INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

IN OUR PREVIOUS PAPER (22), we developed a numerical model of the rat distal tubule to help understand the complex transport phenomena that had been observed in this nephron segment. In that study, we concentrated on transport of Na⁺, K⁺, Cl, and water, because the magnitude of transport flux ascribable to these molecules overwhelms that of other molecules (such as H⁺, HCO, Ca²⁺, Mg²⁺, ammonium, and phosphate). The distal tubule, however, actively participates in acidification of the tubular fluid, and normally it reabsorbs 5-10% of filtered HCO, an amount equal to or greater than that assigned to the cortical and medullary collecting duct (16). In this paper, we extend our previous model to deal with acid-base transport in the rat distal tubule.

Acid-base transport in the rat distal tubule has been extensively studied by free-flow micropuncture and in vivo microperfusion experiments. These experiments have shown that there is an axial heterogeneity in the mechanism underlying transepithelial HCO transport. In the early part of the distal tubule (that is, the distal convoluted tubule), H⁺ is secreted into the tubular fluid via an Na/H exchanger located in the luminal membrane. Closely linked with this process, HCO is transported out of the cytosolic space to the basolateral space, probably via an anion exchanger. In the late part, which is composed of the connecting tubule and the initial collecting tubule, there are distinct types of tubular cells (called intercalated cells) that are specifically involved in acid-base transport. One type of cell (type A intercalated cell) is involved in transepithelial HCO reabsorption by secreting H⁺ via luminal H-ATPase and extruding intracellular HCO via the basolateral anion exchanger. Another type of cell (type B intercalated cell) has these transporters on the opposite side and secretes HCO into the tubular fluid. These features of acid-base transport in the distal tubule are represented in the present model by introduction of new cell types that correspond to intercalated cells and new models of transporters that simulate transport kinetics of the Na/H exchanger, H-ATPase, and the anion exchanger. With the aid of a computer program (21) that solves steady-state equations of transitional state diagrams, we have been able to simulate transport kinetics of these transporters in a consistent manner.

As we did in the previous model, we adjust model parameters so that model predictions simulate the results of micropuncture experiments, because these experiments are the least invasive and yield mutually consistent results. Specifically, we try to fit the model to the results by Capasso et al. (16, 17): an HCO reabsorption rate of ~50 pmol/min in normal rats and ~180 pmol/min in acutely HCO-loaded rats. Additionally, we compare the model with in vivo microperfusion experiments. These experiments yield widely varied and mutually inconsistent results. For example, the HCO reabsorption rate in normal rats was essentially equal to zero in several reports (46, 56, 57), whereas it was comparable to micropuncture experiments in other reports (20, 54). The mechanism underlying the inconsistency has not been resolved and is possibly multifactorial, and we will make only a limited attempt to fit the model to microperfusion experiments.

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

Model geometry and variables. The model tubule has the same diameters (inner, 24 µm; outer, 37 µm) and length (0.23 cm) as in the previous model (22). Reflecting the axial heterogeneity of the actual tubule, the model is divided into two parts. The early (or upstream) part is 0.10 cm long and corresponds to the distal convoluted tubule (53). This portion is composed of a single type of cell (distal convoluted tubule cell). Accordingly, the model has only one cell type. The late part is 0.13 cm long and corresponds to the connecting tubule and the initial collecting tubule (53). This portion is composed of heterogeneous cell types. The predominant number of cells¹ is involved in Na⁺ and K⁺ transport and had been the only cell type incorporated in the previous model. Other cell types are type A intercalated cells and type B intercalated cells, which are involved in H⁺ secretion and HCO secretion, respectively. The present model incorporates both intercalated cell types. To derive the discretized form of the system of model equations that is suitable for numerical solution, the model tubule has been conceptually divided into 23 sections with equal widths (0.01 cm).

Model equations. In the cellular compartments, mass conservation of water or solute is specified as

where J is the volume flux from the luminal space into the cell, J is the volume flux from the serosal space into the cell, J is the flux of solute x from the luminal space into the cell, and J is the flux of solute x from serosal space into the cell. The iteration procedure of the Newton method (22), which has been used to solve the system of model equations, is continued until the absolute magnitude of the difference between the left-hand side and the right-hand side of each model equation becomes smaller than the tolerance value that is predetermined for each equation. The tolerance values for the above equations have been 1.42 × 10⁷ ml · s¹ · cm² and 1.42 × 10¹⁰ mmol · s¹ · cm². These values have been selected to ensure that the sum of errors of all sections is, at most, 0.023 nl/min and 0.023 pmol/min. Conservation of solutes that constitute buffer systems is handled differently, as described later.

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Transporters

View larger version (19K): [in this window] [in a new window]   Fig. 1.   Transport mechanisms of the early distal tubule. Distal convoluted tubule cell is represented as a large rectangle. In the luminal cell membrane, there are (from top to bottom) thiazide-sensitive Na-Cl cotransporter, Na/H exchanger, K-Cl cotransporter, Na+ channel, K+ channel (also permeable for NH), Clchannel, permeability for H2CO3 (not listed), permeability for NH3, and H-ATPase. In the basolateral cell membrane, there are Na-K-ATPase, anion exchanger, K+ channel (also permeable for NH), Cl channel, and permeability for NH3. In the tight junction (paracellular pathway), there are conductances for Na+, K+, NH, Cl, and HCO. Alongside of circles indicating individual transport mechanisms, transport velocities in the basic state are expressed as pmol/min. , Electrical potential.

View larger version (22K): [in this window] [in a new window]   Fig. 2.   Transport mechanisms of the late distal tubule. From the top of the figure, principal cell, type A intercalated cell, and type B intercalated cell are represented by rectangles. In the luminal cell membrane of principal cells, there are (from top to bottom) thiazide-sensitive Na-Cl cotransporter, K-Cl cotransporter, Na+ channel, K+ channel (also permeable for NH), Cl channel, permeability for H2CO3 (not listed), and permeability for NH3. In the basolateral side of principal cells, there are Na-K-ATPase, Na/H exchanger, anion exchanger, K+ channel (also permeable for NH), Cl channel, and permeability for NH3. In the tight junction, there are conductances for Na+, K+, NH, Cl, and HCO. In type A intercalated cells, there is H-ATPase on the luminal side; and there are anion exchanger and Cl channel on the basolateral side. In type B intercalated cells, there is anion exchanger on the luminal side; and there are H-ATPase and Cl channel on the basolateral side. Alongside of circles indicating individual transport mechanisms, transport velocities in the basic state are expressed as pmol/min.

Transport velocities via the K-Cl cotransporter, ion channels, and paracellular conductances have been calculated as in the previous model (22). Briefly, transport velocity via the K-Cl cotransporter has been calculated by an equation derived from a kinetic diagram that accounts for apparent dissociation constants for K⁺ and Cl (38); and transport velocities via ion channels and paracellular conductances have been calculated by the Goldman-Hodgkin-Katz current equation (41). In the present model, we have assumed that the K⁺ channel is also permeable for NH, because the ROMK channel, which is the native K⁺ channel in principal cells of the cortical collecting duct, has been demonstrated to be permeable for NH (66). On the basis of the single-channel conductances and channel open probabilities for K⁺ and NH, we have assumed that the magnitude of NH permeability, P_NH4, is 20% of K⁺ permeability, P_K (66, 78). We have also assumed that NH is permeable through the tight junction with permeability similar to that for other cations. Transport rates of water across cell membranes have been assumed to be proportional to the difference in osmolality across the membrane.

Calculation of transport velocity via the Na-Cl cotransporter [thiazide-sensitive Na-Cl cotransporter (TSC)] has been extended from the previous model to simulate a wider range of experimental observations as described in Chang and Fujita (21). The extended model simulates those experimental data of TSC, such as binding of thiazide in the absence of substrates, inhibitory effect of Cl on thiazide binding, stimulatory effect of Na⁺ on thiazide binding, combined effects of Na⁺ and Cl on thiazide binding, dependence of Na⁺ influx on extracellular Na⁺ and Cl, and inhibition of Na⁺ influx by extracellular thiazide (21). To be consistent with the previous model parameters, transport velocity via TSC (J_TSC) has been represented in the following form
where J_TSC, max (mmol · s¹ · cm²) is a model parameter, j_TSC is the transport velocity via a single TSC molecule, and j_TSC, max is the maximal transport velocity via a single TSC molecule. In this way, model parameter J_TSC, max represents the maximal transport rate that is achievable via the TSC transport mechanism as in the previous presentation. This convention has also been followed by other transporters that have been newly introduced.

Calculation of transport velocities via Na-K-ATPase has been extended to include NH transport (75). Specifically, transport velocities of NH (J_NH4-ATPase) and K⁺ (J_K-ATPase) have been calculated from the following equations (78)

where J_a is the rate of ATP hydrolysis, C and C are basolateral K⁺ and NH concentrations, and K_K and K_NH4 are kinetic constants with the ratio (K_NH4/K_K) of 5.8 (75). With typical values of basolateral K⁺ concentration (4.25 mM) and NH concentration (0.068 mM; Ref. 44), J_NH4-ATPase/ J_K, ATPase becomes 0.0028. The rate of ATP hydrolysis (J_a) and transport velocity of Na⁺ (J_Na-ATPase) have been calculated as before (22, 83)

where J_a, max is a model parameter, C is cytosolic Na⁺ concentration, and K_Na-ATPase is a kinetic constant with a value of 12 mM.

Transport velocities via NH₃ permeability and H₂CO₃ permeability have been assumed to be proportional to concentration differences

Transport via the Na/H exchanger has been assumed to obey a kinetic diagram (Fig. 3 and Table 1) that is based on the one elaborated by Weinstein (77). In this diagram, the Na/H exchanger has a single binding site to which Na⁺, H⁺, and NH competitively bind, and only the bound forms of the transporter are able to cross the membrane. One noticeable feature of this model is that transitional rate constants are symmetrical with respect to the membrane (for example, k₁ = k₇, k₂ = k₈, and k₁₃ = k₁₄; Fig. 3 and Table 1). Besides the benefit of decreasing the number of independent parameters, this feature ensures that thermodynamic requirements (55) such as k₁k₄k₈k₉k₁₃k₁₆ = k₂k₃k₇k₁₀k₁₄k₁₅ are automatically fulfilled. Another feature (described in Table 1) is that rates of translocation (from k₁₃ through k₁₈) are affected by cytosolic H⁺ concentration (for quantitative description, see the legend for Table 1). This aspect of the model (internal modifier site) was necessary (77) to account for the observed complex effect of internal H⁺ on net transport, with intracellular alkalosis shutting off Na/H exchange more sharply than a simple substrate depletion effect (4).

View larger version (14K): [in this window] [in a new window]   Fig. 3.   State diagram of Na-H exchanger. The model Na-H exchanger (E) has a single binding site to which Na+, H+, and NH bind competitively. Only loaded transporters (ENa, EH and ENH4 in the extracellular side; and ENa*, EH*, and ENH4* in the intracellular side) can cross the membrane. Bracketed symbols (such as [H] and [Na*]) indicate substrate concentrations.

Transport velocity via the Na/H exchanger can be calculated by numerically solving the steady-state equations of this diagram by using the program that we had developed (21). With an internal pH of 6.0 (and no internal Na⁺), Na⁺ influx of the model Na/H exchanger has an apparent K_Na of 58.8 mM when external pH is 6.6 and 11.9 mM when external pH is 7.5, whereas the corresponding experimental values are 54 and 13 mM (5), respectively. The inhibition constant (K_i) for inhibition of Na⁺ influx by external H⁺ (external Na⁺ 0.1 mM; internal Na⁺ 0 mM; and internal pH 6.0) is 38 nM, and the corresponding experimental value is 35 nM (5). K_i for inhibition of Na⁺ influx by external NH (external Na⁺ 0.1 mM; internal Na⁺ 0 mM; and internal pH 6.0) is 50 mM when external pH is 6.6 and 11 mM when external pH is 7.5, both of which are identical to experimental values (5).

The kinetic model of H-ATPase is from Andersen et al. (3). This model had been developed to explain the relationship between transport velocity via H-ATPase and luminal pH in the turtle bladder (see Fig. 5 in Ref. 3). According to this model, H-ATPase consists of two components (Fig. 4): a catalytic unit at the cytoplasmic side that mediates the ATP-driven H⁺ translocation, and a transmembrane channel that mediates the transfer of H⁺ from the catalytic unit to the extracellular solution. Between these two compartments there exists a buffer compartment (antechamber; Fig. 4), in which H⁺ is nearly in equilibrium with extracellular H⁺. The catalytic unit has two binding sites for H⁺, and only the fully loaded form can translocate H⁺ from the cytosolic space to the antechamber (Fig. 5). Therefore, stoichiometry is strictly 2H⁺:1ATP.

View larger version (15K): [in this window] [in a new window]   Fig. 4.   Conceptual diagram of H-ATPase. The transporter consists of 2 components: a membrane channel and a catalytic unit. Between these components, there is a buffer space (antechamber), in which hydrogen ion (Ha) is essentially in equilibrium with extracellular hydrogen ion (H) owing to a large conductance of the membrane channel. Hydrogen ion in the antechamber is also interchangeable with cytosolic hydrogen ion (H*) through the catalytic unit. This process is coupled with ATP hydrolysis/synthesis with a stoichiometry of 2H+:1ATP.

View larger version (10K): [in this window] [in a new window]   Fig. 5.   State diagram of the catalytic unit of H-ATPase. The catalytic unit (E) has 2 binding sites for H. Symbols with asterisk (*), such as EH*, indicate conformations of the catalytic unit in which binding sites face the cytosolic space, and symbols without asterisk (e.g., EH) indicate conformations in which binding sites face the antechamber. Transition between the unloaded conformations (EE*) is coupled with ATP hydrolysis/synthesis. The label [Ha] indicates H+ concentration in the antechamber, and other bracketed labels, such as [ATP*] and [H*], indicate substrate concentrations in the cytosolic space. H+ in the antechamber is assumed to be in equilibrium with extracellular H+, and electrical potential of the antechamber is assumed to be equal to that of cytosolic space. Therefore where [H]o is extracellular H+ concentration, and o and i are electrical potentials of the extracellular space and the intracellular space, respectively.

Values of rate constants for the kinetic model of H-ATPase are listed in Table 2, which are essentially identical to the ones reported by Andersen et al. (3). These values fulfill the thermodynamic requirement
where G is the standard free energy change of ATP hydrolysis (33 kJ/mol). We have assumed that cytosolic ATP concentration ([ATP]_c) = 2.5 × 10³ M, and [ADP]_c · [P_i]_c = 10⁷ M² (3). Transport velocity through H-ATPase has been calculated by solving the steady-state equations of the kinetic diagram. Transport velocity of the model H-ATPase is plotted in Fig. 6 as a function of luminal pH. The continuous line represents the calculation that simulates the experiments by Andersen et al. (3). The model prediction fits well with the experimental data ( in Fig. 6). Transport velocity becomes half-maximal near the luminal pH of 6.0, and it becomes essentially undetectable at a luminal pH of 4 (0.5% of the maximal rate). At a luminal pH of 3.2, H⁺ transport reverses its direction.⁶ In Fig. 6, transport velocities under the conditions simulating distal convoluted tubule cells and type A intercalated cells are also shown (dashed lines).

View larger version (10K): [in this window] [in a new window]   Fig. 6.   Transport velocity of the model H-ATPase. Transport velocity of the model H-ATPase is plotted as a function of luminal pH. The solid continuous line is calculated with the conditions (cytosolic pH 7.5; luminal potential 0 mV; and cytosolic potential 30 mV) that simulate experiments by Andersen et al. (3). Solid circles represent corresponding experimental values (3). Transport velocity is normalized by the transport velocity with luminal pH of 9.0 (J). The dashed line represents a calculation with the conditions simulating distal convoluted tubule cells (cytosolic pH 7.0; luminal potential 0 mV; and cytosolic potential 90 mV). The dashed-dotted line represents a calculation with the conditions simulating type A intercalated cells (cytosolic pH 7.4; luminal potential 17 mV; and cytosolic potential 26 mV). JH, rate of H+ transport via H-ATPase.

A state diagram for the anion exchanger, which catalyzes one-for-one exchange of anions such as Cl and HCO, is illustrated in Fig. 7. In this diagram, termed the "ping-pong" mechanism by Gunn and Frölich (43), the anion exchanger has a single binding site (transport site) to which substrates (Cl and HCO) competitively bind, and only loaded transporters can cross the membrane. Additionally, there is an internal modifier site to which cytosolic Cl and HCO competitively bind. Binding to this modifier site is independent of the state of the transport site, and the anion exchanger with the modifier site occupied cannot participate in ion transport. Therefore, the transport rate is decreased by a factor of (1+[Cl*]/K + [HCO^*₃]/K)¹, where [Cl*] is the cytosolic Cl concentration, [HCO^*₃] is the cytosolic HCO concentration, K is the dissociation constant of the modifier site for Cl, and K is the dissociation constant for HCO.

View larger version (10K): [in this window] [in a new window]   Fig. 7.   State diagram of anion exchanger. The model transporter (E) has a single binding site to which Cl and HCO competitively bind. Only loaded transporters (ECl and EHCO3 in the extracellular side; and ECl* and EHCO3* in the intracellular side) can cross the membrane. Brackets indicate substrate concentrations.

Values of rate constants for anion exchanger are listed in Table 3. We have assumed that rate constants involved in substrate binding (k₁, k₃, k₅, and k₇) are diffusion limited (45). We have also assumed that affinities of the transport site for Cl and HCO are symmetrical with respect to the cell membrane, according to the report by Liu et al. (59). Other rate constants have been optimized by Powell's method (69) to fit the experimental results of Gasbjerg et al. (36) and Knauf et al. (49), who investigated the kinetics of the anion exchanger in human red blood cells at body temperature (38°C). We note that rate constants for Cl dissociation (k₂ and k₆) are consistent with the lower limits of these values determined by ³⁵Cl NMR (4.5 × 10⁵ s¹ for k₂ and 1.3 × 10⁵ s¹ for k₆; Ref. 31 ). Model predictions with these rate constants, together with experimental data, are plotted in Figs. 8 and 9. The model anion exchanger simulates dependency of transport velocity on extracellular and intracellular HCO concentrations (Fig. 8) and dependency of transport velocity on extracellular and intracellular Cl concentrations (Fig. 9). A similar kinetic model of the anion exchanger that also fits well to these experimental results has been recently reported by Weinstein (79).

View larger version (15K): [in this window] [in a new window]   Fig. 8.   Transport velocity of the model anion exchanger: HCO dependency. Transport velocity of the model anion exchanger is calculated as a function of HCO concentration and is plotted as a solid line. Experimental data (36) are plotted together as solid circles. A: unidirectional HCO efflux (Jeff) as extracellular (o) HCO concentration is varied from 0 to 54 mM. Internal (i) HCO concentration is 50 mM. There are no Cl ions. B: unidirectional HCO efflux as extracellular HCO is varied from 0 to 250 mM. Internal HCO concentration is 165 mM. There are no Cl ions. C: unidirectional HCO efflux as internal HCO concentration is varied from 0 to 640 mM. External HCO concentration is 50 mM. There are no Cl ions. D: HCO exchange flux (Jeff) as extracellular and intracellular HCO concentrations are varied simultaneously (no HCO gradient) from 0 to 640 mM. There are no Cl ions. Experimental data are from Fig. 5 in Ref. 36.

View larger version (13K): [in this window] [in a new window]   Fig. 9.   Transport velocity of the model anion exchanger: Cl dependency. Transport velocity of the model anion exchanger is calculated as a function of Cl concentration and is plotted as a solid line. Experimental data (49) are plotted together as solid circles. A: unidirectional Cl efflux as extracellular Cl concentration is varied from 0 to 160 mM. Internal Cl concentration is 105.4 mM. There are no HCO ions. B: Cl exchange flux as extracellular and intracellular Cl concentrations are varied simultaneously (no Cl gradient) from 0 to 650 mM. There are no HCO ions. Experimental data are representative points of Figs. 1 and 4 in Ref. 49.

From Table 3, we can see that loaded transporters translocate faster to the intracellular side than to the extracellular side (that is, k₉ > k₁₀ and k₁₁ > k₁₂). This indicates that the anion exchanger is more stable in the intracellular side than in the extracellular side when substrate concentrations are equal on both sides of the membrane. Quantitatively, this is represented by the asymmetry factor (A) that is defined as the ratio of unloaded outward-facing sites to unloaded inward-facing sites, [E]/[E*], with equal concentrations of substrates. According to the ping-pong model (as is the present model), A should be the same regardless of concentrations and species of the substrate that are used to measure it, because A reflects the free energy difference (G) between the unloaded inward- and outward-facing forms (E and E*). A of the model exchanger is 0.18, which is within the range of experimental estimations (0.03-0.37; Refs. 36 and 49). Molar Gibbs free energy change (G) of the transition between the two unloaded forms is estimated from A (G = RT log A) to be 4.4 kJ/mol.

Parameter assignment. Values of model parameters are listed in Table 4. Values directly related to acid-base transport are as follows. Permeabilities of NH₃ are 0.0113 cm/s and 0.0036 cm/s in the luminal and the basolateral membrane, respectively. These values are from the measurement in the rabbit cortical collecting duct (82). Permeability of H₂CO₃ through luminal membranes is 1.28 × 10³ cm/s, which is estimated from the apical membrane formic acid permeability (4.6 × 10² cm/s; Ref. 68) in the rat proximal tubule and its 36-fold amplification of the luminal surface area by microvilli (52). Permeabilities of Cl via basolateral conductances in type A and type B intercalated cells are 7.40 × 10⁶ and 1.80 × 10⁵ cm/s, respectively. These values are from the basolateral conductances of the intercalated cells in the model of cortical collecting duct by Strieter et al. (73).

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	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

Comparison with free-flow micropuncture experiments. We first examine whether incorporation of the new cell types and transporters has not affected the goodness of fit of the previous model. Model predictions of transport of Na⁺, K⁺, Cl