A mathematical model of the outer medullary collecting duct of
the rat
Alan M.
Weinstein
Department of Physiology and Biophysics, Weill Medical College of
Cornell University, New York, New York 10021
 |
ABSTRACT |
A mathematical model of the outer medullary
collecting duct (OMCD) has been developed, consisting of
-intercalated cells and a paracellular pathway, and which includes
Na+, K+, Cl
,
HCO3
, CO2, H2CO3,
phosphate, ammonia, and urea. Proton secretion across the luminal cell
membrane is mediated by both H+-ATPase and H-K-ATPase, with
fluxes through the H-K-ATPase given by a previously developed kinetic
model (Weinstein AM. Am J Physiol Renal Physiol 274:
F856-F867, 1998). The flux across each ATPase is substantial, and
variation in abundance of either pump can be used to control OMCD
proton secretion. In comparison with the H+-ATPase, flux
through the H-K-ATPase is relatively insensitive to changes in lumen
pH, so as luminal acidification proceeds, proton secretion shifts
toward this pathway. Peritubular HCO3
exit is via a
conductive pathway and via the Cl
/HCO3
exchanger, AE1. To represent AE1, a kinetic model has been developed based on transport studies obtained at 38°C in red blood cells. (Gasbjerg PK, Knauf PA, and Brahm J. J Gen Physiol 108:
565-575, 1996; Knauf PA, Gasbjerg PK, and Brahm J. J
Gen Physiol 108: 577-589, 1996). Model calculations indicate
that if all of the chloride entry via AE1 recycles across a peritubular
chloride channel and if this channel is anything other than highly
selective for chloride, then it should conduct a substantial fraction
of the bicarbonate exit. Since both luminal membrane proton pumps are
sensitive to small changes in cytosolic pH, variation in density of
either AE1 or peritubular anion conductance can modulate OMCD proton secretory rate. With respect to the OMCD in situ, available buffer is
predicted to be abundant, including delivered HCO3
and HPO42
, as well as peritubular NH3.
Thus, buffer availability is unlikely to exert a regulatory role in
total proton secretion by this tubule segment.
proton-potassium-activated adenosinetriphosphatase; AE1; urine acidification; ammonia transport
 |
INTRODUCTION |
INACCESSIBLE TO MICROPUNCTURE,
transport by the outer medullary collecting duct (OMCD) has been
inferred from in vitro studies of rabbit and rat tubules. Such
experiments have established that the OMCD is a proton-secreting
nephron segment with a lumen-positive electrical potential (5,
43, 59). For most of the tubule (inner stripe), there is no
discernible active sodium transport (58),
although at least 60% of the OMCD cells resemble principal cells from
the cortical collecting duct (26, 51, 57). Acid secretion
occurs via electrogenic H+-ATPase and electroneutral
H-K- ATPase (3, 22, 64), with all of the intercalated
cells of this segment (and none of the principal cells) displaying
luminal membrane staining for the H+-ATPase
(1) and H-K-ATPase (6). The luminal membrane
of the intercalated cells has virtually no electrical conductance, whereas that of the peritubular cell membrane is dominated by a
chloride pathway (33, 46). Proton secretion is contingent upon the presence of peritubular chloride (60), presumably
the result of peritubular HCO3
exit in exchange for
Cl
. The anion exchanger specific to the erythrocyte, AE1,
has been identified as that of the peritubular cell membrane of OMCD
intercalated cells (54, 66). Indeed, mutations of AE1 have
recently been associated with a clinical defect in urinary
acidification (62).
A mathematical model of the OMCD provides a means for considering
the cellular interaction of the membrane components of acid secretion.
It also provides a means of extrapolating from in vitro observations to
the likely conditions in vivo. The transport characteristics of the
H+-ATPase have been known for some time (2)
and have been used in a mathematical model of the cortical collecting
tubule (61). More recently, construction of a model of the
inner medullary collecting duct (IMCD) (71) required
revision of a full kinetic model of the H-K-ATPase (10).
In the present work, transport properties of AE1 in erythrocytes at
38°C (20, 31) have been used to fashion a kinetic model
of this peritubular anion exchanger. These components, along with a
peritubular anion channel, provide the critical elements for simulation
of the
-intercalated cell, or equivalently, the OMCD. In what
follows, each of these four transporters appears to be quantitatively
important in intercalated cell proton secretion and thus could be a
suitable candidate for regulation of OMCD acidification. For the tubule
in vivo, the model provides a means of resolving luminal proton
secretion into its three components: titration of luminal
HCO3
, titration of secreted NH3, and
titration of luminal HPO42
. Calculations suggest that
for OMCD in vivo, changes in buffer availability may shift the luminal
composition but are not likely to have a substantial effect on net acid
excretion by this segment.
 |
MODEL AE1 |
Figure 1 depicts a scheme for a
carrier, X, which may be oriented toward the external
(X') or internal (X") membrane face, where either
HCO3
or Cl
may be bound. In this
scheme, it is assumed that anion exchange proceeds via sequential
translocation, the so-called "ping-pong" mechanism (12,
25). It is also assumed that anion binding is rapid relative to
translocation, so that the concentration of bound carrier at each face
is determined from equilibrium binding constants,
KCl and KHCO3. With
respect to chloride, there is NMR evidence supporting this assumption
(13). The carrier is not assumed to be symmetric, so that
1) distinct binding constants are specified for each
membrane face, and 2) the translocation constant for outside
to inside flux (P') will not necessarily be equal to that
for inside to outside flux (P"). Denote b', c', b", and c" as the concentrations of bicarbonate and
chloride within each bath, and bx', cx', and x'
and bx", cx", and x" are the concentrations of
bound and free carrier on each membrane face. Then, the equilibrium condition implies that the ratios of bound to free carrier may be
represented
|
(1)
|
Corresponding to the two unknowns, x' and
x", are the model equations for conservation of total
carrier, xT
|
(2)
|
and for zero net flux of carrier
|
(3)
|
In Eq. 3, the left-hand and right-hand sides represent
the unidirectional inward and outward fluxes of the carrier. There is
no flux of unloaded carrier, corresponding to strict 1:1 stoichiometry for a two-ion system. Using the equilibrium conditions of Eq. 1,
Eqs. 2 and 3 may be rewritten
|
(4)
|
|
(5)
|
This linear system is solved for x' and x"
|
(6)
|
where
+(1+
"+
")(P'b
'+P'c
')

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Fig. 1.
Kinetic scheme for AE1. Carrier X may be
oriented toward the external (X') or internal
(X") membrane faces, where either HCO3 or
Cl are bound. Anion binding is rapid relative to
translocation, so that the concentration of bound carrier at each face
is determined from equilibrium binding constants,
Kc and Kb. The
carrier is not assumed to be symmetric, so that the translocation
constants for outside to inside flux (P') will not
necessarily be equal to those for inside to outside flux
(P"). There is no slippage of empty carrier.
|
|
Thus one obtains expressions for the unidirectional influx and
efflux of bicarbonate and chloride
|
(7)
|
|
(8)
|
and the net efflux for bicarbonate which must be equal and
opposite to that for chloride
|
(9)
|
It should be observed that for the net flux to equal zero when
bathing media are equal (b' = b" and c' = c")
|
(10)
|
The two sides of Eq. 10 have been recognized by
Fröhlich and Gunn (16) as the ratio of the
concentrations of unbound carrier, outward:inward facing, when the
bathing media are either all chloride or all bicarbonate. This ratio
has been denoted the "asymmetry factor," A, and by
virtue of Eq. 10, must be independent of the identity of the
ambient anion.
Recently, Brahm and coworkers (20, 31) investigated the
kinetics of AE1 at 38°C in erythrocytes in a system that could be
used to examine either bicarbonate or chloride self-exchange. When
bicarbonate self-exchange is under consideration, the ambient chloride
concentrations are zero, and unidirectional fluxes of bicarbonate must
be equal. This restricts the representation of the experiment to the
top half of Fig. 1, and thus only four of the eight model
parameters are relevant. According to Eq. 7 the unidirectional efflux of bicarbonate must be
|
(11)
|
so that
|
(12)
|
For the bicarbonate studies, three protocols were used: changing
extracellular bicarbonate only (with cytosolic bicarbonate fixed),
changing cytosolic bicarbonate only, and changing both symmetrically.
Corresponding to each of these experiments are maximal self-exchange
rates (in their notation, Jbmo,
Jbmi, and
Jbms) and apparent affinities
(Kbmo,
Kbmi, and
Kbms). With reference to
Eq. 12, the model defines these measured quantities
|
(13)
|
Equation 13 indicates that the three self-exchange
experiments depend upon only three composite parameters, namely, the
geometric mean of the translocation constants
and the ratios of the affinities to the translocation constants
K'b/xTP'b
and
K
b/xTP
b.
Although the experimental studies of Brahm and colleagues
(31) can provide six observations, if the model is
applicable, then three dependence relations among these observations
should be satisfied. Even when only two of the experiments are
performed, variation of the external anion and symmetric variation of
the anions (31), the model predicts
|
(14)
|
where b" is the constant internal bicarbonate
concentration used in the study. Furthermore, this analysis indicates
that these three experiments cannot suffice to solve for all four model parameters, P'b,
P
b, K'b,
and K
b. Indeed, within the
constraints of the experimental data, one is free to select the ratio
K'b/K
b arbitrarily, and then using the three composite parameters, the four
model parameters are determined. Finally, Eq. 10 provides a
relationship between the bicarbonate parameters and the chloride parameters. Although the two sets of parameters were obtained from
independent self-exchange experiments, the absence of metabolic coupling requires equality of the asymmetry factors
|
(15)
|
In Table 1, AE1
self-exchange parameters have been abstracted from the work of Brahm
and colleagues (20, 31). For both bicarbonate and chloride
studies, data from variation of external anion concentration and
symmetric variation of anion concentration have been used. For
consistency with the bicarbonate study, the chloride data obtained from
red cell ghosts was selected. From the ratios
Kbms/Jbms
and
Kbmo/Jbmo,
the ratio
Kbmi/Jbmi
was obtained as a difference (Eq. 13) and was obtained
similarly for chloride. It is immediately apparent that the data
selected do not satisfy the equilibrium Eq. 15. This
discrepancy was noted by the authors, who preferred to attribute it to
experimental error, rather than inapplicability of the ping-pong scheme
(31). Indeed, Eq. 15 can be satisfied by
choosing different values for the affinities, all still within the
published standard errors. These modified values appear in the second
column of each section of Table 1, and the computation showing
satisfaction of Eq. 15 is indicated there. With respect to
the consistency of the model data with the scheme of Fig. 1 (i.e.,
satisfaction of Eq. 14), both the original and modified
values for both anions give decent agreement, and this computation is
also included. As indicated above, a kinetic model consistent with
these experimental data could be built with any value for the ratio of
affinities for either anion. In the case of chloride, the study of Liu
et al. (42) suggests that the ratio of internal and
external affinities is close to 1.0. Assuming a similar ratio for
bicarbonate, values for the translocation constants and affinities are
indicated in Table 1. Finally, the study of Gasbjerg et al.
(20) indicated that internal bicarbonate appeared to
inactivate the anion exchanger in a noncompetitive way, with a
half-maximal inhibitory concentration, KI,
of 172 mmol/l. In the context of this model, this inhibition is
represented as an effect on the transporter abundance,
xT, relative to a maximal abundance,
xTm
|
(16)
|
In Fig. 2, the kinetic model of AE1
is used to simulate several self-exchange experiments from which the
input parameters were generated. The four top panels of Fig.
2 illustrate HCO3
self-exchange, for which ambient
Cl
= 0, and the calculations correspond to the
experiments displayed in figure 5 of Gasbjerg et al.
(20). In Fig. 2, A and B, external HCO3
is varied while the internal concentration is
fixed, either at 50 or 165 mmol/l. The slightly smaller values for the
efflux rates obtained from the model (more apparent in Fig.
2B), derive from the higher value taken for
Kbms. In Fig. 2, D and
E, internal HCO3
is varied, either alone
or symmetrically. The appearance of a maximal efflux rate in a
neighborhood of 100 mmol/l HCO3
is a consequence of
the internal site for noncompetitive inhibition of the exchanger. The
two bottom panels of Figure 2 illustrate Cl
self-exchange, in which ambient HCO3
= 0, and the
calculations correspond to experiments in red cell ghosts shown in
figures 2 and 4 of Knauf et al. (31). In Fig. 2C external Cl
is varied, with internal
Cl
= 175 mmol/l, and in Fig. 2F, the
concentrations on both sides of the membrane are varied symmetrically.
For these simulations, there is little discrepancy with the
experimental data.

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Fig. 2.
Model simulations of self-exchange of
HCO3 or Cl via AE1. In A and
B, model Eq. 7 (using parameters from Table 1) is
evaluated over a range of external (CE)
HCO3 , while the internal concentration
(CI) is fixed, either at 50 or 165 mM; ambient
Cl is absent. In D and E, model
Eq. 7 is solved while internal HCO3 is
varied, either alone or symmetrically. In C, model Eq. 8 is evaluated over a range of external Cl , with
internal Cl = 175 mM; ambient HCO3
is absent. In F, the Cl concentrations on both
sides of the membrane are varied symmetrically.
|
|
Figure 3 displays calculations
illustrating Cl
/HCO3
flux by the model
AE1 operating as an exchanger in the neighborhood of a reference
condition: internal HCO3
and Cl
concentrations of 26 and 29 mmol/l, and external concentrations of 26 and 114 mmol/l, respectively. This reference is approximately that of
the model tubule developed below. Each panel of Fig. 3 illustrates the
variation of a single internal (A and B) or
external (C and D) anion concentration (solid
curves). The most obvious feature of Fig. 3 is the greater sensitivity
of model fluxes with variation of cytosolic concentrations (compared
with variation of external concentrations), with the greatest
sensitivity to changes in internal HCO3
. The numbers
C(dJ/dC) are the derivatives of the fluxes
with respect to the fractional change in ion concentration, taken at the reference, and are essentially derivatives with respect to chemical
potential. For each panel of Fig. 3 and each value of the logarithmic
derivative, a dotted curve is drawn to approximate the exchange rate as
a linear function of the logarithm of the abscissa.

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Fig. 3.
Cl /HCO3 flux by the model
AE1 operating as an exchanger (Eq. 9) in the neighborhood of
a reference condition: internal HCO3 and
Cl concentrations 26 and 29 mM, and external
concentrations 26 and 114 mM, respectively. In A and
B, internal HCO3 and internal
Cl are varied; in C and D, the
external anion concentrations are the independent variables. The dashed
curves are best-fit single exponentials through the reference
condition.
|
|
 |
MODEL OMCD |
The model outer medullary collecting duct formulated here will be
essentially that found in the inner stripe, in which transport activity
appears to be that of the intercalated cells. With this simplification,
all transport pathways will be ascribed to either a transcellular
pathway across intercalated cells or to a paracellular pathway. The
model will be formulated both as an OMCD epithelium, with specified
luminal and peritubular conditions, or as a tubule, in which luminal
concentrations vary axially. Figure 4
displays both models, in which cellular and intercellular compartments line the tubule lumen. Within each compartment the concentration of
species i is designated
C
(i), where
is lumen (M),
interspace (E), cell (I), or peritubular solution (S).
Within the epithelium the flux of solute i across membrane

is denoted J
(i) (mmol · s
1 · cm
2), where 
may refer to
luminal cell membrane (MI), tight junction (ME), lateral cell
membrane (IE), basal cell membrane (IS), or interspace
basement membrane (ES). Along the tubule lumen, axial flows of
solute are designated FM(i) (mmol/s). The 12 model solutes are Na+, K+, Cl
,
HCO3
, CO2, H2CO3,
HPO42
, H2PO4
,
NH3, NH4+, H+, and urea, as
well as an impermeant species within the cells and possibly within the
lumen. These are the minimal set of solutes that will permit
representation of net acid excretion.

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Fig. 4.
Schematic representation of outer medullary collecting
duct (OMCD) epithelium, consisting of intercalated cell and lateral
intercellular space (LIS), and tubule model, whose lumen is lined by
this epithelium. Intraepithelial fluxes are designated
J (i), where the subscript  refer to
luminal cell membrane (MI), tight junction (ME), lateral cell membrane
(IE), basal cell membrane (IS), or interspace basement membrane (ES).
Along the tubule lumen, axial flows are designated
FM(i).
|
|
To formulate the equations of mass conservation with multiple reacting
solutes, consider first an expression for the generation of each
species within each model compartment. Within a cell or interspace, the
generation of i (s
(i)) is equal to its net export plus its accumulation
|
(17)
|
|
(18)
|
where V
is the compartment volume (in
cm3/cm2). Within the tubule lumen, solute
generation is appreciated as an increase in axial flux, as transport
into the epithelium, or as local accumulation.
|
(19)
|
where BM is the tubule circumference,
and AM is the tubule cross-sectional area.
With this notation, the equations of mass conservation for the
nonreacting species (Na+, K+, Cl
,
and urea) are written
|
(20)
|
where
= E, I, or M. For the
phosphate and for the ammonia buffer pairs, there is conservation of
total buffer
|
(21)
|
|
(22)
|
Although peritubular PCO2 will be
specified, the CO2 concentrations of the cells, interspace,
and lumen are model variables. The relevant reactions are
where dissociation of H2CO3 is rapid, and
assumed to be at equilibrium. Since HCO3
and
H2CO3 are interconverted, mass conservation
requires
|
(23)
|
for
= I or E, whereas for the tubule
lumen
|
(24)
|
In each compartment (
= I, E, or
M), conservation of total CO2 is expressed as
|
(25)
|
Corresponding to conservation of protons is the equation for
conservation of charge for all the buffer reactions
|
(26)
|
where zi is the valence of species
i. In this model, conservation of charge for the buffer
reactions takes the form
|
(27)
|
The solute equations are completed with the chemical equilibria of
the buffer pairs:
HPO42
:H2PO4
,
NH3:NH4+, and
HCO3
:H2CO3. Corresponding to
the electrical potentials, 
, for
= E, I, or M, is the equation for electroneutrality
|
(28)
|
With respect to water flows, volume conservation equations for
lumen, interspace, and cell can be used to compute the three unknowns:
luminal volume flow, lateral interspace hydrostatic pressure, and cell
volume. (Cell hydrostatic pressure is set equal to luminal pressure;
total cell impermeant content is assumed fixed.) This approach has been
adopted for the epithelial model with fixed peritubular conditions but
is not satisfactory for the tubule for which the large variations in
peritubular osmolality would impact unrealistically on cytosolic
electrolytes. As utilized previously in modeling the IMCD
(70), the approach to the tubule model has been to
restrict simulations to steady-state problems and to assume that cell
volume homeostasis has been achieved by adjustment of an impermeant
osmolyte, b. Thus with cell volume specified and fixed,
CI(b) is the model variable used to satisfy the equations for fluid balance across the luminal and peritubular cell
membranes. Across each cell membrane, the volume fluxes are proportional to the hydrosmotic driving forces. With respect to the
lateral interspace, its volume, VE, and its basement
membrane area, AES, are functions of
interspace hydrostatic pressure, PE
|
(29)
|
where VE0 and
AES0 are reference values for volume
and outlet area, respectively, and
E is
a compliance.
Solute transport is either electrodiffusive (e.g., via a channel),
coupled to the electrochemical potential gradients of other solutes
(e.g., via a cotransporter or an antiporter), or coupled to metabolic
energy (via an ATPase). This is expressed in the model by the flux
equation
|
(30)
|
In Eq. 30, the first term is the Goldman relation for
ionic fluxes, where h
(i) is a solute
permeability, and C
(i) and
C
(i) are the concentrations of i
in compartments
and
, respectively. Here
|
(31)
|
is a normalized electrical potential difference, where
zi is the valence of i, and


is the potential
difference between compartments
and
. The second term of the
solute flux equation specifies the coupled transport of species
i and j according to linear nonequilibrium
thermodynamics, where the electrochemical potential of j in
compartment
is
|
(32)
|
For each of these transporters, the assumption of fixed
stoichiometry for the coupled fluxes allows the activity of each transporter to be specified by a single coefficient. The exception to
this representation of coupled fluxes is that of
Cl
/HCO3
exchange across the peritubular
membrane, referable to AE1. Here the kinetic model developed above has
been used, so that a single transporter density parameter suffices to
represent its activity.
In this model, there are two proton ATPases within the luminal cell
membrane. The H-K-ATPase is identical to that which has been developed
for the model of the IMCD (71), with only the transporter
density adjusted to suit the change in context. Also in earlier work,
an empiric expression representing the H+- ATPase was
devised by Strieter et al. (61), approximating data of
Andersen et al. (2) for turtle bladder
|
(33)
|
where J(H+)max is the maximum
proton flux, and
MI(H+) is the
electrochemical potential difference of H+ from the cytosol
to the lumen;
MI defines the steepness
of the function, and
0 defines the point of
half-maximal activity. The important finding of Andersen et al.
(2) was that the proton flux depended upon both electrical
and chemical components of the proton potential and that the flux went
from maximal to zero over a range of the proton potential of 180 mV (or
3 pH units or 17.5 J/mmol). The data of Andersen et al. (figure 9 in
Ref. 2) are approximately represented by choosing
= 0.4 and
0 =
4.0 J/mmol. Figure
5 illustrates the response of each of
these proton pumps to changes in luminal and cytosolic conditions in a
neighborhood of a reference condition: lumen and cell pH, 7.34, lumen
and cell K+, 45 and 130 mmol/l, and transmembrane potential
difference (PD), 42 mV. The pump densities were taken so that at the
reference point, the contributions of each transporter were equal. In
Fig. 5, left, luminal pH is varied while cytosolic
conditions are fixed. Transport by the H+-ATPase increases
with luminal alkalinization and decreases nearly 90% with
acidification of the lumen by 1 pH unit. In this model H-K-ATPase,
transport is predicted to be quite insensitive to luminal pH near the
reference, only declining after a 2 pH unit reduction. In Fig. 5,
right, cytosolic pH has been varied, and it is apparent that
both ATPases are relatively sensitive to small changes in cell pH.

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Fig. 5.
Proton fluxes as a function of luminal and cytosolic
conditions in the neighborhood of a reference condition: lumen and cell
pH, 7.34; lumen and cell K+, 45 and 130 mM, respectively;
and transmembrane potential difference (PD), 42 mV. Pump densities were
chosen so that at the reference point, fluxes through each transporter
were equal. Left: luminal pH is varied while cytosolic
conditions are fixed. Right: cytosolic pH is the independent
variable.
|
|
Within the peritubular membrane, the Na-K- ATPase is represented by
the expression
in which the half-maximal Na+ concentration,
KNa, increases linearly with internal
K+, and the half-maximal K+ concentration,
KK, increases linearly with external
Na+ (19). The pump flux of K+ plus
NH4+ reflects the 3:2 stoichiometry
|
(35)
|
with the transport of either K+ or
NH4+ determined by their relative affinities,
KK and
KNH+4
|
(36)
|
Analogous expressions are written for active transport at the
basal cell membrane, JISact.
 |
MODEL PARAMETERS |
The parameters displayed in Table
2 were selected so that
the model tubule might correspond most closely to the OMCD of the rat.
Where rat data were not available, rabbit measurements were used for
guidance. With respect to acidification, there seems to be little to
distinguish the outer and inner stripes of the rat OMCD: reported
proton secretory rates in vitro (in pmol · mm
1 · min
1) for the outer stripe
[10.2 (5), 22.1 (18), and 37.6 (22)] and for the inner stripe [24.4 (15)
and 13.1 (47)] are similar and cover a broad range; the
fractional content of intercalated cells appears to be about 35% for
both segments (26, 51, 57); and there is no evidence in
the rat for the presence of membrane-bound carbonic anhydrase (CA-IV),
either from histochemical (9) or functional studies
(15). [This is in contrast to the rabbit, for which
membrane-bound CA appears to be present in the inner stripe but not the
outer stripe (55)]. Thus, in view of the relatively short
length of the outer stripe of rat OMCD (0.5 mm), compared with the
inner stripe (1.5 mm) (32), the whole tubule has been
approximated as a uniform 2-mm segment with a 30-µm inner diameter.
For a tubule thickness of 9 µm, the 35% intercalated cell fraction
corresponds to an intercalated cell volume (VI) of
about 0.3 × 10
3 cm3/cm2 of
epithelium. Estimates of intercalated cell surface area suggest a ratio
of peritubular to luminal membrane of between 3 and 4 to 1 and an
absolute luminal membrane area of about 2 cm2/cm2 epithelium (45, 51, 57).
The volume of the lateral intercellular space was taken to be about
10% of the epithelial volume (with a relatively small compliance), a
value comparable to that observed in cortical collecting duct
(72).
Figure 6 depicts several of the important
cellular transport pathways. Both H-K-ATPase and
H+- ATPase are contained within the luminal membrane of
OMCD (73), and the rates of proton secretion by H-K-ATPase
relative to H+-ATPase have been identified in both rat
[2.5 (22)] and rabbit [0.7 (4), 0.8 (64), 1.0 (68), and 2.0 (3)].
To select pump densities for the model OMCD, proton transport via the
two ATPases was set approximately equal at neutral luminal pH, and relative activity of the pumps as a function of luminal pH is explored
in the model calculations. There is no evidence for any other coupled
transport pathway within the luminal membrane, and in the rabbit OMCD,
electrophysiological study indicates no significant luminal membrane
conductance (33, 34, 46). Furthermore, there are no
detectable aquaporin AQP-2 water channels in the luminal cell membrane
of intercalated cells in the rat (17, 48). Accordingly,
the total luminal membrane water permeability was set at 1% of the
peritubular membrane water permeability. In view of the intense
staining for carbonic anhydrase within OMCD cells (44),
the rate constants for full catalysis (10,000-fold increase) were
assumed for the cytosolic compartment.

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Fig. 6.
OMCD cellular transport pathways, along with model cell
fluxes (pmol · s 1 · cm 2)
computed for luminal and peritubular conditions representing
corticomedullary junction (Table 3).
|
|
The peritubular membrane of rat OMCD contains Na-K-ATPase
(52), and its density was selected to obtain a suitably
low cytosolic Na+ concentration. In addition to the
Cl
/HCO3
exchanger, a peritubular
Na+/H+ exchanger is present in rabbit OMCD
(8). It has been demonstrated in intercalated cells (in
addition to principal cells), where it is capable of proton extrusion
rates comparable to that of the proton pumps (41, 69). Its
density coefficient was selected to yield fluxes comparable to those of
the H+-ATPase. The model peritubular membrane also contains
a coupled phosphate transporter, which maintains a small entry flux.
Although the total conductance of the peritubular membrane in rabbit
OMCD intercalated is unknown, it has been established that the
principal ion permeability is that for chloride (33, 34),
while that for potassium is much smaller (46). Whereas a
variety of chloride channels show substantial bicarbonate conductance
(39, 50), bicarbonate conductance of the intercalated cell
peritubular membrane has not been demonstrated. Koeppen
(33) did find significant steady-state membrane
depolarization with reduction in peritubular HCO3
,
but the time course was slow, and no rapid depolarization was evident.
The peritubular chloride permeability for the model cell was estimated
from the constraints of cell PD (
30 to
40 mV), a suitable cell
chloride concentration, and the need to recycle all of the
Cl
uptake through AE1 back out through this channel.
Potassium permeability was taken as
that of chloride,
bicarbonate permeability as
that of chloride, and
NH4+ permeability as 1/4 that of potassium.
Overall epithelial electrical conductance (in mS/cm2) has
been measured in OMCD only for rabbit and was found to be slightly higher in outer stripe [3.7 (34) and 5.7 (36)] than in inner stripe [1.9 (33), 2.2 (36), and 3.4 (46)]. These conductances are
compatible with estimates of ion permeability,
PNa:PK:PCl = 3.9:5.9:4.8 × 10
6 cm/s (27, 38, 58).
In the rat, OMCD NH4+ permeability is 1.3 × 10
5 cm/s (14). Presumably, these epithelial
ion permeabilities reflect the properties of the OMCD tight junctions.
For the selection of model tight junction solute permeabilities, it has
been assumed that OMCD K+ permeability is approximately
that of NH4+ and that the relative ion permeabilities
in rat are comparable to those of rabbit. This yields an overall
epithelial conductance for rat OMCD about twice that of rabbit. The
interspace basement membrane conductance was assumed to be about two
orders of magnitude greater than that of the tight junction, and solute
permeabilities were proportional to diffusivity in free solution.
Membrane permeabilities have also been assigned for the non-ionic
species: water, urea, NH3, CO2, and
H2CO3. In the rabbit, antidiuretic hormone
(ADH)-stimulated OMCD water permeability has been reported as 0.046 cm/s, an increase about 30-fold above the unstimulated permeability
(29). Although a value for rat OMCD is not available,
water permeabilities for the two species are comparable in cortical
collecting tubule. For the model calculations, a water permeability
about half-maximal was assumed and referred entirely to the
"paracellular" pathway (which includes the principal cell-lateral
interspace route). All membrane and tight junction reflection
coefficients are assumed to be 1.0, while those for interspace basement
membrane are 0.0. The overall urea permeability has been measured for
rat OMCD (3.5 × 10
5) and is about 10-fold greater
than that for rabbit (23). In the absence of information
about the transepithelial route for urea permeation, for this model,
45% of the epithelial permeability has been ascribed to the
intercalated cell (with uniform unit membrane urea permeability), and
the remainder paracellular. With respect to NH3, the rat
OMCD permeability, 0.012 cm/s (14), is sufficiently high
to reflect diffusion limitation across the cellular layer, rather than
membrane limitation. Within the scope of this model, it suffices to
ascribe this permeability to cell membranes, with uniform unit membrane
permeability. This avoids creating a paracellular NH3
pathway wherein one presumes the (unrealistic) routing of the bulk of
the NH3 flux through the lateral interspace. Similar
concerns apply to CO2, so that CO2
permeabilities have been assumed equal to those of NH3.
H2CO3 has been assumed to permeate at 1% the
rate of CO2.
 |
MODEL CALCULATIONS |
Table 3 and Fig. 6 display
the solution of the equations for the epithelial model of
OMCD with lumen and bath conditions suggestive of the corticomedullary
junction. Overall, the lumen is isotonic to blood, with a urea
concentration comparable to that from a cortical nephron
(23). The concentrations of Na+,
K+, and Cl
are within the range reported for
the last accessible micropuncture site (7). The higher
values used here reflect the transition to isotonicity (via water
abstraction) within the cortex of the antidiuretic kidney. From another
perspective, in later calculations the total volume flow into the model
OMCD will be assumed to be 7.2% of glomerular filtration rate (GFR),
or 36 µl/min. With this assumption, the concentrations chosen
correspond to Na+ and K+ delivery to this
segment of 3.6% and 65% of filtered loads. The HCO3
concentration, 10 meq/l, corresponds to a delivery of 2.9% of filtered
load, which may be compared with 6.4% delivery found at the last
micropuncture site (11). The NH4+
concentration, 2 meq/l, also yields a delivered load close to that
reported for the rat (53). The luminal total phosphate concentration corresponds to approximately 85% fractional reabsorption in proximal nephron.
The computed concentrations of Na+, K+, and
Cl
within the model cell (Table 3) are within the range
of values obtained by microprobe for intercalated cells of the cortical
collecting duct (7, 21). The peritubular membrane PD,
35.6 mV, is comparable to values obtained in intercalated cells of
the OMCD of the rabbit (30, 36). The open-circuit PD of
the model, +1.2 mV, is low, but this PD also reflects the negative
Na+ diffusion potential (which dominates the positive
K+ diffusion potential). In a calculation in which the
luminal ion composition is identical to that of the bath, the
open-circuit PD is +4.8 mV. These values are consistent with the PD
determinations in rat OMCD [+2.5 mV (22) and +1.05 mV
(15)] as well as some measurements in rabbit
(4), albeit lower than other observations in rabbit OMCD
[+16.5 mV (36) and +10.6 mV (59)]. The
total proton secretory rate, 594 pmol · s
1
· cm
2 (34 pmol · min
1 · mm
1), increases when the composition of the luminal
perfusion solution is identical to that of the peritubular bath
(characteristic of in vitro tubule studies), 722 pmol · s
1 · cm
2 (41 pmol · min
1 · mm
1). These values are high
even for the rat [24.4 pmol · min
1 · mm
1 (15)], but as will be indicated below,
this rate of proton secretion is only just capable of titrating the
base delivery to the OMCD.
In view of the large transepithelial solute concentration gradients
across OMCD, the overall epithelial solute permeabilities have a
substantial impact on luminal solute flows. Table
4 displays the results of simulating
idealized epithelial permeability determinations. For these
calculations, a short-circuited tubule epithelium in vitro was
represented, bathed by equal luminal and peritubular solutions of
composition (mmol/l) 140 Na+, 10 K+, 119 Cl
, 25 HCO3
, 1.5 CO2, 3.9 total phosphate, 5.0 urea, 1.0 NH4+, and 0.1 impermeant. A series of calculations were performed in which each
luminal solute concentration in turn was lowered and then raised by 0.1 mmol/l. The change in solute flux relative to the change in
concentration is listed in Table 4 as the permeability, HM(i) (in cm/s), and is the average
of the two determinations. Alternatively, epithelial ion permeability
was determined by imposing a transepithelial voltage (positive and
negative 0.1 mV). The change in ion flux relative to voltage, when
multiplied by z(i)F is the partial conductance,
GM, shown on the right (in
mS/cm2). The permeabilities displayed are, by design,
comparable to those cited in the previous section. In particular, these
permeabilities predict a substantial secretory flux of sodium, as well
as significant reabsorption of potassium and urea by OMCD.
Table 5 and Figure
7 display the predictions of
the model OMCD configured as a 2-mm tubule within the renal medulla.
Initial conditions are those already specified at the corticomedullary junction, with an inlet volume flow of 5 nl/min for each tubule, or 36 µl/min for the 7,200 OMCD of the rat. The peritubular composition at
the endpoint includes a doubling of Na+ and K+
concentrations, along with an increase of NH4+ to 3 mmol/l and an increase of urea to 20 mmol/l. Peritubular composition at
intermediate points are determined by linear interpolation. The
differential equations of mass conservation were cast as a centered
difference scheme and integrated with a mesh of 40 points; accuracy was
confirmed by refining this spacing. Table 5 indicates that osmotic
equilibration was nearly complete with reabsorption of about half the
entering volume flow. There is secretion of Na+ equal to
25% of the delivered load, with reabsorption of approximately 40% of
entering potassium and 60% of entering urea. [Fortuitously, the
absolute rates of Na+ absorption and K+
secretion are nearly equal, and thus resemble Stokes'
(58) observation of OMCD Na+ and
K+ fluxes, but bath and perfusion conditions here are quite
different from those experiments.] Proton secretion along the OMCD
results in reabsorption of 50% of delivered HCO3
(24 pmol/min), more than doubling of luminal NH4+ (13 pmol/min), and titration of luminal HPO42
(8 pmol/min). The axial profiles of pH, HCO3
, and
NH4+ are shown in Fig. 7. Within the first 0.6 mm of
OMCD, a disequilibrium pH of about 0.4 units is developed, and
NH3 influx into this portion of the tubule is sluggish. In
the latter portion of OMCD, where the acid lumen is fully developed,
NH3 influx is linear, reflecting the increase in
peritubular concentration. Luminal acidification results in a small
decrease in the rate of proton secretion, largely due to its impact on
transport by the H+-ATPase.

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Fig. 7.
Predicted acidification along the OMCD configured as a
2-mm tubule within the renal medulla. Initial conditions are those of
the corticomedullary junction, with an inlet volume flow of 5 nl/min
(Table 5). Left: calculations using baseline parameters, in
which hydration of CO2 within the lumen is uncatalyzed.
Right: there is a 10,000-fold increase in this reaction
rate. For all panels, the abcissa is position along the tubule.
Top: lumen pH. Middle: luminal concentrations of
HCO3 and NH4+. Bottom:
total proton secretion and that referable to the H+-ATPase.
CA, carbonic anhydrase.
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The case in which luminal CA is present is considered in Fig. 7,
right. Proton secretion now consumes 75% of delivered
HCO3
(38 pmol/min), but makes a smaller contribution
to increase luminal NH4+ (8 pmol/min) and titrates
luminal HPO42
(6 pmol/min). Thus the rate of net acid
secretion in the presence of luminal CA (52 pmol/min) exceeds that in
the absence of CA (45 pmol/min), but the differences are not predicted
to be striking. As indicated in the previous section, the rate
coefficient for the hydration of CO2 in the cytosol has
been set at 10,000 times the uncatalyzed value. For the rate of proton
secretion by OMCD, calculations with this model indicate that any value
for the hydration rate coefficient greater than 1% of the selected
value (i.e., anything greater than 100 times the uncatalyzed value)
yields virtually identical cytosolic pH, and thus no impact on luminal proton secretion. Within the lateral interspace, no catalysis of
CO2 hydration has been assumed. With reference to Table 3, an acid disequilibrium pH of 0.14 units develops, and this is due to
the activity of the Na+/H+ exchanger within the
peritubular membrane. Model calculations in which full CA catalysis is
extended to the lateral interspace have no significant impact on
luminal acid secretion but do increase NH3 secretion by
removing an intraepithelial acid compartment. The NH3
fluxes of Table 3 describe a situation in which acidification of the
interspace shunts NH3 taken up across the basal cell
membrane and returns NH4+ back to the peritubular
blood. This effect appears to be largely due to the small
NH3 fluxes under the assumed corticomedullary junction
conditions, but its impact diminishes rapidly in importance as
peritubular NH4+ concentrations increase along the medulla.
In view of the uncertainty of the outer medullary interstitial
NH4+ concentration profile, tubule transport is
examined over a a range of values for peritubular NH4+.
This is done in the calculations of Fig.
8, which utilize the tubule model with
the same initial conditions as in Fig. 7. In these calculations,
peritubular NH4+ remains at 1 mmol/l at
x = 0, while the concentration at
x = 2 mm is varied from 1 to 9 mmol/l. For each
simulation, interstitial NH4+ concentration along the
tubule is the linear interpolation between start and endpoints. Figure
8, top, displays the end-luminal HCO3
and
NH4+ concentrations; the middle shows
end-luminal pH; and the bottom contains the three components
of acid excretion by a single OMCD segment: HCO3
reabsorption, NH4+ addition, and
HPO42
titration. It is clear that with the increase
in peritubular NH4+, there is progressive buffer
addition to the tubule lumen and progressive tubule fluid
alkalinization. At all but the lowest peritubular NH4+
concentrations, there is an increase in end-luminal
HCO3
concentration above that at the inlet (10 mmol/l). The curve labeled "total proton secretion" is the sum of
the three buffer changes, and is relatively insensitive to changes in
peritubular NH4+. Thus, availability of ammonia buffer
can shift the luminal composition substantially but appears to have
little impact on total proton secretion by OMCD. There is also
uncertainty regarding the outer medullary interstitial urea profile,
and Table 6 displays the results of
exploring OMCD urea reabsorption over a range of conditions. As with
the NH4+ calculations, the initial conditions remain
those of Fig. 7, while the peritubular urea concentration at
x = 2 mm is varied from 20 to 100 mmol/l;
interstitial urea concentrations are the linear interpolation. It is
clear that at all concentrations, the model predicts substantial outer
medullary urea reabsorption.

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Fig. 8.
Acid/base transport by OMCD: effect of peritubular
NH4+. The model OMCD is solved as terminal
(x = 2 mm) peritubular NH4Cl is varied from
1 to 9 mmol/l, and this concentration is taken as the abcissa. Other
conditions for lumen and peritubular bath are as in Fig. 7.
Top: end-luminal concentrations of HCO3
and NH4+. Middle: pH. Bottom:
resolves total proton secretion into its component buffer titrations:
changes in axial flow of HCO3 ,
HPO42 , and NH4+ along the tubule.
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Although pH dependence of the important OMCD transporters has been
illustrated above (Figs. 3 and 5), in the calculations that follow, the
epithelial model has been used to examine interactions of these
components, with emphasis on possible regulatory sites. The impact of
luminal HCO3
concentration on OMCD proton secretion
is displayed in Fig. 9. The bath
conditions are those of the corticomedullary junction (Table 3), with
the exception of luminal HCO3
-for-Cl
substitution. Figure 9 includes peritubular PD, cytosolic pH, and the
components of luminal proton secretion and peritubular base exit
(lateral plus basal terms). By design, the contributions of
H+-ATPase and H-K- ATPase are nearly equal under baseline
conditions (HCO3
= 10 mmol/l). The curve labeled
"base exit" is equal in magnitude to the "total" proton
secretion, but it is clearly less than the sum of
HCO3
fluxes via AE1 and the conductive pathway. This
is due to the presence of peritubular proton reabsorption by the
Na+/H+ exchanger. Over the range of luminal
HCO3
concentrations (1.0 to 50 mmol/l), there is a
relatively steep dependence of flux through the H+-ATPase.
With luminal acidification, H+-ATPase flux is nearly shut
off, resulting in cell acidification and peritubular depolarization.
Since the H-K-ATPase is stimulated by the decrease in cytosolic pH and
is relatively insensitive to lumen pH, the decrease in lumen
HCO3
actually results in a small increase in flux
through this transporter. At the peritubular membrane,
HCO3
fluxes through the two exit pathways move
(fortuitously) in parallel, although the driving forces for each
pathway are distinct. In Figs. 10-13, the density of each of the
four important components has been varied over approximately 100-fold
range, from 0.03 to 3.0 times the baseline value, and a transport
tableau similar to that of Fig. 9 has been generated. In all of these
calculations, the bath conditions are those of the corticomedullary
junction. In Fig. 10, the density of
the luminal membrane H+-ATPase has been varied. The curves
are virtually identical to those of Fig. 9, confirming the impact of
luminal pH as solely through this transporter. In Fig.
11, the density of the H-K-ATPase has
been varied, and the effect is more complex. Although total proton
secretion is nearly identical, in the absence of the electrical impact
of the H+-ATPase, there is peritubular membrane
hyperpolarization with the decrease in H-K-ATPase activity. This is due
to the decrease in cytosolic Cl
concentration coincident
with the decrease in cell HCO3
. The hyperpolarization
exacerbates the cytosolic acidosis associated with the decrease in
luminal proton secretion. In short, modulating the
H+-ATPase results in less derangement of cell composition
than modulating the H-K-ATPase. Figure
12 contains results of calculations in
which the AE1 density is varied. As expected, decreasing AE1
alkalinizes the cell and shifts HCO3
exit to the
conductive pathway. Since both of the luminal proton transporters are
sensitive to cytosolic pH, changes in AE1 activity modulate total
proton secretion. Coincident with the decrease in AE1, there is a
decrease in cytosolic Cl
, and as in Fig. 11, peritubular
hyperpolarization. Finally, Fig. 13 examines the effect of
modulating the conductive pathway. Since there is no
evidence for separate HCO3
and Cl
channels, the peritubular anion permeability for each species has been
varied proportionally. As with changes in AE1, decreases in
HCO3
conductance alkalinize the cell and decrease
proton secretion. The peritubular membrane hyperpolarizes because of
the decrease in Cl
conductance (despite the increase in
cell Cl
), due to the relatively greater contribution of
K+ to total membrane conductance. Ultimately, increases in
cell Cl
start to diminish the AE1 flux.

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Fig. 9.
-Intercalated cell function: variation of luminal
HCO3 . Calculations use the OMCD epithelial model.
Bath conditions are those of the corticomedullary junction (Table 3),
with the exception of luminal
HCO3 -for-Cl substitution. Peritubular
PD, cytosolic pH, and the components of luminal proton secretion and
peritubular base exit (lateral plus basal terms) are shown.
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Fig. 10.
-Intercalated cell function: variation of luminal
H+-ATPase. Calculations use the OMCD epithelial model with
corticomedullary junction bath conditions. Luminal membrane
H+-ATPase density, relative to control, is plotted on the
abcissa, and the tableau of dependent cellular variables is as in Fig.
9.
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Fig. 11.
-Intercalated cell function: variation of luminal
H-K-ATPase. Calculations use the OMCD epithelial model with
corticomedullary junction bath conditions. Luminal membrane H-K-ATPase
density, relative to control, is plotted on the abcissa, and the
tableau of dependent cellular variables is as in Fig. 9.
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Fig. 12.
-Intercalated cell function: variation of peritubular
AE1. Calculations use the OMCD epithelial model with corticomedullary
junction bath conditions. Peritubular membrane AE1 density, relative to
control, is plotted on the abcissa, and the tableau of dependent
cellular variables is as in Fig. 9.
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Fig. 13.
-Intercalated cell function: variation of peritubular
bicarbonate conductance, g(HCO3 ), and
chloride conductance, g(Cl ). Calculations use
the OMCD epithelial model with corticomedullary junction bath
conditions. Peritubular membrane Cl and
HCO3 permeabilities are varied proportionally, and
their value, relative to control, is plotted on the abcissa. The
tableau of dependent cellular variables is as in Fig. 9.
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This epithelial model affords the opportunity to consider the impact of
modulating proton secretion on the volume of the intercalated cells.
For the calculations of Figs. 10-13, the predicted cell volumes (relative to baseline) are displayed in Fig.
14. It is clear that changing the rate
of transport by changing the H-K-ATPase density buys a new set of
problems. Doubling this transporter density (about a 20% increase in
total proton transport) results in a near doubling of cell volume. In
contrast, changing the flux through the H+-ATPase leaves
cell volume virtually unchanged. Indeed, the hyperpolarization associated with the increase in flux through this transporter produces
a slight decrease in cell volume. Changes in AE1 density produce
relatively modest parallel changes in cell volume. Modulating peritubular anion permeability shows less of an effect on cell volume
when this pathway is increased, due to the concomitant Cl
exit. However, conductances below about 30% of baseline start to yield
steep increases in cell volume. In proximal tubule, it has been shown
that changes in flux through the Na-K-ATPase modulate the (dominant)
peritubular K+ conductance via changes in cytosolic ATP
concentration and the effect of ATP to shut the K+ channel
(63). In Fig. 14, top left, the effect of
coordinate regulation of H-K-ATPase and peritubular K+
permeability are considered. For these calculations, the three-fold increase in H-K-ATPase density is accompanied by a two-fold increase in
peritubular K+ permeability, and for each 17% decrease in
H-K-ATPase density there is a concomitant decrease of about 10% in the
K+ permeability. The result is nearly perfect cell volume
homeostasis on the side of increasing H-K-ATPase. This modulated
K+ permeability also proves to be homeostatic with respect
to cytosolic pH, as seen by comparing the acid/base tableau of Fig.
15 with that of Fig. 11. It also shifts
the H-K-ATPase increase from depolarizing to hyperpolarizing and thus
enhances total base exit.

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Fig. 14.
-Intercalated cell volume: modulation of luminal and
peritubular membrane transporters. For the calculations of Figs.
10-13, the predicted cell volumes (relative to baseline) are
displayed in the respective panel. The additional curve in the
H-K-ATPase panel [modulated g(K+)] is the
result of a calculation in which peritubular K+
permeability is varied in proportion to the change in H-K-ATPase
density (Fig. 15).
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Fig. 15.
-Intercalated cell function: variation of
luminal H-K-ATPase and peritubular K+ permeability.
Calculations use the OMCD epithelial model with corticomedullary
junction bath conditions. Luminal membrane H-K-ATPase density (abcissa)
is varied as in Fig. 11, and the variation in peritubular membrane
K+ permeability is two-thirds that of the pump density. The
tableau of dependent cellular variables is as in Fig. 9.
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To this point, all of the calculations have examined steady-state
behavior of the OMCD. The model has been formulated, however, with
inclusion of time-dependent accumulation terms in the mass balance
Eqs. 17-19, and the computer code represents these
terms in a centered difference scheme. This permits calculation of OMCD transients, and in particular, simulation of experiments in which cell
pH is measured following acid loading. Figure
16 illustrates a protocol similar to
that used by Kuwahara et al. (40), in which the
contributions of the H+-ATPase and H-K-ATPase toward pH
recovery are compared. In these calculations, the initial condition is
a steady state in which ambient ammonia has been raised to 20 mM,
luminal K+ has been reduced to near zero, and peritubular
Na+/H+ exchange is blocked. Specifically, the
peritubular solution is that of Table 3, with the exception of 19 mM
NH4+ for Na+ substitution; the luminal
solution is identical to the bath, with the exception of 4.95 mM
Na+ for K+ substitution; and the
Na+/H+ coefficient has been reduced to 0.1% of
its value in Table 2. Since two of three of its acid extruders are
nonfunctional, the OMCD pH is low (6.8). At t = 0, the
ambient ammonia is restored to 1 mM, and the cell acidifies further, to
nearly pH 6.0. At this point, the H+-ATPase is solely
responsible for the observed recovery of approximately 0.01 pH unit per
second. At t = 90, luminal K+ is restored,
luminal H-K-ATPase is activated, and the recovery rate increases
fivefold. The greater contribution of the H-K-ATPase could have been
predicted from its greater sensitivity to cytosolic acidification (Fig.
3). Furthermore, in these calculations, the H+-ATPase is
additionally hindered by cellular hyperpolarization to nearly
70 mV
(due to low cell Cl
, consequent to low cell
HCO3
). It is clear from these considerations that a
number of factors (including the cell pH at which one chooses to
restore luminal K+) can impact on the apparent contribution
of the two proton pumps.

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Fig. 16.
Time course of cellular pH recovery from an acid load.
The initial condition is a steady-state solution in which ambient
ammonia has been raised to 20 mM, luminal K+ is near zero,
and peritubular Na+/H+ exchange is blocked. At
t = 0, ambient ammonia is restored to 1 mM. The cell
rapidly acidifies and then, by t = 10, starts its pH
recovery due to action of the H+-ATPase. The value of
dpH/dt was computed as the slope at t = 30.
At t = 90, luminal K+ is restored, luminal
H-K-ATPase is activated, and the recovery rate increases fivefold (the
slope at t = 93).
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 |
DISCUSSION |
This model, the first to represent OMCD, is, in essence, a model
of the proton-secreting
-intercalated cell, in parallel with a
lateral interspace. Although OMCD contains both principal cells and
intercalated cells, no active transport activity has been ascribed to
the principal cell. It must be acknowledged that although these two
cell types are distinguished morphologically and immunocytochemically,
only a single cell type has been recognized electrophysiologically in
the inner stripe of OMCD (33, 46). For the purposes of
this model, their data has been assumed applicable to the intercalated
cell. The principal cell is most certainly the water pathway and should
also serve as a route for CO2,
H2CO3, NH3, and urea. To represent
the epithelial permeabilities for these species, a full principal cell
model was not considered necessary: water and urea permeabilities could
be lumped within the paracellular pathway, and CO2,
H2CO3, and NH3 fluxes could be
attributed to a route through the intercalated cell without significant
distortion of cytosolic solute concentrations. The four crucial
membrane transporters for this model are the two luminal proton
ATPases, H+-ATPase and H-K-ATPase, and the two peritubular
base exit pathways, the Cl
/HCO3
exchanger and a Cl
channel that admits
HCO3
. Proton transport by the H+-ATPase
is a function of transmembrane proton-motive force, as described by
Andersen et al. (2), and was cast into functional form by
Strieter et al. (61) for a model of the
-intercalated cell of cortical collecting tubule. The H-K-ATPase of this model is
that which had been developed for the IMCD (71) and had
been an adaptation of the gastric H-K-ATPase by Brzezinski et al.
(10). The representation of the
Cl
/HCO3
exchanger of OMCD is new.
The peritubular membrane Cl
/HCO3
exchanger of the
-intercalated cell has been identified with the
erythroid band 3 anion exchanger, AE1 (1, 54). Although
this transporter has been the object of extensive experimental
investigation and theoretical analysis (16, 49), it
appears that no computable kinetic model for this transporter has, as
yet, been formulated. This refers specifically to assigning parameter
values within the scheme of binding constants and translocation rates
depicted in Fig. 1. The present model was enabled by the recent
experimental observations of Brahm and colleagues (20,
31), who systematically delineated the 38°C kinetics of the
upper limb (HCO3
self-exchange) and the lower limb
(Cl
self-exchange) of the ping-pong mechanism. Their work
provided estimates of internal and external anion affinities along with maximal fluxes. Even with these extensive data, the model AE1 remains
incompletely determined. Each limb of the scheme is defined by four
parameters: two affinities plus forward and backward translocation rates. The experimental observations were sufficient to determine the
model coefficients up to a single free parameter in each limb. For the
parameter selection here (Table 1), the ratio of internal and external
binding affinities was set at 1. This was based on information obtained
from other chloride studies (42) and is presumed to be
true for bicarbonate as well. The resulting model is thus completely
compatible with all of the experimental observations (Fig. 2), but a
family of such models could have been generated, by varying the
bicarbonate affinity ratio. Parenthetically, in calculations not shown,
exchange of internal Cl
for external
HCO3
was remarkably insensitive to this ratio, and no
useful experiments to determine this parameter could be devised.
In this model OMCD, peritubular base exit proceeds via both AE1 and via
a conductive pathway, nearly equally under baseline conditions. The
magnitude of the conductive chloride pathway is bounded from below by
measurements of intercalated cell chloride concentration and
peritubular PD and by the requirement that at least 50% of luminal
proton secretion be matched by peritubular chloride flux (recycled from
entry via AE1). The observation made in developing this model is that
even if HCO3
permeability of the chloride channel is
one-eighth of that for chloride, the driving forces are such that about
one-half of the generated HCO3
should be reabsorbed
via this pathway. Where specific measurements of single channel
HCO3
-to-Cl
permeability ratios have
been obtained, a figure of 1:8 appears to be a conservative
underestimate (39, 50). Furthermore, in cultured OMCD
cells, a HCO3
conductance has been identified
(37). Thus in this model, even when peritubular AE1
activity is reduced to near zero, model proton secretion decreases by
only a one-third (Fig. 12). This model prediction, however, is at odds
with the experimental observation that OMCD proton secretion is
eliminated by removal of peritubular chloride or by application of a
stilbene inhibitor of AE1 (60). It is also at odds with
another study in which OMCD cell pH was monitored and removal of
ambient chloride reduced peritubular HCO3
permeability by 90% (28). A number of explanations could
be invoked to rationalize this important discrepancy. It is possible that the OMCD peritubular chloride channel is much more selective than
others in favor of chloride. Alternatively, with peritubular chloride
removal, either cell shrinkage or alkalinization might inhibit anion
channel activity. It is also possible that much of the peritubular exit
of chloride occurs in an electroneutral manner (e.g., KCl cotransport)
so that the conductive pathway is much smaller in magnitude than
estimated here. Resolution of this issue could be achieved with direct
determination of whole cell conductance with and without the presence
of peritubular chloride, and in the presence of AE1 inhibition. Indeed,
Koeppen (35) speculated that the slow depolarization of
SITS-inhibited OMCD cells might be due to loss of cell chloride via a
KCl cotransporter.
Another aspect of these model calculations has been the different
character of luminal membrane proton transport exhibited by the two
proton ATPases. The H+-ATPase appears to be more sensitive
to luminal pH so that flux through this transporter is nearly shut off
when the lumen falls below pH 6.5 (Figs. 5 and 9). The model H-K-ATPase
functions virtually undiminished until lumen pH falls below 5.0. Both
pumps are sensitive to relatively small changes in cytosolic pH. Model
calculations suggest that the H+-ATPase transport density
could be modulated over a broad range without serious derangement of
cell pH or cell volume (Figs. 10 and 14). In this model, the
hyperpolarization associated with increased H+-ATPase
activity acts directly to enhance peritubular Cl
and
HCO3
exit and indirectly to enhance
Cl
/HCO3
exchange. In contrast,
modulating the H-K-ATPase density led to substantial swings in cell
volume, becoming quite marked when the transport activity was
increased. The difficulties associated with changes in H-K-ATPase
activity derive from the relatively small peritubular K+
permeability of this model cell. However, this permeability is congruent with the experimental finding of a small, if not vanishing, K+ conductance (33, 46). One possible
resolution to this difficulty was illustrated in model calculations
(Figs. 14 and 15), namely, parallel activation of peritubular
K+ permeability with changes in H-K-ATPase activity.
Although this has precedent in proximal tubule pump-leak coupling via
ATP, there appear to be important differences in OMCD. In OMCD, the
H-K-ATPase is spatially removed from the K+ channel, and if
ATP were the mediator, then one might expect confusing signals with
changes in H+-ATPase activity. Alternatively, the
derangements in cell volume with changing H-K-ATPase activity might be
substantially mitigated if there were an important electroneutral
K+ exit pathway (i.e., KCl cotransport).
Several calculations examined the impact of CA activity on OMCD
transport. It was found that cytosolic CA was critical to maintain
proton secretion at normal rates. At reaction rates less than 100-fold
greater than the uncatalyzed rate of CO2 hydration, the
cell alkalinized and both luminal proton pumps showed diminished transport. Catalysis was assumed to be absent in the lateral
intercellular space, and by virtue of the peritubular
Na+/H+ exchanger, the interspace became a
region with an acid disequilibrium pH. In comparison with a model in
which CO2 hydration in the interspace was fully catalyzed,
the absence of CA here had no appreciable effect on OMCD proton
secretion. This acidic intraepithelial compartment could, however,
shunt cytosolic NH3 back to the peritubular bath. This
effect was substantial only at low peritubular NH3
concentrations. In the rat, luminal CA is absent (9, 15),
and the uncatalyzed rate of CO2 hydration was used in the
model lumen. This is in contrast to the inner stripe OMCD of the
rabbit, where CA is present and where inhibition abolishes the bulk
[62% (55)], if not all (65) of bicarbonate
reabsorption. The data of Tsuruoka and Schwartz (65) are
particularly strong on this point, showing nearly full inhibition of
HCO3
reabsorption with either benzolamide or a
lumen-restricted CA inhibitor and restoring reabsorption by adding CA
to the luminal perfusate. The present model OMCD cannot reproduce those
results (Fig. 7). This is because the predicted luminal acid
disequilibrium (0.4 pH unit) has no effect on H-K-ATPase and
approximately a 55% effect on transport by the H+-ATPase.
Luminal H-K-ATPase is found in rabbit OMCD (4, 64). It is
possible that the reported effect of CA inhibition derives from proton
pump kinetics different from those assumed here, or perhaps from a
direct effect of the inhibitor on the pumps.
In the model calculations of this report, buffer was abundant,
including delivered HCO3
and HPO42
,
as well as peritubular NH3. As indicated by Flessner et al. (15), the OMCD is sufficiently permeable to
NH3 that high rates of proton secretion can be sustained by
virtue of NH3 availability alone. Peritubular
NH4+ concentrations within outer medulla have not been
determined, but measurements within papilla include 2.1 mmol/l
(24) and 9.2 mmol/l (56), suggesting ambient
concentrations well above those of systemic plasma. In this model,
increasing peritubular NH4+ simply shifted the buffer
composition of the end-OMCD urine but had little impact on the rate of
total proton secretion (Fig. 8). This is in contrast to inner medullary
acidification where NH4+ can serve as a proton donor,
and increases in peritubular NH4+ enhance luminal
proton secretion. In inner medulla, a significant fraction of proton
secretion is dependent on NH4+ availability and thus
independent of cytosolic CA (67, 71). Overall, total
proton secretion by this model OMCD was approximately 50 pmol/min, or
for a single kidney with 7,200 OMCD, 6.3 nmol/s. This may be compared
with the estimate for total inner medullary proton secretion by the rat
kidney of 5.2 nmol/s (71). The HCO3
delivery to IMCD from the model outer medulla is 3.1 nmol/s, with a
volume delivery rate of 0.27 µl/s (Table 5). This base delivery is
within the capability of the model IMCD to acidify the urine. When the
end-luminal solution from this OMCD (Table 5) is deployed as initial
conditions for the IMCD model using the same papillary interstitial
values (Table 2 in Ref. 70), luminal acidification proceeds to
pH 5.7 with a HCO3
concentration 1.3 mmol/l.
In summary, the use of mass balance equations that can accommodate
several buffer systems has permitted formulation of a model of the
acid-secreting intercalated cell of the rat outer medulla and thus a
representation of OMCD. For this model, the luminal membrane proton
pumps have been adapted from the IMCD, and a kinetic model for the
peritubular Cl
/HCO3
exchanger is newly
devised. Although this model AE1 is fully compatible with erythrocyte
data at 38°C, it is not unique, and the means to create a family of
compatible anion exchangers has been indicated. With respect to the
intercalated cell, there are two model observations that suggest
further investigation. 1) If all of the chloride entry via
AE1 recycles across a peritubular chloride channel, and if this channel
is anything other than highly selective
(HCO3
:Cl
permeability ratio <1:8),
then it should conduct a substantial fraction of the bicarbonate exit.
2) If all of the peritubular K+ exit is
conductive, then variation in luminal membrane H-K-ATPase activity is
predicted to result in significant derangement of cell volume. Both of
these model conclusions could be invalidated if peritubular KCl
cotransport were present. With respect to the OMCD in situ, available
buffer appears to be present well in excess and is unlikely to exert a
regulatory role in total proton secretion by this tubule segment.
 |
ACKNOWLEDGEMENTS |
I thank Dr. Philip A. Knauf for several very helpful discussions
regarding AE1 function and for critical reading of portions of this manuscript.
 |
FOOTNOTES |
This investigation was supported by National Institute of Arthritis,
Diabetes, and Digestive and Kidney Disease Grant 1-R01-DK-29857.
Address for reprint requests and other correspondence: A. M. Weinstein, Department of Physiology and Biophysics, Weill Medical College of Cornell University, 1300 York Avenue, New York, NY 10021 (E-mail: alan{at}nephron.med.cornell.edu).
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.
Received 27 October 1999; accepted in final form 27 January 2000.
 |
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