SPECIAL COMMUNICATION
A kinetic model of the thiazide-sensitive Na-Cl cotransporter
Hangil
Chang1 and
Toshiro
Fujita2
1 Health Service Center, and
2 Fourth Department of Internal Medicine,
University of Tokyo, Tokyo 153-8902, Japan
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ABSTRACT |
The aim of
this study was to construct a numerical model of the thiazide-sensitive
Na-Cl cotransporter (TSC) that can predict kinetics of thiazide binding
and substrate transport of TSC. We hypothesized that the mechanisms
underlying these kinetic properties can be approximated by a state
diagram in which the transporter has two binding sites, one for sodium
and another for chloride and thiazide. On the basis of the state
diagram, a system of linear equations that should be satisfied in the
steady state was postulated. Numerical solution of these equations
yielded model prediction of kinetics of thiazide binding and substrate
transport. Rate constants, which determine transitional rates between
states, were systematically adjusted to minimize a penalty function
that was devised to quantitatively estimate the difference between model predictions and experimental results. With the resultant rate
constants, the model could simulate the following experimental results:
1) dissociation constant of thiazide in the absence of sodium
and chloride; 2) inhibitory effect of chloride on thiazide binding; 3) stimulatory effect of sodium on thiazide binding; 4) combined effects of sodium and chloride on thiazide binding; 5) dependence of sodium influx on extracellular sodium and
chloride; and 6) inhibition of sodium influx by extracellular
thiazide. We conclude that known kinetic properties of TSC can be
predicted by a model which is based on a state diagram.
sodium-chloride cotransporter; thiazide diuretics; electrolyte
metabolism; kinetic model; computer program
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INTRODUCTION |
THIAZIDE-SENSITIVE Na-Cl cotransporter
(TSC) is present in the renal distal tubule where ~10% of
filtered sodium and chloride are reabsorbed (6). In the distal tubule,
TSC is localized to the luminal cell membrane and contributes to
reabsorption of sodium and chloride by facilitating the entry of these
ions into the cell. Transport mediated by TSC is blocked by thiazide,
and physiological significance of TSC in the maintenance of body sodium balance can be inferred by the efficacy of thiazide in the treatment of
human hypertension. Physiological significance of TSC has been further
illustrated recently by the finding that genetic defects in TSC in
human subjects caused derangement in electrolyte metabolism (hypokalemia and metabolic alkalosis) known as Gitelman's syndrome (9).
Fanestil and coworkers (1, 13) reported detailed experiments addressing
the kinetics of thiazide binding of TSC using renal cortical membranes.
They found that 1) metolazone, a diuretic with a thiazide-like
mechanism of action, had a high affinity to TSC; 2) chloride
inhibited metolazone binding (i.e., the fraction of TSC bound with
metolazone); and 3) sodium stimulated metolazone binding. To
explain these results, they proposed a conceptual model of TSC, in
which there are two binding sites, one that is selective for sodium and
another that recognizes chloride and metolazone in a mutually exclusive
fashion and in which occupancy of the former site by sodium increases
the affinity of the latter site for metolazone.
Success in cDNA cloning of TSC (3, 4) has enabled direct measurement of
sodium influx through TSC expressed in Xenopus laevis oocytes.
Gamba et al. (4) measured sodium influx with various extracellular
concentrations of sodium and chloride, and they found that sodium
influx was a saturable process and was consistent with a
carrier-mediated mechanism cotransporting sodium and chloride with 1:1
stoichiometry. Dependence of sodium influx on extracellular sodium and
chloride could be described by Michaelis-Menten equations with
Michaelis constants of 25.0 and 13.6 mM, respectively.
Thiazide, when present in the extracellular fluid, inhibits transport
by TSC in a dose-dependent manner. The effective concentration of
thiazide is significantly higher than the apparent dissociation constant reported in binding studies. For example, hydrochlorothiazide at 100 µM inhibited sodium influx by 59% (4), whereas the apparent dissociation constant of hydrochlorothiazide is ~100 nM (1). Similar
dose dependency of hydrochlorothiazide in inhibiting sodium influx was
reported in the urinary bladder of the winter flounder, which is rich
in TSC; half-maximal effective concentration was 20-50 µM
(11). Thus effective affinity (10) of thiazide measured by
functional studies appears to be 200- to 1,000-fold lower than the
apparent affinity measured by binding studies.
The aim of the present study was to develop a numerical model of TSC
that predicts quantitatively and coherently these experimental findings. The strategy was 1) to construct a model based on a state diagram, 2) to calculate the predictions of the model by numerically solving the steady-state equations, and 3) to find the set of rate constants with which the model simulates the
experimental results best. We have assumed a state diagram that is
similar to the one proposed by Fanestil and coworkers,
(13) except that in our model thiazide binding can be
possible both from the extracellular and intracellular sides. This
assumption was considered to be necessary, since our preliminary
examination had revealed that a model without intracellular thiazide
binding could not simulate the observed discrepancy between effective
and apparent affinities of thiazide. Optimal values of rate constants
were determined as a minimization problem of a penalty function that
was devised to quantitatively estimate the difference between model
predictions and experimental results. Systematic search procedure
yielded a set of rate constants with which the model could simulate all the experimental results listed above.
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METHODS |
Experimental data to simulate.
We constructed the present model to simulate the published experiments
of TSC that addressed kinetics of thiazide binding (1, 13) and
substrate transport (4). Specifically, we considered experimental
results on kinetics of metolazone binding in the absence of sodium and
chloride (1), inhibitory effect of chloride on metolazone binding (see
figure 5B in Ref. 13), stimulatory effect of sodium on
metolazone binding (see figure 2A in Ref. 13), combined effects
of sodium and chloride on metolazone binding (see figure 5A in
Ref. 13), dependence of sodium influx on extracellular sodium (see
figure 2B in Ref. 4) and chloride (see figure 2C in
Ref. 4), and inhibition of sodium influx by extracellular thiazide (4).
Values of data points were obtained by tracing the figures in the publications.
State diagram.
The state diagram of TSC, which we hypothesized in this study, is
illustrated in Fig. 1. TSC molecule ("E") has two binding sites,
one for sodium ("Na") and another for chloride ("Cl"). The
latter site can also bind thiazide ("D"), but bindings of chloride and thiazide are mutually exclusive. The two sites can be
occupied in any order, and occupancy of one site can affect the binding
or dissociation processes of the other site. For example, the
dissociation rate constant of extracellular sodium when the second site
is unoccupied (k2) can be different from that when the second site is occupied by chloride (k8). TSC
exists either on the extracellular side or on the intracellular side,
and states on the intracellular side are indicated by prime mark (').
Only unloaded and fully loaded TSC (E, E', ENaCl, and ENaCl') can cross the cell membrane, and TSC loaded with thiazide cannot cross the membrane.
Steady-state equations.
In the steady state, probabilities of finding a TSC molecule in
individual states are time-independent and satisfy the system of linear
equations (steady-state equations) listed in Table
1. In these equations, the probabilities
are represented by bracketed symbols1 such as [ENa].
Equations M-1 to M-11 indicate that rates of
generation and destruction of each state balance each other out.
Equation M-12 derives from the definition of these
probabilities and indicates that a TSC molecule should be in one of
these states in a given instant. Although similar equations for simpler
diagrams with less than six states might be solved manually (8), the
steady-state equations for TSC were too complex to be solved in closed
form. To circumvent the problem, we developed a computer program that represents and solves the steady-state equations in numerical form.
Solution obtained by the program yielded the proportion of TSC bound
with metolazone ([ED] + [ENaD] + [ED'] + [ENaD']), and
experiments addressing kinetics of metolazone binding could be
simulated by altering the concentrations of sodium, chloride, and
metolazone. Specifically, experiments examining the effect of chloride
on metolazone binding were simulated with [Na] = [Na'] = 0, [Cl] = [Cl'] = 0 ~ 150 mM, and [D] = [D'] = 1 nM. Experiments examining the effect of sodium on metolazone binding
were simulated with [Na] = [Na'] = 0 ~ 150 mM, [Cl] = [Cl'] = 0, and [D] = [D'] = 1 nM. Finally, experiments examining
metolazone binding in the presence of both sodium and chloride were
simulated with [Na] = [Na'] = [Cl] = [Cl'] = 0 ~ 150 mM, and [D] = [D'] = 1 nM. We have made [D] = [D'] in these
calculations, because experiments by Fanestil and colleagues (1,
13) were conducted with membrane preparations.
Calculation of unidirectional sodium influx.
Unidirectional sodium influx was calculated by extending the original
state diagram with radioactive sodium to conform to the actual
experimental setup. There were six additional states representing TSC
molecule loaded with radioactive sodium (Na*). Specifically, they were
ENa*, ENa*D, ENa*Cl, ENa*', ENa*D', and ENa*Cl'. These states were
connected to other states similarly as their radioinactive
counterparts. For example, ENa* was connected with E, ENa*D, and
ENa*Cl. Forward and backward rate constants between E and ENa* were
k1[Na*] and k2, respectively.
We note that, in this extension of the diagram, no additional rate
constants were introduced, since we assumed that bindings and
dissociations of radioinactive and radioactive sodium were governed by
the same set of rate constants. The ratio of [Na*] to [Na]
(specific activity in the extracellular space) was set to 0.001 throughout the study. The actual magnitude of this ratio is irrelevant
as long as it is sufficiently small. The concentration of radioactive
sodium in the intracellular space was set to zero. Steady-state
equations were postulated and solved similarly as in the original state diagram, and unidirectional sodium influx was calculated as
k19 [ENa*Cl]
k20
[ENa*Cl']. Experiments examining transport kinetics of TSC could be
simulated by altering the concentrations of extracellular sodium and
chloride. Specifically, experiments that measured unidirectional sodium
influx with various extracellular sodium concentrations were simulated
with [Na] = 1,000 × [Na*] = 0 ~ 96 mM, [Cl] = 96 mM, [Na'] = 10 mM, [Na*'] = 0, [Cl'] = 40 mM, and [D] = [D'] = 0. Experiments that measured unidirectional sodium influx with various
extracellular chloride concentrations were simulated with [Na] = 1,000 × [Na*] = 96 mM, [Cl] = 0 ~ 96 mM, [Na'] = 10 mM, [Na*'] = 0, [Cl'] = 40 mM, and [D] = [D'] = 0. Unidirectional efflux of sodium and unidirectional influx or efflux of chloride could
be calculated similarly by introducing appropriate radioactive substrates.
Parameter search.
If we assume the state diagram presented in Fig.
1, then kinetic properties of the model TSC
are completely determined by the values of 32 rate constants
(k1 to k32). Ideally these
values should be specified on the basis of direct experimental
measurements. However, pertinent data about individual rate constants
are not available in the case of TSC. Therefore, as an alternative
choice, we deduced values of a portion of rate constants from
theoretical considerations and indirect experimental data, and we
systematically adjusted remaining rate constants to make model
predictions as close to experimental results as possible.

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Fig. 1.
State diagram for the thiazide-sensitive Na-Cl cotransporter (TSC).
Transporter (E) has binding sites for sodium (Na) and chloride (Cl).
Thiazide diuretics (D) are assumed to compete for the same binding site
with chloride. Sodium and chloride can bind to the transporter in any
order. Rate constants of transitional steps are indicated beside
arrows. Symbols with brackets, such as "[Na]," stand for
concentrations of substrates or thiazide diuretics. Prime mark (') is
used to indicate which symbols belong to the intracellular side.
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Rate constants that are involved in substrate and inhibitor binding
(k1, k3,
k5, k7, k9,
k11, k13,
k15, k21,
k23, k25,
k27, k29, and
k31) were assumed to be diffusion limited (5) and were set to 1.0 × 108
l · mol
1 · s
1
(2). Value of k30 was determined to conform to the
apparent dissociation constant of metolazone measured in the absence of sodium and chloride (1). Specifically, the steady-state equations in
the absence of substrates become sufficiently simple to be solved in
closed form, and the value of k30 was determined
using the following equation
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(C-1)
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where KD is the apparent dissociation constant
of metolazone in the absence of substrates
(4.27 × 10
9 mol/l, Ref. 1). Values of
k8, k16,
k20, k22, and
k28 were determined from the following
thermodynamic requirements
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(C-2)
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(C-3)
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(C-4)
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(C-5)
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(C-6)
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Values of 12 remaining rate constatnts (k2,
k4, k6,
k10, k12,
k14, k17,
k18, k19,
k24, k26, and
k32) were adjusted to minimize a penalty function
that was defined as
where
Mi are model predictions,
Ei are experimental results, and
SDi are standard deviations. Experimental results
were comprised of values of data points reproduced in Figs. 2-6 as
solid circles (1, 4, 13) and effective affinity of extracellular
metolazone in inhibiting substrate transport. The latter value (4.27 µM) was deduced from experiments with hydrochlorothiazide (1, 11),
since a relevant experiment is not available with metolazone. Standard
deviations of experimental results were from representative values
found in the original studies (1, 4, 13).
Minimization of the penalty function was conducted using the algorithm
of Powell's direction set method (7). This method has the advantage of
being applicable to functions whose differentials are difficult to
evaluate (as in the present case). It is an iterative method and
requires an initial value for each variable to start with. The initial
values we used were as follows: 1.0 × 106
s
1 for k2, k4,
k6, k10,
k12, and k14; 1.0 × 105 s
1 for k17,
k18, and k19; 2.0 × 103 s
1 for k24 and
k26; and 2.14 × 10
1
s
1 for k32. Thus we started with
intrinsic dissociation constants of 10 mM for sodium and chloride on
both sides and 2 µM for metolazone binding on the extracellular side.
Preliminary attempts of minimization revealed that inclusion of
k17, k18,
k20, k24, and
k26 in optimizing variables resulted in convergence
to unrealistic values (including negative values for several rate
constants). To circumvent this problem, we imposed the following
constraints during the optimization procedure: values of
k18 and k19 should be one of
the predefined values (102, 103,
104, 105, and 106
s
1); and k24 should be equal to
k26. In the initial step of optimization, an
additional constraint was imposed, which preserved the symmetry of
intrinsic dissociation constants of sodium and chloride (i.e., k2 = k10,
k4 = k12, and
k6 = k14). This constraint,
which will be referred to as symmetrical substrate binding hereafter,
is a frequently used assumption in modeling transporters, and it simplifies the present model by reducing the number of variables by
three. Next, we relaxed the constraint of symmetrical substrate binding
and tested whether substantial further reduction in the value of
penalty function could be brought about.
Computer hardware and software.
The program that solved the steady-state equations and
optimized the rate constants was developed for this project using C++ programming language (12). To invert the matrix representing the linear
equations, the method of LU decomposition was applied (7). To minimize
the penalty function, the function powell in Press et al. (7), which
implements the Powell's direction set method in multidimensions, was
used after modification for use in C++ language. All the calculations
were done in double precision (64-bit IEEE format). The source code was
compiled with Metrowerks C/C++ compiler (version 1.8) running under
MacOS. All the development and calculations were conducted with a
desktop computer equipped with 180 MHz PowerPC 604e CPU.
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RESULTS AND DISCUSSION |
The program we developed for this study required 0.2 megabytes of heap
memory and less than 0.2 ms to solve the steady-state equations, and it
had completed optimization of the rate constants within 3 min. In these
calculations most of the computational time was spent in inverting the
matrix with LU decomposition. This algorithm is known (7) to require
computational time in proportion to the cubic of the dimension of the
matrix (i.e., to the cubic of the number of states). Consequently, we
speculate that our program would solve the steady-state equations with
17,000 states in less than 10 min. On the other hand, we estimate that the size of heap memory required to store the matrix would become approximately 100 megabytes in the case of a diagram with 3,500 states.
Thus, limitation of available computer memory would become a
restrictive factor if we apply the program to diagrams with an
extremely large number of states. However, for the analysis of most
transporters and ionic channels, the present program would be
sufficiently fast and compact.
Before optimization of the rate constants, model prediction deviated
from the experimental results to a great extent, and the penalty
function amounted to 7,943. To improve the model, we first conducted
the optimization, which preserved the constraint of symmetrical
substrate binding. Despite the constraint, this step of optimization
turned out to be highly effective, and the penalty function decreased
to as low as 25.813 (a 308-fold reduction). In the actual
calculations, values of k18 and
k19 were fixed to one of the predefined values to
avoid unrealistic outcome, and the minimal values of the penalty
function for all combinations of k18 and
k19 were listed in Table
2. Acceptable outcomes were associated only
with restricted combinations of k18 and
k19, and, among them, the best result was attained
when k18 and k19 were set to
105 and 103 s
1, respectively.
Next, we proceeded with the optimization by removing the constraint of
symmetrical substrate binding. However, it was found that the penalty
function decreased only marginally (to 25.801), and we judged it
worthless to increase the complexity of the model in terms
of the number of independently adjustable rate constants. Thus we
adopted the set of rate constants with symmetrical substrate binding.
Their values were tabulated in Table 3 and
were used in all calculations presented below.
We confirmed that the rate constants in Table 3 indeed satisfy the
constraints of Eqs. C-1 to C-6. Furthermore, the
constraint of Eq. C-1 predicts that model prediction of the
apparent dissociation constant of metolazone in the absence of
substrates is identical to the experimental measurement (4.27 nM), and
the constraints of Eqs. C-2 to C-6 predict that the net
flux becomes zero when [Na] × [Cl] = [Na'] × [Cl']. To
confirm that these predictions were really fulfilled, we calculated the
dissociation constant of metolazone and the net flux in various
substrate concentrations and verified that the former was equal to 4.27 nM in all cases and the latter was equal to zero within roundoff errors
when [Na] × [Cl] = [Na'] × [Cl'].
The result of the simulation that examined the effect of chloride on
metolazone binding is presented in Fig. 2.
The model predicted a dose-dependent inhibition of metolazone binding
by chloride (continuous line), which was in good agreement with
experimental results (solid circles). Half-maximal inhibitory
concentration of chloride was 61 mM with the model TSC,
which was close to the experimental estimate of 60 mM (13). Inhibitory
effect of chloride on metolazone binding is comprehensible from the
structure of the state diagram, in which chloride and metolazone bind
competitively and exclusively to the same binding site. We note that,
in these experiments and the simulation, concentration of metolazone (1 nM) was much lower than the intrinsic dissociation constant of metolazone on the extracellular side
(k24/k23, 319 nM) and binding of metolazone took place mostly on the intracellular side.

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Fig. 2.
Inhibitory effect of chloride on metolazone binding. Metolazone binding
in various chloride concentrations was examined in absence of sodium.
Metolazone was present at 1 nM. Ordinate represents the percent of
metolazone bound compared with control, which contained no chloride.
Solid circles are experimental results obtained from Tran et al. (see
figure 5B in Ref. 13). Solid line shows the model prediction.
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The result of the simulation that examined the effect of sodium on
metolazone binding is presented in Fig. 3.
The model predicted a dose-dependent stimulatory effect of sodium on
metolazone binding (continuous line), which was in good agreement with
experimental results (solid circles). Half-maximal
stimulatory concentration of sodium was predicted to be 3.1 mM, which
was close to the experimental estimate of 3.7 mM (13). This effect of
sodium on metolazone binding could be attributable to the higher
affinity of sodium-loaded TSC to metolazone
(k32/k31, 1.17 nM) compared
with that of unloaded TSC
(k30/k29, 3.51 nM) on the
intracellular side. Higher sodium concentration promotes transition
from unloaded to sodium-loaded TSC and results in higher apparent
affinity to metolazone. We note that binding of sodium to TSC on the
extracellular side would have an inhibitory effect on metolazone
binding due to the low intrinsic affinity to metolazone on this side.
However, because of the favorable stability of the model TSC on the
intracellular side (see k17/k18
and k19/k20), the effect of
sodium binding on the intracellular side predominated.

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Fig. 3.
Stimulatory effect of sodium on metolazone binding. Metolazone
binding in various sodium concentrations was examined in absence of
chloride. Metolazone was present at 1 nM. Ordinate represents the
percent of metolazone bound compared with control, which
contained no sodium. Solid circles are experimental results obtained
from Tran et al. (see figure 2 in Ref. 13). Solid line shows the model
prediction.
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The result of the simulation that examined the combined effects of
sodium and chloride on metolazone binding is presented in Fig.
4 (continuous line), which was in agreement
with experimental results (solid circles). In this simulation,
concentrations of sodium and chloride were changed simultaneously from
0 to 150 mM. At the lower range of concentration, stimulatory effect of sodium prevailed, whereas at the higher range of concentration, inhibitory effect of chloride was predominant. Peak stimulatory effect
was observed at 4.4 mM with the model TSC, which was close to the
experimental estimate of 4.2 mM (13). Half-maximal inhibitory effect
(using the metolazone binding in the absence of sodium and
chloride as control) was observed at 62 mM, which was also close to the
experimental estimate of 61 mM (13).

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Fig. 4.
Combined effects of sodium and chloride on metolazone binding.
Metolazone binding was studied in various concentrations of sodium and
chloride. Both were present at the same concentrations ranging from 0 to 150 mM. Metolazone was present at 1 nM. Ordinate represents the
percent of metolazone bound compared with control, which contained
neither sodium or chloride. Solid circles are experimental results
reported by Tran et al. (see figure 5A in Ref. 13). Solid line
shows the model prediction.
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Results of the simulations that examined the effects of extracellular
sodium and chloride on unidirectional sodium influx are presented in
Figs. 5 and
6. Model predictions were plotted in
continuous lines, together with experimental results (solid circles)
obtained from Gamba et al. (4). Model predictions were in good
agreement with experimental results. They reproduced the
unidirectional sodium influx as a sodium-dependent (Fig. 5), chloride-depenent (Fig. 6), and saturable process.
Half-maximal saturating effects were observed at sodium and chloride
concentrations of 29.6 and 31.6 mM, respectively. Whereas the former
value was comparable to the Michaelis constant for sodium (25.0 mM)
reported by Gamaba et al. (4), the latter value was higher than the reported Michaelis constant for chloride (13.6 mM). The cause of the
discrepancy is unclear but may be due to the difference in fitting
method (nonlinear least square vs. Eadie-Hofstee plot), fitting
function (function derived from the steady-state equations vs. function
defined by Michaelis-Menten equation), and data to fit to (all data
incorporated in the penalty function vs. data plotted in Fig. 6).

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Fig. 5.
Dependence of unidirectional sodium influx on extracellular sodium.
Unidirectional sodium influx was examined in presence of various
concentrations of extracellular sodium. Chloride concentrations in
extracellular fluid and intracellular fluid were 96 and 40 mM,
respectively. Sodium concentration in intracellular fluid was 10 mM.
Metolazone was absent. Sodium influx was normalized by the value
observed at 96 mM extracellular sodium. Solid circles are experimental
results reported by Gamba et al. (see figure 2B in Ref. 4).
Solid line shows the model prediction.
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Fig. 6.
Dependence of unidirectional sodium influx on extracellular chloride.
Unidirectional sodium influx was examined in presence of various
concentrations of extracellular chloride. Sodium concentrations in
extracellular fluid and intracellular fluid were 96 and 10 mM,
respectively. Chloride concentration in intracellular fluid was 40 mM.
Metolazone was absent. Sodium influx was normalized by the value
observed at 96 mM extracellular chloride. Solid circles are
experimental results reported by Gamba et al. (see figure 2C in
Ref. 4). Solid line shows the model prediction.
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To further illustrate the transport kinetics of the model TSC, we
plotted transport velocities in both directions as functions of
substrate concentrations (Fig.
7). In these calculations, the whence
side (e.g., the extracellular side in the case of substrate influx)
contained variable concentrations of sodium and chloride, whereas the
whither side contained no substrates. When extracellular chloride was
fixed to 96 mM, sodium (and chloride) influx was dependent on
extracellular sodium (Fig. 7, line A), and half-maximal saturating effect was observed at 3.12 mM.
Similarly, sodium efflux was dependent on intracellular sodium (Fig. 7,
line B), and half-maximal saturating effect was
observed at 1.32 mM. Thus effective affinity for sodium was asymmetric,
and was higher on the intracellular side than on the extracellular
side. Maximal velocity was also asymmetric, and the maximal influx was
4.54-fold higher than the maximal efflux. Similar properties in
transport velocities could be observed when chloride concentrations
were varied (Fig. 7, lines C and D). Effective
affinity for chloride was 12.9 mM on the extracellular side, and it was
11.1 mM on the intracellular side. Maximal influx was 4.54-fold higher
than maximal efflux. These asymmetric transport kinetics in the face of
symmetric substrate binding can be ascribed to the different stability
of the model TSC across the membrane, and they conform to the
theoretical prediction made from analysis of a simple carrier with one
substrate and four states (10). The prediction says that if
interconversion of loaded transporters is the rate limiting step (which
can be seen from Table 3 in the present case), then the side at which the transporter is less stable (extracellular side in the present case)
will be the side at which transport will half-saturate at higher
concentration and from which the maximal velocity of transport will be
greater.2

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Fig. 7.
Asymmetry of transport kinetics of the model TSC. Top:
transport velocities as functions of sodium concentrations. Line
A: unidirectional sodium influx with various extracellular sodium
concentrations. Extracellular chloride concentration was 96 mM. Both
sodium and chloride were absent from intracellular side. Line
B: unidirectional sodium efflux with various
intracellular sodium concentrations. Intracellular chloride
concentration was 96 mM. Both sodium and chloride were absent from
extracellular side. Transport velocities were normalized by the maximal
velocity of sodium influx. Bottom: transport velocities as
functions of chloride concentrations. Line C: unidirectional
chloride influx with various extracellular chloride concentrations.
Extracellular sodium concentration was 96 mM. Both sodium and chloride
were absent from intracellular side. Line D: unidirectional
chloride efflux with various intracellular chloride concentrations.
Intracellular sodium concentration was 96 mM. Both sodium and chloride
were absent from extracellular side. Transport velocities were
normalized by the maximal velocity of chloride influx.
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Model prediction of half-maximal effective concentration of metolazone
to inhibit sodium influx was 4.27 µM, which was identical to the
experimental value we used to calculate the penalty function. This
value was 1,000-fold higher than the apparent dissociation constant for
metolazone in the absence of substrates. One of the causes of this
difference would be the asymmetry of intrinsic dissociation constant
for metolazone: 0.32 µM on the extracellular side vs. 3.51 nM
(k30/k29) on the intracellular
side. In the case of the experiments and simulations studying effective
affinity to metolazone, metolazone is present only on the extracellular side and cannot bind to the high-affinity binding site present on the
intracellular side.
In conclusion, a model of TSC that was based on a state diagram with
one cation-binding site and one anion-binding site can reproduce the
known kinetics of inhibitor binding and substrate transport of TSC.
This model is useful in summarizing the experimental data, as well as
in predicting transport velocities in various physiological conditions.
The computer program we developed is estimated to be sufficiently
general and efficient to be applied to transporters and ionic channels
involved in renal epithelial transport.
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APPENDIX |
Solution of the Steady-State Equations in a Closed Form
Inspection of Table 3 reveals that magnitudes of rate constants
involved in interconversions of unloaded and fully loaded transporters
(k17, k18,
k19, and k20) are smaller than
most of the rate constants involved in substrate binding and
dissociation. This finding suggests that the steady-state equations may
be simplified on the basis of rapid equilibrium, which implies that all
the processes of binding and dissociation are in equilibrium on both sides of the membrane. The simplified steady state equations can be
solved in a closed form as
follows3
where
and
When substrate concentrations are sufficiently high, these
formulas provided a good approximation. For example, if we assume [Na] = [Cl] = 40 mM, [Na'] = 10 mM, and [Cl'] = 40 mM, then
values calculated with these formulas were identical to the true values with less than 1% of relative error. However, as substrate
concentrations on either side of the membrane decrease, binding
processes of substrates become slower and the assumption of rapid
equilibrium becomes less appropriate. For example, if we assume [Na] = [Cl] = 1 mM, [Na'] = 10 mM, and [Cl'] = 40 mM, then relative
error of [ENaCl] calculated with the formula is as high as 29.2%.
Thus, although the formulas derived from the assumption of rapid
equilibrium may be accurate at the physiological range of substrate
concentrations, they should be used with caution when substrate
concentrations are low on either side of the membrane.
 |
FOOTNOTES |
1
It should be noted that solute concentrations are
also indicated by bracketed symbols (for example, [Na], [Cl], and
[D] in Eq. M-1).
2
In his original report, Stein (10) actually did an
analysis of the case in which interconversion of unloaded transporters was the rate limiting step, and he predicted higher effective affinity
and lower maximal velocity on the less stable side. However, equation 1 in his report (10), from which the prediction
originates, is valid irrespective of which step is the rate limiting
step, and we can derive, based on the same equation, the predictions appropriate when interconversion of loaded transporters is the rate-limiting step.
3
In these calculations, states bound with thiazide
were neglected. However, extending the formula by taking these states
into account is straightforward.
The costs of publication of this
article were defrayed in part by the
payment of page charges. The article
must therefore be hereby marked
"advertisement"
in accordance with 18 U.S.C. §1734 solely to indicate this fact.
Address for reprint requests and other correspondence: H. Chang,
Health Service Center, Univ. of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo
153-8902, Japan.
Received 27 April 1998; accepted in final form 20 January
1999.
 |
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