mM, respectively. Ambient PCO2 was assumed to be 40 mmHg, and H2CO3 was taken to be in equilibrium for this baseline calculation. With these choices, luminal pH was 6.93; when the calculations were extended to the full 1-mm tubule, the rapid development of a disequilibrium pH, along with HCO3 reabsorption depressed luminal pH toward 6.4. With regard to selecting the ambient PCO2, these luminal pH values seem more compatible with micropuncture measurements compared with an ambient PCO2 of 50 mmHg, which had been used previously in proximal tubule models (71). There is experimental support for selecting cortical PCO2 close to that of arterial plasma (13), although this value remains a point of controversy (15). Total luminal phosphate is 4 mM, total luminal ammonia 3.2 mM, and luminal urea 30 mM. Peritubular ammonia was assumed to be 0.2 mM, as done previously for cortical capillaries, and rationalized as being midway between observed rat renal artery and renal venous ammonia concentrations (72). Under these conditions, luminal PD is 5 mV, dependent largely on the small inward Na+ current across the luminal cell membrane. This current corresponds to a flux of 21 pmol?mm1?min1, and should be seen in relation to the TSC and NHE fluxes of 95 and 125 pmol?mm1?min1, respectively. This NHE flux is high but drops sharply with luminal acidification, so that by midtubule, the magnitudes of TSC and NHE fluxes are 127 and 31 pmol?mm1?min1. These midtubule fluxes are displayed in Fig. 5. At the tubule inlet, backflux through the luminal KCl cotransporter is small, 4.4 pmol?mm1?min1, but not an insignificant fraction of the secretory flux through the K+ channel, 8.9 pmol?mm1?min1.
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In the baseline calculation, peritubular transport of Na+ by the Na-K-ATPase is 295 pmol?mm1?min1, composed of the 241 pmol?mm1?min1 that entered across the luminal membrane plus peritubular entry from Na+/H+ exchange (48 pmol?mm1?min1) and a small phosphate flux (6 pmol?mm1?min1). The cation uptake by the Na-K-ATPase for K+ and NH4+, respectively, is 180 and 17 pmol?mm1?min1. Of this K+ entry, 168 pmol?mm1?min1 returns across the peritubular KCl cotransporter, accompanied by Cl that came from both luminal TSC and peritubular Cl/HCO3 in nearly equal measure. Even in midtubule (Fig. 5), where there is substantially less luminal NHE flux, there is still a major contribution from peritubular anion exchange to the KCl flux, due to cytosolic acidosis and activation of peritubular NHE. Of note, the K+ flux across the peritubular K+ channel is close to zero, corresponding to the fact that K+ is nearly at electrochemical equilibrium across the peritubular membrane. In addition to the 17 pmol?mm1?min1 of active peritubular NH4+ uptake, there is reabsorption of 5.6 pmol?mm1?min1 across the luminal membrane, due mostly to NH4+/H+ exchange across the NHE. These together (plus a small luminal NH3 entry) yield a peritubular efflux of NH3 of 24 pmol?mm1?min1, a small net ammonia reabsorption, but a more significant cellular acidification. By midtubule, with considerably diminished fluxes through luminal NHE, net ammonia reabsorption is reduced by 90% and the acidification due to NH4+/NH3 exchange is reduced by two-thirds.
The epithelial model of DCT can be used to simulate experimental determination of the epithelial solute and water permeabilities under ideal conditions. For these calculations, luminal and peritubular solute concentrations were taken close to physiological values (luminal Na+ = 70, K+ = 5.0, Cl = 49.1, HCO3 = 25, urea = 5.0, and NH4+ = 1.0 mM; peritubular Na+ = 140, K+ = 5.0, Cl = 119.1, HCO3 = 25, urea = 5.0, and NH4+ = 1.0 mM), and a neutral luminal impermeant at 140 mM was added to minimize convective fluxes. Then, for the solute species, Na+, K+, Cl, HCO3, urea, NH4+, and the impermeant, the luminal concentration was first increased and then decreased by 0.1 mM. The resulting flux changes (computed under short-circuit conditions) were averaged (for increase and decrease) and used to compute the solute permeabilities, hMS(i), or, in the case of the impermeant, the DCT water permeability. Finally, the transepithelial PD was increased and decreased by 0.1 mV, and the flux changes in charged species yielded the partial conductance, gMS(i) (mS/cm2) or the conductive permeability (cm/s). The results of these calculations are contained in Table 4. Here, one sees the concordance of K+ permeabilities, whether determined from a concentration or voltage perturbation, and similarly for HCO3. This is different from the case of Na+ or Cl or NH4+, in which concentration perturbations yield greater permeability values, due to the presence of electroneutral luminal entry pathways. The discrepancy in permeabilities is a factor of 3 for Na+, but only 50% for Cl, reflecting the relative permeabilities of the NHE and TSC pathways. As the model was configured, the overall epithelial conductance is 6.7 mS/cm2, with a Na+:Cl partial conductance ratio of 1.5; the epithelial K+ permeability largely reflects the transcellular pathway; and the epithelial urea permeability, 1.1 x 105 cm/s, is close to the measured value. In these calculations, an epithelial NH3 permeability is computed from the NH3 flux across the luminal membrane and tight junction, during the perturbations of luminal NH4+, using the increment in luminal NH3.
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Figures 6 and 7 display the results of calculations from the DCT tubule model, in which the abcissa is tubule length over a 1-mm segment, and the initial conditions are identical to those in Table 3. In Fig. 6, the panels on the left show the luminal PD and the concentrations of Na+, K+, Cl,and urea; on the right are axial volume and solute flows.The figure illustrates that this tubule is a site for NaCl reabsorption and K+ secretion, with little change in volume or urea flow. Table 5 quantifies these changes, indicating reabsorption of 40% of entering Na+ (155 pmol?mm1?min1) with 34% of entering Cl (114 pmol?mm1?min1). Thus with a 10% decrease in volume flow, there is an overall decrease in the luminal concentrations of Na+ and Cl from 65 to 43 mM and from 56 to 41 mM, and these changes occur uniformly over the tubule length. A departure from this uniformity appears in the small length near the inlet where the DCT hyperpolarizes. With reference to Fig. 7, this region is the locus in which the lumen acidifies with the development of a disequilibrium pH. In Fig. 7, the left panels include the luminal pH and the concentrations of HCO3, titratable acid (TA), and NH4+, and the panels on the right show the axial flows of these species plus "net acid flow" (NH4+ + TA HCO3 flows). The bottom left panel shows the disequilibrium pH calculated either for an open system, with the PCO2 fixed at 40 mmHg, or for a closed system, in which there is conservation of total CO2. The open system calculation is likely the one that corresponds more closely to the measurements by DuBose et al. (16). For the closed system, there is a prediction of the equilibrium PCO2, and that is shown in the bottom right panel. After the initial development of an acid disequilibrium pH of 0.4 units, the model predicts a relatively constant luminal pH of 6.4 with progressivere absorption of 51% of delivered HCO3 (25 pmol?mm1?min1). There is negligible change in the flow of ammonia or TA (after the initial acidification). Corresponding to the early luminal acidification, there is cytosolic acidification (not shown), which alters cell HCO3 and Cl concentrations, and thus impacts cytosolic and ultimately luminal PD in this region.
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Figure 8 uses the epithelial model to address the issue of the magnitude of peritubular KCl cotransport. In these calculations, the permeability coefficient of the cotransporter was decreased from the control value, in steps, and then the peritubular permeabilities for K+ and Cl were adjusted (keeping the ratio constant) to maintain the baseline transcellular epithelial Na+ flux. The point of full KCl cotransport (abcissa = 1.0) corresponds to the parameters of this model; the point of zero KCl cotransport corresponds to the choice Chang and Fujita (7, 9) made for their DCT model. The luminal and peritubular conditions for these calculations correspond to those of the mid-DCT segment, in which the luminal acid disequilibrium is fully developed. The bottom panel displays the peritubular conductances for K+ and Cl, which satisfy the constraint on Na+ flux. The top panels show the fluxes of Cl and H+ across the luminal cell membrane, the cell Cl and HCO3 concentrations, and the peritubular PD for these calculations. It is clear that with the exception of peritubular PD, there is virtually no difference in cytosolic conditions or in transcellular fluxes with the replacement of the cotransporter by parallel conductances. The major difference is in the total conductance of the peritubular cell membrane, which varies from 18.7 mS/cm2 (53 ?cm2) with the KCl cotransport coefficient of this model, to 124 mS/cm2 (0.8 ?cm2) when only conductive pathways are present. The actual value for the conductance of this membrane is not known; however, it is known that in rat proximal tubule the peritubular resistance is 90 ?cm2 (23), and in rat CCT principal cells 24 ?cm2 (54), so the choice for this model DCT is within that range. A value of 24 ?cm2 for this model would correspond to a KCl permeability coefficient 80% of that selected as the baseline.
In contrast to proximal tubule, the DCT cell routinely faces wide variations in luminal Na+ concentration. This poses a challenge to the epithelium to vary Na+ reabsorption in response to load, while maintaining cell volume within reasonable bounds and avoiding derangements of acid-base balance. Figure 9 contains results from the DCT epithelial model predicting the impact of varying luminal NaCl on cell volume, cytosolic concentrations, and solute fluxes in an early DCT cell. Boundary conditions are as in Table 3, with the exception of luminal Na+, which is varied along with Cl, and which appears on the abcissa. The top left panel shows the DCT cell volume, which remains within a 10% band around its midpoint value (6.1 x 104 cm3/cm2). The notable finding is in the right panels in which the Cl flux displays a transport maximum, with an apparent half-maximal Na+ concentration of 21 mM (luminal Cl concentration = 12 mM), compatible with the kinetics of the TSC. (Of note, the Cl flux corresponding to luminal Na+ = 15 mM is secretory and is not shown. The corresponding luminal Cl = 6 mM, a value that could not be reached in this DCT, because Cl reabsorption ceases at Cl = 7 mM.) At the higher values of luminal Na+, reabsorptive Cl flux saturates while proton secretion via NHE increases, yielding an increase in Na+ reabsorption over the full range of the abcissa. When these calculations are repeated using midpoint DCT boundaryvalues (luminal acid disequilibrium), the qualitative picture remains the same, in this case with half-maximal Cl flux achieved at luminal Na+ of 14 mM (and Cl of 9 mM).
The impact of delivered load on overall DCT Na+ reabsorption can be examined using the tubule model, and the results are shown in Fig. 10. In each panel, the abcissa is entering Na+ concentration (NaCl variation), and with a constant inlet flow of 6 nl/min, the delivered load (solid line in each panel) varies from 90 to 570 pmol/min. The top left panel shows Na+ reabsorption in relation to the load, using baseline model parameters. It is apparent that reabsorption keeps pace with load at low luminal Na+, but then falls off at higher entering concentrations. The maximal Na+ reabsorption rate for this 1-mm segment is 200 pmol/min, well below that observed in micropuncture studies of the load-dependence of Na+ reabsorption by total DCT (11, 33). Nevertheless, the transport rate here for early DCT (120 pmol/min for a delivered load of 400 pmol/min) is compatible with data for this segment (see APPENDIX 1 data from Refs. 10 and 11). A proper comparison with DCT micropuncture requires inclusion of connecting tubule Na+ transport, with reabsorption through the (nonsaturable) luminal Na+ channels.With reference to Fig. 9, luminal NHE varied over the full range of luminal Na+ concentrations; however, in the calculations of Fig. 10, the absolute magnitude of this flux is small relative to Cl and varied from 16 to 30 pmol/min from lowest to highest entering Na+ (4063% of delivered HCO3). In considerations of homeostasis of renal epithelia, a number of control mechanisms have been considered as modulators of either peritubular or luminal transporters (75). These include activation of peritubular KCl cotransport or ion channels by cell volume,or inhibition of luminal entry steps by either cell volume or cell Na+ (as a surrogate for cytosolic calcium concentration). The impact of such controllers is examined in the remaining panels of Fig. 10, and for these calculations, variable transporter densities have been included which have the generic form
(34)
in which P is a parameter of interest (P0, its reference value), u is a cellular variable (u0, its reference value), and confers the sensitivity of this permeability to relative changes in the control variable. For each of the remaining panels in Fig. 10, the reference parameters are equal to the baseline values used in the top left panel, and the value of the reference cell variable is that of the initial DCT cell under baseline flows (Table 3). The bottom left panel shows the effect of volume-activated peritubular KCl cotransport (with sensitivity 4.0); the top right panel shows volume-inhibited TSC density (sensitivity 4.0); and the bottom right panel shows the effect of TSC inhibition by cell Na+ (sensitivity 4.0). Thus for each mechanism, a 10% change in the cell variable will produce a permeability change of 40%. The calculations demonstrate virtually no significant impact of any of these controllers on overall DCT Na+ reabsorption. When examined in the DCT epithelial model (initial or midtubule), these control mechanisms had inconsequential impact on Na+ fluxes, and only volume-dependent KCl blunted the small perturbations of cell volume.
The response of the DCT to changes in peritubular K+ concentration is of interest in two contexts, cell volume homeostasis and modulation of potassium excretion. Predictably, increases in peritubular K+ will increase cell volume, either by decreasing the driving force for exit across the KCl cotransporter or by depolarizing the peritubular membrane and thus decreasing the force for Cl exit across its channel. Regulated transporters should be able to blunt this volume challenge, and in prior models have either involved activation of a peritubular exit pathway or inhibition of an important luminal entry pathway (75). In proximal tubule, upregulation of exit permeability was attractive insofar as it enabled the cell to protect its volume while maintaining transepithelial Na+ flux. However, in the context of K+ excretion, there should be a benefit for mechanisms that blunt luminal Na+ uptake in DCT in response to peritubular K+ increases, and thus enhance Na+ delivery to the connecting tubule, where it can drive K+ secretion. Figure 11 uses the DCT epithelial model with baseline parameters and boundary conditions at the tubule entry (Table 3); the abcissa marks variation in peritubular KCl concentration from 2 to 8 mM. The panels on the left show changes in cell volume and cytosolic Na+ and Cl concentrations, and those on the right the net fluxes of Na+, Cl, and H+. Over this range of peritubular K+, cell volume varies from 4.9 to 7.5 x 104 cm3/cm2, in parallel with an increase in cytosolic Cl from 10 to 25 mM. The graph of cytosolic Na+ shows a sharp increase with low peritubular K+, and this is due to the inhibitory effect of low K+ on Na-K-ATPase activity. With respect to solute transport, the effects of peritubular K+ are small, namely, a small decrease in Cl reabsorption, and by virtue of peritubular Cl/HCO3 exchange, a small cytosolic alkalinization that blunts luminal NHE.These calculations were repeated with the transport controllers of Fig. 10, namely, volume-activated peritubular KCl cotransport, volume-inhibition of TSC, or cytosolic Na+ inhibition of TSC. The only significant benefit to cell volume came with KCl regulation, in which the range of volumes was halved, from 5.6 to 6.8 x 104 cm3/cm2.The impact on Na+ flux is best examined in the DCT tubule model, because fluxes through the entry pathways shift along the tubule length. These calculations are illustrated in Fig. 12, in which the abcissa is peritubular K+ and each panel shows DCT Na+ delivery to the CNT, for the model without control (top left) or with parameter modulation. For these calculations, only luminal Na+ and Cl have been changed from their baseline values (65 and 56 mM) to 40 and 21 mM to simulate hydropenia; entering volume flow remains at 6 nl/min. The figure shows the model prediction that with baseline parameters there is a small (13%) increase in luminal Na+ flow, consistent with the blunting of Na+ flux seen in Fig. 11. This increase in CNT Na+ delivery is amplified by volume-inhibited TSC to a 30% effect. For both volume-activated KCl or Na+-inhibited TSC, distal Na+ flow increases with low peritubular K+ and fails to rise with hyperkalemia.
Bicarbonate reabsorption by DCT has been studied to determine the contribution of this segment to net acid excretion. It has also been examined in experimental models of metabolic alkalosis to define its role in the maintenance of Cl-depletion alkalosis, as well as in the correction phase. In theory, shifts in the luminal Na+ entry from NHE to TSC could help mediate correction of alkalosis. These issues are considered in the model calculations shown in Fig. 13, using the DCT tubule model in which peritubular HCO3 is varied (via HCO3-for-Cl exchange) over the range 1050 mM. The three panels on the left show Na+, Cl, and HCO3 reabsorption by the 1-mm tubule segment using baseline inlet conditions (Na+ = 65, Cl = 56, and HCO3 = 8 mM), and the three panels on the right show these same fluxes when inlet NaCl is reduced (Na+ = 25, Cl = 16, and HCO3 = 8 mM). The salient findings are that with parallel entry pathways for Na+, there is little impact of peritubular HCO3 on overall Na+ reabsorption. In both control and low-sodium states, Cl reabsorption by DCT and HCO3 delivery to CNT vary directly with peritubular HCO3. Under the most acid condition, reabsorption of HCO3 is 75% complete under high entering Na+ and 57% complete under low entering Na+. With alkalosis, HCO3 falls to 23 or 6% of delivered HCO3, for high or low Na+ perfusates, respectively. These fluxes imply that the model DCT functions in a corrective way to reclaim or excrete delivered HCO3 appropriately. However, the increase in luminal NaCl acts to enhance HCO3 reabsorption at all peritubular HCO3 concentrations, due to enhanced driving force for Na+ across luminal NHE. Thus the model provides no support for the hypothesis that the segment might act as a mediator of bicarbonaturia during chloride repletion.
DISCUSSION
The DCT model of Chang and Fujita (9) and the current work obviously share a number of features, notably identical choice of model solutes, but there are important differences, some in the basic structure of the model. In the DCT of Chang and Fujita, the cells are rigid and of constant volume. This must imply that small changes in cell solute concentrations yield unrealistically large changes in cellular hydrostatic pressure, rather than changes in impermeant solute concentration. This may be of little consequence with respect to representing water movement, but for other solutes it should be kept in mind that when an anionic impermeant is concentrated within the cell, Donnan potentials may shift. Furthermore, the assumption of constant cell volume precludes using their model to study challenges to cell volume, attendant with changing solute fluxes. Another difference with the work of Chang and Fujita is their assumption that the lateral intercellular space is completely equilibrated with the peritubular conditions. With reference to Table 3, it is clear that the interspace conditions in this model are not identical to those of blood, even though the basement membrane is a high-conductance barrier (total electrical resistance is 0.8 ?cm2). In this model, for example, lateral cell membrane Na-K-ATPase activity is predicted to deplete the interspace of K+ and thus blunt its own transport rate. It should also be noted that Chang and Fujita elaborated kinetic models for each cotransporter of their tubule. In the present model, only the luminal cotransporters TSC and NHE are given by kinetic representations, whereas peritubular cotransporters use a linear nonequilibrium thermodynamic formulation. The approach here has the advantage of simplicity (a single parameter representing transporter permeability) in the face of scant data. The disadvantage of this approach is that transporter saturation is not represented, but this should be of little consequence in the membrane between cell and blood for which ambient concentrations are generally constrained.
The present work updates the model of the TSC presented by Chang and Fujita (8), for whom the only available kinetic data were the two uptake curves for TSC from flounder (Fig. 2 from Ref. 24). The present model was able to draw on the subsequent study of rat TSC in oocytes by Monroy et al. (49), which contains one set of uptake curves (their Fig. 3) plus a total of 11 values for Na+ or Cl Km under differing bath conditions. With regard to Fig. 2, the coefficients of Chang and Fujita (8) yield a much lower transport affinity than would fit the data of Monroy et al (49). It may be the case that differences between coefficients derived from the two experiments reflect experimental differences (e.g., temperature), or more likely, differences between flounder and rat TSC. Indeed, when Vazquez et al. (60) examined flounder TSC under conditions comparable to those in which rat TSC had been studied, they found Km values for Na+ and Cl of 58 and 22 mM, in contrast to the higher affinities of rat TSC (Na+ and Cl Km both 6 mM). The kinetic parameters of the present work are by no means definitive. All of the experiments are single sided, so there is no information about transporter symmetry. Consequently, the kinetic model was simplified with a symmetry assumption, which along with an equilibrium binding assumption, yielded a model of only five independent parameters. It is noteworthy that two different fitting strategies to the Fig. 3 data of Monroy et al. (49) provided two very different sets of coefficients (Table 1) that appear indistinguishable with respect to the goodness of fit (Fig. 2). When all of the Km data from Monroy et al. are included in the fit, the Na+ and Cl affinities are greater than when Fig. 3 data alone are used. With respect to cellular function, Fig. 9 shows DCT epithelium Cl reabsorption as a saturating function of inlet Na+ concentration, and half-maximal flux occurs when luminal Na+ = 21 and Cl = 12 mM. In their study of DCT NaCl transport in rat, Velazquez et al. (62) observed Cl reabsorption to be a saturable function of luminal Na+, and when luminal Cl was close to 100 mM, the Km for luminal Na+ was 9 mM. In contrast to Cl, the epithelial Na+ flux is more complex, and even in the present DCT model there appears to be a component that is not saturable (Figs. 9 and 10), due to enhanced Na+/H+ exchange at higher luminal Na+ (Fig. 9). In the perfused DCT, which included early and late segments, Velazquez et al. (62) observed an approximately linear dependence of Na+ reabsorption on luminal Na+ concentration. This confirmed the earlier observation of Costanzo and Windhager (11) in the microperfused DCT, that Na+ reabsorption was about a third of delivered load over a very broad range. This linear response to luminal Na+ concentration is too large to depend on NaHCO3 transport and likely reflects channel-mediated reabsorption in late DCT (CNT).
Perhaps the most important difference between this model and that of Chang and Fujita (9) is the inclusion here of a peritubular KCl cotransporter. Its presence (KCC4) was recently confirmed in the peritubular membrane of rabbit DCT (64), and what the present work adds to this discussion is t