## Abstract

A simulation of the rat distal convoluted tubule (DCT) is completed with a model of the late portion, or connecting tubule (CNT). This CNT model is developed by relying on a prior cortical collecting duct (CCD) model (Weinstein AM. *Am J Physiol Renal Physiol* 280: F1072–F1092, 2001), and scaling up transport activity of the three cell types to a level appropriate for DCT. The major difference between the two tubule segments is the lower CNT water permeability. In early CNT the luminal solution is hypotonic, with a K^{+} concentration less than that of plasma, and it is predicted that osmotic equilibration requires the whole length of CNT, to end with a nearly isotonic fluid, whose K^{+} concentration is severalfold greater than plasma. With respect to potassium secretion, early CNT conditions are conducive to maximal fluxes, whereas late conditions require the capacity to transport against a steep electrochemical gradient. The parameter dependence for K^{+} secretion under each condition is different: maximal secretion depends on luminal membrane K^{+} permeability, but the limiting luminal K^{+} concentration does not. However, maximal secretion and the limiting gradient are both enhanced by greater Na^{+} reabsorption. While higher CNT water permeability depresses K^{+} secretion, it favors Na^{+} reabsorption. Thus in antidiuresis there is a trade-off between enhanced Na^{+}-dependent K^{+} secretion and the attenuation of K^{+} secretion by slow flow. When the CNT model is configured in series with the early DCT, thiazide diuretics promote renal K^{+} wasting by shifting Na^{+} reabsorption from early DCT to CNT; they promote alkalosis by shifting the remaining early DCT Na^{+} reabsorption to Na^{+}/H^{+} exchange. This full DCT is suitable for simulating the defects of hyperkalemic hypertension, but the model offers no suggestion of a tight junction abnormality that might contribute to the phenotype.

- sodium reabsorption
- flow-dependent transport
- collecting duct
- thiazide diuretic

this paper describes the development of a mathematical model of the second half of the distal convoluted tubule (DCT), the connecting tubule (CNT). The first hurdle is to respect the complex anatomy of this nephron region, and the differences among superficial and midcortical nephrons (30). Between the early DCT and the cortical collecting duct (CCD), the cell type changes from a uniform population of early DCT cells to a mix of CNT cells and intercalated cells. In the superficial cortex, the CNT is relatively short and flows into CCD, whereas in midcortex, CNT is longer and joins with other CNT segments in arcades before transitioning to CCD. In the rat, this coalescing within arcades reduces the single kidney tubule number from ∼36,000 DCT to ∼7,200 CCD (23). Virtually all of the transport data pertinent to CNT come from micropuncture and in vivo microperfusion of superficial nephrons (see appendix in Ref. 44). From those data, it is clear that the essential function of CNT is potassium secretion, (27, 28, 35, 39). With CNT K^{+} transport rates at least twice those measured in rat CCD (see Table 6 in Ref. 42), the fivefold greater tubule number suggests an order of magnitude difference between the K^{+} secretion rates of these two nephron regions. Although CNT transport is hormonally regulated, it is also modulated by physical factors, including ambient K^{+} concentrations, luminal Na^{+} concentration, and the flow rate of tubule fluid. The interaction of these factors, and their modification along the tubule length, are ideally suited for model exploration.

Several models are formulated: an epithelial CNT model, in which luminal and bath conditions are specified; a CNT tubule model, in which luminal conditions develop axially; and a full DCT composed of early DCT and CNT in series, to assess compatibility with the micropuncture picture. For this full DCT, the CNT will be unbranched and 1 mm in length. (With respect to available transport area, this system should approximate a coalescing arcade of twice the length, in which both lumen cross section and circumference decrease linearly to 20% of their initial value.) The CNT model will rely heavily on the prior model of the CCD, with its representations of principal and α and β intercalated cells. Perhaps the major difference in function between these two models is the fact that the CNT begins with a dilute luminal fluid, whose K^{+} concentration is substantially less than that of plasma, and ends with a nearly isotonic fluid, whose K^{+} concentration is severalfold greater than plasma. The CNT epithelial model will be used to illustrate shifting transporter dependence of K^{+} secretion, as one moves from low to high luminal K^{+}. Flow-dependence of CNT K^{+} secretion is an obvious consequence of the axial concentration gradient. Moreover, it is also clear from the impact of CNT water reabsorption on luminal solute concentrations that lower water permeability (*P*_{f}) should favor K^{+} secretion. Complexity arises, however, from the fact that CNT water reabsorption favors Na^{+} reabsorption, which by itself will enhance K^{+} secretion. The ultimate balance struck by these competing effects will be examined in the CNT tubule model. Finally, the full DCT model will be used to simulate generation of hypokalemic alkalosis by a thiazide, and to consider possible DCT pathophysiology in hyperkalemic disorders.

## MODEL FORMULATION AND PARAMETER SELECTION

The model of the micropuncture-accessible DCT consists of a 1-mm early DCT (44) in series with a 1-mm connecting segment (CNT). The anatomical scheme for the CNT is identical to that used previously for the rat cortical collecting duct (CCD) and is shown in Fig. 1 (42). The epithelium contains three cell types and a common intercellular space, all bounded by luminal and peritubular solutions. In the CNT tubule model, the four epithelial compartments line the tubule lumen, and luminal concentrations vary axially as a consequence of transport. Within each compartment the concentration of species *i* is designated C_{α}(*i*), where α is lumen (M), interspace (E), principal cell (P), α intercalated cell (A), β intercalated cell (B), or peritubular solution (S). Within the epithelium, the flux of solute *i* across membrane αβ is denoted *J*_{αβ}(*i*) (mmol·s^{−1}·cm^{−2}), where αβ may refer to tight junction (ME), interspace basement membrane (ES), any of the luminal cell membranes (MP, MA, or MB), lateral cell membranes (PE, AE, or BE), or basal cell membranes (PS, AS, or BS). Along the tubule lumen, axial flows of solute are designated F_{M}(*i*). As previously, the 12 model solutes are Na^{+}, K^{+}, Cl^{−}, HCO_{3}^{−}, CO_{2}, H_{2}CO_{3}, HPO_{4}^{2−}, H_{2}PO_{4}^{−}, NH_{3}, NH_{4}^{+}, H^{+},and urea. For each cell, there is an impermeant species whose concentration varies with changes in cell volume, and within the lumen and the lateral interspace, hydrostatic pressure is an additional model variable.(Cell pressures are assumed equal to luminal pressure.) Including the voltage of each compartment, there are 14 unknowns per compartment. This yields 57 model variables for the CNT epithelial model [open-circuit potential difference (PD) is the additional unknown], and 71 variables at each luminal step along the tubule model (axial volume flow is the additional unknown).

The equations of mass conservation with multiple reacting solutes are identical to those specified for the early DCT, with the exception that the bookkeeping for mass conservation within the lumen and within the lateral intercellular space must include exchange with each of the three cell types. In brief, the mass conservation equations are constructed beginning with expressions for mass generation within each model compartment, and allowing for the possibility that compartmental volume may also vary. For the cells, I (I = P, A, or B), the generation of volume, *s*_{I}(v), or solute *i*, [*s*_{I}(*i*)], is equal to its net export plus its accumulation (1) where V_{I} is the cell volume (cm^{3}/cm^{2}). The interspace exchanges with all of the model compartments, so that mass generation is written (2) Within the tubule lumen, mass generation is appreciated as an increase in axial flux, as transport into the epithelium, or as local accumulation. (3) where *B*_{M} is the tubule circumference, and *A*_{M} is the tubule cross-sectional area. With this notation, conservation of volume and solute are formulated and constitute eight equations for each of the compartments (volume, Na^{+}, K^{+}, Cl^{−}, urea, total phosphate, total ammonia, and total CO_{2}). The pH equilibria for carbonic acid, phosphate, and ammonia, the finite rate of hydration of CO_{2}, and charge conservation of the buffer reactions provide an additional five equations for each model compartment. Finally, the equations are completed with the condition of local electroneutrality. Thus for the epithelial model, the zero-current open-circuit condition completes the equation count at 57, and for the tubule model, the Poiseuille flow pressure drop brings the equation count to 71.

The model parameters for rat CNT are displayed in Table 1 and include the permeabilities or transporter densities for each of the intraepithelial barriers: tight junction, interspace basement membrane, plus luminal, lateral, and basal cell membranes of principal and intercalated cells. As in early DCT, lateral interspace (LIS) volume and basement membrane area have been assumed to vary linearly with the hydrostatic pressure difference between LIS and cells, and the single compliance constant is indicated. Basement membrane water and solute permeabilities are specified for the reference area. The parameters are designed to represent the epithelium under aldosterone-stimulated conditions, and individual permeation pathways are shown in Fig. 2. In general, the CNT parameters were obtained from the CCD model by scaling up permeabilities and transporter densities. Specifically, interspace basement membrane permeabilities, and principal cell and β-intercalated cell transporters were scaled up by a factor of 4; CNT tight junction permeabilities were scaled up by a factor of 2; and α-intercalated cell parameters were scaled up by a factor of 6–10 above those for CCD. The net result is an epithelium that transports Na^{+} and K^{+} at a rate that is fourfold greater than the CCD, is relatively hyperpolarized, and has relatively greater acid excretion.

To rationalize the scaling, recall that distal microperfusion data indicate that CNT Na^{+} reabsorption is comparable to that of the early DCT (5, 9), and depending on the extent of AVP and aldosterone stimulation, absolute transport rates are several-fold greater than observed in isolated, perfused rat CCD (33, 37). The only electrophysiology of rat CNT principal cells comes from the work of Frindt and Palmer (13), who have identified maximal (aldosterone-stimulated) whole cell Na^{+} conductance as 1,180 pA·cell^{−1}·80 mV^{−1}, or 15 pA·cell^{−1}·mV^{−1}. Their tubules had 516 principal cells/mm, so that for a tubule diameter of 18 μm, this indicates luminal membrane Na^{+} conductance, *g*_{MP} = 13.5 mS/cm^{2}. Because their bath contained 135 mM Na^{+}, this conductance translates into a principal cell Na^{+} permeability, *h*_{MP} =2.7 ×10^{−5} cm/s [using the formula *h*_{MP} = *RT*·*g*_{MP}/*F*^{2}·C(Na^{+})]. This number is close to the 3.9 ×10^{−5} cm/s for *h*_{MP}(Na) selected in Table 1. The smaller value obtained by Frindt and Palmer is consistent with the fact that the rats used by these workers weighed 100–150 g and thus were smaller than those used for micropuncture. With respect to CNT K^{+} secretion, most of the K^{+} transport observed in distal micropuncture is referable to the CNT (35, 39), and the absolute transport rates are several-fold greater than those of perfused rat CCD (33, 37).

Luminal acidification in CNT has also been assessed by microperfusion. With stationary microperfusion, CNT proton secretion has been found to be greater than (15), equal to (11), or less than (1) proton secretion in early DCT. With continuous microperfusion, using perfusate with 25 mM HCO_{3}^{−} and 14 mM chloride, acidification by early and late segments was comparable (60 pmol·mm^{−1}·min^{−1}) (41), and this rate was about five times that reported for isolated perfused rat CCD (16, 36). The relative magnitude of β cell HCO_{3}^{−} secretion in DCT has been estimated from the effect of luminal Cl^{−} restoration on measured HCO_{3}^{−} flux during microperfusion. In studies in whole DCT, HCO_{3}^{−} reabsorption fell by 30–50% with luminal Cl^{−} addition (45). By itself, this would suggest a β cell contribution of this magnitude; however, there was also substantial HCO_{3}^{−} secretion in the absence of luminal Cl^{−}. Wang et al. (41) perfused late DCT in isolation and found that addition of luminal Cl^{−} reduced HCO_{3}^{−} secretion by 65%. In these experiments, however, the absolute magnitude of luminal acidification under Cl^{−}-free conditions was not large (29 pmol·mm^{−1}·min^{−1}), so that the HCO_{3}^{−} secretory flux was only 18 pmol·mm^{−1}·min^{−1}. These observations were used to inform the model scaling of α- and β-cell transporters indicated above, and the model fluxes are presented below. As in the CCD, carbonic anhydrase activity of CNT intercalated cells was assumed to be full (10,000-fold greater than the uncatalyzed reaction rate), and that of principal cells 10-fold greater than uncatalyzed. As in the DCT model, CO_{2} hydration and dehydration rates for the lateral interspace were assumed 1,000-fold greater than uncatalyzed, to avoid developing a significant disequilibrium pH within the interspace; luminal rates were 10-fold greater than uncatalyzed, to achieve a realistic luminal disequilibrium pH value (see parameter justification in Ref. 44).

The overall electrical conductance of the distal tubule epithelium increases as one proceeds from early to late DCT, with measured values in the CNT of 4.0 (26), 25 (7), and 16.1 mS/cm^{2} (6). In the early DCT, the epithelial conductance could be referred almost entirely to the tight junction, but in CNT, principal cell channels for Na^{+} and K^{+} are likely the dominant conductive pathways. When the principal cell luminal membrane Na^{+} permeability (3.9 × 10^{−5} cm/s) is chosen to yield the desired Na^{+} flux, its Na^{+} conductance is determined, 4.3 mS/cm^{2}. Then, the observation that the CNT Na^{+}-to-Cl^{−} conductance ratio is in the range of 2–3 (26) provides a means for estimating the tight junction chloride permeability. Assuming the junction Na^{+} and Cl^{−} permeabilities nearly equal, 1.0 and 1.2 × 10^{−5} cm/s, the respective conductances are 3.6 and 3.7 mS/cm^{2} (Table 1), and the desired ratio for epithelial Na^{+}-to-Cl^{−} conductance is achieved (see Table 3). These estimates for tight junction Na^{+} and Cl^{−} permeabilities were the source of the scaling of the CNT tight junction permeabilities, relative to those chosen for CCD.

The *P*_{f} of the connecting segment assumes additional importance by virtue of the fact that the luminal K^{+} concentration profile is determined by the rate of epithelial K^{+} secretion in conjunction with the rate of water reabsorption. The earliest observations of DCT osmolality showed equilibration of luminal and peritubular tonicity within two-thirds of the length of accessible DCT (19). Additional micropuncture studies revealed reabsorption of at least half of the water delivered to DCT (24). Microperfusion measurements of DCT *P*_{f} (summarized by Ullrich) (38) yielded values for *P*_{f} of ∼ 0.044 cm/s (*RTL*p ∼ 7. 9 × 10^{−4}·ml·s^{−1}·cm^{−2}·osmol^{−1}). When subsegments of DCT were microperfused by Costanzo (5), the observed volume fluxes indicated that late DCT *P*_{f} was about twice that of the early segment. Assuming a CNT luminal diameter of 18 μm (10), the observed volume flux translates into a *P*_{f} = 0.013 cm/s for late DCT (*RTL*p = 2.3 × 10^{−4}·ml·s^{−1}·cm^{−2}·osmol^{−1}). The volume fluxes measured by Wang and Giebisch (40) in late distal microperfusion were comparable and yield a *P*_{f} = 0.016 cm/s, which increased with angiotensin infusion to 0.035 cm/s. The discrepancy between whole DCT *P*_{f} and subsegment perfusions is informed by morphological investigation in the rat, which revealed differences between the early and late DCT response to vasopressin (48). In that work, Woodhall and Tisher identified structural changes associated with ADH-mediated water transport in CCD (cell swelling and dilated intercellular spaces) and noted similar changes in late DCT, a region they termed initial collecting tubule. These changes were absent in early DCT. Dissection of rat arcades and CCD and quantification of aquaporin-2 content of these segments revealed water channels in both segments, with arcade channel density ∼60% of that in CCD (22). Subsequent immunohistochemical investigation of rat CNT identified water channels in CNT, with luminal expression apparently responsive to vasopressin (4). Taken together, these data suggest that over the length of the micropuncture-accessible DCT there is a progressive increase in epithelial *P*_{f} and that subsegmental *P*_{f} determinations (over 0.5-mm tubule length) would be predicted to vary with the location of the perfusion site. With respect to choice of water permeabilities for CNT components, values for principal cell membrane permeabilities have been taken that just yield osmotic equilibration near end-CNT. The value of overall CNT *P*_{f} corresponds to that of Ullrich (38).

## MODEL CALCULATIONS

Table 2 contains the results of epithelial model calculations for the open-circuited CNT, with predicted solute concentrations within each of the three cells and the lateral interspace. The solute fluxes are in picmoles per second per square centimeter, and these are displayed in Fig. 2, where they have been translated to picomoles per millimeter per minute, assuming a CNT diameter of 18 μm (1 nmol·s^{−1}·cm^{−2} = 34 pmol·mm^{−1}·min^{−1}). The luminal solution is identical to that used to illustrate DCT function (44), to facilitate comparison between the two segments. This luminal solution does not represent the disequilibrium pH (see below), but in contrast to DCT, the impact of acidification on CNT proton secretion is small, and there is no impact on Na^{+} reabsorption. Volume reabsorption is 100 nl·s^{−1}·cm^{−2} (3.4 nl·min^{−1}·mm^{−1}) all via the principal cell, with the osmotic pressure drop from lumen to blood nearly fully developed across the luminal cell membrane. With luminal Na^{+} concentration of 65 mM, the open-circuit potential is −24 mV and overall Na^{+} reabsorption is 172 pmol·mm^{−1}·min^{−1}, composed of principal cell uptake of 231 and tight junction secretion of 60 pmol·mm^{−1}·min^{−1}. The uptake of Na^{+} is largely in exchange for luminal K^{+}, 103 pmol·mm^{−1}·min^{−1}, derived from principal cell secretion, 109 pmol·mm^{−1}·min^{−1}, offset by a small flux through the α-cell H-K-ATPase, 8 pmol·mm^{−1}·min^{−1}; tight junction K^{+} secretion is trivial, 2 pmol·mm^{−1}·min^{−1}. In this segment, Cl^{−} reabsorption is small, 26 pmol·mm^{−1}·min^{−1}, with two-thirds via the β-cell luminal Cl^{−}/HCO_{3}^{−} exchanger, and the remainder divided evenly between tight junction reabsorption and a principal cell NaCl cotransporter. Luminal acidification derives mostly from the α-cell, with a total proton secretion of 69 pmol·mm^{−1}·min^{−1} (62 from H^{+}-ATPase, and 8 from H-K-ATPase). β-Cell HCO_{3}^{−} secretion is 17 pmol·mm^{−1}·min^{−1}, or ∼25% of proton secretion. Of note, there are transepithelial ammonia gradients from lumen to blood (3 mM for NH^{+} and 15 μM for NH_{3}), which produce reabsorptive ammonia fluxes (6 pmol·mm^{−1}·min^{−1} for NH_{4}^{+} and 12 pmol·mm^{−1}·min^{−1} for NH_{3}), that on balance, acidify the tubule lumen. This mechanism for effective proton secretion, without local proton ATPase activity, had been identified previously in conjunction with the CCD (42). However, with luminal acidification, NH_{3} concentration drops substantially, and this contribution to net acid excretion remains small.

The epithelial model of CNT was used to simulate experimental determination of solute permeabilities and *P*_{f}, and these are displayed in Table 3. These calculations were done exactly as for the early DCT (44), in which luminal and peritubular solute concentrations were taken close to physiological values and a neutral luminal impermeant at 140 mM was added to minimize convective fluxes. Then, for the solute species, Na^{+}, K^{+}, Cl^{−}, HCO_{3}^{−}, urea, NH_{4}^{+}, 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, *h*_{MS}(*i*), or in the case of the impermeant, the CNT *P*_{f}. Finally, the transepithelial PD was increased and decreased by 0.1 mV, and the flux changes of charged species yielded the conductive permeability, expressed as centimeters per second, or as the partial conductance, *g*_{MS}(*i*), in millisiemens per square centimeter. The results of these calculations include a Na^{+} permeability that is 33% larger than that of DCT, indicating only a slightly larger dependence of CNT Na^{+} reabsorption on luminal concentration. The CNT Na^{+} conductance, however, is nearly threefold greater than DCT, reflecting the dominance of rheogenic transport in CNT. With respect to K^{+}, the CNT permeability is 2.3 × 10^{−4} cm/s, referable to the K^{+} channels in series across the principal cell. It may be noted, however, that this overall permeability is 2.4-fold higher than the principal cell luminal membrane K^{+} permeability shown in Table 1. This increased permeability derives from the use of the Goldman equation to represent K^{+} fluxes: (4) Here *J*_{MP} is the luminal K^{+} flux, and *h*_{MP} the permeability value in Table 1. The transmembrane PD, ψ_{M} −-ψ_{P}, is 75 mV, so that with *F* = 97 C/mmol and *RT* = 2.57 J/mmol (*RT*/*F* = 26.4 mV), this translates into a normalized potential ζ_{MP} = 2.8. With these values, the change in luminal membrane flux with a change in C_{M} should be ∼2.8-fold greater than *h*_{MP}, although the peritubular membrane in series blunts this slightly. For this channel within the luminal cell membrane, the Goldman equation represents both diffusive and voltage-dependent fluxes, and the voltage effect is formally identical to a convective term. Thus, in varying the upstream ion concentration, the pure diffusive effect (embodied in *h*_{MP}) is amplified. The Cl^{−} conductance in the model CNT is about half that of Na^{+} and concordant with experimental observation (26). Overall, the total conductance for this model CNT is ∼18.5 mS/cm^{2}.

Figures 3 and 4 display the results of calculations from the CNT tubule model, in which the abcissa is tubule length over a 1-mm segment, and the initial conditions are identical to those in Table 2. The values of the tubule net solute fluxes are displayed in Table 4. In Fig. 3, 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. Over the length of the tubule, the lumen hyperpolarizes from −24 to −35 mV. With the osmotic reabsorption of 42% of the delivered volume, there is an increase in all four solute concentrations. With virtually no change in axial urea flow, its concentration increase from 30 to 50 mM along the tubule accurately reflects the change in volume flow. The overall tubule fluid osmolality changes only 44%, from 169 to 244 mosmol/kgH_{2}O (data not shown), due to net solute reabsorption (NaCl) by CNT. For this segment, the Na^{+} reabsorption is 141 pmol·mm^{−1}·min^{−1}, 36% of the delivered load and approximately equal to the model calculation for DCT Na^{+} flux under similar perfusion conditions (155 pmol·mm^{−1}·min^{−1}). About half of this Na^{+} flux is in exchange for K^{+} (67 pmol·mm^{−1}·min^{−1}), about one-third as NaCl reabsorption (49 pmol·mm^{−1}·min^{−1}) and the remainder as NaHCO_{3} (26 pmol·mm^{−1}·min^{−1}). Figure 4 displays the variables relevant to CNT acid-base transport: the *left* panels include the luminal pH and the concentrations of HCO_{3}^{−}, titratable acid (TA), and NH_{4}^{+}, and the panels on the *right* show the axial flows of these species plus “net acid flow” (NH_{4}^{+} + TA − HCO_{3}^{−} flows). The *bottom left* panel shows the disequilibrium pH calculated for either an open system, with the Pco_{2} fixed at 40 mmHg, or for a closed system, in which there is conservation of total CO_{2}. The open system calculation is likely the one that corresponds more closely to the measurements by DuBose et al. (8). For the closed system, there is a prediction of the equilibrium Pco_{2}, 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 ∼53% of delivered HCO_{3}^{−}. This luminal pH and HCO_{3}^{−} flux are nearly identical to values computed for the DCT model (44), so that with identical phosphate flows, axial change in TA was also identical. With respect to net acid flow, however, DCT acid excretion was ∼20% higher than in CNT, and this is reflected in the difference in ammonia flows along the two segments.

CNT K^{+} secretion is modulated by ambient solute concentrations, as well as the transepithelial electrical potential. In the calculations in Fig. 5, the CNT epithelial model predicts the impact of luminal and peritubular K^{+} concentrations on K^{+} flux. In the *left* panel, peritubular and luminal solutions are as in Table 2, with the exception that luminal K^{+} is varied from 2 to 7 mM (via KCl addition) whereas peritubular K^{+} is 5 mM; on the *right*, peritubular K^{+} is varied from 2 to 7 mM (via KCl addition) whereas luminal K^{+} is 2 mM. Each panel shows the transepithelial K^{+} fluxes in both open-circuit and short-circuit conditions. What is apparent is that over the 5 mM range of input K^{+} concentrations, the magnitude of the variation in K^{+} secretion depends on the side of the concentration change and the transepithelial PD: for the open-circuited epithelium, with peritubular K^{+} ranging over 5 mM, the variation in K^{+} secretion is 1.4 nmol·s^{−1}·cm^{−2}; for the open-circuited epithelium, with luminal K^{+} ranging over 5 mM, the variation in K^{+} secretion is 0.7 nmol·s^{−1}·cm^{−2}; and for both short-circuit conditions, the variation in K^{+} secretion is ∼1.2 nmol·s^{−1}·cm^{−2}. With these fluxes, the apparent CNT K^{+} permeabilities in the open-circuited epithelium are 14 and 28 × 10^{−5} cm/s, for respective changes in luminal and peritubular K^{+} concentration; for the short-circuited epithelium, the apparent permeability is 24 × 10^{−5} cm/s, similar to that calculated for Table 3. The difference between the luminal and peritubular K^{+} application is not due to perturbation of the transepithelial potential by the solution change: when the PD is clamped to −25 mV, then over the 5 mM K^{+} range, the fluxes change by 0.8 and 1.9 nmol·s^{−1}·cm^{−2} for luminal and peritubular variation, respectively. Instead, this asymmetry derives from Goldman-type conductance applied to the whole epithelium. (When ζ = 1.0, as in this case, Goldman asymmetry predicts a permeability ratio of 2.7.) From the perspective of CNT physiology, the effect of the asymmetry is that variations in plasma K^{+} concentration are amplified in modulating tubular K^{+} secretion.

Changes in luminal K^{+} concentration derive from the combination of K^{+} secretion and water reabsorption. In Fig. 3, for example, K^{+} concentration increased 11-fold along the CNT, when the axial flow increase was 6.6-fold. In turn, concentration of luminal K^{+} by water reabsorption blunts further K^{+} secretion. In view of the hypotonic solution entering CNT, these considerations suggest a sharp dependence of K^{+} excretion on CNT *P*_{f}. What is shown in Fig. 6 are calculations using the CNT tubule model in which CNT *P*_{f} has been varied over a 20-fold range with coordinate changes in permeability of both luminal and peritubular membranes of the principal cell. To compute the abcissa for each calculation, the unit membrane permeabilities were varied in the CNT epithelial model over a 25-fold range, and then whole epithelial *P*_{f} values were computed as in Table 3. For each tubule calculation, the entering solution was taken to be the end-DCT solution computed using the DCT model, for which Na^{+} = 43.4, K^{+} = 4.1, Cl^{−} = 40.9, HCO_{3}^{−} = 4.3, HPO_{4}^{2−} = 1.2, H_{2}PO_{4}^{−} = 3.1, urea = 31.9, and NH_{4}^{+} = 3.4 mM, at entering flow of 5.4 nl/min. The *top* panels displaythe end-tubule (*x* = 1 mm) osmolality and total CNT volume reabsorption. It is apparent that at this relatively sluggish flow, volume reabsorption shows *P*_{f} dependence below∼0.07 cm/s. Recall that with the baseline model parameters, CNT *P*_{f} was 0.042 cm/s, so that for the control case, most of the volume reabsorption would be expected. The *middle* panel on the *left* shows the end-luminal K^{+} concentration, and its *P*_{f} dependence parallels that for volume reabsorption. For this model CNT, the maximum luminal K^{+} concentration is ∼35 mM. The *middle* panel on the *right* shows the total K^{+} secretion for the 1-mm CNT. From the most water-permeable CNT to the least permeable, there is a 50% increase in K^{+} secretion. In the *bottom* panels are the end-tubule Na^{+} concentration and Na^{+} reabsorption. In parallel with osmotic equilibration, there is concentration of luminal Na^{+}, which occurs progressively earlier along the CNT (not shown). With this increase in luminal Na^{+} concentration, there is a 20% increase in reabsorption from the smallest *P*_{f} to the greatest.

One important message from Fig. 6 is that overall CNT K^{+} secretion depends not only on the maximal rate of epithelial K^{+} secretion but also on the maximal luminal K^{+} concentration against which secretion can proceed. The implication for epithelial model calculations of K^{+} transport is that both of these features deserve consideration. This is illustrated in Fig. 7, which examines the impact of ambient conditions on CNT K^{+} secretion. In the panels on the *left*, the lumen and bath solute concentrations are those in Table 2, and panels show CNT K^{+} secretion. To obtain the panels on the *right*, the CNT epithelial model is called as a subroutine of a Newton iteration in which luminal K^{+} concentration is varied (by changes in KCl concentration) to achieve zero net K^{+} transport. It is this equilibrium K^{+} concentration that is plotted as a function of the boundary condition. In the *bottom* panels, the peritubular K^{+} concentration is varied from 2 to 7 mM, as in Fig. 5, and the curve of K^{+} secretion under open-circuit conditions is identical to that in Fig. 5. Under these conditions, the limiting luminal K^{+} concentration approximately doubles over the range of peritubular K^{+} concentrations, from 17 to 31 mM. In the *middle* panels, the concentration of luminal Na^{+} is varied by varying NaCl concentration, and the open-circuit model is used. The bath conditions are those in Table 2, with K^{+} = 5 mM. When luminal Na^{+} varies between 10 and 60 mM, both K^{+} secretion and the limiting K^{+} concentration increase in parallel over a factor of 2. For higher luminal Na^{+} concentrations, the predicted changes in K^{+} transport are negligible. In the *top* two panels, the closed-circuit epithelial model is used, and transepithelial PD appears on the abcissa. Luminal and bath solute concentrations are those in Table 2 (luminal K^{+} = 2 mM, bath K^{+} = 5 mM). The curve of K^{+} secretion is linear and shows a variation in K^{+} flux of 3.7 nmol·s^{−1}·cm^{−2} (0.36 mA/cm^{2}), over the 60-mV range. This translates into an epithelial K^{+} conductance of 6 mS/cm^{2}, in agreement with that calculated in Table 3. In contrast, the equilibrium K^{+} concentration is an exponential function of luminal PD, rising sharply when the epithelium is hyperpolarized below −30 mV, and reaching ∼80 mM when luminal PD is −60 mV.

The determinants of the equilibrium K^{+} concentration were previously examined in association with a CCD model (42), where it was shown that when only K^{+} fluxes through the principal cell are considered (ignoring tight junction and intercalated cells) (5) where C_{M}/C_{S} is the ratio of the luminal equilibrium K^{+} to peritubular concentration, and *J* is K^{+} uptake by the peritubular Na-K-ATPase. *L*_{PS} is a permeability coefficient defined by (6) where *h*_{PS} and *g*_{PS} are peritubular membrane K^{+} permeability and conductance, respectively, and c̄_{PS} is the mean membrane K^{+} concentration. From the expression for the limiting K^{+} concentration, the exponential dependence on epithelial PD is evident. There is also the prediction that the limiting K^{+} concentration will be independent of the luminal membrane K^{+} permeability. For the CNT model peritubular K^{+} permeability of 3.2 × 10^{−4} cm/s (Table 1), the K^{+} conductance is 43 mS/cm^{2}, so that the value of *L*_{PS} is 4.6 × 10^{−6} mmol^{2}·J^{−1}·s^{−1}·cm^{−2}. When the peritubular K^{+} uptake is 5.1 ×10^{−6} mmol·s^{−1}·cm^{−2} (Table 2), the ratio *J*/*RT*·*L*_{PS} is 0.43, and when the transepithelial PD is −25 mV, its contribution (*F*/*RT*) is ∼1.0. Thus the expectation is for an equilibrium ratio C_{M}/C_{S} ∼4.2, or a limiting K^{+} concentration of 22 mM.

Some of these factors are illustrated in Fig. 8, in which the CNT epithelial model is used to predict the effect of parameter variation on K^{+} transport. As in Fig. 7, the *left* panels show epithelial K^{+} flux when luminal and bath composition are as in Table 2, and the *right* panels show the equilibrium luminal K^{+} concentration. The parameter of interest in the *bottom* panels is principal cell luminal Na^{+} permeability; in the *middle* panels, principal cell luminal K^{+} permeability; and in the *top* panels, tight junction solute permeability (all varied proportionally). In each case, the parameter is varied over a factor of 5 above and below baseline, and the results are plotted against the logarithm (base 10) of the ratio to baseline. Three cases are distinguished: when luminal membrane K^{+} permeability is increased, there is strong enhancement of early K^{+} secretion, but little change in equilibrium K^{+} concentration; when tight junction permeability is decreased, there is only a modest increase in early K^{+} secretion, but a substantial increase in equilibrium K^{+}; and when luminal membrane Na^{+} permeability is increased there are prominent increases in both early K^{+} secretion and the limiting K^{+} concentration.

In Fig. 9, these parameter variations are examined using the CNT tubule model. For these calculations, the entering solution is that in Table 2 in which Na^{+} = 65 mM (rather than end-DCT fluid where Na^{+} = 43 mM) to ensure a nonlimiting supply of Na^{+} (390 pmol/min). The panels on the *left* show total CNT Na^{+} reabsorption and on the *right*, K^{+} secretion. In the calculations in the *top* panels, the principal cell luminal Na^{+} permeability has been varied over a factor of 10, from half of its baseline value to five times that value. The results are plotted as a function of the logarithm (base 10) of the ratio of parameter to baseline. In the *middle* panels, both luminal membrane Na^{+} and K^{+} permeabilities are varied in parallel over the same range. The *bottom* panels feature the same luminal membrane parameters as in the *middle* panels, with the addition that tight junctional permeabilities (all in proportion) vary reciprocally. Thus for the fivefold increase in luminal Na^{+} and K^{+} permeabilities, tight junctional permeabilities are 20% of baseline. [This last simulation is motivated by the observation that chronic mineralocorticoid treatment of rabbit CCD decreases tight junction conductance (31).] When only the luminal membrane Na^{+} permeability is varied, Na^{+} reabsorption varies over 190 pmol·mm^{−1}·min^{−1}, from 88 to 278, and K^{+} secretion varies over 120 pmol·mm^{−1}·min^{−1}. At the highest value for Na^{+} permeability, end-CNT Na^{+} flow is 110 pmol/min and K^{+} flow is 170 pmol/min. When luminal K^{+} permeability is added, the total variation in Na^{+} reabsorption is 220 pmol·mm^{−1}·min^{−1}, and K^{+} secretion varies over 170 pmol·mm^{−1}·min^{−1}. At the highest permeabilities, end-CNT Na^{+} flow is 85 pmol/min and K^{+} flow is 208 pmol/min. When tight junction effects are included, the total flux variation for Na^{+} is 280 pmol·mm^{−1}·min^{−1}, and that for K^{+} secretion is 230 pmol·mm^{−1}·min^{−1}. At the highest values for the luminal membrane permeabilities (and the least-permeable tight junction), end-CNT Na^{+} flow is 50 pmol/min and K^{+} flow is 260 pmol/min.

DCT microperfusion has established that K^{+} secretion is enhanced by increasing luminal perfusion (17, 18). The underlying mechanism could involve enhanced Na^{+} delivery (preserving the luminal Na^{+} concentration along a greater length of CNT), and/or greater dilution of secreted K^{+} (preserving a favorable K^{+} concentration gradient). Good and Wright (18) concluded that the latter effect was dominant, and that is borne out in the calculations of this model. In Fig. 10, the CNT tubule model is used, and overall K^{+} secretion is plotted as a function of perfusion rate. The inlet solute concentrations are those in Table 2, with the exception of perfusion Na^{+}, which has been varied by changing NaCl concentration, so that entering Na^{+} concentrations are 12, 18, 30, or 60 mM. For each curve, there is a sharp flow-dependent increase in K^{+} secretion at low flows, which diminishes at the higher flows. The four curves are distinct due to the differences in luminal Na^{+} concentration, but they are nearly parallel. Indeed, at the lowest perfusion rate (4 nl/min), the difference in K^{+} secretion between the lowest and highest perfusate Na^{+} is 36 pmol/min, and at the highest perfusion rate (20 nl/min), this difference is 39 pmol/min. This suggests that the flow-dependent change in K^{+} secretion depends primarily on the change in flow rather than the change in Na^{+} delivery. As is evident in Fig. 3, the concomitant absorption of Na^{+} and water leaves the luminal Na^{+} concentration little changed along the tubule length, and this is true as well in the calculations for Fig. 10 (not shown). In this regard, the impact of luminal Na^{+} on tubule K^{+} secretion parallels the Na^{+} effects demonstrated with the epithelial model in the calculations of Fig. 7. A more comprehensive view of the combination of Na^{+} and flow on CNT K^{+} secretion is displayed in Fig. 11, *A* and *B*. Again, the CNT tubule model is used, and luminal Na^{+} is incremented (NaCl variation) in steps of 2 mM between 12 and 68 mM; for each perfusate, inlet flow is varied between 4 and 20 nl/min. These calculations determine K^{+} secretion at 493 gridpoints, and the values are indicated by the shading from blue (lowest) to green (highest), along with level curves of constant secretion. What this view shows is that in the neighborhood of physiological Na^{+} concentrations (20–50 mM) and flows (4–10 nl/min), the level curves are close and more horizontal than vertical, illustrating the greater impact of flow on CNT K^{+} secretion. This impression is confirmed in the log-log plot of the same data shown in Fig. 11*B*. If one considers the level curves of 50 pmol/min and below, the slopes are close to −2; at the higher rates of K^{+} secretion, the fractional effects of flow and Na^{+} concentration are more nearly equal.

The simulations of early DCT and CNT provide the components of the DCT available for micropuncture. Figure 12 and Table 5 display the results of calculations with these segments in series for which the luminal perfusate is that in Table 2, with the exception that Na = 45 mM (removal of 20 mM NaCl). The abcissa for each panel is distance along the tubule, composed of two 1-mm segments. Panels on the *left* show axial flows of volume, Na^{+}, K^{+}, and Cl^{−}; on the *right*, the axial flows are relevant to acid-base excretion. In each panel, the dotted curves show the tubule under baseline conditions, and the solid curves simulate the case in which 99% of luminal thiazide-sensitive NaCl cotransporter (TSC) has been inhibited in early DCT. Under baseline conditions, with an entering flow of 6 nl/min, there is little volume reabsorption in DCT, but water flux in CNT reduces end-tubule flow to 2 nl/min. Overall, reabsorption of Na^{+} is ∼80% of delivered load, with substantial reabsorption in both segments. The greater Na^{+} flux in early DCT reflects its higher concentration there. Secretion of K^{+} occurs in CNT, and its flux is equal to about half of the CNT Na^{+} reabsorption. Acidification is uniform across both segments, with reabsorption of ∼90% of the delivered HCO_{3}^{−} overall. With application of the thiazide, the DCT Na^{+} reabsorption falls from 50 to 12% of delivered load. However, with the elimination of NaCl reabsorption, flux through the Na^{+}/H^{+} exchanger is favored, and HCO_{3}^{−} reabsorption through early DCT nearly doubles, while titration of luminal phosphate is also increased. Although there is no change in DCT volume flow with the thiazide, there is increased Na^{+} delivery to CNT (50% above baseline) and this enhances CNT K^{+} secretion from 36 to 50 pmol/min. Overall, the thiazide increases the end-tubule flows of both net acid (TA + NH_{4}^{+} + HCO_{3}^{−}) and K^{+} by ∼30% in this 2-mm segment, the former largely in early DCT, the latter in CNT. One might expect that the increased Na^{+} and fluid delivery to the collecting duct would further augment K^{+} excretion, although differences in net acid excretion might well be blunted.

It had been speculated that a distal nephron “chloride shunt” could be responsible for a disorder that was marked by Na^{+} retention (hypertension) and defective K^{+} excretion that was not remediable by aldosterone (34). It was envisioned that enhanced chloride reabsorption in the tubule segment responsible for K^{+} secretion would diminish luminal electronegativity, thus blunting K^{+} secretion and enhancing Na^{+} reabsorption. Although a specific defect was not proposed, this model offers the opportunity to consider a pure chloride shunt in CNT, namely, increased tight junction Cl^{−} permeability. Figure 13 and Table 5 display calculations using the full DCT model in conjunction with the high Na^{+} perfusate of Table 2, and where tight junction Cl^{−} permeability has been increased by a factor of 10. With this chloride shunt, the overall CNT electrical resistance has been reduced to 24 Ω·cm^{2}. In the figure, results with the abnormal junction are the solid curves and those with the baseline parameters are dotted. The most important observation is that the CNT lumen is only mildly depolarized, on average from about −35 (baseline) to −30 mV (chloride shunt). Although the shunt does act to blunt the PD generated by cellular Na^{+} current, it also enhances the negative Cl^{−} diffusion potential across the tight junction. With this shunt, there is a 34% decrease in CNT K^{+} secretion, from 50.7 to 33.4 pmol/min and an 88% increase in CNT Cl^{−} reabsorption, from 44.7 to 84.2 pmol/min. With this Cl^{−} shunt, the net change in Na^{+} reabsorption is only 7%, from 263 to 281 pmol/min, and the decrease in CNT K^{+} secretion can be remediated with 50% increases in principal cell luminal membrane Na^{+} and K^{+} permeabilities (not shown).

A role for DCT in the correction of metabolic alkalosis was considered in conjunction with the model of early DCT. Using simulations in which peritubular HCO_{3}^{−} was raised as high as 50 mM, differences between high and low perfusate NaCl (Cl^{−} = 56 and 16 mM) were examined (44). It was found that although high luminal Cl^{−} would act to diminish NHE proton secretion, the high luminal Na^{+} acted to enhance it, so that DCT HCO_{3}^{−} reabsorption was little influenced by changing luminal NaCl. In short, there was no support for an early DCT contribution to the correction of metabolic alkalosis. The full DCT model of this paper allows the simulation to be extended to include the CNT, and those results are included in Table 5. Only calculations with the highest peritubular HCO_{3}^{−} (50 mM) are shown for the two luminal perfusates. With low luminal NaCl, there is virtually no early HCO_{3}^{−} reabsorption (3 pmol/min), but almost 70% of the delivered load is reabsorbed in CNT (33 pmol/min). Luminal Cl^{−} is still reabsorbed from DCT, but early in CNT luminal Cl^{−} is near 4 mM, and there is a paracellular secretory flux. With CNT volume reabsorption, luminal Cl^{−} increases to allow late reabsorption, but overall CNT Cl^{−} flux is secretory and small. When luminal NaCl is increased, early DCT HCO_{3}^{−} reabsorption increases to 11 pmol/min, as expected, but CNT HCO_{3}^{−} flux falls to 18 pmol/min. The decrease in CNT HCO_{3}^{−} absorption occurs in conjunction with an increase in Cl^{−} reabsorption to 44 pmol/min, mediated, in part, by β-intercalated cell luminal Cl^{−}/HCO_{3}^{−} exchange. Overall, with the provision of luminal NaCl, full DCT HCO_{3}^{−} reabsorption goes from 35 to 28 pmol/min, and the difference appears to be small. This difference corresponds to a change in the delivered HCO_{3}^{−} load to the collecting duct from 13 to 20 pmol/min. It must be noted, however, that with the sluggish flow of the low NaCl perfusion, there is enhanced ammonia reabsorption along the full DCT. In particular, high NaCl augments collecting duct NH_{4}^{+} delivery from 4.9 to 10.4 pmol/min. From the perspective of net acid delivery to the collecting duct, the difference between high- and low-NaCl perfusates is only 0.3 pmol/min. This difference is trivial, and it must be acknowledged that this model has failed to represent recovery from metabolic alkalosis.

## DISCUSSION

The formulation of the CNT model developed here began as a straightforward adaptation of the model CCD. Principal cell, α- and β-intercalated cells, and tight junction and basement membranes, could be considered as five distinct components, which could be scaled independently to accommodate to measured fluxes, permeabilities, and potentials of the CNT in vivo. In essence, it became a problem of a five-parameter fit, rather than the problem that is 10–20 times larger, of configuring all new cells. In contrast to the CCD model, in which the data derive from studies in vitro, all of the supporting CNT data come from micropuncture and in vivo microperfusion, albeit, from superficial nephrons. While there is an abundance of distal tubule transport experiments from which to draw, there are relatively few segmental perfusions of early and late DCT (see the appendix in Ref. 44). On balance, the constraint of overall DCT function is more secure than the data for either individual segment. Nevertheless, it is clear that the essential function of CNT is potassium secretion (27, 28, 35, 39), and that has been the focus of the model calculations. Several models were used in the simulations: an epithelial CNT model, in which luminal and bath conditions are specified, a CNT tubule model, in which luminal conditions develop axially, and the CNT tubule in series with the early DCT model, to assess compatibility with the micropuncture picture.

The major departure from the CCD model, and the one aspect of the principal cell in which simple scaling from CCD was abandoned, is in the CNT *P*_{f}. As indicated in the selection of model *P*_{f}, experimental data suggest that in antidiuresis, CNT is the locus of the transition from the nearly water-impermeable DCT to the highly water-permeable CCD (48). Furthermore, the *P*_{f} along CNT is likely to increase over time during sustained antidiuresis (4). Ultimately, the value of overall CNT *P*_{f} corresponded to that of Ullrich (38), and in the simulation of the 2-mm full DCT (Fig. 12) yielded reabsorption of 60% of the delivered water. The selection of the value for CNT *P*_{f} is particularly important by virtue of its impact on K^{+} secretion. In exploring the CCD model, it had been observed that despite its capacity to secrete K^{+} in vitro, simulations of CCD in vivo showed relatively little K^{+} secretion (42). This was due to rapid osmotic equilibration and concentration of luminal K^{+} to its equilibrium value. It was suggested that the bulk of K^{+} secretion was best left to a relatively water-impermeable tubule region (i.e., the upstream CNT). In that case, CCD water reabsorption could actually raise luminal K^{+} above its equilibrium concentration. The simulations of this model confirmed that view, and K^{+} secretion increased monotonically as CNT *P*_{f} decreased (Fig. 6).

From the calculations presented here, that observation now needs to be more nuanced. CNT K^{+} secretion is an increasing function of both luminal volume flow and luminal Na^{+} concentration (Fig. 11). If secretion were to depend only on delivered Na^{+} load, then the level curves of Fig. 11*A* would be hyperbolas and water reabsorption (which leaves Na^{+} load unchanged) would have no effect. In that case, the effect of increased water reabsorption to concentrate luminal K^{+} would be offset by its simultaneous concentration of luminal Na^{+}. In the log-log plot, if secretion were an exact function of delivered Na^{+} load, then the level curves would be lines of slope −1. In fact, the level curves in Fig. 11*A* are not perfect hyperbolas, and the figure illustrates that, at the lower perfusion rates, the flow-effect is more powerful than the Na^{+} effect. Nevertheless, even at the lower perfusion rates, the level curves of the log-log plot are not horizontal, and although water reabsorption does diminish K^{+} secretion, this effect is blunted in a meaningful way by the increase in luminal Na^{+} concentration. It is true that at the higher flows, there is a smaller impact of flow on K^{+} secretion, so in that region, sacrifice of flow for luminal Na^{+} concentration may be of little consequence to K^{+} secretion. From the perspective of Na^{+} conservation, however, it must be recognized that when there is volume reabsorption, and concentration of luminal Na^{+}, there is enhanced Na^{+} reabsorption (Fig. 6). Thus if the optimal system performance values both K^{+} secretion and Na^{+} conservation, a tunable CNT *P*_{f} would allow compromise between these two objectives, and indeed selection from one to the other. All of these considerations are complicated by recent observations that certain luminal membrane transporters of the principal cell may be activated by flow. These include the Ca^{2+}-activated K^{+} channel (47) and the Na^{+} channel (32). Such activation could be a direct effect of flow or stretch on the channels, or due to deformation of luminal cilia, with secondary Ca^{2+} release (25). In data plotted as in Fig. 11*B*, such flow-dependent model parameters should produce level curves that are more horizontal than those shown. It has not been established, however, that this flow dependence of luminal transporters extends into the region of higher flows considered in that figure.

With respect to cellular determinants of K^{+} secretion, two regimes have been identified (42). One is “level flow,” in which the luminal K^{+} concentration is little different from the peritubular side, and the other is “static head,” in which luminal K^{+} is at equilibrium and secretion has been brought to a halt. For the CCD, the level flow condition really applies only to the tubule studied in vitro; for the tubule in vivo, equilibrium is rapid and static head is the relevant regime. For the CNT in vivo, however, both regimes apply, with the transition point dependent on tubular *P*_{f} and entering flow rate. In level flow, K^{+} secretion is sensitive to changes in luminal cell membrane K^{+} permeability, luminal Na^{+} permeability, and to a lesser extent tight junction parameters. For static head, the limiting K^{+} concentration is dependent on luminal Na^{+} permeability, peritubular K^{+} permeability, and tight junction parameters. There is virtually no dependence on luminal membrane K^{+} permeability. In short, increasing ROMK density within CNT luminal membrane diminishes the time to equilibrium, but not the ultimate equilibrium K^{+} concentration. One observation that had not been made in prior considerations of distal K^{+} transport was the asymmetry of bath K^{+} concentration to influence level flow secretion. It was noted here that peritubular K^{+} changes were about twice as effective as those from the luminal side (Fig. 5). This derived from negative transepithelial PD and the applicability of the Goldman equation (rather than an ohmic resistor) to model passive fluxes. The Goldman equation includes both the effect of concentration and the presence of the electric field on local K^{+} fluxes. The latter is formally equivalent to a convective term and provides the asymmetry to the overall concentration dependence. From a physiological point of view, this aspect of CNT K^{+} transport renders K^{+} secretion more sensitive to changes in plasma concentration. Beyond this asymmetry, additional effects of peritubular K^{+} to upregulate the ensemble of transporters that mediates principal cell K^{+} secretion have recently been reviewed (14).

The microperfusion observations of Good and Wright (18) are relevant to the model predictions of the DCT-limiting K^{+} concentration. In their work, luminal perfusion with low- and high-K^{+} solutions yielded DCT K^{+} fluxes that spanned the range from secretory to reabsorptive. This yielded a plot of K^{+} flux as a function of mean luminal K^{+} concentration that bracketed the point of zero-flux (their Fig. 7). A linear fit to their data yielded a maximal K^{+} secretory rate (*y*-intercept) of −75 pmol/min and a slope of 5.7 pmol·min^{−1}/mM luminal K^{+}, so that their estimate of a DCT-limiting K^{+} concentration (*x*-intercept) was 13 mM. When peritubular K^{+} concentration is 4 mM (the concentration in plasma of the rats studied by Good and Wright), the estimate from this model for the equilibrium concentration is 25 mM, and so this difference from their finding needs to be understood. Good and Wright perfused a 1.2-mm tubule, and assuming a luminal diameter of 18 μm, the slope of their linear fit translates into a luminal K^{+} permeability of 1.7 × 10^{−4} cm/s. The model permeability of the early DCT was 0.9 × 10^{−4} cm/s and of the late segment, 2.3 × 10^{−4} cm/s, so that value of Good and Wright corresponds to a model segment overlapping both regions, but more toward the late DCT. The higher permeability of the CNT used here would, in itself, bias the limiting K^{+} concentration toward a lower value. The major departure of the model is the level-flow K^{+} flux, ∼3 nmol·s^{−1}·cm^{−2} (Fig. 5), or 100 pmol·mm^{−1}·min^{−1}, and this is ∼50% higher than the 63 pmol·mm^{−1}·min^{−1} of Good and Wright (75 pmol/min for a 1.2-mm segment). With reference to Fig. 8, resetting the model luminal membrane Na^{+} permeability to ∼40% of its baseline value would yield a level-flow K^{+} secretion comparable to that of Good and Wright. Then, reading across the figure to the corresponding equilibrium K^{+} concentration, one finds a limiting value nearly identical to theirs. The justification for the higher Na^{+} permeability for this model CNT lies in the comparison of the Na^{+} flux measured by Good and Wright (∼100 pmol/min) with Na^{+} fluxes measured by other workers (see the appendix in Ref. 44). In sum, the difference between model and experiment is the simulation of a slightly more Na^{+}-avid tubule.

Chang and Fujita (2, 3) have provided the only other model of CNT, their “late DCT,” which they have presented in series with early DCT. The differences between their model and that developed here lie largely in the choice of parameters, rather than basic structure. Chang and Fujita selected a *P*_{f} for the luminal membrane of principal cell of 7.5 × 10^{−8} cm·s^{−1}·mmHg^{−1} (equivalently 1.4 × 10^{−3} cm·s^{−1}·osmol^{−1}, or *P*_{f} = 0.08 cm/s), about twice the permeability used here, so that osmotic equilibration occurs more rapidly in their tubule. Their principal cell luminal Na^{+} permeability was 1.3 × 10^{−5} cm/s, or about a third of that used here. Because their late DCT diameter was 33% greater than that in this model, and their tubule length 20% longer, the overall reduction of CNT Na^{+} transport capacity was ∼50%. Similarly, their luminal cell membrane K^{+} permeability, 3.4 × 10^{−5} cm/s, was also about a third of that used here. Their late DCT tight junction had permeabilities about half those of the CNT tight junction here, so that with the reduced Na^{+} flux, transepithelial PDs were similar. In the calculations of Chang and Fujita, their baseline Na^{+} delivery to the full DCT was ∼400 pmol/min (50 mM in 8 nl/min flow), nearly identical to the 390 pmol/min of this model (65 mM in 6 nl/min flow). However, their early Na^{+} reabsorption was substantially greater, 327 compared with 155 pmol/min, and their late DCT reabsorption smaller, 62 compared with 108 pmol/min (Table 10 in Ref. 2). Thus their overall DCT K^{+} secretion was ∼22 pmol/min, compared with 61 pmol/min in the present model. One way to put these numbers into perspective is to consider the distal K^{+} delivery of 36,000 nephrons, 0.8 vs.2.2 μmol/min for the two models. The collecting ducts have been demonstrated to be a site of K^{+} reabsorption (27, 28), and in model calculations the reabsorption was generally greater than 50% (43). Obviously, urinary K^{+} excretion is quite variable, with Malnic et al. (28) measuring 1.6 μmol/min under control conditions, and a number of other workers reporting values in the range 0.4–1.0 μmol/min (summarized in Table 4 in Ref. 43). From these considerations, parameters favoring the higher rate of DCT K^{+} secretion were adopted here. One point of agreement between the two models is identification of the late DCT (or CNT) as the site of thiazide-induced K^{+} wasting. In each model, the diuretic produced a sharp increase in luminal Na^{+} at the late DCT and led to 2.5-fold increases in K^{+} secretion.

It has been recognized that mutations of WNK kinase produce coordinated transport defects along DCT that result in Na^{+} retention and hyperkalemia, type 2 pseudohypoaldosteronism (46). Within early DCT, there is defective retrieval of TSC from the luminal membrane (49), and within CNT, there is excessive retrieval of ROMK (21). Of note, WNK was found prominently associated with DCT tight junctions (46), but no functional changes in the paracellular pathway have yet been identified. From the perspective of this model, opening of the tight junction would be expected to lead to both decreased level-flow K^{+} secretion and decreased equilibrium K^{+} (Fig. 8). However, this nonspecific (both cation and anion) increase in junctional permeability produced a decrease in Na^{+} reabsorption (Fig. 9) and would be expected to blunt DCT Na^{+} retention. When the tight junction was modified by selective enhancement of Cl^{−} permeability, the depolarization was not large, due to the blood-to-lumen Cl^{−} concentration gradient. With a10-fold increase in tight junction Cl^{−} permeability, there was decreased CNT K^{+} secretion, but a trivial impact on DCT Na^{+} flux. From these simulations, it is difficult to generate a guess as to what tight junctional abnormality might be present in this disorder, or to speculate on its contribution to the overall phenotype.

Although not considered in detail here, the acid-base bookkeeping of this model CNT should be acknowledged. With delivery of HCO_{3}^{−} at the rate of 48 pmol/min to the DCT, early and late reabsorption accounts for 25 and 16 pmol/min (Table 5). Of this CNT reabsorption, almost half is attributable to NH_{3} reabsorption (with subsequent dissociation of NH_{4}^{+} and availability of a new luminal proton). Obviously, this ammonia flux makes no contribution to net acid excretion, but it is a means of reclaiming filtered HCO_{3}^{−} without the metabolic cost of new proton secretion by CNT. This mechanism had first been noticed in connection with the CCD model (42). Its operation is dependent on a lumen-to-blood ammonia concentration difference, plus adequate epithelial NH_{3} permeability. The tubule permeability used here was 0.024 cm/s, identical to that measured for rat CCD (12). This process looks similar from the perspective of the model of Chang and Fujita (3). They had used a principal cell NH_{3} permeability of 0.011 cm/s, but with a 50% larger tubule. In their model, late DCT HCO_{3}^{−} reabsorption was 18 pmol/min, with NH_{3} exit accounting for 5.2 pmol/min. In one micropuncture study of DCT ammonia handling, there was a substantial lumen-to-blood concentration gradient, but little reabsorption was found except in the setting of a higher delivered load (20). These considerations highlight the question raised by Nakhoul et al. (29), who documented NH_{3} permeation of aquaporin-1 and asked whether aquaporin-2 might be a route for membrane transport of NH_{3}. In that case, one might expect CNT NH_{3} permeability to correlate with tubule *P*_{f} and thus vary with the antidiuretic state of the animal.

In sum, this mathematical model of rat CNT has been developed by relying heavily on a previous CCD model and scaling up transport activity of the three cell types to a level appropriate for DCT. The major difference between the two tubule segments is the lower CNT *P*_{f}. In early CNT, the luminal solution is hypotonic, and the length of tubule over which osmotic equilibration occurs is critically dependent on the tubule *P*_{f}. With the reported DCT *P*_{f}, it is predicted that osmotic equilibration requires the whole length of CNT. One implication of this axial concentration profile is the flow dependence of solute transport. With respect to K^{+} secretion, early CNT conditions are conducive to maximal fluxes, whereas late conditions require the capacity to transport against an electrochemical gradient. The parameter dependence of each of these regimes is different, but both are enhanced by greater Na^{+} reabsorption. While higher CNT *P*_{f} depresses K^{+} secretion, it favors Na^{+} reabsorption. Thus in antidiuresis, one finds a complex trade-off between enhanced Na^{+}-dependent K^{+} secretion and the attenuation of K^{+} secretion by slow flow. When the CNT model is configured in series with the early DCT, it provides a simulation of a full DCT. In this model, thiazide diuretics promote renal K^{+} wasting by shifting Na^{+} reabsorption from early DCT to CNT; they promote alkalosis by shifting the remaining early DCT Na^{+} reabsorption to Na^{+}/H^{+} exchange. This full DCT is suitable for simulating the defects of hyperkalemic hypertension, but there is no suggestion of a tight junction abnormality that might contribute to the phenotype.

## GRANTS

This investigation was supported by Public Health Service Grant RO1-DK-29857 from the National Institute of Diabetes and Digestive and Kidney Diseases.

## Footnotes

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- Copyright © 2005 the American Physiological Society