A numerical model of acid-base transport in rat distal tubule
Hangil
Chang1 and
Toshiro
Fujita2
1 Health Service Center and 2 Department
of Internal Medicine, University of Tokyo, Tokyo 153-8902, Japan
 |
ABSTRACT |
The purpose of this study is to develop a numerical model that
simulates acid-base transport in rat distal tubule. We have previously
reported a model that deals with transport of Na+,
K+, Cl
, and water in this nephron segment
(Chang H and Fujita T. Am J Physiol Renal Physiol 276:
F931-F951, 1999). In this study, we extend our previous model by
incorporating buffer systems, new cell types, and new transport
mechanisms. Specifically, the model incorporates bicarbonate, ammonium,
and phosphate buffer systems; has cell types corresponding to
intercalated cells; and includes the Na/H exchanger, H-ATPase, and
anion exchanger. Incorporation of buffer systems has required the
following modifications of model equations: new model equations are
introduced to represent chemical equilibria of buffer partners [e.g.,
pH = pKa + log10 (NH3/NH4)], and the formulation of mass
conservation is extended to take into account interconversion of buffer
partners. Furthermore, finite rates of
H2CO3-CO2 interconversion (i.e.,
H2CO3
CO2 + H2O) are taken into account in modeling the bicarbonate
buffer system. Owing to this treatment, the model can simulate the
development of disequilibrium pH in the distal tubular fluid. For each
new transporter, a state diagram has been constructed to simulate its
transport kinetics. With appropriate assignment of maximal transport
rates for individual transporters, the model predictions are in
agreement with free-flow micropuncture experiments in terms of
HCO
reabsorption rate in the normal state as well as
under the high bicarbonate load. Although the model cannot simulate all
of the microperfusion experiments, especially those that showed a
flow-dependent increase in HCO
reabsorption, the
model is consistent with those microperfusion experiments that showed
HCO
reabsorption rates similar to those in the
free-flow micropuncture experiments. We conclude that it is possible to
develop a numerical model of the rat distal tubule that simulates
acid-base transport, as well as basic solute and water transport, on
the basis of tubular geometry, physical principles, and
transporter kinetics. Such a model would provide a useful means
of integrating detailed kinetic properties of transporters and
predicting macroscopic transport characteristics of this nephron
segment under physiological and pathophysiological settings.
bicarbonate transport; hydrogen ion transport; anion
exchanger; hydrogen adenosinetriphosphatase; sodium-hydrogen
exchanger
 |
INTRODUCTION |
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 |
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 variables are composed of electrical potential, flow rate, and
concentrations of Na+, K+, Cl
,
H+, HCO
, H2CO3,
NH
, NH3,
H2PO
, HPO
, and urea
in the luminal space; electrical potential and concentrations of
Na+, K+, Cl
, H+,
HCO
, NH
, NH3, and
impermeant solute in distal convoluted tubule cells and principal
cells; and electrical potential and concentrations of Cl
,
H+, HCO
, and impermeant solute in
intercalated cells. The total number of model variables (in the
discretized form) has increased from 270 in the previous model to 611 in the present model.
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 × 10
7
ml · s
1 · cm
2 and 1.42 × 10
10
mmol · s
1 · cm
2. 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.
In the luminal compartment, mass conservation includes a convective
term
where Fv is the tubular flow rate,
Ri is the inner radius of the luminal
compartment, Jv is the rate of transepithelial
volume reabsorption, Fx is the flux of solute
x along the tubular axis, and Jx is
the rate of transepithelial reabsorption of solute x. If we
neglect the electrodiffusive movement of solutes along the tubular
axis, Fx is simply
FvC
, where C
is the
concentration of solute x in the luminal compartment. The
tolerance values have been 1.7 × 10
11 ml/s (0.001 nl/min) and 1.7 × 10
14 mmol/s (0.001 pmol/min).
Conservation of solutes that constitute buffer systems are handled
differently, as described later.
Electroneutrality within each compartment requires
where zx is the valence of solute
x, Cx is the concentration of solute
x within the compartment, and the sum is of all the solutes
within the compartment. The tolerance value has been 0.01 mM.
In the present model, introduction of buffer systems has required the
following modifications of model equations. First, concentrations of
the acid form ([Acid]) and the base form ([Base]) of a buffer system should fulfill the following condition
where pKa = 10
Ka
(Ka: dissociation constant of the buffer
system). The tolerance value has been 0.0001 pH unit.
Second, conservation of the total number of molecules that constitute a
buffer system is considered, instead of individual molecular species.
For example, in the case of the ammonium buffer system in the cytosolic
space, the following equation has been postulated
where J
and
J
are fluxes of
NH
and NH3 from the luminal space into
the cell, and J
and
J
are fluxes from the
serosal space, respectively. The above equation is derived from the
fact that the amount of NH
generated by the chemical reaction (NH3 + H+
NH
) is exactly the same as the amount of
NH3 consumed in the reaction.
Third, the model equation that represents mass conservation of
H+ includes the term of the rate of H+
generation via interconversion of buffer partners (Base + H+
Acid). For example, the equation of mass
conservation of H+ in the luminal space becomes
|
(1)
|
where FH is the flux of H+ along the
tubular axis, JH is the rate of transepithelial
efflux of H+, and GH is the rate of
H+ generation via interconversion of buffer partners. This
equation can be used as a model equation without the introduction of a new model variable (GH), because
GH can be expressed by model variables as
follows.2 When there is only
one buffer system, it can be easily deduced that
GBase = GH =
GAcid. When there are more than one buffer system, the relationship becomes
where the sum on the right-hand side encompasses all
the bases (that is, HCO
, NH3, and
HPO
in this model). GBase
can be expressed in terms of model variables using the equation similar
to Eq. 1. Consequently, GH can be
expressed in terms of model variables.
Last, finite rates of H2CO3 dehydration and
CO2 hydration are taken into account in modeling the
bicarbonate buffer system. The bicarbonate buffer system is composed of
HCO
, H2CO3, and
CO2, which are interconverted according to the following reaction3
The left part of the reaction is a rapid process and
is essentially at equilibrium. Therefore, given the
pKa of 3.57 for carbonic acid, the following
relationship can be assumed
|
(2)
|
In contrast, the right part of the reaction is a
slower process,4 and
H2CO3 concentration can deviate significantly
from its equilibrium value when there is an H+ load. For
example, when there is an H+ load of 0.8 × 10
3
mmol · s
1 · cm
3 (which is
equivalent to 50 pmol · min
1 · distal
tubule
1) in the absence of carbonic anhydrase activity,
luminal H2CO3 concentration should
increase5 by 16 µM from its
equilibrium value
(kh/kd[CO2] = 3.6 µM). Owing to this increase in H2CO3
concentration, the pH value becomes lower (Eq. 2) than the
value observed when there is no H+ load. The difference is
called "disequilibrium pH" and is demonstrated experimentally in
the distal tubule (62). Disequilibrium pH is simulated in
the present model by formulating the following model equation that
relates the rate of H2CO3 generation and
the rate of HCO
generation, which is equivalent to
Eq. 7 in Ref. 78
|
(3)
|
The tolerance value has been 3.7 mmol · s
1 · cm
3
(equivalent to 0.0001 pmol · min
1 · section
1). In
the present model, disequilibrium pH exists only in the luminal space.
In the cytosolic space, we have assumed that
H2CO3 concentration is essentially equal to the
equilibrium value (3.6 µM) due to the carbonic anhydrase in the
distal tubular cells (27). Consequently, cytosolic
H2CO3 concentration is equal to the serosal
H2CO3 concentration (which is also equal to the
equilibrium value), and there would be no H2CO3
transport across basolateral cell membranes even if there is
H2CO3 permeability. Therefore, basolateral
H2CO3 permeability is not incorporated in the
present model.
The model equations have been transformed to a system of
difference equations and solved numerically. The derivation of
difference equations has been conducted as in the previous study
(22). Briefly, the model tubule has been divided into 23 sections, and the continuous model variables have been replaced by
discrete ones that represent the values at the center of each section. Previously, we had compared two separate derivations of difference equations: one taking into account the axial electrodiffusive movement
of molecules and another neglecting it. The derivation with
electrodiffusive terms had the advantage of being a more realistic
prediction of luminal electrical potential profile at the junction of
the early and late distal tubules but had the disadvantage of demanding
an ~35 times longer computational time. In the present study, we have
employed only the derivation without electrodiffusive terms, because
solution of the equations with electrodiffusive terms has become
prohibitively time-consuming due to the increased number of model
variables and more involved calculations of transport velocities
through transporters (as described below). Fortunately, model
predictions relevant to the present study (that is, reabsorption rates
of solute and water) had been essentially unaffected by the choice of
the discretization schemes (22). The present program
solves the model equation in ~910 ms when run on a machine with
PowerPC 604e (180-MHz clock cycle).
Transporters
Transporters that have been incorporated into the
present model are illustrated in Figs. 1
and 2. In the early distal tubule (Fig. 1), distal convoluted tubule cells have a Na-Cl cotransporter, Na/H exchanger, K-Cl cotransporter, Na+ channel,
K+ channel (also permeable for NH
),
Cl
channel, NH3 permeability,
H2CO3 permeability (not listed in Fig. 1), and
H-ATPase in the luminal membrane; and Na-K-ATPase, an anion exchanger,
K+ channel (also permeable for NH
),
Cl
channel, and NH3 permeability in the
basolateral membrane. The paracellular pathway has conductances for
Na+, K+, NH
,
Cl
, and HCO
ions. In the late distal tubule (Fig. 2), principal cells have the Na-Cl
cotransporter, K-Cl cotransporter, Na+ channel,
K+ channel (also permeable for NH
),
Cl
channel, NH3 permeability, and
H2CO3 permeability (not listed in Fig. 2) in
the luminal membrane; and Na-K-ATPase, an Na/H exchanger, anion
exchanger, K+ channel (also permeable for
NH
), Cl
channel, and NH3
permeability in the basolateral membrane. Type A intercalated cells
have H-ATPase in the luminal membrane and an anion exchanger and
Cl
channel in the basolateral membrane. Type B
intercalated cells have an anion exchanger in the luminal membrane and
H-ATPase and Cl
channel in the basolateral membrane. The
paracellular pathway has the same conductances as in the early distal
tubule.

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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 ), Cl channel,
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.
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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.
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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,
PNH4, is 20% of K+
permeability, PK (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
(JTSC) has been represented in the following
form
where JTSC, max
(mmol · s
1 · cm
2) is
a model parameter, jTSC is the transport
velocity via a single TSC molecule, and jTSC, max is the maximal transport velocity via
a single TSC molecule. In this way, model parameter
JTSC, 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
(JNH4-ATPase) and
K+ (JK-ATPase) have been calculated
from the following equations (78)
where Ja is the rate of ATP hydrolysis,
C
and C
are
basolateral K+ and NH
concentrations, and
KK and KNH4 are
kinetic constants with the ratio
(KNH4/KK) of 5.8 (75). With typical values of basolateral K+
concentration (4.25 mM) and NH
concentration (0.068 mM; Ref. 44),
JNH4-ATPase/ JK, ATPase becomes 0.0028. The rate of ATP hydrolysis (Ja)
and transport velocity of Na+
(JNa-ATPase) have been calculated as before
(22, 83)
where Ja, max is a model
parameter, C
is cytosolic Na+
concentration, and KNa-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).

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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.
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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 KNa 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
(Ki) 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).
Ki 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.

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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.
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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 (E E*) 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.
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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 × 10
3 M, and
[ADP]c · [Pi]c = 10
7 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).

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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.
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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
.

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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.
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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 s
1 for k2 and
1.3 × 105 s
1 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).

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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.
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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.
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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 × 10
3 cm/s, which is estimated from the apical membrane
formic acid permeability (4.6 × 10
2 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 × 10
6 and 1.80 × 10
5 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).
We have assumed that NH
permeability through the
paracellular pathway is the same as those of other cations, because no
experimental information is available about its value. Similarly,
permeability of HCO
through the paracellular pathway
(P
) has been assumed
to be equal to that of Cl
. This value (2.40 × 10
6 cm/s), however, is significantly smaller than the
value reported by Chan et al. (20). They measured
HCO
backflux during the perfusion of the rat distal
tubule with nominally HCO
-free solution and deduced
a P
value of
2.32 × 10
5 cm/s. However, this value is larger than
the estimated paracellular HCO
permeability of the
rat proximal tubule (1.77 × 10
5; Ref.
23), a "leaky" epithelium, and should be regarded, as they had pointed out (20), as the estimation of the upper
limit of paracellular HCO
permeability in the distal
tubule, because processes not directly related to paracellular HCO
entry, such as the diffusion of ammonia from
blood into the tubule lumen, can affect the measurement of
HCO
backflux. Consequently, instead of adopting
their value, we have assumed that paracellular HCO
permeability is equal to that of Cl
, which had been
determined from the transepithelial conductance of the distal tubule
(22, 63). If we recalculate the model predictions with a
P
value of 2.32× 10
5 cm/s, HCO
reabsorption in the
distal tubule decreases by 9.2 pmol/min in the basic state due to an increased backflux of HCO
through the paracellular pathway.
The value of the rate constant of H2CO3
hydrolysis in the luminal space (k
) has
been chosen to be considerably larger than the value reported for
carbonic anhydrase-free solution (35). This is based on
the experiments by Malnic et al. (62), who showed that to
explain the relationship among measurements of luminal pH, luminal
HCO
concentration, and magnitude of H+
secretion in the distal tubule, k
should be from 2.4- to 64.1-fold higher than the value in the carbonic anhydrase-free solution. In this model, we have assumed that
k
as well as k
is 10-fold larger than the corresponding values in carbonic
anhydrase-free solution (35). The origin of this higher
k
is not known. An authoritative immunocytochemical study (15) did not detect the
membrane-bound form carbonic anhydrase in the distal tubule. However, a
recent study has shown that carbonic anhydrase immunoreactivity is
detected on apical membranes of type A intercalated cells in the rabbit distal tubule (72). Therefore, a modest increase in
k
in the distal tubule might be due to
carbonic anhydrase activity in apical membranes of a restricted group
of tubular cells.
Other parameters that are related to acid-base transport have been
chosen to fit free-flow micropuncture experiments by Capasso et al.
(16, 17): that is, a distal tubular HCO
reabsorption rate of 53.2 pmol/min in normal
rats7 and 179 pmol/min in
acutely bicarbonate-loaded rats. To find those values, we have employed
an epithelial model, which neglects axial changes of solute
concentrations and greatly simplifies model equations. After
determining the parameters that are directly related to acid-base
transport as above, we have redone the parameter optimization procedure
that had been conducted in the previous study (22), using
the Powell method, to fit the model with experimental data of
Na+, K+, Cl
, and water transport
in the rat distal tubule.
 |
RESULTS |
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
, and water and luminal electrical
potentials in the basic state are listed in Table
5 (details of the boundary conditions are described in the legend). As in the previous model, they are consistent with free-flow micropuncture experiments in the following conditions: in the basic state, in the presence of thiazide, in the presence of
amiloride, and under the high sodium load with increased flow rate.
Thus the extensions that have been introduced in the present model
preserve the goodness of fit of the previous model.
Acid-base transport in the model tubule is summarized in Table
6. The rate of HCO
reabsorption is 62.4 pmol/min in the basic state. This value is
comparable to the experimental estimation of 53.2 pmol/min
(17). Both early and late subsegments contribute to this
HCO
reabsorption, with the early subsegment
reabsorbing at 44.3 pmol/min and the late subsegment at 18.2 pmol/min.
This is also comparable to the results of Wang et al.
(76), in which HCO
reabsorption rates
were 32.9 pmol/min in the early subsegment and 13.9 pmol/min in the
late subsegment.8
One of the benefits of modeling the renal tubule is that it allows for
determination of transport velocities through all of the transporters
(Figs. 1 and 2). On the basis of these velocities, we can
quantitatively estimate the relative importance of each transport
mechanism. In the case of HCO
reabsorption, two
separate mechanisms can be involved: addition of H+ into
the tubular fluid and transport of HCO
itself out of
the tubular fluid. The former mechanism can be divided further into
secretion of H+ via luminal transporters and generation of
H+ through chemical reactions. In the early distal tubule,
the model shows that secretion of H+ is the major mechanism
of HCO
reabsorption (Table 6). In total,
H+ is secreted at 48.0 pmol/min, in which 79% is ascribed
to Na/H exchanger and 21% to H-ATPase. The magnitude of other
components is much smaller: chemical generation of H+ is
1.1 pmol/min (that is, H+ is consumed at 1.1 pmol/min),
and HCO
is secreted via the paracellular pathway at
the rate of 2.6 pmol/min. With regard to the origin of secreted
H+, all of the ions are generated by chemical reactions in
distal convoluted tubule cells: 97% from dissociation of carbonic acid (H2CO3
+ HCO
) and 3%
from dissociation of ammonium ion
(NH
H+ + NH3). In the
late distal tubule, luminal secretion of H+ becomes much
smaller (15.2 pmol/min) due to the absence of luminal Na/H exchanger in
this subsegment. On the other hand, amount of H+ generated
by ammonium and phosphate buffer systems becomes larger (8.0 pmol/min
in total).
Solute concentrations and electrical potentials in the basic state are
listed in Tables 7 and
8. In the early distal tubule (Table
7), intracellular potential is highly hyperpolarized (on average,
89.8 mV) and is consistent with experimental measurements (80). Intracellular Na+ concentration is low
(on average, 9.5 mM), and intracellular K+ concentration is
high (on average, 175 mM), reflecting the high Na-K-ATPase activity in
these cells (22, 48). These values as well as the
intracellular Cl
concentration are consistent with the
results of electron microprobe analysis on distal tubular cells
(7). Mean intracellular pH is 6.94. Together with the
luminal pH and potential difference, it can be calculated that
H+ is secreted against the electrochemical gradient of 10.6 kJ/mol at the apical membrane of the distal convoluted tubule cells. This is accomplished via the Na/H exchanger by utilizing the
electrochemical gradient of Na+ (11.3 kJ/mol) across the
apical cell membrane and via H-ATPase by utilizing the free energy
change of ATP hydrolysis (59 kJ/mol, coupled with transport of 2 mol of
H+).
In the late distal tubule (Table 8), principal cells show
intracellular potential and concentrations of Na+,
K+, and Cl
that are consistent with
experimental measurements (7, 80). Intracellular pH of
principal cells is 6.59, which is determined mainly by the Na/H
exchanger and the anion exchanger in the basolateral membrane and is
not directly related to transepithelial acid-base transport. In
intercalated cells, intracellular potentials are
26 and
24 mV for
type A and type B intercalated cells, respectively. These values are
consistent with experimental measurements (from
19 to
26 mV) with
intercalated cells of the rabbit cortical collecting duct
(34). Intracellular Cl
concentrations are
near the equilibrium values (with respect to the bath
Cl
), which is due to the Cl
- selective
basolateral membrane of the intercalated cells (34, 50, 51,
64). Intracellular pH in the type A intercalated cells is 7.35;
and H+ is secreted against the gradient of 4.25 kJ/mol
across the apical membrane via H-ATPase.
The axial profile of luminal HCO
concentration is
plotted in the upper panel of Fig. 10.
In the early distal tubule, luminal HCO
concentration decreased steadily from its inlet value of
9.8-5.2 mM, whereas in the late distal tubule it increased again
up to 9.3 mM and then decreased to 6.6 mM. As a whole, this is
compatible with the experimental observation that there was no fixed
trend of change in luminal HCO
concentration along
the distal tubule (62). Apparently, the paradoxical
increase in HCO
concentration in the late
subsegment, in the face of active H+ secretion, is
ascribable to prominent water reabsorption in this part of nephron that
concentrates tubular fluid (22). The axial profile of
luminal pH is plotted in the middle panel of Fig. 10 (solid line).
Values of luminal pH range from 6.47 to 6.84 (mean 6.69) and are
comparable to experimental measurement (range 5.75-6.91; mean 6.35 in Ref. 62). The initial increase in luminal pH in the
late distal tubule is, again, ascribable to water reabsorption. Concentrated tubular fluid tends to have higher levels of
HCO
, H2CO3, and
CO2. However, permeability for CO2 is so high
that CO2 diffuses out quickly and CO2
concentration remains essentially unchanged. Consequently, the
following chemical reaction is driven rightward: HCO
+ H
H2CO3
CO2 + H2O. Disequilibrium pH is represented in Fig. 10
(middle panel) as the difference between the solid line and
the broken line. Disequilibrium pH is larger in the early distal tubule
(range 0.23-0.30; mean 0.29) than in the late distal tubule (range
0.11-0.23; mean 0.14), reflecting the larger H+
secretion rate in the former. These values are comparable to the level
of disequilibrium pH observed in micropuncture experiments (for
example, 0.37 in Ref. 62). The profile of
HCO
reabsorption rate is plotted in the
bottom panel of Fig. 10. When the early and late subsegments
are compared, reabsorption rate in the early subsegment is larger than
that in the late subsegment. However, within each subsegment, the
HCO
reabsorption rate is relatively constant, which
is in contrast to profiles of reabsorption rates of other solutes
(22).

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Fig. 10.
Profiles of luminal HCO
concentration, luminal pH, and HCO reabsorption rate
along the tubular axis. Top: profile of luminal
HCO concentration. Middle: profile of
luminal pH (solid line). To illustrate disequilibrium pH, equilibrium
pH (dashed line) is plotted together. The difference between the 2 lines indicates the magnitude of disequilibrium pH. Bottom:
profile of transepithelial HCO reabsorption rate.
The x-axis indicates the coordinate along tubular axis, with
x = 0.1 cm corresponding to the junction between early
and late distal tubules. The figure was made from calculations with
section width of 0.00125 cm.
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|
The axial profile of luminal NH
, NH3, and
total phosphate (H2PO
+ HPO
) concentrations are plotted in Fig.
11. The luminal NH
concentration is 3.2 mM at the inlet, which is 47-fold higher than the
basolateral NH
concentration. It is essentially
constant in the early distal tubule. In the late distal tubule, it
increases transiently up to 6.3 mM in the first half, which is the
consequence of volume reabsorption. In the second half of the late
distal tubule, luminal NH
concentration decreases
again, and it becomes 3.5 mM at the end of the distal tubule. This
decrease is due to chemical conversion to NH3, reabsorption
via luminal NH
conductance, and reabsorption via the
paracellular pathway. The axial profile of the luminal NH3
concentration is similar to that of luminal NH
concentration. This reflects the profile of luminal pH that shows only
modest changes along the tubular axis. In total, ammonium (including
both NH
and NH3) is reabsorbed at the
rate of 17 pmol/min in the distal tubule. This appears to be
inconsistent with micropuncture experiments that show no significant
ammonium reabsorption in the rat distal tubule (39, 44,
47). Possible reasons for this inconsistency are the following.
First, direct measurement of NH3 permeability in the rat
distal tubule is not available. We have assumed that it is similar to
that in the rabbit9 cortical
collecting duct (82), but actual permeability in the rat
distal tubule in vivo might be smaller. Second, we have assumed that
NH
permeability via K channel is 20% of
K+ permeability on the basis of the data of single-channel
conductances and mean open probabilities (66). However,
K+ -to- NH
permeability ratio estimated from the difference in reversal potentials is ~10 (66).
Therefore, actual NH
permeability in the distal tubule may be smaller than our estimation, resulting in smaller NH
reabsorption rate. Third, ammonia production by
metabolism is not incorporated in the present model. According to Good
and Burg (42), the rate of ammonia production in the
distal tubule is comparable to the proximal tubule (in terms of
production per unit length) and is 5.5-6.7
pmol · min
1 · mm
1. Assuming
the total length of 2.3 mm, these values are translated to 13-15
pmol/min. If the fraction of ammonia thus produced is secreted into the
tubular fluid, it would have the effect of reducing the net
reabsorption rate of total ammonia. The axial profile of total
phosphate concentration in the luminal compartment is strictly
determined by the profile of water reabsorption, because phosphates are
assumed to be impermeable in this model. A steeper change in total
phosphate concentration implies greater water reabsorption. Inspection
of Fig. 11 shows that water reabsorption is essentially confined in the
first half of the late distal tubule.

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Fig. 11.
Profiles of luminal NH ,
NH3, and phosphate
(H2PO +HPO )
concentrations along the tubular axis. Top: profile of
luminal NH concentration. Middle: profile
of luminal NH3 concentration. Bottom: profile of
luminal concentration of phosphates (sum of
H2PO and HPO ). The
figure was made from calculations with a section width of 0.00125 cm.
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|
Distal HCO
reabsorption under high
HCO
load has been studied by Capasso et al.
(16). They infused intravenously a solution containing 260 mM HCO
, which brought about an approximately sixfold
increase in HCO
load to the distal tubule (480 pmol/min). In this condition, they observed that HCO
reabsorption is markedly increased to the level of 179 pmol/min
(control state: 53.2 pmol/min). When this experiment is simulated by
altering the boundary conditions (Table 6; boundary conditions are
described in the legend), the model predicts a similar enhancement of
HCO
reabsorption, amounting to 217.7 pmol/min. Both
the early and the late distal tubules contribute to this increase in
HCO
reabsorption. In the early distal tubule, the
HCO
reabsorption rate increases from 44.3 pmol/min
in the basic state to 140.1 pmol/min under the bicarbonate load.
Inspection of Table 6 reveals that this increase is largely due to the
increase in H+ secretion via the luminal Na/H exchanger.
From the solute concentrations and electrical potentials under the
bicarbonate load that are listed in Table
9, we can see that the increased level of
luminal Na+ (72.6 mM) is the cause of the increased Na/H
exchanger activity. In the late distal tubule, HCO
reabsorption rate is increased from 18.2 pmol/min in the basic state to
77.5 pmol/min under the bicarbonate load. From Table 6, it is evident that this change is mostly due to the increase in
HCO
reabsorption via the paracellular pathway. From
the values of solute concentrations and electrical potentials (Table
10), we can see that more
hyperpolarized luminal potential (
40.8 mV) and higher level of
luminal HCO
(95.9 mM) are responsible for the
increased HCO
current.
Comparison with microperfusion experiments.
There are many microfusion experiments that examined
HCO
reabsorption in the rat distal tubule. In Table
11, we have listed them in two
groups: those addressing the effect of perfusate HCO
concentration on HCO
reabsorption rate
(top) and those addressing the effect of tubular flow rate
(bottom). In the former group, HCO
reabsorption rates were highly variable both in direction and in
magnitude (Table 11). For example, with perfusate
HCO
concentration of ~10 mM, Iacovitti et al.
(46) reported net HCO
secretion of 12 pmol · min
1 · mm
1, whereas
Chan et al. (20) reported net HCO
reabsorption of 26.1 pmol · min
1 · mm
1, which is
comparable to micropuncture experiments (23.1 pmol · min
1 · mm
1). With a
similar boundary condition (described in the legend for Table 11), the
model predicts net reabsorption of 31.9 pmol · min
1 · mm
1, which is
consistent with the experiment of Chan et al.
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Table 11.
In vivo microperfusion experiments vs. model predictions: dependence of
net HCO reabsorption on perfusate
HCO concentration and flow
rate
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|
We tried to simulate the net HCO
secretion, as
observed by Iacovitti et al. (46), by adjusting model
parameters. When we modified single model parameter separately (keeping
the other parameters constant), all attempts were unsuccessful. This
suggests that changes restricted in single transport mechanism are not
sufficient to account for the variation in microperfusion experiments.
However, when all the activities in type B intercalated cells are
increased, we have been able to reproduce net HCO
secretion. Specifically, if parameters related to all the transporters in type B intercalated cells (H-ATPase, anion exchanger, and
Cl
channel) have been increased simultaneously by a
factor of 30, the model has come to predict net HCO
secretion of 11.1 pmol · min
1 · mm
1. We have
thought such manipulation to be worth consideration, because it has
been suggested that apparently inconsistent results in microperfusion
experiments might be due to the different ways of feeding the
experimental animals (54, 57). Such difference can affect
the acid-base status of these animals and alter the activity of
intercalated cells (71).
When perfusate HCO
concentration was increased to
~25 mM (Table 11), all investigators had reported a higher
HCO
reabsorption rate. The actual magnitude,
however, was again highly variable, ranging from 6 to 75.9 pmol · min
1 · mm
1
(20, 57). The model prediction (74.2 pmol · min
1 · mm
1) agrees
with the experiments that reported the highest reabsorption rate
(20). We could not fit the model to experiments with low HCO
reabsorption rates, either by adjusting
individual model parameters or by increasing the activity of type B
intercalated cells.
Before we proceed to the effect of flow rate (Table 11), we compare the
model with an experiment that separately perfused early and late distal
tubules with normal flow rate (76). Varying the perfusate
HCO
concentrations, Wang et al. (76)
observed positive correlations between perfusate HCO
concentration and HCO
reabsorption rate both in the
early distal tubule (Fig.
12A,
) and in the late
distal tubule (Fig. 12B). The present model also predicts a
higher HCO
reabsorption rate as the perfusate
HCO
is raised (solid
lines10 in Fig. 12).
However, quantitative agreement is poor, especially in the late distal
tubule. This may suggest that 1) our formulation of
transporter kinetics (e.g., HCO
conductance in the
tight junction) is inadequate in the case of unphysiologically high
HCO
concentration in the tubular fluid; or
2) a transport mechanism exists that is not incorporated in
the present model, but its transport rate is not negligible with a very
high luminal HCO
concentration.

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Fig. 12.
Effect of luminal HCO concentration
on HCO reabsorption rate. A:
HCO reabsorption rate
(JHCO3) in the early distal tubule
when it is perfused with various HCO concentrations
(from 15 mM to 75 mM). Model prediction (solid line) is plotted
together with experimental measurements ( ; Ref. 76).
B: results for the late distal tubule. The late distal
tubule has been separated from the early distal tubule and has been
perfused directly with test solutions (76). Constituents
of the perfusate solution are 70 mM Na+, 1.86 mM
K+, 56.86 mM Cl , 15 mM
HCO , pH 7.20, and 78 mM urea; 70 mM Na+,
1.86 mM K+, 46.86 mM Cl , 25 mM
HCO , pH 7.42, and 78 mM urea; 70 mM Na+,
1.86 mM K+, 21.86 mM Cl , 50 mM
HCO , pH 7.72, and 78 mM urea; 146 mM
Na+, 1.86 mM K+, 87.86 mM Cl , 60 mM HCO , pH 7.80, and 78 mM urea; and 146 mM
Na+, 1.86 mM K+, 72.86 mM Cl , 75 mM HCO , pH 7.89, and 78 mM urea. Flow rate is 8 nl/min, and the bath solution has the same composition as in the legend
for Table 5. It should be noted that Na+ concentration is
abruptly increased from 70 to 146 mM when HCO
concentration is changed from 50 to 60 mM. This change in perfusate
Na+ concentration is probably responsible for the
deflections in the model prediction in this HCO
range.
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|
When the flow rate was increased from a lower range (5-10 nl/min)
to a higher range (20-30 nl/min), microperfusion experiments revealed an increased (20, 57), unchanged (20,
57), or decreased (46, 56, 57)
HCO
reabsorption rate (Table 11). The model predicts
only small changes in HCO
reabsorption rate as flow
rate is increased (Table 11). The flow-dependent increase in
HCO
reabsorption rate in several experiments could
not be simulated by adjusting model parameters. The flow-dependent
decrease, however, can be simulated by increasing the activity of type
B intercalated cells. By increasing the transporter activities of type
B intercalated cell by 30-fold, the model tubule shows an
HCO
reabsorption rate of
11.1,
20.6, and
28.9
pmol · min
1 · mm
1 for flow
rates of 8, 12, and 24 nl/min, respectively (perfusate HCO
concentration is 10 mM; for other boundary
conditions, see the legend for Table 11).
 |
DISCUSSION |
We have developed a numerical model to simulate acid-base
transport in the rat distal tubule. The model is based on our previous model that dealt with Na+, K+,
Cl
, and water transport in this nephron segment
(22). Modifications have included incorporation of buffer
systems, intercalated cells, and relevant transporters and adaptation
of model equations to molecular interconversion in buffer systems and
establishment of disequilibrium pH due to H+ secretion. The
extended model has simulated the results of micropuncture experiments
addressing the transepithelial HCO
reabsorption both
in the normal state and under the bicarbonate load. The model has also
simulated the results of a set of in vivo microperfusion experiments.
Experimental evidence supporting the selection of transporters.
There is pharmacological evidence for the existence of an Na/H
exchanger in the luminal membrane of the early distal tubule. For
example, Wang et al. (76) reported that luminal
application of ethyl-isopropylamiloride (EIPA; an inhibitor of the Na/H
exchanger) reduced HCO
reabsorption by 65% in the
early distal tubule, whereas it did not affect HCO
reabsorption in the late distal tubule. Claveau et al.
(25) also demonstrated EIPA-sensitive Na+
uptake in the luminal membrane vesicles prepared from rabbit distal
tubules. They further suggested, based on pharmacological characteristics, that this Na/H exchange activity is mediated by the
type 1 Na/H exhanger isoform (NHE1) among the six isoforms of Na/H
exchanger (NHE1-NHE6) (65). Unfortunately, this
suggestion is not supported by histological studies (9),
and the identification of the isoform remains controversial (11,
18, 19, 67). However, investigators appear to agree on the point
that NHE3 is not the major isoform in the distal tubule (2,
10). In the present model, the kinetic model of the Na/H
exchanger has been constructed to fit the kinetics of the NHE3 isoform,
because this is the most extensively studied isoform (5,
77). Therefore, it may be argued that this kinetic model is
inadequate to represent the Na/H exchanger in the distal tubule because
of the possible heterogeneity in transport kinetics of the individual
isoforms. In actuality, however, such heterogeneity appears to be
small. According to a study that compared basic kinetics of NHE1, NHE2, and NHE3 isoforms (58), affinity for external
Na+ and cooperativity of internal H+ (in the
internal modifier site) are similar among these isoforms. Hence, usage
of kinetics of NHE3 isoform in the present model would be justified.
Incorporation of H-ATPase in the luminal membrane of the distal tubule
is based on pharmacological evidence that luminal application of
bafilomycin (a specific inhibitor of vacuolar H-ATPase) reduced HCO
reabsorption by 36 and 47% in the early and
late distal tubules, respectively (76). Vacuolar H-ATPase
is an electrogenic proton translocating ATPase that was originally
identified in the intracellular vacuolar systems of eukaryotic cells.
In the renal distal tubule, Brown et al. (13, 14) have
demonstrated the presence of vacuolar H-ATPase immunocytochemically. They localized vacuolar H-ATPase in the luminal membrane of the early
subsegment; in the late subsegment, they found immunoreactivity in the
luminal membrane of type A intercalated cells and in the basolateral
membrane of type B intercalated cells. Unfortunately, studies
specifically addressing the transport kinetics of vacuolar H-ATPase in
the distal tubule are not available. Therefore, we have adopted the
model of H- ATPase in turtle bladder (3), the kinetics
of which have been extensively studied. This approach is supported by
the results of Gluck and Caldwell (40) that vacuolar
H-ATPase purified from renal medulla (which contains intercalated
cells) had basic kinetics that are similar to H-ATPases from other
tissues in terms of pH dependency, substrate specificity, and
Michaelis-Menten constant (Km) for ATP.
Schwartz et al. (71) demonstrated
Cl
/HCO
exchange activity in the
basolateral side of type A intercalated cells and in the apical
side of type B intercalated cells. In type A intercalated cells,
pharmacological and immunocytochemical evidence demonstrates that the
activity is mediated by AE1 isoform of the anion exchanger gene family
(1, 70). Transport kinetics of the AE1 isoform had been
extensively studied using human red blood cell membranes. Most
experiments had been conducted at low temperatures (~0°C), because
transport velocity becomes smaller and, therefore, suitable for
measurements. Those experiments had revealed features such as
asymmetric substrate affinities (43), self inhibition
(i.e., inhibition of transport velocity at high substrate
concentrations; Ref 43), and tunneling (i.e., movement of the anion
through the exchanger without a conformational change; Ref.
33). We have been able to incorporate the feature of
asymmetric substrate affinities and self inhibition in the present
model owing to the recent experiments that addressed these features at
body temperature (36, 49). However, similar data for
tunneling are lacking, and extrapolation from low-temperature data is
difficult because transport via anion exchanger exhibit complicated
temperature dependency (12, 24). Therefore, the small
effect of tunneling with physiological range of substrate
concentrations (33) is not taken into account in the
present model.
In type B intercalated cells, immunocytochemical studies detect none of
the anion exchanger isoforms (AE1-AE3), and the molecular nature that
mediates the apical Cl
/HCO
exchange is
presently unclear (1, 70). Although detailed kinetic
information about this Cl
/HCO
exchange
is not available, experiments by Furuya et al. (34) showed
in rabbit cortical collecting duct that the transport rate is dependent
on luminal Cl
with Km of 18.7 mM
(when luminal HCO
is 25 mM). This
Km is comparable to the value of 16.6 mM that is
predicted11 from the
present model. Therefore, we have assumed that transport kinetics of
apical Cl
/HCO
in type B intercalated
cells can also be approximated by this diagram.
Incorporation of Cl
channel in basolateral membranes of
intercalated cells is based on electrophysiological studies with rabbit distal tubule (64) and collecting duct (34, 50, 51,
64). For example, Muto et al. (64) showed that
reducing the bath Cl concentration by 10-fold (from 120 to 12 mM)
depolarized basolateral potential difference by 37.9 and 25.7 mV in
type A and type B intercalated cells, respectively. On the other hand,
raising the bath K concentration by 10-fold (from 5 to 50 mM)
depolarized basolateral potential only by 4.7 mV (in type A
intercalated cells) and 3.1 mV (in type B intercalated cells). These
results indicate that basolateral membranes of both types of
intercalated cells are predominantly Cl
selective. In
contrast to basolateral membranes, apical membranes of intercalated
cells are devoid of significant ion conductances, as indicated by
fractional resistance of apical membranes
[fRa = Ra/(Ra + Rbl)] being near unity (50, 51,
64).
The basolateral Na/H exchanger and anion exchanger in the model tubule
have been incorporated for practical reasons. Without these activities,
solution of the model equations often led to unrealistic results, such
as negative cytosolic HCO
concentrations, when
boundary conditions are varied to match experimental conditions.
Although functional evidence for these transport activities is not
available (probably for technical reasons), there is immunocytochemical evidence for the presence of NHE1 in distal convoluted tubule cells
(9). However, immunoreactivity for AE1 is absent in both distal convoluted tubule cells and principal cells (74).
Therefore, in the future it may become necessary to replace these anion
exchangers by other HCO
-extruding mechanisms (such
as the Na-HCO3 cotransporter) as more experimental data accumulate.
We have not incorporated H-K-ATPase activity in the present model,
because luminal application of SCH-28080 (an inhibitor of H-K-ATPase)
does not affect HCO
reabsorption in either early or
late distal tubules (76). Although H-K-ATPase plays an
important role in increasing the H+ secretion in
pathological states such as K depletion (81), it appears
to have an insignificant role in acid-base transport in the normal state.
Modeling the bicarbonate buffer system.
There is controversy about PCO2 levels in renal
cortical structures. Early works on urinary acidification mechanism
took it for granted that PCO2 in renal cortex
is equal to that in systemic blood (62). However, this
assumption was challenged by subsequent studies that directly measured
cortical PCO2 with micro-Severinghaus electrodes (29, 30, 37). For example, DuBose et al.
(30) reported that PCO2 levels in
distal tubular fluid (67.1 Torr), peritubular vessels (64.8 Torr), and
proximal tubular fluid (65.1 Torr) were significantly higher than those
in systemic artery (39.2 Torr) and renal vein (41.1 Torr). Such a
prominent PCO2 gradient between renal cortex
and systemic blood appears difficult to explain, if we recall the
highly diffusible nature of the CO2 molecule and the
liberal blood supply to the kidney. Actually, quantitative arguments
had excluded simple explanations such as 1) generation of
CO2 by the addition of reabsorbed HCO
to
peritubular blood, 2) generation of CO2 in the
tubular fluid by secreted H+, and 3) generation
of CO2 by tissue metabolism (8). More involved mathematical models had been required to account for the presumed PCO2 difference (6, 8). In the
present model, we have followed the results of a more recent study by
de Mello-Aires et al. (26), who reinvestigated the issue
and confirmed the earlier assumption that cortical
PCO2 is equal to that of systemic blood. By
using micro-Severinghaus electrodes that contained carbonic anhydrase in the inner buffer solution, these investigators showed that PCO2 levels in cortical structures (for
example, 41.5 Torr in proximal tubule) were not different from those in
renal vein (40.6 Torr) and carotid artery (38.8 Torr). When carbonic
anhydrase was removed from electrodes, PCO2
values in cortical structures apparently increased, and a significant
portion of the measurements fell into the range of 50-60 Torr,
consequently resembling the results of DuBose et al. (30).
On the basis of these results, they argued that 1) when
carbonic anhydrase is absent, there may exist a
PCO2 gradient in the inner buffer solution of
the electrode due to its small tip size; 2) this
PCO2 gradient can deteriorate the reliability
of PCO2 measurement with these electrodes; and 3) this problem is circumvented by incorporating carbonic anhydrase in
the electrode, because carbonic anhydrase is known to facilitate the
diffusion of CO2 in fluid layers (60). We have
considered that these experimental results and arguments are convincing
and have assumed that PCO2 values in cortical
structures are equal to that of systemic blood.
Estimation of disequilibrium pH is dependent on the level of
PCO2.12
Therefore, it is understandable that conflicting results of
PCO2 led to conflicting estimation of
disequilibrium pH. Investigators who maintain the position that there
is no significant PCO2 gap between tubular
fluid and systemic blood supported the presence of disequilibrium pH in
the distal tubular fluid (62), whereas investigators
taking the opposite position denied the establishment of disequilibrium
pH (28). The present model is consistent with the
conclusion of the former investigators, due to our assumption about
PCO2 in cortical structures.
Comparison with microperfusion experiments.
Given the greatly diversified results of microperfusion experiments, it
is certain that no model tubule would simulate all of them with a
single set of model parameters. The present model, whose parameters
have been selected to fit free-flow micropuncture experiments,
simulates properly a subset of microperfusion experiments (20,
54) that report rates of HCO
reabsorption
similar to those in micropuncture experiments. As an attempt to fit the
other groups of microperfusion experiments, we have heuristically
adjusted model parameters and found that by increasing the activities
of transporters in type B intercalated cells we could simulate those
experiments (46, 56, 57) that showed
HCO
secretion in the normal state and flow-dependent
decrease in HCO
reabsorption (equivalently, increase
in HCO
secretion). However, it is clear that
modification of parameters only in type B intercalated cells cannot
resolve all the diversities among microperfusion experiments. For
example, it cannot explain the flow-dependent increase in
HCO
reabsorption in some experiments (20,
57). Furthermore, it cannot explain the more general observation
that in microperfusion experiments transepithelial transport of
Na+, K+, and water is considerably diminished
compared with micropuncture experiments (61). Therefore,
we speculate that the origin of these diversities is most likely
multifactorial, one of the mechanisms being altered activity of type B
intercalated cells. Other proposed factors13 are different
levels of hydration and protein intake of experimental animals
(54) and an unidentified component present in native tubular fluid but missing from artificial perfusate (61).
Obviously, further experimental data addressing the nature of these
mechanisms and their quantitative effects on individual transporters
are required to update the model to simulate all of the details of microperfusion experiments.
 |
FOOTNOTES |
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 (E-mail: hchang-tky{at}umin.ac.jp).
1
They are connecting tubule cells in the
connecting tubule and principal cells in the initial collecting tubule.
Hereafter, they are collectively referred to as "principal cells."
2
FH and JH can
be expressed in terms of model variables as shown in our previous study
(22).
3
The model does not incorporate
CO
, because this component is negligible at
physiological pH.
4
According to Garg and Maren (35),
the rate constant of carbonic acid dehydration
(kd) is 49.6 s
1, and the rate
constant of CO2 hydration (kh) is
0.145 s
1 at 37°C.
5
Throughout the study, CO2
concentraton ([CO2]) is assumed to be 1.2 mM,
corresponding to PCO2 of 40 Torr. This
assumption is based on the experimental observation that there is no
detectable difference in PCO2 between cortical
structures of the kidney and the arterial blood (26). See
DISCUSSION for details.
6
This property of the model H-ATPase is consistent
with the following calculation. Free energy change of ATP hydrolysis
(
G) is calculated to be
G
+ RT log ( [ADP][Pi]/[ATP]) =
59
kJ/mol, which is similar to the values measured in various tissues
including the kidney (55). Due to the stoichiometry of
2H+:1ATP, ~30 kJ is used to transport 1 mole of
H+. Therefore, the lower limit of luminal pH is estimated
to be ~3.2 from the equation, RT log
([H+]m/[H+]c) =
G2
F(
m
c), where
[H+]m and [H+]c are
H+ concentrations in luminal space and intracellular space,
F is the Faraday constant, and
m and
c are luminal and cytosolic electrical potential,
respectively. In this calculation, we have assumed (3)
m = 0;
c =
30 mV; and
log
[H+]c = 7.5.
7
Actually, they reported that
HCO
concentrations of the distal tubular fluid were
9.8 and 9.2 mM at the inlet and the outlet, respectively. Using
estimated tubular fluid flow rates (7.99 nl/min at the inlet and 2.73 nl/min at the outlet; Ref. 22), we have estimated
HCO
reabsorption rate to be 9.8 × 7.99
9.2 × 2.73 = 53.2 pmol/min.
8
Reabsorption rates per unit length have been
converted to total reabsorption rates, assuming the lengths of the
early and the late portions to be 1 and 1.3 mm, respectively.
9
In the rat (32), reported
NH3 permeability in microperfused cortical collecting duct
(0.024 cm/s) is one order of magnitude larger than that in the rabbit.
If we adopt this value, with similar apical-to-basolateral permeability
ratio as reported by Yip and Kurtz (82), total ammonium
reabsorption becomes 24.1 pmol/min (95% of the inlet load).
10
Deflections in HCO
reabsorption rate observable at the perfusate HCO
concentration of 50-60 mM are probably due to the abrupt change in
the perfusate Na+ concentration between the lower
HCO
experiments (70 mM Na+, when
HCO
50 mM) and the higher HCO
experiments (146 mM Na+, when HCO
60
mM). See the legend for Fig. 12 for details.
11
Transport velocities have been calculated with
luminal HCO
of 25 mM, cytosolic Cl
of
46 mM, cytosolic HCO
of 18 mM, and varying luminal
Cl
concentrations; and the Km has
been determined as the luminal Cl
concentration at which
transport velocity becomes one-half of the value with luminal
Cl
concentration of 150 mM.
12
We can see this by examining Eq. 3.
On the left-hand side,
GH2CO3 is actually
negligible due to low concentration of luminal
H2CO3 (<8 µM; see Tables 7 and 8) and small
transepithelial H2CO3 reabsorption rate (0.06 pmol · min
1 · distal
tubule
1). Specifically, H2CO3 is
0.41 pmol · min
1 · tubule
1,
and GH2CO3 is
70.0
pmol · min
1 · tubule
1 in
the basic state. Therefore, we can drop
GHCO3 in Eq. 3, and after
rearrangement we have
Incorporating this equation into Eq. 2
The last term in the above equation,
log10
shows the difference between the disequilibrium pH and the
equilibrium pH. Therefore, discussion on the existence of
disequilibrium pH is greatly affected by the estimation of
[CO2] (or PCO2).
13
Although it is possible that absence of
phosphate and ammonium buffer systems in these experiments had
contributed to the reduced rate of HCO
reabsorption
(17), the present model estimates that fractions of
HCO
reabsorption ascribable to these buffer systems
are 1.0 pmol/min for phosphate buffer and 5.9 pmol/min for ammonium
buffer (GHPO4 and GNH3 in
Table 6). Therefore, it is unlikely that absence of these buffer
systems accounts for the large difference (~50 pmol/min) of
HCO
reabsorption rate between microperfusion and
free-flow micropuncture experiments.
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. Section 1734 solely to indicate this fact.
Received 10 July 2000; accepted in final form 22 February 2001.
 |
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