INTRODUCTION

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The electrogenic Na-HCO₃ cotransporter comprises the main exit pathway for HCO₃ across the basolateral membrane in the proximal tubule (Yoshitomi and Fromter, 1984; Burckhardt et al., 1984; Yoshitomi et al., 1985; Biagi, 1985; Sasaki et al., 1985; Alpern, 1985; Jentsch et al., 1985, 1986; Biagi and Sohtell, 1986; Alpern and Chambers, 1986; Jentsch et al., 1986; Akiba et al., 1986; Soleimani et al., 1987; Lopes et al., 1987). Cotransporter activity results in the net reabsorption of HCO₃ from the tubule lumen to the blood. Both Na⁺ and HCO₃ are cotransported across the basolateral membrane of the proximal tubular cell against their respective concentration gradients. The driving force for the process is the (inside negative) membrane potential. Thus any change in membrane potential will also affect the turnover of the cotransporter. Thermodynamically, the electrical potential difference is like a driving force fully exchangeable with an equivalent chemical potential difference. This is not necessarily so for the rates of cotransport. The effectiveness of the membrane potential as a driving force depends on molecular details, such as which steps in the transport cycle are voltage dependent (binding or dissociation of the substrates, translocation of the loaded and unloaded carrier), whether charge transport is by the loaded or unloaded form of the carrier, and whether the rate-limiting step is voltage sensitive. The data obtained from transport studies, in which the cotransporter is driven by either membrane potential or chemical gradients, complement each other. Obviously, any realistic model of a given cotransporter should be able to describe the results obtained under the two sets of conditions.

The whole-cell recording technique has been used to study the activity of the Na-HCO₃ cotransporter and has provided direct evidence for the potential dependence of this cotransporter. Two experimental systems have been used to measure electrogenic properties of cotransporter activity. In one case whole-cell recordings from dissected tubules were used to measure cotransporter activity in the native state (Boron and Boulpaep, 1983; Yoshitomi et al., 1985; Coppola and Frömter, 1994). A quantitative study of cotransporter activity with the whole-cell model is complicated by its inability to vary intracellular substrate concentrations, and hence its inability to vary the thermodynamic driving force. More recently, activity of the cotransporter from the rabbit proximal tubular clone (Burckhardt et al., 1994) and the salamander kidney clone (Romero et al., 1997) expressed in Xenopus oocytes has been measured. This is an exogenous expression system that lacks the native intracellular environment and potential regulatory accessory proteins that may be important determinants of the kinetic properties of the cotransporter in the native state. Nevertheless, both types of studies have provided direct evidence for the rheogenicity of the transport cycle and indicated the extent to which its turnover is dictated by the membrane potential.

In a previous study, we measured the stoichiometry of the Na-HCO₃ cotransporter in a proximal tubular cell line and found it to be 3 HCO₃:1 Na⁺ (Gross and Hopfer, 1996). The Na-HCO₃ cotransporter is expressed endogenously in this cell line. In the present study, we used the same cell line to study the kinetic properties of the cotransporter. We measured the cotransporter's current-voltage relationships under symmetrical and zero-trans conditions. Under symmetrical conditions, i.e., identical solutions on either side of the monolayer, the membrane potential is the only driving force for the cotransport, whereas under zero-trans conditions both the concentration gradient and the electrical potential drive the flux through the cotransporter. We used a proximal tubular cell line that can be grown on a permeable support to form a polarized cell monolayer. The monolayer exhibits high electrical resistance and can thus be studied in a Ussing chamber by electrophysiological techniques. By selective permeabilization of the apical membrane, with an appropriate agent (e.g., amphotericin B), one can control, maintain, and manipulate the intracellular concentrations of the cotransporter's substrates, through free diffusion exchange between the apical perfusion solution and the cell cytoplasm. Thus the technique allows one to control the thermodynamic driving force acting on the cotransporter, thereby overcoming some of the limitations encountered in isotope influx experiments. The ability to control and modify the intracellular substrate concentrations as well as the membrane potential allows one to describe the free energy of the system and, therefore, to analyze the potential dependence of the cotransport mechanism. The experimental data in this study were fitted to mathematical transport models. The models could account for the global electrical properties of the cotransporter. A single set of numerical rate constants was found to account for the cotransporter's current-voltage relationships under various experimental conditions. From these numerical values, one can identify which steps in the transport cycle are rate-limiting under a defined set of conditions, and which steps are potential sensitive.

	MATERIALS AND METHODS

Top Abstract Introduction Materials & Methods Results Discussion Appendix References

Cell culture

Experiments were carried out with the rat proximal tubular cell line SKPT-0193 C1.2 (Woost et al., 1996). The line is derived from microdissected primary cultures of the S1 region of the proximal tubule. Passages 50-70 were used for the reported experiments. Cells were grown on collagen-coated (20% bovine hoof collagen in 60% ethanol) Millicell-CM filters (area = 0.6 cm²) in a 1:1 mixture of Dulbecco's modified essential medium and Ham's F12, supplemented with 15 mM HEPES, 1.2 mg/ml NaHCO₃, 5 µg/ml insulin, 5 µg/ml transferrin, 5 ng/ml epithelial growth factor, 4 µg/ml dexamethasone, and 10% fetal bovine serum. Typically, 3 × 10⁵ cells were seeded and grown to confluence in 5 days. Light microscopy showed a "cobblestone" appearance that is typical of the morphology of epithelial cells.

Electrophysiology

Confluent SKPT-0193 C1.2 cells have a low basal monolayer conductance of 0.5-1 mS/cm², indicating the poor ion permeability of tight junctions. This low baseline conductance allows the detection of electrical signals from cellular transporters that make only small contributions to the overall monolayer conductance. Filters with cells were mounted horizontally in a Ussing-type chamber (Analytical Bioinstrumentation, Cleveland, OH) equipped with voltage and current electrodes. Only cell monolayers with an initial conductance of 1 mS/cm² or less were used in the experiments described here. Electrophysiological measurements were made with a voltage-clamp module (model 558-C-5; Bioengineering, University of Iowa, IA) controlled by an IBM PC via the DATAQ software package (Dataq Instruments, Akron, OH). Current and voltage were recorded with a strip-chart recorder and in parallel through an A/D converter on a microcomputer. The apical and basolateral compartments of the Ussing chamber have a volume of 0.5 ml each. The cells are perfused separately on each side of the monolayer with a peristaltic pump at a rate of ~2 ml/min. The chamber and all solutions were maintained in a heated incubator allowing control of CO₂ pressure (pCO₂) and temperature.

For experiments that were carried out under zero-trans conditions, apical solutions contained (in mM) 2.5 Ca gluconate, 1.1 Mg gluconate, 100.0 HEPES, 25.0 D-glucose, 80 N-methyl-D-glucamine (NMDG), and 0.1% bovine serum albumin (BSA). Basolateral solutions contained (in mM) 2.5 Ca gluconate, 1.1 Mg gluconate, 100.0 MES, 25.0 D-glucose, 80 NMDG, and 0.1% BSA. For experiments in which the kinetics of the cotransporter to Na⁺ were studied by varying the apical Na⁺ concentration, [Na⁺]_ap, at a fixed [HCO₃]_ap, NMDG was replaced with Na gluconate, in an isoosmotic manner. The pH of the apical and basolateral solutions was adjusted to 7.5 and 6.0, respectively, with acetic acid. Bicarbonate concentration was determined from the Henderson-Hasselbalch relation, [HCO₃] = 3 × 10⁵ · pCO₂ · 10^(pH 6.1) (pCO₂ is given in mm Hg), by adjusting the CO₂ pressure to the appropriate value. From the above equation it can be seen that the difference of 1.5 pH units between the apical and basolateral solutions results in a 31-fold (=10^1.5) higher HCO₃ concentration in the apical solution compared to that in the basolateral solution, at any CO₂ pressure.

For experiments carried out under symmetrical conditions, apical and basolateral solutions contained (in mM) 2.5 Ca gluconate, 1.1 Mg gluconate, 100.0 HEPES, 25.0 D-glucose, 80 NMDG, and 0.1% BSA. In experiments in which the concentration of Na⁺ was varied, NMDG was replaced with Na gluconate in an isoosmotic manner. The pH of the apical and basolateral solutions was adjusted to 7.5 with acetic acid. The bicarbonate concentration was set and determined from the Henderson-Hasselbalch relation, by adjusting the CO₂ pressure to the appropriate value, as described above.

All solutions were first adjusted for pH with acetic acid and then preequilibrated with CO₂ at the appropriate pCO₂ for 1 h. The pH of the solutions was measured and adjusted again before the beginning of the experiment and was measured once more at the end of each experiment. The solutions were maintained at the appropriate pCO₂ throughout the entire experiment, and the pH was found to change by less than 0.1. Experiments were carried out at 37°C. CO₂ pressure was continuously monitored with a CO₂ monitor (Puritan-Bennett, Los Angeles, CA).

To determine the cotransporter's current-voltage (I-V) relation, cell monolayers were permeabilized with 10 µM apical amphotericin B as described previously (Gross and Hopfer, 1996). After I_sc had leveled off, an I-V relation was obtained by running a custom program that steps a voltage between 160 and +160 mV in 20-mV increments. The basolateral compartment was then perfused with 1 mM 4,4'-dinitrostilbene-2,2'-disulfonic acid (DNDS) for 10 min to ensure full equilibration of DNDS with the cotransporter (see also Fig. 2), and a second I-V relation was obtained. The difference current (I) was plotted against voltage to obtain the voltage dependence of the cotransporter. The fit of the data to the Michaelis-Menten and the Hill equations (i.e., Figs. 4 B and 7 B) revealed that although the I_max values obtained from different cells were different from each other, there was no statistical difference between the K_0.5 values obtained from these cell monolayers. We therefore assumed that the variations in I_max are not due to a change in the mechanism of cotransport, but rather reflect a difference in the number of functional cotransporters inserted in the membrane, probably due to the difference in expression levels between the different cell monolayers. We thus normalized the different sets of I-V relationships obtained from different cell monolayers by introducing a "standard" set of conditions as described below. For the three sets of I-V relations, as a function of [HCO₃]_i, collected at three different fixed [Na⁺]_i, an additional I-V relation was taken at the end of each experiment under "standard zero-trans" conditions of [HCO₃]_i = 57 mM and [Na⁺]_i = 20 mM (except for the experiment performed at the fixed [Na⁺]_i = 20 mM, where this "standard" set of conditions was the last leg of the entire experiment). The three curves collected at each of the four [HCO₃]_i were then corrected by normalizing with the current at zero voltage, obtained with the "standard" conditions. The same protocol was used to normalize the three sets of I-V relations, as a function of [Na⁺]_i, collected at three different fixed [HCO₃]_i. All four I-V relations collected under the "symmetrical" conditions were obtained from the same cell preparation (filter). Thus no normalization was performed for these I-V sets.

All fitting procedures were performed with the SCoPFit simulation package (SCoP Simulations, Berrien Springs, MI). The program uses the "principal axis" (praxis) algorithm to automatically search for a global minimum in the error function, by changing the initial value of each parameter by a predetermined fraction (the maximum step). A multistage approach was employed, in which the magnitude of the maximum step was progressively decreased to fine-tune the search process. The algorithm also includes occasional random jumps to avoid confinement to a local minimum. Typically, ~3000 iterations were performed, during which translocation and binding/dissociation rate constants were allowed to vary over a wide range, until a minimum in the error function (²) was reached. In addition to the statistical tests, the quality of the model was also evaluated by calculating the parameter identifiability matrix and parameter sensitivity, over the tested voltage range.

Materials

Amphotericin B, bovine serum albumin, MES, HEPES, D-glucose, N-methyl-D-glucamine (NMDG), gluconic acid, and all salts were purchased from Sigma Chemical Co. (St. Louis, MO). Acetic acid was from Fisher Scientific. 4,4'-Dinitrostilbene-2,2'-disulfonic acid (DNDS) was obtained from Pfaltz and Bauer (Waterbury, CT). Bovine hoof collagen was a generous gift from Ethicon (Somerville, NJ).

Statistics

All experiments were repeated three times. The probability distribution for the reduced ² (i.e., P_n²; Bevington, 1969) was used to assess the goodness of fit of model equations to experimental data, where n is the degree of freedom (i.e., number of data points  number of fitted parameters). An F-test analysis was used to compare different models.

	RESULTS

Top Abstract Introduction Materials & Methods Results Discussion Appendix References

Experimental strategy

Fig. 1 illustrates the primary transporters involved in Na⁺ and HCO₃ reabsorption in the proximal tubule (Emmett et al., 1992). The transporters associated with electrical charge movement are located in the basolateral membrane. These include the Na,K-ATPase (the Na⁺ pump) and the Na-HCO₃ cotransporter. To vary the concentration of intracellular Na⁺ and HCO₃ in a controlled manner, we added 10 µM amphotericin B to the apical solution. Amphotericin B is a polyene ionophore that renders the membrane permeable to small monovalent ions (Na⁺, K⁺, Cl), but not to those with higher valences, such as Ca²⁺ (Kirk and Dawson, 1983), and stays restricted for several hours to the plasma membrane to which it was added. This property is a result of a requirement for cholesterol in the membrane, the relatively high cholesterol content of the plasma membrane, and the relatively low content of intracellular membranes (Kirk and Dawson, 1983). Permeabilization of the apical membrane with amphotericin B "removes" the electrical resistance of that membrane and reveals the basolateral electrogenic processes to external electrodes. The use of amphotericin B for the above-mentioned purposes is a common practice in studies of epithelial transport (Kirk and Dawson, 1983; Backman et al., 1992; Illek et al., 1993; Acevedo, 1994; Gross and Hopfer, 1996). In a previous study, we found that the current generated by Na,K-ATPase under short-circuit conditions interferes with measurements of currents related to Na-HCO₃ cotransporter activity (Gross and Hopfer, 1996). In the same study we also found a spontaneous increase in Cl conductance of the basolateral plasma membrane, which interfered with the measurement of the cotransporter conductance. Therefore, to eliminate these interference, all solutions in the present study were K⁺-free (K⁺ replaced by NMDG⁺) and Cl-free (Cl replaced by gluconate).

View larger version (20K): [in this window] [in a new window]   FIGURE 1   Schematic presentation of the premises on which the experiments were based. The Na-HCO3 cotransporter transports two net negative charges. Apical application of amphotericin B functionally "removes" the apical membrane for electrical measurements.

Na-HCO₃ cotransporter activity

Sodium bicarbonate-dependent current that is sensitive to dinitrostilbene disulfide (DNDS) is taken as flux through the cotransporter. Flux through the Na-HCO₃ cotransporter can be driven by applying Na⁺ and/or HCO₃ gradients across the basolateral membrane. Fig. 2 (upper trace) shows the short-circuit current (I_sc) obtained by permeabilization of a cell monolayer preparation and subsequent experimental maneuvers. Application of a HCO₃ gradient across the monolayer resulted in a 1.0 µA/cm² decrease in I_sc, which mainly reflects the flux of HCO₃ from the apical to the basolateral side, and of H⁺ from the basolateral to the apical side, through the tight junctions (not shown). Upon permeabilization of the apical side with 10 µM amphotericin B, a further decrease of ~2 µA/cm² in I_sc is observed. This additional decrease in I_sc reflects the flux of Na⁺ and HCO₃ through the electrogenic basolateral Na-HCO₃ cotransporter, as judged by its sensitivity to 1 mM DNDS on the basolateral side (shown in the last segment of the experiment in Fig. 2). The inhibition by DNDS is reversible. In a separate experiment we measured a K_i of 0.11 mM for the inhibition of the cotransporter by DNDS (not shown).

View larger version (10K): [in this window] [in a new window]   FIGURE 2   (Upper trace) Short-circuit current (Isc) of Na-HCO3 cotransporter driven by a HCO3 gradient in an apically permeabilized cell monolayer. For set-up, see Materials and Methods. Apical and basal compartments were initially perfused with a solution containing (in mM) 18.0 HCO3, 20.0 Na gluconate, 100.0 NMDG, 2.5 Ca gluconate, 1.1 Mg gluconate, 50.0 HEPES, 25.0 D-glucose, and 0.1% BSA, pH 8.0. The solution was equilibrated with 1% CO2. The cotransporter was activated by establishing a HCO3 gradient (apical 18.1 mM HCO3 to basal 1.8 mM) by lowering the pH on the basal side to 7.0. The resulting negative current returns to baseline levels upon basal application of 1 mM DNDS, as expected for flux through the cotransporter. (Lower trace) Imposition of a HCO3 gradient in the absence of Na+ (Na+ replaced by NMDG+) caused a smaller decrease in Isc, which was not affected by DNDS, and which probably reflects H+/OH movements across the cell monolayer's tight junctions.

In control experiments in the absence of Na⁺ (Na⁺ replaced with NMDG on both sides), lowering [HCO₃]_bl by a similar manipulation generated a smaller decrease in I_sc of ~0.5 µA/cm² (lower trace). This small current was not sensitive to 1 mM DNDS (lower trace), suggesting that it was not due to the Na-HCO₃ cotransporter. More importantly, this control experiment demonstrates the requirement for Na⁺ to observe a DNDS-sensitive current under our experimental conditions (absence of K and Cl) and thus the equivalence of DNDS-sensitive current with Na-HCO₃ cotransporter activity.

Current-voltage relationship

Kinetics of the cotransporter to Na⁺ and HCO₃ were demonstrated under two distinct experimental conditions: 1) "Zero-trans," with finite concentrations of Na⁺ and HCO₃ on the apical/intracellular side and nominally Na⁺-free on the basolateral side. Furthermore, the concentration of HCO₃ on the basolateral side was 31-fold smaller than on the apical side. 2) "Symmetrical," where the concentrations of Na⁺ and HCO₃ on one side of the basolateral membrane were equal to the concentrations of the same ions on the other side.

Zero-trans conditions

View larger version (10K): [in this window] [in a new window]   FIGURE 3   Steady-state current-voltage relationships of the outward Na-coupled HCO3 currents in apically permeabilized proximal tubule cells taken under "zero-trans" conditions. The current values were averaged from 10 points between 0.1 and 1.0 s at each voltage step, to eliminate possible contributions from any capacitative current transients. (A) Cells were perfused, on the apical side, with (in mM) 57 HCO3, 20 Na gluconate, 60 NMDG, 2.5 Ca gluconate, 1.1 Mg gluconate, 100.0 HEPES, 25.0 D-glucose, and 0.1% BSA, pH 7.5. The basolateral solution contained 1.8 HCO3, 80 NMDG, 2.5 Ca gluconate, 1.1 Mg gluconate, 100.0 MES, 25.0 D-glucose, and 0.1% BSA, pH 6.0. Both solutions were kept at 10% CO2 atmosphere during the experiment, in the absence () and in the presence () of 1 mM basolateral DNDS. (B) Difference current-voltage relationship.

View larger version (14K): [in this window] [in a new window]   FIGURE 4   Steady-state DNDS-sensitive currents as a function of [Na+]ap obtained under zero-trans conditions. (A) I-V relationships were obtained for each sodium concentration. Basolateral solutions contained (in mM) 0.6 HCO3, 80 NMDG, 2.5 Ca gluconate, 1.1 Mg gluconate, 100.0 MES, 25.0 D-glucose, and 0.1% BSA, pH 6.0. The apical solution contained 18 HCO3, 2.5 Ca gluconate, 1.1 Mg gluconate, 100.0 HEPES, 25.0 D-glucose, and 0.1% BSA, and varying concentrations of Na+ (NMDG replaced by Na gluconate), pH 7.5. Solutions were kept at 3.1% CO2 atmosphere during the experiment. The I-V currents were sigmoidal, approaching zero at high positive potentials and saturation at high negative potentials. The rectification at high positive potentials suggests that there is no accumulation of Na+ and HCO3 on the basolateral side. Solid lines are model predictions (Eq. A13). (B) DNDS-sensitive currents are plotted as a function of [Na+]ap at 0, 60, and 120 mV. The curves were fitted to Eq. 1 with the following parameters values: at 0 mV: K0.5Na = 22 ± 2 mM, ImaxNa = 32 ± 4 µA/cm2; at 60 mV: K0.5Na = 14 ± 2 mM, ImaxNa = 53 ± 6 µA/cm2; at 120 mV: K0.5Na = 7.5 ± 0.9 mM, ImaxNa = 60 ± 6 µA/cm2.

(1)

View larger version (15K): [in this window] [in a new window]   FIGURE 5   Dependence of K0.5Na, calculated from Eq. 1, on membrane potential, measured at three different HCO3 concentrations (A), and on [HCO3]i, measured at 60 mV (B), under zero-trans conditions. Solid lines represent model predictions; they were generated using Eq. A28 and the numerical values listed in Table 2.

View larger version (11K): [in this window] [in a new window]   FIGURE 6   Dependence of ImaxNa, calculated from Eq. 1, on membrane potential, measured at 18 and 57 mM HCO3 (A), and on [HCO3]i, measured at 60 mV (B), under zero-trans conditions. Solid lines represent model predictions; they were generated using Eq. A27 and the numerical values listed in Table 2.

(2)

View larger version (11K): [in this window] [in a new window]   FIGURE 7   Steady-state DNDS-sensitive currents as a function of [HCO3]ap obtained under zero-trans conditions. (A) I-V relationships were obtained with a fixed [Na+]ap of 10 mM, and 18 mM (), 30 mM (), and 57 mM HCO3 (). Basolateral solutions contained (in mM) 80 NMDG, 2.5 Ca gluconate, 1.1 Mg gluconate, 100.0 MES, 25.0 D-glucose, and 0.1% BSA, pH 6.0. The apical solution contained 2.5 Ca gluconate, 1.1 Mg gluconate, 100.0 HEPES, 25.0 D-glucose, and 0.1% BSA, and varying concentrations of HCO3, pH 7.5. [HCO3]ap was varied from 18 mM to 57 mM by increasing pCO2 and allowing the solutions to equilibrate for 1 h before the next I-V relation was taken ([HCO3] was calculated from the Henderson-Hasselbalch relation as explained in Materials and Methods). Solid lines are model predictions (Eq. A13). (B) DNDS-sensitive currents are plotted as a function of [Na+]ap at 0, 60, and 120 mV. The curves were fitted to Eq. 2 with the following parameters: at 0 mV: K0.5Bic = 21 ± 2 mM, ImaxBic = 27 ± 2 µA/cm2; at 60 mV: K0.5Bic = 19 ± 2 mM, ImaxBic = 56 ± 5 µA/cm2; at 120 mV: K0.5Bic = 17 ± 2 mM, ImaxBic = 65 ± 5 µA/cm2. (C) Voltage dependence of the Hill coefficient, m, for bicarbonate as calculated from fitting the data described in Fig. 7 B to Eq. 2. m was voltage independent, with values ranging from a minimum of 2.9 ± 0.3 at 100 mV to a maximum of 3.3 ± 0.3 at 60 mV.

View larger version (14K): [in this window] [in a new window]   FIGURE 8   Dependence of K0.5Bic, calculated from Eq. 2, on membrane potential, measured at 5, 10, and 20 mM Na+ (A), and on [Na+]i, measured at 60 mV (B), under zero-trans conditions. Solid lines represent model predictions; they were generated using Eq. A31 and the numerical values listed in Table 2.

View larger version (13K): [in this window] [in a new window]   FIGURE 9   Dependence of ImaxBic, calculated from Eq. 2, on membrane potential, measured at 10 and 20 mM Na+ (A), and on [Na+]i, measured at 60 mV (B), under zero-trans conditions. Solid lines represent model predictions; they were generated using Eq. A30 and the numerical values listed in Table 2.

Symmetrical conditions

Transport model

Assumptions and rationale

View larger version (41K): [in this window] [in a new window]   FIGURE 10   Different binding schemes of a six-state ordered-binding transport model of the Na-HCO3 cotransporter. (a) Nai+ first on-last off. (b) Nai+ first on-first off. (c) Nai+ last on-first off. (d) Nai+ last on-last off. The rate constants for the forward (fi) and backward (bi) reactions are modulated by voltage and/or ligand concentration as described by Eqs. 3-14. The binding of three HCO3 anions to the carrier is described as a single, lumped step (see text).

Formal model description

Effect of membrane potential on transport rate constants

xu/2

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

Current equation

(15)

Method I: King-Altman diagrams

This is a diagrammatic method that can be used to solve, analytically, for the steady-state concentration of individual carrier state concentration (C_x) in terms of the transport cycle rate constants and the total concentration of the carrier (C_T). The concentration of C_x is given by the sum of six King-Altman terms, each of which is the product of five different rate constants (see Appendix).

Method II: Integration of the rate equations

The rate at which each of the carrier species concentration changes is given by the difference of the forward and reverse reactions that lead to and from that state. The set of differential equations that describe the change in concentration of each state can be solved, numerically, by numerical solver routines for any desired time interval. The steady-state solution of the set was obtained using the "kinetic" solver of the SCoP simulations package, by assigning a large value (10⁹ s) to the time step. The set of differential equations that describe the rate of change in the concentration of a carrier state, for the F-L scheme, is given below:

(16)

(17)

(18)

(19)

(20)

(21)

Simulation and fitting procedures

Numerical values of rate constants and electrical coefficients were obtained by simultaneously fitting the I-V relationships collected under "zero-trans" and "symmetrical" conditions to Eq. 15. Eleven of the 12 rate constants were assigned an initial guess value; the 12th constant (b₆) was calculated from the other 11 constants by using the mass action (microscopic reversibility) law:

(22)

The electrical coefficients ', ", ', and " were allowed to vary between 0 and 1 during the iterative process. n and m were assigned the values 1 and 3, respectively, based on the 1 Na⁺:3 HCO₃ stoichiometry determined previously (Gross and Hopfer, 1996), and were not allowed to vary during iterations. C_T was assigned a fixed value of 2 pmol/cm² and was not allowed to vary during iterations. This corresponds to a surface density of ~10,000 molecules/µm². For comparison, the surface density of the anion transporter (band 3) of erythrocytes was estimated to be 8000 carriers/µm² (Stein, 1990, p. 156). Carrier translocation rate constants were assigned initial values of 1 s¹, and binding and dissociation rate constants were assigned initial values of 10,000 s¹, s¹ M¹, or s¹ M³. Simulation were performed until one set of rate constants was found that simultaneously fit all of the I-V sets, rather than aiming at multiple solutions that could account for only one I-V set. The numerical values of the rate constants obtained by fitting the I-V curves to the F-L model (i.e., Na⁺ first on and last off) are listed in Table 2. All four ordered binding schemes gave similar numerical values. The numerical values, listed in Table 2, were then used to fit the data sets of I_max and K_0.5, for a given substrate as a function of the concentration of the cosubstrate and as a function of membrane potential. This can be done by rearranging the current equation in a phenomenological (Michaelis-Menten) form (see the Appendix). It should be noted that because the model was solved numerically, rather then analytically, the set of numerical values listed in Table 2 might not represent a unique solution of the model, and other solutions may exist.

Submodels

The model presented above was obtained by making the least number of simplifying assumptions that are practical. Thus we will refer to it as the general model. In the discussion that follows we will consider some simple cases (or submodels) of the general model. The validity of the assumptions used to derive the submodels will be assessed by comparing the error function (_n²) for each case with that of the general model.

Rapid equilibrium models

If intracellular Na⁺ or HCO₃ binding to the cotransporter is rapid compared to the translocation steps, then the binding reactions of these ions to the cotransporter are said to be in "rapid equilibrium." Because it reduces the complexity of the algebra required to model cotransport activity, rapid-equilibrium assumptions of substrate binding have been the most enduring working hypotheses in the analysis of other Na⁺-dependent cotransport systems (Goldner et al., 1969; Restrepo and Kimmich, 1985; Jauch and Läuger, 1986). The rapid equilibrium assumption has been questioned by many in the field (Sanders et al., 1984; Sanders, 1986; Schultz, 1986; Weirzbicki et al., 1990). As discussed above, rapid binding/dissociation of substrate was not assumed in deriving our general model. Thus, all of the 11 rate constants were allowed to vary within the same range (i.e., 0-10⁹) during the fitting. To evaluate the effect on our model of assuming that Na⁺ and HCO₃ binding/dissociation is fast compared to the membrane translocation steps, the lower limit of the range over which the rate constants for substrate binding/dissociation could vary was set higher (10⁴ to 10⁹) than the corresponding range for the translocation rate constants (0 to 10³). The latter range encompasses the translocation rate constants predicted by the general model (see Table 2). The experimental data were then refitted to the model with the new range limits, and the error (_n²) was calculated. These manipulations test the effect of "forcing" rapid binding/dissociation rates, but, in effect, without restricting the translocation rates.

If the rapid binding/dissociation assumption is valid, then this manipulation should not affect, or even improve, the goodness of fit of the data to the submodel and, hence, the value of the error function. Table 3 compares _n² obtained for assuming rapid binding/dissociation for Na⁺, HCO₃, or Na⁺ and HCO₃. As Table 3 suggests, even for "partial rapid equilibrium" submodels (i.e., when only one of the substrates is assumed to rapidly bind/dissociate), the goodness of fit is compromised compared to that obtained with the general model. Thus we conclude that the rapid binding/dissociation assumption, for Na⁺, HCO₃, or both, is not justified.

View this table: [in this window] [in a new window]   TABLE 3   Effect of forcing rapid binding/dissociation of Na+ and HCO3 on the goodness-of-fit (n2)

Voltage-independent binding of HCO₃

Our general model predicts that all binding steps, except that of intracellular HCO₃ to the cotransporter, are voltage-independent. The binding of intracellular HCO₃ "senses" about 12% of the membrane's electric field (i.e., ' = 0.12). In this submodel we test the effect of assuming that all binding steps, including that of intracellular HCO₃, are voltage independent. This restriction is equivalent to the assumption that the translocation of the bound ions, by the cotransporter, through the membrane "senses" 100% of the transmembrane electric field. This was done by setting ', ", ', " = 0 and refitting the data to the submodel. These settings were held fixed during the fitting process. _n² obtained for this model was 1.34 compared to 0.81 for the general model. Therefore the assumption that all binding steps are voltage independent is not justified.

Other ordered-binding models

In deriving the general model we assumed that the binding of 3 HCO₃ on either side of the membrane can be lumped into one step, which can occur either after (F-L) or before (L-F) the binding of Na⁺. To check whether the lumping assumption was justified, we fitted the I-V data to the following ordered-bind