A mathematical model of rat ascending Henle limb. III. Tubular function

Abstract

K⁺ plays a catalytic role in AHL Na⁺ reabsorption via Na⁺-K⁺-2Cl⁻ cotransporter (NKCC2), recycling across luminal K⁺ channels, so that luminal K⁺ is not depleted. Based on models of the ascending Henle limb (AHL) epithelium, it has been hypothesized that NH₄⁺ may also catalyze luminal Na⁺ uptake. This hypothesis requires that luminal NH₄⁺ not be depleted, implying replenishment via either direct secretion of NH₄⁺, or NH₃ in parallel with a proton. In the present work, epithelial models of rat medullary and cortical AHL (Weinstein AM, Krahn TA. Am J Physiol Renal Physiol 298: F000–F000, 2009) are configured as tubules and examined in simulations of function in vitro and in vivo to assess the feasibility of a catalytic role for NH₄⁺ in Na⁺ reabsorption. Modulation of Na⁺ transport is also examined by peritubular K⁺ concentration and by Bartter-type transport defects in NKCC2 (type 1), in luminal membrane K⁺ channels (type 2), and in peritubular Cl⁻ channels (type 3). It is found that a catalytic role for NH₄⁺, which is significant in magnitude (relative to K⁺), is quantitatively realistic, in terms of uptake via NKCC2, and in terms of luminal membrane ammonia backflux. Simulation of a 90% NKCC2 defect is predicted to double distal Na⁺ delivery; it is also predicted to increase distal acid delivery (principally as NH₄⁺). With doubling of medullary K⁺, the model predicts a 30% increase in distal Na⁺ delivery, but in this case there is a decrease in AHL acidification. This effect of peritubular K⁺ on proton secretion appears to be akin to type 3 Bartter's pathophysiology, in which there is decreased peritubular HCO₃⁻ exit, cytosolic alkalinization, and a consequent decrease in luminal proton secretion by NHE3. One consequence of overlapping and redundant roles for K⁺ and NH₄⁺, is a blunted impact of luminal membrane K⁺ permeability on overall Na⁺ reabsorption, so that type 2 Bartter pathophysiology is not well captured by the model.

rat ascending henle limb (AHL), over its 4-mm length, absorbs ∼25% of filtered Na⁺ and ∼10% of filtered HCO₃⁻. This information is indirect, in that it derives from coordinate micropuncture of proximal and distal nephron (DN), and from microperfusion of superficial loops of Henle. In vitro perfusion of rat and rabbit AHL has revealed basic transport mechanisms, but in preparations in which overall fluxes are substantially less than those in vivo; transport by mouse AHL is about an order of magnitude greater, although for the mouse, conditions in vivo have not been as well defined. In the isolated perfused tubule, the peritubular environment is uniform, and axial solute gradients along the lumen are small, so that an epithelial model may be appropriate to assessing this preparation. In vivo, the peritubular environment is variable along the medulla, and perhaps within the cortical labyrinth, but the most important difference from the tubule in vitro is that luminal conditions are determined by tubular solute transport. This aspect becomes critical in consideration of K⁺ fluxes, in which entering concentrations of several millimoles support Na⁺ reabsorption, without gross depletion of luminal K⁺; in this sense, K⁺ has been termed “catalytic” for AHL Na⁺ transport. K⁺ has also been implicated in the regulation of AHL Na⁺ reabsorption from the peritubular side (18). An increase in peritubular K⁺ is expected to increase cytosolic Cl⁻ (whether through peritubular depolarization or by an effect on KCC transport), and luminal Na⁺ entry via Na⁺-K⁺-2Cl⁻ cotransporter (NKCC2) has a quadratic dependence on cell Cl⁻. From the perspective of the epithelial model, however, peritubular K⁺ would also be expected to increase the lumen potential difference (PD) and thus enhance paracellular Na⁺ reabsorption (33). These roles for K⁺ in AHL Na⁺ transport define quantitative questions that must be addressed with a tubular model.

AHL is also the site of ammonia reabsorption, giving NH₄⁺ a short circuit from proximal delivery across medullary AHL, and into the collecting duct (9, 26). This short circuit was originally envisioned as a product of freely diffusible NH₃ into acidic regions of the nephron, but has since been refined to take cognizance of multiple AHL and collecting duct transport pathways for NH₄⁺ per se. It must be acknowledged, however, that the significance, or advantage of such a short circuit for overall acid/base metabolism has not been articulated. Net ammonia flux across AHL is trivial in relation to Na⁺ reabsorption, although the capability of NKCC2 to mediate brisk NH₄⁺ transport has been employed to advantage in delivering an acid challenge to AHL cells in vitro. An observation from models of NKCC2 and AHL epithelium (32, 33) is that NH₄⁺ also has the potential to be catalytic for AHL Na⁺ reabsorption. The important constraint on any such proposal is that like K⁺, luminal NH₄⁺ is not depleted in AHL, but resurfaces in early distal convoluted tubule (DCT) in millimolar concentrations. Recycling of reabsorbed NH₄⁺ back across the luminal cell membrane can occur either as NH₄⁺ [e.g., as Na⁺/NH₄⁺ exchange via type 3 Na⁺/H⁺ exchanger (NHE3)] or as NH₃ in parallel with proton secretion. Either mechanism poses important quantitative questions, as to whether the luminal membrane is sufficiently permeable to NH₃, and whether luminal NHE3 activity is large enough. This potential role for NH₄⁺ in AHL Na⁺ transport will also be addressed in this tubule model.

MODEL FORMULATION

In the preceding paper (33), models for rat medullary and cortical AHL epithelia were developed, employing respectively F- and B-isoforms of NKCC2 (32). In the present work, these epithelial models are configured as tubules of 20-μm diameter and 2.0 mm in length, so that luminal composition is modified by tubular transport and varies with distance, x, from the tubule inlet. Four tubule configurations are examined: the first is the tubule with uniform peritubular solute composition, suggestive of perfusion in vitro; the second features variable peritubular concentrations (varying linearly with x), suggestive of the environment in vivo; and in the third configuration medullary and cortical tubules are in series, with variable peritubular concentrations. As indicated in the model description (33), volume conservation in AHL cells is achieved by solving for a variable cell-impermeant concentration: for tubules in vitro, total cell-impermeant content is fixed, so that changes in impermeant concentration derive from changes in cell volume; for tubules in vivo, cell volume is fixed, and impermeant concentrations change as the result of changes in total impermeant content. This formulation had been used in prior steady-state medullary models, in which external osmolality varied widely (27, 28). For the fourth configuration, medullary and cortical tubules in series feed into a DN, and this is displayed in Fig. 1. With one exception, this is the same DN figure that was recently published (31), in which 36,000 DCT (all 1 mm in length) lead to the connecting tubule (CNT), which coalesces over 2 mm to 7,200 cortical collecting ducts (CCD; 2 mm in length), then 7,200 outer medullary collecting ducts (OMCD; 2 mm in length), and finally the inner medullary collecting duct (IMCD), which coalesces over 5 mm to 112 papillary collecting ducts emptying into the renal pelvis. (It is acknowledged that the model representation of uniform coalescing of CNT segments ignores local concentration differences at mixing points, variations in CNT length, and the possibility of different solute loads from cortical and juxtamedullary nephrons, so that the output from these calculations is best considered as an average within the full CNT ensemble.)

Fig. 1.

Schematic representation of ascending Henle limbs in series with the distal nephron. There are 36,000 nephrons: medullary ascending Henle limb (AHL; 2 mm in length), cortical AHL (2 mm), and distal convoluted tubule (DCT; 1 mm). These lead to the connecting tubule (CNT), which coalesce over 2 mm to 7,200 cortical collecting ducts (CCD; 2 mm), 7,200 outer medullary collecting ducts (OMCD; 2 mm), and finally the inner medullary collecting duct (IMCD), which coalesce over 5 mm to 112 papillary collecting ducts emptying into the renal pelvis. Medullary AHL and OMCD share a common interstitium, and cortical AHL and CCD also have identical peritubular conditions; peritubular DCT and CNT are bathed by the cortical blood.

The one change in the DN model was to the OMCD, which now includes a population of principal cells that is identical to those of CCD. In its original form (28), this OMCD had only functional alpha cell activity, respecting the failure to detect any Na⁺ transport by OMCD perfused in vitro. The current adjustment reflects the recent finding that OMCD principal cells have luminal epithelial Na⁺ channels (ENaC) and that whole cell patch-clamp studies have identified Na⁺ currents that are comparable to those of CCD principal cells (6). In the calculations using this AHL-DN model, peritubular conditions of medullary AHL and OMCD are identical, and peritubular conditions of cortical AHL and CCD are identical. In the reporting of model calculations, the fluxes of AHL in vitro or in vivo are reported for single tubules; for the AHL-DN ensemble, fluxes are reported for the whole kidney. With respect to the method of solving the model equations, all tubule segments, except for IMCD, are discretized to 80 mesh points; IMCD has a chop of 500. Spatial derivatives are approximated by a first-order backward difference scheme, with the exception of OMCD, in which a centered difference scheme has remained satisfactory. For each tubule segment, the steady-state equations were written in Fortran and solved iteratively to double precision machine accuracy using Newton's method.

MODEL CALCULATIONS

The first of the in vitro simulations is a medullary AHL (F-isoform), bathed in a low-ammonia (0.2 mM) solution, and perfused at 6 nl/min with the same solution (Table 1). For this condition, Fig. 2A displays the luminal concentrations predicted by the model, and because there is little change in axial volume flow (∼5% over 2 mm), these can also be viewed as the relative axial solute flows; Table 2 contains absolute and relative solute flows for this calculation. Figure 2B shows transcellular fluxes through the important transporters of luminal (left panes) and peritubular membranes (right panes). The first observation is that over the length of the tubule there is Na⁺ reabsorption, ∼367 pmol/min, or 44% of delivered load. The bulk of this flux is via the NKCC, with a much smaller contribution from NHE3; overall equality of epithelial Na⁺ and Cl⁻ fluxes reflects the substantial component of paracellular Na⁺ flux driven by the transepithelial PD. With low NHE3 activity, HCO₃⁻ reabsorption (35 pmol/min for the tubule) is an order of magnitude less than Na⁺, and addition of titratable acid and loss of NH₄⁺ are both trivial amounts. There is an initial 10-mV lumen positive PD, determined largely by the cell (with only a small tight junctional diffusion potential); over the tubule length, this is augmented by hyperpolarization of the luminal cell membrane from −83 to −112 mV due to the luminal K⁺ profile. Along the tubule, there is a prompt decline in K⁺ concentration, which stabilizes near 0.8 mM; at that value, luminal uptake via NKCC and secretion via K⁺ channels are balanced. At the peritubular membrane, K⁺ exit via KCC largely balances K⁺ uptake via the Na-K-ATPase; and there is only minor flux through peritubular K⁺ channels. With respect to overall Cl⁻ transport, however, KCC accounts for somewhat more than half of the flux, with a substantial contribution via Cl⁻ channels.

Table 1.

Perfusate and Bath Concentrations for In Vitro Simulations

	Low Ammonia Lumen + Bath	2 mM Ammonia Lumen + Bath	2 mM Ammonia + Peritubular K+ Bath
Na+ (mM)	140.00	140.00	140.00
K+	5.00	5.00	25.00
Cl−	115.60	117.37	137.37
HCO3−	25.00	25.00	25.00
H2CO3	4.41 × 10−3	4.41 × 10−3	4.41 × 10−3
CO2	1.50	1.50	1.50
HPO4=	2.00	2.00	2.00
H2PO4−	0.60	0.60	0.60
Urea	5.00	5.00	5.00
NH3	2.94 × 10−3	29.4 × 10−3	29.4 × 10−3
NH4+	0.20	1.97	1.97
Imper.	0.00	0.00	0.00

Fig. 2.

Perfusion in vitro of medullary AHL [Na⁺-K⁺-2Cl⁻ cotransporter (NKCC2) F-isoform]. Perfusate and bath are low-ammonia (0.2 mM) solutions (Table 1), and perfusion is at 6 nl/min along a 2-mm tubule. A: solute concentrations (mM) and transepithelial electrical potential difference (PD). Because there is no significant water transport, luminal concentrations are indicative of axial solute flows. B: transcellular fluxes through the important transporters of luminal (left panes) and peritubular membranes (right panes). G_K refers to luminal or peritubular membrane K⁺ channels; G_Cl to peritubular Cl⁻ channels; and NH₃ is its diffusive flux through either membrane. TA, titratable acid.

Table 2.

AHL (F-Isoform) Solute Reabsorption-In Vitro Simulations

	Low Ammonia Lumen + Bath	2 mM Ammonia Lumen + Bath	2 mM Ammonia + Peritubular K+ Bath
Absolute Reabsorption (pmol/min per 2 mm tubule)
Na+	367.00	474.00	340.90
K+	25.38	16.56	−51.00
Cl−	358.60	411.60	278.50
HCO4=	34.74	76.98	14.40
HPO4=	0.30	1.92	−0.24
H2PO4−	−0.72	−2.16	−0.18
TA	−0.60	−2.10	−0.12
NH4+	0.72	−0.36	2.40
Urea	0.06	0.06	0.06
Reabsorption Relative to Segmental Delivery
Na+	0.437	0.564	0.406
K+	0.846	0.552	−1.700
Cl−	0.517	0.585	0.395
HCO3−	0.232	0.513	0.096
HPO4=	0.025	0.160	−0.020
H2PO4−	−0.200	−0.600	−0.050
TA	−1.250	−4.375	−0.250
NH4+	0.600	−0.031	0.203
Urea	0.002	0.002	0.002

For the calculations of Figs. 3, A and B, 2 mM ammonia has been added to both lumen and perfusate (Table 1): Fig. 3A displays luminal concentrations, and Fig. 3B the luminal and peritubular membrane fluxes. With 2 mM ammonia, Na⁺ reabsorption is 474 pmol/min, an increase of 29% over the case with 0.2 mM ambient ammonia (Table 2), and this is almost exclusively transcellular. Over the length of the tubule, NKCC Na⁺ flux varies from 170 to 90 pmol·mm⁻¹·min⁻¹, accompanied in large part by K⁺ (125 to 43 pmol·mm⁻¹·min⁻¹), but with a substantial contribution from NH₄⁺ (∼50 pmol·mm⁻¹·min⁻¹) (Fig. 3B). This component of Na⁺ uptake with NH₄⁺ is about equal to the overall increase in Na⁺ reabsorption (107 pmol/min for a 2-mm tubule). With high ammonia, there is cytosolic acidification, and NHE3 responds by increasing Na⁺ transport to ∼120 pmol·mm⁻¹·min⁻¹, ∼70 pmol·mm⁻¹·min⁻¹ in exchange for protons and ∼50 pmol·mm⁻¹·min⁻¹ in exchange for NH₄⁺ (Fig. 3B). There is also an increase in diffusive backflux for NH₃, 30 pmol·mm⁻¹·min⁻¹ across the luminal membrane, which decreases net HCO₃⁻ reabsorption to 38 pmol·mm⁻¹·min⁻¹, about half of the proton secretion via NHE3 (Table 2). The combination of NHE3 proton secretion and NH₃ backflux is equivalent to NH₄⁺ recycling, and together with frank NH₄⁺ secretion on NHE3, bring net ammonia transport to near zero (Fig. 3A). The overall K⁺ transport by this model tubule, 17 pmol/min, is also minute in relation to Na⁺ flux, so that both K⁺ and NH₄⁺ may be viewed as “catalytic” in mediating Na⁺ reabsorption. In quantitative terms, the importance of K⁺ in this role is a bit more than twice that of NH₄⁺. Of note, the presence of ammonia blunts luminal K⁺ depletion, so that the large positive lumen PD does not develop (Fig. 3A), with the consequence in this case that paracellular Na⁺ flux is reduced. Finally, the application of ammonia appears to do little to alter peritubular membrane transport: most of the K⁺ exit is via KCC, and Cl⁻ exit is approximately split between KCC and Cl⁻ channels.

Fig. 3.

Perfusion in vitro of medullary AHL (NKCC2 F-isoform). Perfusate and bath are 2.0 mM ammonia solutions (Table 1), and perfusion is at 6 nl/min along a 2-mm tubule. A: solute concentrations (mM) and transepithelial electrical PD. B: transcellular fluxes through the transporters of luminal (left panes) and peritubular membranes (right panes).

The impact on AHL transport of increasing peritubular K⁺ to 25 mM is examined in Fig. 4, A and B; in these calculations, ambient ammonia is maintained at 2 mM (Table 1). The overall impact on Na⁺ reabsorption is a decrease to 340 pmol/min (Table 2), with dramatic effects on both components of transcellular Na⁺ uptake, NKCC and NHE3 fluxes. Over the length of the tubule, NKCC Na⁺ flux varies from 110 to 90 pmol·mm⁻¹·min⁻¹ (Fig. 4B), and this Na⁺ flux is almost exclusively with K⁺ (from 95 to 80 pmol·mm⁻¹·min⁻¹), in view of the impact of high peritubular K⁺ to prevent luminal K⁺ depletion (and increase luminal K⁺ concentration) (Fig. 4A). With the increase in K⁺ in both peritubular and luminal solutions, there is blunted cellular NH₄⁺ uptake by both Na-K-ATPase and NKCC. As a consequence, ambient ammonia does not acidify the cell, there is trivial NHE3 Na⁺ flux (∼30 pmol·mm⁻¹·min⁻¹), and virtually no net HCO₃⁻ reabsorption along AHL. As was the case with increased ambient ammonia, there is still little net ammonia flux, but now both the reabsorptive and secretory components of the net flux are reduced (Fig. 4B). With the high peritubular K⁺, there is a large positive transepithelial PD (Fig. 4A) and early transcellular K⁺ secretion (Fig. 4B). There is now a paracellular component to net Na⁺ reabsorption, whose profile balances the transcellular K⁺ secretion shown in Fig. 4B; there is also paracellular Cl⁻ secretion, which is relatively constant over the tubule length (∼60 pmol·mm⁻¹·min⁻¹). Finally, the impact of peritubular KCl on Cl⁻ exit appears to be greatest on the electrogenic component of transport, so now KCC is the dominant pathway for Cl⁻ exit. This derives from depolarization of the peritubular cell membrane by ∼30 mV.

Fig. 4.

Perfusion in vitro of medullary AHL (NKCC2 F-isoform). Perfusate and bath are 2.0 mM ammonia solutions, with peritubular K⁺ increased to 25 mM by KCl addition (Table 1); perfusion is at 6 nl/min along a 2-mm tubule. A: solute concentrations (mM) and transepithelial electrical PD. B: transcellular fluxes through the transporters of luminal (left panes) and peritubular membranes (right panes).

An important difficulty which haunts simulations of AHL in vivo is uncertainty of luminal and peritubular composition. Nevertheless, it is possible to make reasoned guesses, and these are displayed in Table 3, for medullary and cortical AHL tubular fluid, and for the interstitium at outer-inner medullary junction (OIMJ), corticomedullary junction (CMJ), and at the outlet of cortical AHL. With respect to fluid flow entering medullary AHL, it is assumed that an end-proximal flow of 10 nl/min (one-third of single-nephron glomerular filtration rate) has been reduced to 6 nl/min (early DCT flow). By itself, water abstraction of this magnitude would raise luminal Na⁺ concentration from 144 to 240 mM, and this has been increased to 250 mM, to reflect likely diffusive Na⁺ entry along proximal straight tubule and descending Henle limb. Overall the Na⁺ flow entering medullary AHL is 35% of filtered load. The entering concentrations of K⁺ (10 mM) and HCO₃⁻ (25 mM) are also about twice what is likely to be their end-proximal values. The entering phosphate concentration, 7.5 mM, corresponds to a flow of 45 nmol/min, and is comparable to the luminal flow of phosphate observed in late proximal tubule of phosphate-infused rats (34). At 4 mM, luminal ammonia concentration is also approximately twice its late-proximal value (7, 34). For all of the remaining calculations, medullary gradients of CO₂ have been ignored, and a uniform Pco₂ of 50 mmHg has been assumed. The cortical AHL-entering fluid in Table 3 is derived from solution of the medullary AHL model, and ultimately, with the concatenation of the two models, these cortical-entering conditions will be superfluous. Interstitial solute concentrations at the OIMJ are identical to those used previously, specifically with an ammonia concentration of 4 mM (31). However, at the CMJ, the ammonia concentration has now been increased to 2 mM, and at the cortical outlet to 1 mM. The rationale for these increases will be examined in the calculations that follow. Peritubular solute concentrations at points between OIMJ and CMJ, and between CMJ and the exit cortex are determined by linear interpolation.

Table 3.

Perfusate and Bath Concentrations for In Vivo Simulations

Medullary Segment (NKCC2 F-Isoform)
	Lumen	Bath (x = 0 mm)	Bath (x = 2 mm)
Na+ (mM)	250.00	284.00	144.00
K+	10.00	10.00	5.00
Cl−	225.04	265.94	121.37
HCO3−	25.00	25.00	25.00
H2CO3	4.41 × 10−3	4.41 × 10−3	4.41 × 10−3
CO2	1.50	1.50	1.50
HPO4=	6.00	3.00	2.00
H2PO4−	1.80	0.90	0.60
Urea	10.00	20.00	5.00
NH3	57.3 × 10−3	57.3 × 10−3	29.4 × 10−3
NH4+	3.84	3.84	1.97
Imper.	0.00	2.00	2.00

Cortical Segment (NKCC2 B-Isoform)
	Lumen	Bath (x = 0 mm)	Bath (x = 2 mm)
Na+ (mM)	145.00	144.00	144.00
K+	5.00	5.00	5.00
Cl−	131.62	121.37	120.38
HCO3−	10.00	25.00	25.00
H2CO3	4.41 × 10−3	4.41 × 10−3	4.41 × 10−3
CO2	1.50	1.50	1.50
HPO4=	4.46	2.00	2.00
H2PO4−	3.34	0.60	0.60
Urea	10.00	5.00	5.00
NH3	23.1 × 10−3	29.4 × 10−3	14.7 × 10−3
NH4+	3.88	1.97	0.99
Imper.	0.00	2.00	2.00

Figure 5 displays the concentrations of the important luminal solutes predicted by the medullary AHL model (F-isoform of NKCC), using entering conditions and peritubular composition specified in Table 3. With these conditions, reabsorption of Na⁺, K⁺, and HCO₃⁻ over the 2-mm segment are 596, 32, and 88 pmol/min, corresponding to 40, 54, and 58% of delivered load; generation of titratable acid is 8 pmol/min. Of note, the backflux of NH₃ is ∼30 pmol·mm⁻¹·min⁻¹, or 60 pmol/min for the whole tubule, so that overall NHE3 proton secretion for the whole tubule must be ∼150 pmol/min. The choice of CMJ NH₄⁺ concentration (2 mM) was motivated by the finding that at this peritubular concentration luminal K⁺ and NH₄⁺ are relatively constant, and their end-luminal concentrations are 4.7 and 3.1 mM, supplying ample substrate for NKCC activity in cortical AHL. When CMJ NH₄⁺ is reduced to 1 mM, end-luminal K⁺ and NH₄⁺ are 3.4 and 1.5 mM, and when CMJ NH₄⁺ is reduced to 0.2 mM, these concentrations are 2.3 and 0.5 mM. Figure 6 displays luminal concentrations for a cortical AHL (B-isoform of NKCC), using luminal and peritubular composition specified in Table 3. Early in the tubule, the transepithelial electrical potential falls: Due to the kinetics ascribed to this NKCC isoform, there is avid uptake of NH₄⁺ to the exclusion of K⁺ (note the small increase in luminal K⁺ concentration). This produces a sharp increase in cytosolic NH₄⁺ and Na⁺, at the expense of cytosolic K⁺, and thus a fall in the luminal membrane potential. Over distance, with the decline in luminal NH₄⁺, there is restoration of K⁺ uptake and of the luminal PD. For this tubule, reabsorption of Na⁺, K⁺, and HCO₃⁻ are 471, 16, and 39 pmol/min, corresponding to 54, 54, and 64% of delivered load; generation of titratable acid is 11 pmol/min. Backflux of NH₃ is ∼35 pmol·mm⁻¹·min⁻¹, so that the major portion of proton secretion actually goes to ammonia titration; NHE3 proton secretion is estimated to be 120 pmol/min for this 2-mm tubule.

Fig. 5.

Conditions along medullary AHL (NKCC2 F-isoform) in vivo. Solute concentrations (mM) and transepithelial electrical PD are displayed. Entering conditions and peritubular composition are specified in Table 3: initial luminal concentrations are approximately twice those of cortical interstitium; peritubular concentration gradients of Na⁺, K⁺, Cl⁻, and NH₄⁺ are intended to represent outer medullary interstitium. Entering flow is 6 nl/min for a 2-mm tubule.

Fig. 6.

Conditions along cortical AHL (NKCC2 B-isoform) in vivo. Solute concentrations (mM) and transepithelial electrical PD are displayed. Entering conditions and peritubular composition are specified in Table 3: initial luminal concentrations are approximately those of cortical interstitium, except for low HCO₃⁻ (10 mM) and increased NH₄⁺ (3.9 mM); there is a peritubular concentration gradient for NH₄⁺ (2.0 → 1.0 mM). Entering flow is 6 nl/min for a 2-mm tubule.

Concentrations of K⁺ and NH₄⁺ in early distal tubular fluid have been measured by several investigators, and those values were compiled in development of a model DCT (see appendix in Ref. 30). Acknowledging some variability, the reported values for both K⁺ and NH₄⁺ are ∼2 mM, and this provides an important constraint on peritubular conditions for cortical AHL. [In AHL microperfusion, this value for distal K⁺ is also obtained (e.g., Ref. 19), but no NH₄⁺ measurements have been made.] This is addressed in the calculations of Fig. 7, in which the cortical AHL model is solved over a range of values of peritubular ammonia at the AHL outlet. Using the values in Table 3 for luminal inlet and peritubular CMJ, the peritubular ammonia concentration is varied from 0.1 mM (renal arterial ammonia concentration) to 2.0 mM (the selected CMJ ammonia concentration); luminal outlet concentrations are plotted. The figure shows that AHL Na⁺ reabsorption varies little (470 pmol/min, ±7%), and acidification not at all, as a function of peritubular outlet ammonia concentration. As peritubular NH₄⁺ increases, there is a progressive increase in both end-luminal K⁺ and NH₄⁺. Although there is no value of peritubular ammonia that yields end-luminal K⁺ and NH₄⁺ concentrations both 2 mM, it is clear that an end-luminal concentration of 0.2 mM is grossly unsatisfactory, with end-luminal concentrations of 0.9 and 0.06 mM for K⁺ and NH₄⁺. The value of outlet peritubular ammonia selected for baseline calculations, 1.0 mM, yields end-luminal K⁺ and NH₄⁺ of 2.5 and 0.7 mM. Although end-luminal ammonia is low, selection of a higher end-luminal peritubular ammonia is tempered by an unsettling apparent necessity for positing a cortical locus of relatively high ammonia concentration, compared with renal venous ammonia (0.3 mM) (4).

Fig. 7.

End-luminal cortical AHL (NKCC2 B-isoform) solute concentrations (mM) as a function of interstitial ammonia. Entering conditions and peritubular composition are as in Table 3, except for end-tubular interstitial ammonia, which varies from 0.1 to 2.0 mM; initial luminal NH₄⁺ is fixed at 2.0 mM, and the peritubular interstitial ammonia concentration varies linearly. The panes show end-luminal tubular concentrations and electrical PD, as a function of end-tubular interstitial ammonia. Entering flows are 6 nl/min for a 2-mm tubule.

Using medullary luminal entry concentrations and peritubular solute concentration profiles in Table 3, Fig. 8 displays concentrations of the model AHL for medullary and cortical segments in series; corresponding solute reabsorption rates are listed in Table 4. The boundary of medullary and cortical segments is marked by the luminal depolarization as the NKCC2 isoform shifts from F to B, and which has been described above. Overall, there is reabsorption of 72% of delivered Na⁺ (corresponding to 26% of filtered Na⁺), and 80, 85, and 87% of delivered K⁺, HCO₃⁻, and NH₄⁺. The end-luminal pH is 6.54 with end-luminal HCO₃⁻ 4.1 mM, and these values appear to be approaching an equilibrium. Net acid flow is the sum of titratable acid and ammonium, less HCO₃⁻, so that for this tubule, net acid entry is −126 pmol/min and net acid exit is near zero (1 pmol/min). Several disorders of thick ascending limb, collectively termed Bartter's syndrome, have been identified as molecular defects of AHL transporters: NKCC2 (Bartter's type 1), ROMK (type 2), and peritubular chloride channel function (types 3 and 4). With this model, simulated defects are applied to transporters within both medullary and cortical segments: the type 1 defect is represented as a 90% reduction in NKCC2 density; the type 2 defect is a 90% reduction in luminal membrane K⁺ and NH₄⁺ permeabilities (envisioning a common channel); and the type 3 defect is a 90% reduction in peritubular membrane Cl⁻ and HCO₃⁻ permeabilities. In the case of the conductance defects, reduction to zero function is unlikely in view of redundant transport pathways. Solute transport with each of these defects is summarized in Table 4 and in Fig. 9. For these calculations, as for those of Fig. 8, medullary and cortical segments in series are solved using medullary entry concentrations and peritubular solute profiles in Table 3, and it is assumed that each Bartter defect is present uniformly in each AHL segment. Figure 9 displays end-luminal flows of Na⁺, K⁺, NH₄⁺, and net acid (flow of titratable acid plus NH₄⁺, less HCO₃⁻ flow), for the normal and Bartter tubules. Patterns specific for each abnormality can be identified: natriuresis is greatest with defects in either NKCC or peritubular Cl⁻ permeability. The NKCC defect is also associated with the largest K⁺ and NH₄⁺ outflows, and is the only abnormality associated with an increase in net acid flow. The K⁺ permeability defect shows the mildest natriuresis, along with negligible outflows of K⁺ and NH₄⁺. Intuitively, this is consistent with preservation of luminal membrane NH₄⁺ cycling to catalyze Na⁺ reabsorption. The peritubular Cl⁻ and HCO₃⁻ permeability defect is the only abnormality that results in a decrease in net acid flow, and this derives in part from the increase in end-luminal HCO₃⁻ concentration to 7.2 mM.

Fig. 8.

Solute concentrations and electrical PD along AHL: in vivo conditions, with medullary and cortical segments in series. Entering conditions are those of medullary AHL; peritubular conditions are the concatenation of medullary and cortical segments (Table 3). Entering flow is 6 nl/min for a 4-mm tubule. Solute delivery, segmental reabsorption, and outlet flows are displayed in Table 4.

Table 4.

AHL Solute Reabsorption-In Vivo Simulations

Baseline
	Absolute Reabsorption (pmol/min)
	Deliv.	AHL-M	AHL-C	Total	Exit
Na+	1500.0	595.7	491.0	1087.0	413.3
K+	60.0	32.2	15.5	47.7	12.3
Cl−	1350.0	535.7	470.2	1006.0	344.3
HCO3−	150.0	87.6	39.9	127.5	22.5
TA	1.4	−8.4	−10.6	−19.0	20.4
NH4+	23.0	4.8	15.2	20.0	3.1
Reabsorption Relative to AHL Delivery
Na+		0.397	0.327	0.724
K+		0.537	0.258	0.795
Cl−		0.397	0.348	0.745
HCO3−		0.584	0.266	0.850
TA		−6.087	−7.696	−13.783
NH4+		0.208	0.659	0.867

Bartter type 1–90% NKCC2 reduction
	Absolute Reabsorption (pmol/min)
	Deliv.	AHL-M	AHL-C	Total	Exit
Na+	1500.0	366.4	398.5	764.9	735.1
K+	60.0	32.1	10.6	42.7	17.3
Cl−	1350.0	307.8	360.0	667.8	682.4
HCO3−	150.0	79.3	47.4	126.7	23.3
TA	1.4	−7.3	−12.2	−19.5	20.9
NH4+	23.0	−2.9	11.6	8.7	14.3
Reabsorption Relative to AHL Delivery
Na+		0.244	0.266	0.510
K+		0.535	0.176	0.711
Cl−		0.228	0.267	0.495
HCO3−		0.529	0.316	0.845
TA		−5.304	−8.826	−14.130
NH4+		−0.125	0.503	0.378

Bartter type 2–90% luminal K+ channel reduction
	Absolute Reabsorption (pmol/min)
	Deliv.	AHL-M	AHL-C	Total	Exit
Na+	1500.0	568.7	471.1	1040.	460.0
K+	60.0	43.3	12.7	56.0	4.0
Cl−	1350.0	520.9	446.8	967.6	382.6
HCO3−	150.0	88.4	40.0	128.3	21.7
TA	1.4	−8.6	−10.9	−19.4	20.8
NH4+	23.0	7.8	14.9	21.7	1.4
Reabsorption Relative to AHL Delivery
Na+		0.379	0.314	0.693
K+		0.722	0.212	0.934
Cl−		0.386	0.331	0.717
HCO3−		0.589	0.266	0.856
TA		−6.217	−7.870	−14.087
NH4+		0.294	0.646	0.940

Bartter type 3–90% peritubular CI− channel reduction
	Absolute Reabsorption (pmol/min)
	Deliv.	AHL-M	AHL-C	Total	Exit
Na+	1500.0	354.8	403.3	758.0	742.0
K+	60.0	31.9	16.4	48.3	11.7
Cl−	1350.0	321.5	384.0	705.5	644.7
HCO3−	150.0	65.2	41.4	106.6	43.4
TA	1.4	−5.3	−7.3	−12.6	14.0
NH4+	23.0	6.7	14.3	20.9	2.1
Reabsorption Relative to AHL Delivery
Na+		0.237	0.269	0.505
K+		0.532	0.273	0.805
Cl−		0.238	0.284	0.523
HCO3−		0.434	0.276	0.710
TA		−3.870	−5.261	−9.130
NH4+		0.289	0.620	0.909

AHLM - Medullary AHL; AHLC - Cortical AHL.

Fig. 9.

Using the concatenated medullary and cortical AHL model of Fig. 8, three Bartter defects are simulated: type 1 is a 90% reduction in NKCC2 density; type 2 is a 90% reduction in luminal membrane K⁺ and NH₄⁺ permeabilities; and type 3 is a 90% reduction in peritubular membrane Cl⁻ and HCO₃⁻ permeabilities. What are shown are AHL outlet solute flows of Na⁺, K⁺, NH₄⁺, and net acid (flow of TA plus NH₄⁺, less HCO₃⁻ flow).

The impact of medullary interstitial K⁺ on overall renal K⁺ excretion is best examined in the AHL-DN model of Fig. 1. The abscissa of Fig. 10, A and B, is the 16-mm distance along the nephron from medullary AHL through the IMCD. Entering luminal solute concentrations, and the peritubular concentrations for the component nephron segments are in Table 5. These are the solute concentrations that have been used previously in the model of distal nephron (Table 1 in Ref. 31), with the exception of the increases in NH₄⁺ concentration to 2 mM at the CMJ, and to 1 mM at the corticocortical junction (CCJ), separating cortical AHL and DCT (and connecting segment and CCD). Figure 10A displays luminal PD and the flows of volume, Na⁺, K⁺, and Cl⁻ computed for the full nephron ensemble of a single kidney; Fig. 10B shows the lumen pH and the flows of net acid, HCO₃⁻, titratable acid, and NH₄⁺. The summary of solute delivery, segmental reabsorption, and solute excretion for the simulation of these figures appears in Table 6. Overall Na⁺ reabsorption by this AHL-DN is 95% of the delivered load, so that excreted Na⁺ is 2.9 μmol/min, and just ∼2% of what would be the estimated filtered load. In absolute terms, Na⁺ reabsorption is most prominent in medullary and cortical AHL, 40 and 33% of AHL delivery, and extends into DCT and CNT, 10 and 7% of AHL delivery; there is little transport in CCD and OMCD, but despite reduced transport area, IMCD still reabsorbs over 4% of Na⁺ delivered to AHL. This Na⁺ transport is “exponential” in character in the sense that when expressed as a fraction of segmental delivery, these segments (excluding CCD and OMCD) all reabsorb ∼40% of the Na⁺ that is presented to them. Potassium secretion occurs in DCT, but primarily in CNT, and the amount is about equal to the K⁺ flow delivered to AHL. About two-thirds of K⁺ flowing out of CNT is reabsorbed along the collecting duct segments, and final urine K⁺ excretion is 0.9 μmol/min (∼17% of estimated filtered load). There is a steady decline in lumen pH along AHL, where luminal carbonic anhydrase is present, but sharp acidification steps in DCT and CNT, where luminal carbonic anhydrase is absent. In association with the acid disequilibrium pH, there is a step up in NH₄⁺ flow in DCT. Along OMCD and IMCD, there is reabsorption of three-fourths of entering volume flow, but no apparent change in net acid flow. In this simulation, renal Pco₂ is constant at 50 mmHg, so that by itself, this volume reabsorption without ongoing acidification would have produced substantial alkalinization of tubular fluid. Specifically, this implies that failure to alkalinize is a consequence of proton secretion along the collecting duct. Final urine pH is 6.27; final urine titratable acid and NH₄⁺ are 56 and 39 mM; and net acid excretion is 1.3 μmol/min.

Fig. 10.

Model calculations for the concatenated AHL-distal nephron model of Fig. 1. The abscissa is the 16-mm distance along the nephron from medullary AHL through IMCD. Initial conditions (medullary AHL) and peritubular interstitial conditions are those of Table 5. A: luminal PD and flows of volume, Na⁺, K⁺, and Cl⁻ computed for the full nephron ensemble of a single kidney. B: lumen pH and the flows of net acid, HCO₃⁻, TA, and NH₄⁺. The summary of solute delivery, segmental reabsorption, and solute excretion for the simulation of these figures appears in Table 6.

Table 5.

Entering AHL and Peritubular Concentrations in Distal Nephron Calculations

	AHL Lumen	Papilla	OIMJ	CMJ	CCJ	Cortex
Na+ (mM)	250.00	284.00	284.00	144.00	144.00	144.00
K+	10.00	20.00	10.00	5.00	5.00	5.00
Cl−	225.04	280.92	265.94	121.37	120.38	119.60
HCO3−	25.00	25.00	25.00	25.00	25.00	25.00
H2CO3	4.41 × 10−3	4.41 × 10−3	4.41 × 10−3	4.41 × 10−3	4.41 × 10−3	4.41 × 10−3
CO2	1.50	1.50	1.50	1.50	1.50	1.50
HPO4−	6.00	3.00	3.00	2.00	2.00	2.00
H2PO4−	1.80	0.90	0.90	0.60	0.60	0.60
Urea	10.00	500.00	20.00	5.00	5.00	5.00
NH3	57.3 × 10−3	131.4 × 10−3	57.3 × 10−3	29.4 × 10−3	14.7 × 10−3	2.9 × 10−3
NH4+	3.84	8.82	3.84	1.97	0.99	0.20
Imper.	0.00	2.00	2.00	2.00	2.00	2.00

Papilla - IMCD outlet. OIMJ - Outer-Inner Medullary Junction, separating IMCD-OMCD, and AHLM inlet. CMJ - Cortico-Medullary Junction, separating AHLM-AHLC and CCD-OMCD. CCJ - Cortico-Cortical Junction, separating AHLC-DCT, and CNT-CCD. Cortex - Separating DCT-CNT.

Table 6.

AHL-Distal Nephron Solute Reabsorption

Baseline
	Absolute Reabsorption (μmol/min)
	Deliv.	AHL-M	AHL-C	DCT	CNT	CCD	OMCD	IMCD	Total	Excr.
Na+	54.00	21.44	17.68	5.62	3.95	0.27	−0.09	2.26	51.13	2.87
K+	2.16	1.16	0.56	−0.35	−1.79	0.23	0.90	0.55	1.26	0.90
Cl−	48.61	19.29	16.93	4.26	2.00	0.58	0.58	2.44	46.07	2.54
HCO3−	5.40	3.15	1.44	0.37	−0.05	0.13	0.14	0.17	5.35	0.05
TA	0.05	−0.30	−0.38	−0.25	−0.14	0.31	0.04	−0.05	−0.77	0.82
NH4+	0.83	0.17	0.55	−0.38	0.02	−0.06	−0.05	0.02	0.26	0.57
Reabsorption Relative to AHL Delivery
Na+		0.397	0.327	0.104	0.073	0.005	−0.002	0.042	0.947
K+		0.537	0.259	−0.164	−0.830	0.107	0.417	0.257	0.583
Cl−		0.397	0.348	0.088	0.041	0.012	0.012	0.050	0.948
HCO3−		0.584	0.266	0.068	−0.009	0.024	0.027	0.031	0.991
TA		−6.012	−7.615	−4.891	−2.843	6.214	0.862	−0.937	−15.222
NH4+		0.208	0.659	−0.456	0.026	−0.075	−0.065	0.018	0.314

High Medullary K+ Concentration
	Absolute Reabsorption (μmol/min)
	Deliv.	AHL-M	AHL-C	DCT	CNT	CCD	OMCD	IMCD	Total	Excr.
Na+	54.00	17.81	16.72	6.60	4.60	0.44	0.16	2.43	48.75	5.25
K+	2.16	0.30	1.40	−0.38	−2.06	0.05	0.59	0.37	0.27	1.89
Cl−	48.61	16.49	16.98	3.86	2.22	0.56	0.52	2.52	43.15	5.46
HCO3−	5.40	1.76	1.40	1.36	0.08	0.16	0.17	0.23	5.16	0.24
TA	0.05	−0.12	−0.17	−0.61	−0.16	0.35	0.07	0.06	−0.58	0.63
NH4+	0.83	0.29	0.46	−0.37	−0.02	−0.08	−0.08	0.01	0.20	0.63
Reabsorption Relative to AHL Delivery
Na+		0.330	0.310	0.122	0.085	0.008	0.003	0.045	0.903
K+		0.139	0.647	−0.175	−0.951	0.023	0.271	0.170	0.124
Cl−		0.339	0.349	0.079	0.046	0.012	0.011	0.052	0.888
HCO3−		0.326	0.260	0.252	0.014	0.031	0.031	0.043	0.956
TA		−2.407	−3.374	−12.173	−3.217	6.992	1.452	1.203	−11.524
NH4+		0.345	0.555	−0.451	−0.021	−0.101	−0.100	0.014	0.240

In Fig. 11, delivery, transport, and excretion by AHL-DN is summarized in bar graph form, with the three panes displaying flows of Na⁺, K⁺, and net acid. The first set of bars in each pane is delivered load to AHL and the last set is renal excretion (flow exiting IMCD). The seven intervening sets of bars represent transport by the model nephron segments, namely, the increments or decrements to delivered load that ultimately sum to excretory flow. Within each set of bars, the first (open) bar represents the baseline conditions that have been plotted in Fig. 10, A and B, in which the cortical-to-medullary K⁺ concentrations are 5 mM (cortex), 5 mM (CMJ), 10 mM (OIMJ), and 20 mM (papillary tip). In the second set of bars (cross-hatch), cortical K⁺ is still 5 mM, but the medullary K⁺ concentrations have been doubled to 10, 20, and 40 mM. The numerical values for these simulations appear in Table 6. Overall, the doubling of medullary K⁺ is natriuretic and diuretic: urinary Na⁺ excretion nearly doubles to 5.2 μmol/min, and urine flow increases from 0.24 to 0.41 μl/min. The natriuresis is due to decreased reabsorption in AHL, although this is blunted by the response of DCT and CNT to increase Na⁺ transport with increased delivery. Increasing interstitial K⁺ concentrations produces a twofold kaliuresis (from 0.9 to 1.9 μmol/min), and the bar graph reveals the contributing loci: an increase in CNT K⁺ secretion, and decreases in OMCD and IMCD K⁺ reabsorption. The decrease in medullary AHL K⁺ reabsorption is nearly balanced by increased cortical AHL reabsorption in response to increased delivery. The 15% increase in CNT K⁺ secretion is due to a 40% increase in CNT Na⁺ delivery; the decreased K⁺ reabsorption in medullary collecting ducts is due to the increase in luminal flow, and this flow effect has been noted previously in the model collecting duct (29). With respect to acidification, the baseline net acid excretion of 1.3 μmol/min is expected to decrease to 1.0 μmol/min. Medullary K⁺ is predicted to decrease AHL proton secretion in both segments, however, within DCT the response of NHE2 to increased Na⁺ delivery compensates for much of this decrease. As noted previously, the effect of flow on collecting duct net acid excretion is minimal, although the urinary composition (HCO₃⁻, titratable acid, and NH₄⁺) changes substantially (29).

Fig. 11.

Delivery, transport, and excretion by AHL-distal nephron of Na⁺, K⁺, and net acid. The first set of bars in each pane is delivered load to AHL and the last set is renal excretion (flow exiting IMCD). The 7 intervening sets of bars represent transport by the model nephron segments: increments or decrements to tubular flow. Within each set of bars, the first (open) bar is baseline conditions that are plotted in Fig. 10, A and B, in which the cortical-to-medullary K⁺ concentrations are 5 mM (cortex), 5 mM (corticomedullary junction), 10 mM (outer-inner medullary junction), and 20 mM (papillary tip). In the second set of bars (cross-hatch), cortical K⁺ is still 5 mM, but medullary K⁺ concentrations have been doubled to 10, 20, and 40 mM.

DISCUSSION

The present work has configured models of medullary and cortical AHL epithelia as tubules for simulation of function in vitro, in which peritubular conditions are fixed, and in vivo, in which there are peritubular concentration gradients. The importance of this step in model development derives from the requirement of matching overall transport rates in a system in which luminal conditions are determined by tubular transport. The only prior AHL model (5) was an epithelial model whose principal focus was local electrophysiology. Parameter selection for the present model (33) was done with an eye toward its function in vivo, in the presence of ambient ammonia, and the sine qua non for a plausible AHL model is matching the rate of Na⁺ reabsorption. In that regard, late proximal micropuncture and early distal micropuncture in the rat have suggested that ∼25% of filtered Na⁺ is reabsorbed in AHL, and this estimate is consistent with direct microperfusion of single loops of Henle, for which Na⁺ reabsorption was 1.5 nmol/min (20). This aspect of fitting the model to rat transport data is straightforward. What is unique about AHL among nephron segments, however, is that lumen K⁺ is catalytic for Na⁺ reabsorption, in the sense that relatively small K⁺ concentrations mediate large Na⁺ fluxes with little change in axial K⁺ flow. This requirement places meaningful constraints on the parameters of K⁺ transport for both luminal and peritubular membranes. Specifically, the luminal membrane must be sufficiently permeable, while the peritubular membrane is sufficiently tight, so that luminal K⁺ is replenished. A novel observation of the present work is that luminal NH₄⁺ may also be catalytic for Na⁺ reabsorption. Quantitatively, this model suggests that this catalytic role for NH₄⁺ may be comparable to that of K⁺.

A catalytic role for NH₄⁺ in AHL Na⁺ reabsorption has not been recognized, and thus needs to be considered with caution. The essentials of this mechanism are rapid uptake of NH₄⁺ with Na⁺ across NKCC2, and rapid secretory return of NH₄⁺ to the luminal fluid. The secretory flux may be direct Na⁺/NH₄⁺ exchange, or diffusive NH₃ exit in parallel with Na⁺/H⁺ exchange, and in either case, the mechanism depends upon sufficient NHE3 activity. With respect to uptake, it is secure that NH₄⁺ can replace K⁺ as a transported cation on NKCC2, with an apparent affinity that is several-fold lower than K⁺ (15). It is also secure that luminal NH₄⁺ entry is furosemide inhibitable, rapid, and acidifies the cell (14, 24). Based on sampling from early DCT, luminal concentrations of K⁺ and NH₄⁺ are comparable, and this has two important implications. The first is that if cytosolic concentration of K⁺ is several-fold higher than that of NH₄⁺, then the driving force for NH₄⁺ entry across NKCC2 will also be several-fold higher; this would increase the rate of NH₄⁺ uptake, perhaps comparable to that of K⁺. The second implication of millimolar concentrations of NH₄⁺ in early DCT is that luminal backflux of NH₄⁺ (or its equivalent) must be substantial. The present model calculations suggest that the reported luminal membrane NH₃ permeability is adequate to sustain much of this backflux. What is required to complete this limb of the argument is documentation that NHE3 activity in AHL is sufficient to sustain this flux.

Estimates of AHL proton secretion derive from both in vitro perfusions and microperfusion of Henle limbs of superficial nephrons. In rat AHL in vitro, Watts and Good (23) found that NHE3 proton efflux increased with cell acidification, and plateaued at 70 pmol·mm⁻¹·min⁻¹, when cell pH was <7.0. This rate of proton efflux is fivefold higher than the rate of HCO₃⁻ reabsorption by rat AHL in vitro (10). This functional dependence of AHL NHE3 on cell pH was different from that described for proximal tubule NHE3 in vesicles, in which increasing turnover persisted as cytosolic pH was reduced down to 6.5 (1). In mouse AHL tubule suspensions, pH recovery from acid challenge did accelerate down through pH 6.5 (13). In the NHE3 of this paper, the functional character of NHE3 is that of proximal tubule, so that increased proton efflux accompanies luminal addition of NH₄⁺ into the millimolar range. In vivo microperfusion of Henle loops of superficial rat nephrons reveals HCO₃⁻ reabsorption that is load dependent: perfusion with a physiological HCO₃⁻ load produced reabsorption of 146 pmol/min (about two-thirds of delivery), and with increasing delivery, reabsorption appeared to plateau at 600 pmol/min (3). In this same study, under baseline perfusion conditions, three-fourths of HCO₃⁻ reabsorption was inhibited by ethylisopropyl amiloride. These observations imply considerably greater luminal proton secretion by AHL in vivo. Of note, all Henle loop microperfusions seem to have been done with artificial late proximal solutions, that contained no ammonia.

A direct consequence of rapid uptake and secretion of NH₄⁺ is rapid adjustment of luminal ammonia concentration to changes in peritubular composition. This was observed in the model calculations (Fig. 7), and it was found that no solution to the tubule model would be compatible with millimolar concentrations of luminal NH₄⁺ in early DCT, unless there was at least a millimolar concentration of NH₄⁺ in the peritubular environment of the tubule outlet. To understand this, consider a solution flowing at rate F_v (ml/s) through a cylinder of diameter, D (cm), and containing a solute at concentration, C, which can exit the cylinder according to its permeability, h (cm/s). The resulting concentration profile is exponential as a function of distance, with a length constant, λ (cm),

$$\lambda =F_{\mathrm{v}}/(\pi Dh)$$ (1)

Thus for slow flows or high permeabilities, the lengths are small and adjustments to peritubular conditions impact on the flow profile. For AHL, with flows of 0.1 nl/s and diameter 20 μm, F_v/πD = 1.6 × 10⁻⁵ cm²/s, and this is the number to be compared with h. For NH₄⁺, the tubule permeability is 6.0 × 10⁻⁵ cm/s (8), giving a long length constant of 2.7 mm. However, with respect to NH₄⁺ uptake by NKCC2, Fig. 6 of the companion manuscript (33) shows the response of the F-isoform to changes in luminal NH₄⁺, and maximal flux (100 pmol·mm⁻¹·min⁻¹, or 2.6 nmol·s⁻¹·cm⁻²) is achieved by luminal NH₄⁺ = 2 mM; for the B-isoform (not shown), this same maximal flux is achieved by a luminal concentration of 1 mM. Over this early portion of the flux/concentration curve therefore, an effective NH₄⁺ permeability is h = 2.6 × 10⁻³ cm/s. For such a permeability, the length constant is 60 μm. For NH₃, the (membrane area-adjusted) permeability is 0.03 cm/s, so that the length constant is an order of magnitude smaller. Thus this model requires that there be regions within the cortical labyrinth in which ambient NH₄⁺ is several-fold higher than in renal venous plasma. Considering that concentrations of NH₄⁺ of 2 mM are found in late proximal tubule (2, 12), this may not be unreasonable. With its focus on luminal NH₃ entry, this model also addresses an unexplained observation of Levine et al. (17a), who microperfused loops of Henle with isotonic saline and found nearly constant HCO₃⁻ concentrations in the collected fluid of the early distal tubule (∼6 mM), over the range of perfusion rates from 10–30 nl/min. They understood that tubule HCO₃⁻ permeability was insufficient for luminal equilibration, but did not consider the possibility that luminal entry of NH₃ and CO₂ would be rapid, and could provide the measured HCO₃⁻.

The model simulations of this paper have included the molecular defects that have been identified in Bartter's syndrome: defects in NKCC2, in luminal membrane K⁺ permeability, and in peritubular Cl⁻ permeability. The isolated decrease in NKCC2 could be realistically considered acute application of a loop diuretic, and what was found is intuitive: near doubling of delivered Na⁺, and increases in delivery of K⁺ and net acid to DCT (Fig. 9). Looking more closely (Table 4), the net acid increase was exclusively an increase in NH₄⁺, with no change in HCO₃⁻ or titratable acid. Blunting luminal Na⁺ entry via NKCC2 would by itself be expected to decrease cytosolic Na⁺, and thus increase NHE3 activity. However, decreasing NH₄⁺ entry via NKCC2 acts to decrease cytosolic acidification, and thus diminish NHE proton secretion. In the model, these two effects balanced. Where this has been examined in rat micropuncture, both furosemide and piretanide produced increases in net acid delivery to early DCT, but this was in the form of reduced luminal pH (from 6.5 to 6.0) along with increased titratable acid; there was no change in luminal ammonia concentration (11). However, in the isolated perfused loop of Henle, application of luminal furosemide produced no change in HCO₃⁻ reabsorption, while bumetanide gave only a small increase (3). In the model, an isolated decrease in luminal K⁺ permeability gave only a modest increase in distal Na⁺ delivery, while predictably eliminating K⁺ delivery; a small increase in ammonia reabsorption was balanced by a similar increase in HCO₃⁻ reabsorption, so that there was little impact on net acid delivery. Acute K⁺ channel blockade has been examined in the perfused loop of Henle, using either glyburide (21) or a K_ATP channel blocker (22). The findings for either blocker were ∼50% reduction in AHL Na⁺ reabsorption, and an even greater reduction in K⁺ reabsorption, so that the K⁺ concentration of early DCT fluid actually increased. These findings are certainly discordant with the present model. It may be possible to argue that in the model, the supply of luminal ammonia masks the natriuretic effect of K⁺ channel blockade, but the effect on K⁺ reabsorption is more problematic. Indeed, the authors (21) noted the need to argue for a separate inhibitory effect of glyburide on NKCC2, beyond diminishing luminal K⁺ availability, since luminal K⁺ was not decreased. They proposed that blocking luminal K⁺ channels might depolarize the cell, thus increasing cytosolic Cl⁻, and thus diminish NKCC function. That mechanism is not supported by this model. With respect to the natriuresis of the type 3 Bartter defect, one notable feature of the model was the decrease in luminal acidification by the AHL. This can be attributed, at least in part, to the HCO₃⁻ permeability assigned to the peritubular Cl⁻ channel. Although a number of epithelial Cl⁻ channels do show substantial HCO₃⁻ permeability (e.g., Ref. 17), the AHL Cl⁻ channel associated with the Bartter defect, CLC-Kb (16), does not appear to have been characterized with respect to HCO₃⁻ permeation. As regards the overall phenotype of type 3 Bartter's, enhanced distal Na⁺ delivery would be expected to increase urinary acidification, so that a decrease in AHL proton secretion may not be appreciated in the final urine.

One of the principal aims of this modeling effort was to examine the proposal that medullary interstitial K⁺ concentration functioned within a positive feedback loop to increase distal Na⁺ delivery, and thus amplify cortical K⁺ secretion (18). The model captures this, so that with a doubling of medullary K⁺, there is a 32% increase in distal Na⁺ delivery, and a near doubling of Na⁺ excretion. In turn, K⁺ secretion is enhanced in CNT and collecting duct K⁺ reabsorption is blunted by the increase in luminal volume flow. Of note, AHL epithelial model calculations failed to predict decreased AHL Na⁺ reabsorption, because the positive transepithelial PD increased paracellular Na⁺ flux. That compensatory mechanism was not as powerful in the tubular model, in which over distance, luminal composition was altered (e.g., decreased medullary AHL K⁺ reabsorption) so as to blunt the increase in the transepithelial PD. Overall, the effect of medullary interstitial K⁺ on net acid excretion was a small reduction, although in AHL the decrease in acid secretion was substantial (Fig. 11). This stands in contrast to a pure decrease in NKCC2 function, which tends to increase AHL acidification. What appears to be happening with peritubular K⁺ is more akin to the type 3 Bartter's pathophysiology, shown in Fig. 9. In that case, there is decreased peritubular HCO₃⁻ permeability, cytosolic alkalinization, and a consequent decrease in luminal proton secretion by NHE3. With high peritubular K⁺, peritubular membrane depolarization acts to blunt HCO₃⁻ exit, and thus impair luminal acidification.

In sum, the transformation of the AHL epithelial models to an AHL tubule is necessary to assess the impact of transport perturbations on luminal composition. One source of difficulty for modeling this particular segment is the limited data set for studies in vivo, and the evident discrepancy between fluxes determined in vitro and those derived from combined proximal and distal micropuncture, or Henle limb perfusions. When the most secure data are early distal tubule concentrations and flows, they give the modeling the character of a boundary value problem, in which the initial conditions must be back-calculated. The added uncertainty here is that the peritubular solute profiles can only be estimated. A second source of uncertainty are NKCC2 kinetics, especially with respect to ammonia transport. In this model, ammonia took on a much more prominent place than anticipated: if there is brisk ammonia uptake on NKCC2, then, by itself, it will produce prompt luminal ammonia depletion. Since luminal ammonia is not depleted, there must be rapid backflux, either as NH₄⁺ or NH₃, and in either case a prominent role for NHE3. This cycle of uptake and secretion renders ammonia catalytic for AHL Na⁺ reabsorption, in parallel with the luminal membrane cycling of K⁺. Since these are two redundant and overlapping systems, their relative importance will certainly be difficult to dissect experimentally. One gap in our database that might be most straightforward to address is to measure early distal ammonia concentrations in Henle limb perfusion experiments. When this is done with varying luminal perfusion rates, then one may be able to document that ammonia secretion is sufficiently rapid to sustain its proposed role in AHL Na⁺ transport.

DISCLOSURES

No conflicts of interest are declared by the author.