A mathematical model of rat cortical collecting duct:
determinants of the transtubular potassium gradient
Alan M.
Weinstein
Department of Physiology and Biophysics, Weill Medical College
of Cornell University, New York, New York 10021
 |
ABSTRACT |
In assessing disorders of
potassium excretion, urine composition is used to calculate the
transtubular gradient (TTKG), as an estimate of tubule fluid
concentration, at a point when the fluid was last isotonic to plasma,
namely, within the cortical collecting duct (CCD). A mathematical model
of the CCD has been developed, consisting of principal cells and
-
and
-intercalated cells, and which includes Na+,
K+, Cl
, HCO
,
CO2, H2CO3, phosphate, ammonia, and
urea. Parameters have been selected to achieve fluxes and permeabilities compatible with data obtained from perfusion studies of
rat CCD under the influence of both antidiuretic hormone and mineralocorticoid. Both epithelial (flat sheet) and tubule models have
been configured, and model calculations have focused on the determinants of the TTKG. Using the epithelial model, luminal K+ concentrations can be computed at which K+
secretion ceases (0-flux equilibrium), and this luminal concentration derives from the magnitude of principal cell peritubular uptake of
K+ via the Na-K-ATPase, relative to principal cell
peritubular membrane K+ permeability. When the model is
configured as a tubule and examined in the context of conditions in
vivo, osmotic equilibration of luminal fluid produces a doubling of the
initial K+ concentration, which, depending on delivered
load, may be substantially greater than the zero-flux equilibrium
value. Under such circumstances, the CCD will be a site for
K+ reabsorption, although the relatively low permeability
ensures that this reabsorptive flux is likely to be small. Osmotic
equilibration may also raise luminal NH3 concentrations
well above those in cortical blood. In this situation, diffusive
reabsorption of NH3 provides a mechanism for base
reclamation without the metabolic cost of active proton secretion.
distal nephron; aldosterone; urine acidification; ammonia transport
 |
INTRODUCTION |
MICROPUNCTURE STUDY
OF potassium transport in the rat kidney has identified the
accessible portion of the distal tubule as the principal site for
potassium secretion (28, 29). Further along the nephron,
there is little change in potassium flow, at least from a comparison of
potassium delivery to the collecting duct with its appearance in the
final urine. These data stand in contrast to a substantial body of
subsequent investigation of the collecting duct in vitro. Examination
of the cortical collecting tubule of the rabbit (35, 46, 52,
54), and of the rat (40, 59), has demonstrated that
this segment is a site of sodium reabsorption and potassium secretion,
with transport activity enhanced by aldosterone and antidiuretic
hormone (ADH). Dispersion in the data does exist with regard to the
component of sodium reabsorption that is matched by potassium
secretion, where a number of reports indicate that this fraction is
about one-half (35, 40, 46), whereas others have observed
that potassium secretion may be three-fourths (35, 52) or
one-fourth (35, 54, 59) of the sodium flux. These
observations have motivated extensive study of the principal cell of
the cortical collecting duct (CCD), in which the luminal membrane
Na+ and K+ channels, in series with peritubular
membrane Na-K-ATPase, mediate the Na+ for K+
exchange by this segment. Indeed, clinical disorders of renal potassium
excretion are implicitly referred back to the cortical collecting
tubule with the calculation of the transtubular potassium gradient
(13, 72).
A mathematical model of the CCD should be the appropriate instrument
for extrapolating tubule function from perfusion conditions to those in
vivo. Such a model must represent the principal cell and both
intercalated cell types of this tubule segment, along with the solute
species to allow simulation of Na+, K+,
Cl
, and acid-base transport. The only previous model of
the cortical collecting duct was that of Strieter et al. (55,
56), which represented the perfused tubule of the rabbit. That
model was used to investigate the factors defining the limiting luminal Na+ concentration, at which Na+ reabsorption
was brought to a halt. One conclusion of that work was that the inverse
dependence of luminal membrane Na+ channel permeability on
luminal Na+ concentration was key to representing the very
low, limiting concentrations that had been observed, and that feature
has been retained in the present model. In general, rat and rabbit
cortical collecting tubules are similar with respect to the overall
rates of transport and transepithelial electrical potentials. Notable differences include the fraction of intercalated cells that are
-cells [60% in rat (1, 60); 30% in rabbit
(33)], and the chloride conductance of the peritubular
membrane of the principal cell [greater in rabbit than rat (33,
44)]. In the CCD model developed here, the
-cell has been
updated to conform to that recently published in a model of the outer
medullary collecting duct of the rat (70), which includes
a kinetic representation of the peritubular
Cl
/HCO
exchanger and a luminal
membrane H-K-ATPase.
The primary focus of the present work is the potassium gradient that
can be established across the cortical collecting tubule. First, the
model CCD is described and is compared with rat tubule transport rates
and permeabilities. Modeled as an epithelium, this CCD is a flat sheet
of cells between specified bathing solutions. This version of the model
predicts K+ fluxes as a function of bath conditions and is
incorporated into a program that computes the limiting luminal
concentration, at which K+ flux ceases. In these
calculations, the zero-flux K+ gradient is found to vary
directly with the rate of Na+ transport, and, inversely,
with the peritubular membrane K+ permeability. Modeled as a
tubule, this CCD predicts the solute flows in vivo, given estimates of
the entering conditions from micropuncture of late distal tubule. Using
a high tubule water permeability typical for the effect of ADH, it is
found that CCD osmolality equilibrates early, and CCD K+
concentration is near its limiting value for most of the tubule length.
Indeed, under conditions of high K+ delivery, CCD luminal
K+ concentration can rise well above its equilibrium value,
and thus the CCD can become a site for K+ reabsorption.
 |
MODEL CCD |
The model CCD is depicted in Fig. 1,
in which the epithelium contains three cell types and a common
intercellular space, all bounded by luminal and peritubular solutions.
In the CCD tubule model, the epithelial compartments line the tubule
lumen, and luminal concentrations vary axially as a consequence of
transport. Within each compartment the concentration of species
i is designated C
(i), where
is
lumen (M), interspace (E), principal cell (P),
-intercalated cell
(A),
-intercalated cell (B), or peritubular solution (S). Within the
epithelium, the flux of solute i across membrane 
is
denoted J
(i)
(mmol · s
1 · cm2), where

may refer to tight junction (ME), interspace basement membrane
(ES), any of the luminal cell membranes (MP, MA, or MB), lateral cell
membranes (PE, AE, or BE), or basal cell membranes (PS, AS, or BS).
Along the tubule lumen, axial flows of solute are designated
FM(i) (mmol/s). The 12 model solutes are
Na+, K+, Cl
,
HCO
, CO2, H2CO3,
HPO
, H2PO
,
NH3, NH
, H+, and urea, as
well as an impermeant species within the cells, and possibly within the
lumen. These comprise the minimal set of solutes that will permit
representation of net acid excretion.

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Fig. 1.
Schematic representation of cortical collecting duct
(CCD) epithelium, consisting of principal and intercalated cells and
lateral intercellular space (LIS). The model tubule lumen is lined by
this epithelium. Intraepithelial fluxes are designated
J (i), where the subscript
 refers to luminal cell membranes (MP, MA, MB), lateral cell
membranes (PE, AE, BE), basal cell membranes (PS, AS, BS), tight
junction (ME), 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 (68, 70).
Within a cell (I; I = P, A, or B), the generation of volume,
sI(v), or of solute i,
[sI(i)], is equal to its net export
plus its accumulation
|
(1)
|
where VI is the compartment volume
(cm3/cm2). The interspace exchanges with all of
the model compartments, so that mass generation is written
|
(2)
|
Within the tubule lumen, mass generation is appreciated as an
increase in axial flux, as transport into the epithelium, or as local
accumulation
|
(3)
|
where BM is the tubule circumference, and
AM is the tubule cross-sectional area. With this
notation, the equations of mass conservation for volume and
for the nonreacting species (Na+,
K+ , Cl
, and urea) are written
|
(4)
|
where
= P, A, B, E, or M. For the phosphate and for the
ammonia buffer pairs, there is conservation of total buffer
|
(5)
|
|
(6)
|
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. Because HCO
and
H2CO3 are interconverted, mass conservation
requires
|
(7)
|
for
= P, A, B, or E, whereas for the tubule lumen
|
(8)
|
In each compartment (
= P, A, B, E, or M), conservation of
total CO2 is expressed
|
(9)
|
Corresponding to conservation of protons is the equation for
conservation of charge for all the buffer reactions
|
(10)
|
where zi is the valence of species
i. In this model, conservation of charge for the buffer
reactions (Eq. 10) may be rewritten
|
(11)
|
The solute equations are completed with the chemical equilibria of
the buffer pairs:
HPO
:H2PO
, NH3:NH
, and
HCO
:H2CO3. Corresponding to
the electrical potentials, 
, for
= P, A, B,
E, or M, is the equation for electroneutrality
|
(12)
|
With respect to water flows, volume conservation equations for
lumen, interspace, and cells can be used to compute the five unknowns:
luminal volume flow, lateral interspace hydrostatic pressure, and the
three cell volumes. (Cell hydrostatic pressure is set equal to luminal
pressure; total cell impermeant content is assumed fixed.) Across each
cell membrane, the volume fluxes are proportional to the hydroosmotic
driving forces. With respect to the lateral interspace, its volume,
VE, and its basement membrane area,
AES, are functions of interspace hydrostatic
pressure, PE
|
(13)
|
where VE0 and AES0 are
reference values for volume and outlet area, respectively, and
E is a compliance. Along the lumen, hydrostatic pressure
changes according to an equation for Poiseuille flow
|
(14)
|
Solute transport across the model membranes is either
electrodiffusive (through a porous matrix or via a channel), coupled to
the electrochemical potential gradients of other solutes (via a
cotransporter or an antiporter), or coupled to metabolic energy (via an
ATPase). This is expressed in the model by the flux equation
|
(15)
|
In Eq. 15, 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
|
(16)
|
is a normalized electrical potential difference (PD), where
zi is the valence of i, F
is the Faraday, RT is the product of gas constant and
temperature, and 


is the PD
between compartments
and
. In this model, all of the
permeabilities, h
(i), are
constant, with the exception of the Na+ permeability of the
luminal membrane of the principal cell. For this channel, an inverse
relationship between Na+ permeability and both luminal and
cytosolic Na+ concentrations has been described in several
epithelia and represented by Civan and Bookman (10) as
|
(17)
|
The second term of the solute flux (in Eq. 15)
specifies the coupled transport of species i and
j according to linear nonequilibrium thermodynamics, where
the electrochemical potential of j in compartment
is
|
(18)
|
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
/HCO
exchange across the peritubular membrane of the
-intercalated cell. Here, the kinetic model for AE1
(70) has been used, with a single transporter density
parameter representing its activity.
In this model, there are three ATPases. Within the peritubular membrane
(both lateral and basal membranes) of all three cell types (I = P,
A, B), the Na-K-ATPase is represented by an expression
|
(19)
|
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+
|
(20)
|
The pump flux of K+ plus NH
reflects
the 3:2 stoichiometry
|
(21)
|
with the transport of either K+ or
NH
determined by their relative affinities,
KK and
KNH
|
(22)
|
Analogous expressions are written for active transport at the
basal cell membranes,
J
(Na+).
Within the luminal membrane of the
-cell and the peritubular membrane of the
-cell, there is a proton ATPase. An empirical expression representing the H+-ATPase was devised by
Strieter et al. (55), approximating data of Andersen et
al. (3) for turtle bladder
|
(23)
|
where J(H+)max is the maximum
proton flux,
MI (H+) is the
electrochemical PD 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. (3) was that the
proton flux depended on 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).
Within the luminal membrane of the
-intercalated cell, there is an
H-K-ATPase, which has been given a full kinetic representation (69).
 |
MODEL PARAMETERS |
The parameters displayed in Table
1 were selected so that
the model tubule would correspond to the CCD of the rat. Where rat data were not available, rabbit measurements were considered for
guidance. The distal nephron of the rat accessible to micropuncture includes a distal convoluted tubule (DCT), connecting segment, and
initial collecting duct. The CCD includes this initial collecting duct
and the cortical collecting tubule within the medullary ray (27). In the rat, the cortical collecting tubule is short,
~1.5 mm, (37), so that a total CCD length of 2.0 mm has
been chosen to represent the nephron segment between the last
accessible micropuncture and the medullary collecting duct. In the
tubule calculations, it will be assumed that all of the coalescing of
nephrons in the arcade is complete, so that the model tubule segment is
unbranched. In the rabbit, measurements of inner and outer tubule
diameters are 25 and 35 µm, giving luminal and epithelial volumes of
490 and 960 pl/mm, respectively, comparable to those reported for the
rat (36). In the rat CCD, intercalated cells comprise
~40% of epithelial volume, with 21.4 and 15.7%
- and
-cells,
respectively (23). In the rabbit, lateral interspace
volume is ~11% of epithelial volume (71). For a total
epithelial volume of ~5 × 10
4
cm3/cm2, this yields a principal cell volume of
3 × 10
4 cm3/cm2 and
-
and
-intercalated cell volumes of 1.2 and 0.8 × 10
4 cm3/cm2, respectively. The
cellular compartments, along with the important transport pathways, are
displayed in Fig. 2.

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Fig. 2.
A: CCD transport pathways along with the model
concentrations (mM) and fluxes
(pmol · mm 1 · min 1)
computed at the initial portion of the tubule. B: CCD
transport pathways along with the model concentrations (mM) and fluxes
(pmol · mm 1 · min 1)
computed at the distal portion of the tubule (x = 2 mm,
corticomedullary junction).
|
|
In rat CCD, the principal cell luminal and peritubular membrane areas
have been reported to be ~4,000 and 25,000 cm2/cm3 cell volume (50, 61),
which translates into 1.2 and 7.5 cm2/cm2
epithelial area, respectively. The important water channels of this
epithelium are restricted to the principal cells (34), in
which the luminal membrane, containing aquaporin-2, is rate limiting
for water flow. For most of the calculations, ADH is assumed to be
present and luminal water permeability has been selected to achieve an
epithelial water permeability (Pf) of ~0.1 cm/s (9, 37). The peritubular membrane unit water
conductance was taken to be two-thirds that of the luminal membrane, so
that with its sixfold area amplification, the peritubular
Pf was four times that of the luminal membrane.
Electrophysiology of the principal cell indicates that the luminal
conductance is enhanced by both aldosterone and ADH. With both hormones
present, the fractional apical resistance may be 0.7-0.8,
indicating a luminal conductance ~30% of that of the peritubular
membrane (42, 44). Under these circumstances, an absolute
luminal conductance has been reported to be 16 mS/cm2
(41). Most of this luminal membrane conductance is due to
potassium, with a transference number of 0.73 (44). For
the model parameters (Table 1), the maximal Na+
permeability was set equal to the K+ permeability, and the
value shown (~22% of the K+ permeability) derives from
the inhibitory effect of ambient Na+ (Eq. 17).
This value of Na+ permeability was found to yield
transepithelial Na+ fluxes in the reported range when both
aldosterone and ADH are present. Luminal membrane NH
permeability was set at 20% of that of K+, comparable to
NH
permeation of other K+ channels.
Luminal membrane Cl
permeability was set (arbitrarily) at
10% of that of K+, and HCO
permeability
to 20% of that of Cl
, rendering conductive anion fluxes
negligible. A luminal membrane NaCl cotransporter has been included, on
the basis of a single report that 50% of Na+ reabsorption
by rat CCD is thiazide sensitive (57). In view of the fact
that this finding has not received confirmation (6, 43),
the baseline flux through this pathway has been set low (3.6% of
Na+ flux), but the impact of greater cotransport is
explored in the model calculations. In the rat, the peritubular
membrane is nearly selective for K+ (42, 44).
In this model, the K+ permeability is such that the
peritubular conductance is 2.5 times that of the luminal membrane. The
peritubular NH
permeability is again 20% of that for
K+; Cl
permeability is 5% that of
K+; and HCO
permeability is 20% that of
Cl
. Within the peritubular membrane of principal cells of
rabbit CCD, there are also electroneutral cotransporters,
Na+/H+ (63, 65), and
Cl
/HCO
(63, 66). The
density of these transporters was adjusted to achieve realistic cell pH
and Cl
[e.g., 13 mmol/l for cell Cl
(5); principal cell pH 7.3-7.4 (48)].
Because of the lack of direct information, the unit membrane
NH3 and urea permeabilities were taken to be equal and
adjusted to achieve agreement with overall epithelial permeabilities
for these solutes.
Membrane areas of intercalated cells of the CCD have been determined
and indicate 4,800 and 14,400 cm2/cm3 cell
volume for the luminal and peritubular membranes of
-cells, and
2,150 and 21,000 cm2/cm3 for
-cell membranes
(60), respectively. When corrected for the volume density
of
- and
-cells (1.2 and 0.8 × 10
4
cm3/cm2, respectively), these membrane areas
are 0.6 and 1.7 cm2/cm2 for the
-cell and
0.2 and 1.7 cm2/cm2 epithelial area for the
-cell, respectively. The important
-cell transport pathways are
shown in Fig. 2 and include two luminal membrane proton ATPases in
series with peritubular Cl
/HCO
(AE1)
and a Cl
channel. The parameters are essentially those
selected for the
-cell of the outer medullary collecting duct
(70), with the exception that the density of the
H-K-ATPase and peritubular K+ channel have been decreased
by 60%, whereas the density of the H+-ATPase has been
increased by 33%. The H-K-ATPase is present in intercalated cells
of rat and rabbit CCD (47, 49), although under control
conditions its proton secretory rate appears to be less than that of
the H+-ATPase (31). The decreased value taken
for peritubular K+ conductance remains compatible with the
electrophysiology of the
-cell (33). The important
-cell transport pathways are a luminal membrane
Cl
/HCO
exchanger in series with
peritubular H+-ATPase and Cl
channel. Under
control conditions, no H-K-ATPase activity has been identified in this
cell, although it may become important in the correction of metabolic
alkalosis (16) or under conditions of low sodium intake
(47). The luminal membrane anion exchanger is different
from AE1 (12), and, to respect this difference, the
nonequilibrium thermodynamic formulation has been used. The luminal
membrane has no significant electrical conductance, and, as in the
-cell, the peritubular membrane is dominated by the chloride
conductance (33). Peritubular
Na+/H+ and
Cl
/HCO
exchangers have been included in view of their presence in
-cells of rabbit CCD (65, 67, 74). The
-cell unit membrane permeabilities for
nonelectrolytes have been assumed to be identical to those chosen for
the
-cell. In the rat, cytoplasm of intercalated cells stains
intensely for carbonic anhydrase (26), although the
membrane-bound isoform (CA-IV) is absent (7). Thus the
rate coefficients for CO2 hydration and dehydration
(Eq. 7) have been taken as those of the uncatalyzed reaction
within the tubule lumen and lateral intercellular space and for full
catalysis (10,000-fold greater) within
- and
-cells. In view of
some staining within principal cells (26), the
coefficients were taken to be 10-fold greater than the uncatalyzed rate.
Values for the tight junctional conductance of the rat CCD have been
found to be 11-13 mS/cm2, perhaps two- to threefold
greater than that for the rabbit tubule (41, 44). As in
rabbit, the Cl
permeability appears to be slightly larger
than that for Na+ (44). In the model of rabbit
CCD (55), it had been noted that the low value of
Na+ permeability was essential to achieving tubule fluid
Na+ concentrations as low as those observed. In preliminary
calculations for this model of the rat tubule, it was observed that if
tight junctional conductance were twice that of rabbit CCD, then
paracellular backflux would be unacceptably large: overall epithelial
Na+ secretion for a "late distal" luminal fluid
composition (35 mM NaCl concentration). Indeed, even with junctional
conductance comparable to rabbit CCD, 5 mS/cm2, the
paracellular backflux of Na+ is still two-thirds of the
reabsorptive Na+ flux across the principal cell (Fig. 2).
That lower conductance has been selected for this model. The tight
junctional Cl
-to-Na+ permeability ratio has
been set at 1.2, consistent with observation in rats (44),
and perhaps somewhat lower than in rabbits (64). Junctional K+ and NH
permeabilities were
assumed to be equal to that of Cl
and that of
HCO
to be 25% of the value for Cl
.
The junctional urea permeability was set equal to half the measured epithelial urea permeability (25). In models of renal
tubule segments, the basement membrane is a permeability barrier to the lateral interspace and allows for the possibility that the interspace can act as an unstirred layer, with solute concentrations distinct from
those of the peritubular bath. In this model, the overall conductance
of the basement membrane is ~1,000 mS/cm2, with relative
solute permeabilities comparable to their mobility in solution.
 |
MODEL CALCULATIONS |
Suitability of the parameter choices is assessed, in part, by
examining predicted fluxes and permeabilities. Table
2 contains the
solutions of the model equations for the open-circuited epithelium, when lumen and bath solutions are equal, comparable to solutions used
in perfusion studies. The computed compartment volumes
(principal:
:
:interspace) are 59, 22, 14, and 5%, respectively,
of a total epithelial volume of 6 × 10
4
cm3/cm2. The electrical PD of tubule lumen
(
18.4 mV) and of the peritubular membrane of the principal cell
(
79.2 mV) are similar to those found in tubules under the influence
of both aldosterone and ADH (42, 43). Within the principal
cell, the Cl
concentration is low (12.7 mM) but still
above its equilibrium concentration of 5.8 mM. This is a consequence
primarily of the luminal NaCl cotransporter, although the peritubular
Cl
/HCO
exchanger contributes ~23%
of the entering Cl
. This cytosolic Cl
concentration is within the range of determinations using
Cl
-sensitive microelectrodes [Cl
activity
~9 mmol/l (43)]. When the transepithelial solute fluxes are scaled to a tubule of 25 µm diameter, principal cell
Na+ reabsorption is 91.6 pmol · mm
1 · min
1 with a
paracellular backflux of 23.0, giving a net reabsorptive Na+ of 68.6 pmol · mm
1 · min
1. Close to
half of this is balanced by principal cell K+ secretion of
30.0 and close to half by Cl
reabsorption of 30.5 pmol · mm
1 · min
1. The
Cl
flux is primarily paracellular (22.2), with smaller
components across the
-cell (5.2) and principal cell (3.1). The
remainder of the Na+ flux is balanced by an equivalent
reabsorptive "HCO
" flux
(pmol · mm
1 · min
1) of
8.2, comprised
-cell HCO
secretion (5.2) in
parallel with
-cell H+ secretion (13.4). These values
for transepithelial ionic fluxes are within ranges that would be
appropriate for tubules under the influence of both aldosterone and
ADH. It may also be noted that with ambient total ammonia
concentrations of 1.0 mM, the model predicts net reabsorption of
ammonia, despite the lumen negative PD. This is due to the acid
disequilibrium within the lateral interspace (due to peritubular
Na+/H+), with diffusion trapping of
NH3.
Table 3 displays the results of
simulating idealized epithelial permeability determinations. For these
calculations, the model represents a short-circuited epithelium in
vitro bathed by the equal luminal and peritubular solutions in Table 2,
plus an additional luminal impermeant at a concentration 0.1 mM.
Calculations were performed in which each luminal solute concentration
in turn was lowered and then raised by 0.1 mM. The change in solute
flux relative to the change in concentration is listed in Table 3 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 shown in column 2 in Table 3
(mS/cm2). The total conductance is ~8 mS/cm2;
by design, it is 30-50% of the measured conductance in rat
tubules (42, 44), more like that in rabbit tubules. For
comparison with the model tubule, permeability measurements in rat CCD
have been made for urea, 0.4 × 10
5 cm/s
(25), for NH
, 2.6 × 10
5 cm/s (14), and for NH3,
0.024 cm/s (14). In Fig.
3, the model of the voltage-clamped
epithelium is used to examine the effect of transepithelial electrical
PD on ion flux. Each panel corresponds to a different solute species,
Na+, K+, Cl
, and
"HCO
," where "HCO
" is the
sum of HCO
reabsorption and H+
secretion. In each panel, both transjunctional and total fluxes are
displayed. It is apparent that throughout an 80-mV variation in
transepithelial PD, nearly all of the Cl
flux is
transjunctional, nearly all of the K+ flux is
transcellular, and the "HCO
" flux is small. The
Na+ flux remains reabsorptive down to
60 mV, due to a
substantial transcellular component that is relatively insensitive to
transepithelial PD. Although the paracellular K+
permeability is slightly greater than that for Na+ (Table
1), the small magnitude of the junctional K+ flux is due to
the small magnitude of the ambient K+ concentration. The
principal cell K+ permeability (luminal and peritubular
membranes in series) is only about threefold greater than that of the
tight junction; the much greater sensitivity of transcellular
K+ flux to PD is due to the high intracellular
K+ concentrations maintained by active peritubular
K+ uptake.

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Fig. 3.
Impact of transepithelial potential difference (PD) on
CCD ion fluxes. Calculations use the epithelial model with equal
luminal and peritubular solutions (Table 2).
"HCO "refers to the sum of proton secretion plus
HCO reabsorption.
|
|
Variations in sodium transport by the model CCD are examined in Figs.
4-6.
In Fig. 4, epithelial PD and ion fluxes are calculated over a range of
variation in luminal Na+ concentrations. The perfusion and
bathing solutions are as in Table 2, with the exception that the
luminal HCO
concentration has been decreased to 5 mM
(replaced by Cl
). Figure 4, left, corresponds
to experiments in which Na+ is replaced by an impermeant
cation, and Cl
is constant (~135.5 mM), whereas NaCl
variation (equal changes in Na+ and Cl
, with
isosmotic replacement by an impermeant) is shown on the right. The curves on the left are similar
to those calculated by Strieter et al. (55; see Fig. 8) in simulating
experiments by Stokes (52). As in the previous model, net
reabsorptive Na+ flux continues down to luminal
Na+ concentrations below 10 mM, and K+
secretion varies over the whole range of Na+
concentrations. With NaCl variation (right), the major
differences in epithelial transport are the smaller excursion in
luminal PD and the nearly constant rate of K+ secretion at
all luminal Na+ concentrations >30 mM. Principal cell
function during this NaCl variation is examined in more detail in Fig.
5. Here, the change in luminal membrane permeability (relative to the
fixed luminal K+ permeability) as luminal Na+
is varied is shown (top left). Thus, even though the luminal membrane Na+ potential increases progressively with
increasing luminal Na+ concentration (middle
left), the decrease in Na+ permeability produces a
luminal membrane electrical potential that is relatively constant at
higher luminal Na+ (top right). This means a
relatively constant luminal membrane K+ potential
(middle right) and thus a stable K+ flux
(bottom right). Variation in CCD Na+
reabsorption can be examined over an even broader range by varying the
density of luminal membrane Na+ channels, and, in the
calculations of Fig. 6, this permeability has been varied from 3 to
300% of control. As is shown on the left, the
Na+ permeability has been varied in isolation, whereas on
the right, there is also proportional variation in the
density of the peritubular Na-K-ATPase. In each set of calculations,
enhanced luminal Na+ entry hyperpolarizes the epithelium
and increases K+ secretion, Cl
reabsorption,
and even "HCO
" reabsorption. It is apparent,
however, that coordination of peritubular exit with luminal entry
amplifies the luminal signal. This is true even for the reabsorptive
Cl
flux, which is primarily paracellular.

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Fig. 4.
Effect of luminal Na variation on CCD function.
Calculations use the open-circuited epithelial model. Left:
luminal Na decreased by substitution with an impermeant cation.
Right: varied luminal NaCl and addition of a neutral
impermeant for osmotic balance.
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Fig. 5.
Principal cell luminal membrane during luminal NaCl
variation. Calculations are those in Fig. 4 (right) using
the open-circuited epithelial model with luminal NaCl variation.
hMP(Na) and hMP(K),
luminal membrane ionic permeabilities, with
hMP(Na) being a function of luminal
Na+ concentration (Eq. 17). Middle:
electrochemical potentials of Na+ and K+,
respectively, across the luminal membrane of the principal cell.
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Fig. 6.
Variation of principal cell luminal membrane
Na+ permeability, hMP(Na).
Left: variation of principal cell luminal membrane
Na+ in isolation. Right: variation of
peritubular Na-K-ATPase density in parallel with
hMP(Na). The open-circuited epithelial model is
used with identical (high-Na+) luminal and
peritubular solutions of Table 2.
|
|
Figure 7 summarizes the impact of
changes in Na+ transport on the other fluxes. Throughout
the figure, the predicted K+ and Cl
transport
are plotted as a function of the rate of Na+ reabsorption.
In the replotting of the data from Fig. 6 (top), it is
apparent that both K+ and Cl
fluxes are
nearly linear functions of Na+ flux and nearly through the
origin. This implies that the relative fraction of
Na+ reabsorption balanced by K+ and by
Cl
fluxes is nearly constant over the full range of
transport [consistent with observations of Stokes (52)].
In contrast, for the simulations of luminal NaCl variation (Fig. 4),
the changes in Na+ flux are nearly completely balanced by
changes in Cl
flux (bottom right). This curve
does not go through the origin, so that for small Na+
fluxes there is essentially KCl secretion. [In the model, this K+ secretion derives from the diffusion potential set up by
the NaCl gradient. Although direct KCl coupling within the luminal membrane of CCD principal cells has been suspected from studies of
perfused rabbit tubules (73), it is not a feature of this model.] Figure 7 (bottom left) corresponds to simulations
in which the luminal membrane NaCl cotransport coefficient is increased over a factor of 20, with a parallel increase in peritubular
Cl
permeability. In these calculations, there is no
change in K+ flux with the variation in Na+
reabsorption. Here, even though the increased transcellular
Na+ flux produces a proportional peritubular K+
uptake (Na-K-ATPase), the increase in Cl
permeability
depolarizes the peritubular membrane and enhances the return of
K+ back across this membrane (not shown).

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Fig. 7.
Epithelial Cl and K+ fluxes as
a function of Na+ flux. Top: replotting of
results from the calculations in Fig. 6, in which principal cell
luminal membrane Na+ permeability is varied alone
(left), or in parallel with peritubular Na-K-ATPase density
(right). Bottom left: luminal NaCl cotransport
coefficient of the principal cell increased over a 20-fold range in
parallel with the peritubular membrane Cl permeability.
Bottom right: replotting of the calculations in Fig. 4, in
which luminal NaCl is varied.
|
|
Beyond the effect of Na+ reabsorption, K+
secretion can be modulated by principal cell membrane K+
permeabilities, as well as luminal fluid K+ concentration.
Figure 8 displays epithelial model
predictions for PD and solute fluxes using the high-Na+,
low-K+ (5 mM) perfusion solution (Table 2) in the
open-circuited epithelium. In Fig. 8 (left) luminal membrane
K+ permeability is varied from 3 to 300% of baseline. As
the luminal K+ permeability increases, K+
secretion is enhanced and the epithelium depolarizes, thus
decreasing paracellular Cl
reabsorption and paracellular
Na+ backleak. Furthermore, with increasing luminal
K+ permeability, the luminal membrane hyperpolarizes (not
shown), thus enhancing transcellular Na+ reabsorption and
peritubular K+ uptake, ultimately augmenting K+
secretion. In Fig. 8 (right) the effect of modulating
peritubular K+ permeability from 3 to 300% of control is
shown. As expected, increasing peritubular K+ permeability
decreases K+ secretion and enhances Cl
reabsorption. However, the striking feature of these calculations is
the lack of effect on Na+ transport. In contrast to luminal
K+ permeability variation, increasing peritubular
K+ permeability hyperpolarizes both the tight junction and
luminal cell membranes. As a consequence, both paracellular backflux
and luminal reabsorptive Na+ flux are enhanced and cancel.
It may also be noted that the overall impact on K+ flux is
smaller with peritubular variation than with luminal variation of
K+ permeabilities.

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Fig. 8.
Variation of principal cell membrane K+
permeabilities. Left: variation of luminal membrane
permeability. Right: variation of peritubular K+
permeability. The open-circuited epithelial model is used, with
identical (high-Na+) luminal and peritubular solutions of
Table 2.
|
|
In Fig. 9, luminal KCl
concentration is varied from 1.0 to 39 mM. In the panels on the
left, the luminal perfusate is the high-Na+
solution in Table 2. Epithelial hyperpolarization with increasing luminal K+ (top) and a depolarization of the
luminal membrane of the principal cell (middle) are shown.
The curve labeled "K Potential" (middle) is the
potential from lumen to cell, so that the crossing point from negative
to positive potential corresponds to the transition from principal cell
K+ secretion to reabsorption. The K+ fluxes are
shown (bottom), where it is evident that most of the variation in epithelial K+ flux is due to changes in
principal cell flux. With respect to flux across the luminal cell
membrane, the transition from secretion to reabsorption occurs at a
luminal K+ concentration of ~25 mM; for the epithelium,
this transition point is ~23 mM. In the panels on the
right in Fig. 9, the perfusion solution is a
low-Na+, low-Cl
(35 mM) solution
characteristic of early CCD conditions (see below). The potentials are
not very different, and the ability of the tubule to secrete
K+ is not affected.

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Fig. 9.
Variation of luminal K+ concentration by
addition of KCl. Baseline perfusion and bath using high-Na+
solutions in Table 2 (left) and luminal perfusate in vivo
condition in Fig. 2 (right). Middle: electrical
PD and K+ potential across the principal cell luminal
membrane, with the cytosol as reference (positive potentials support
reabsorptive fluxes). Bottom: transepithelial K+
fluxes resolved into cellular and paracellular components.
|
|
The maximal luminal K+ concentration that can be sustained
by epithelial K+ secretion, or zero-flux K+
concentration, can be estimated analytically with reference to the
scheme depicted in Fig. 2. Considering only K+ and
combining lateral and basal cell membranes into a peritubular membrane, the fluxes may be estimated from the K+
potentials by
|
(24)
|
where JPS is total peritubular
K+ flux, and µM and µP are
computed relative to peritubular conditions
The coefficients L
(mmol2 · J
1 · s
1)
are related to membrane K+ permeabilities,
h
(cm3/s), and conductances,
g
(mS/cm2)
|
(25)
|
where 

is a (logarithmic) mean membrane
concentration
Because cellular K+ balance requires
JMP = JPS
|
(26)
|
luminal membrane K+ secretion is zero when
|
(27)
|
and, ignoring K+ transport by the
-intercalated
cell, overall epithelial K+ flux is zero when
|
(28)
|
For a relatively nonconductive tight junction, Eq. 28
is approximated by Eq. 27, which indicates that the limiting
K+ concentration is determined largely by the peritubular
membrane K+ permeability relative to the peritubular uptake
of K+ via Na-K-ATPase. It should be influenced little by
luminal membrane K+ permeability, because when luminal
membrane K+ flux is zero, the permeability is irrelevant.
(With reference to Eq. 27, the impact of luminal membrane
K+ conductance is greatest when K+ secretion is
substantial and is an important determinant of both the magnitude of
early tubular K+ secretion and of overall epithelial
K+ conductance.) Table 4 uses
the parameters in Table 1 and the K+ concentrations in
Table 2 to estimate the limiting K+ concentration
(Eq. 28). For luminal electrical PD of
30 mV, this concentration is 23 mM, comparable to the value obtained from Fig. 9.
To compute the limiting K+ concentrations directly, the CCD
epithelial model has been configured as a subroutine in a program that
uses Newton iteration to identify the luminal K+
concentration (via KCl adjustments) at which secretion is zero, and the
results are displayed in Fig. 10. The
top two panels compute limiting K+
concentrations over the range of variation in luminal Na+
permeability considered in Figs. 6 and 7; the middle panels
show the limiting concentrations computed for variation in membrane K+ permeability considered in Fig. 8; the bottom
left panel contains limiting K+ concentrations when
coupled NaCl transport is varied 20-fold as in Fig. 7; and the
bottom right panel examines the effect of varying luminal
NaCl concentration, as in Figs. 4 and 7. Consistent with Eq. 27, increasing Na+ flux two- to threefold via
increasing Na+ channel permeability increases the limiting
K+ gradient two- to threefold. Also consistent with
Eq. 27 is the insensitivity of the limiting K+
gradient to luminal K+ permeability and a hyperbolic
sensitivity of the limiting gradient to the peritubular permeability.
Figure 10 (bottom) demonstrates that it is possible to
modulate Na+ transport without any effect on the limiting
K+ gradient: in the case of coupled NaCl transport, there
is parallel peritubular Cl
channel activation, and in the
case of luminal Na+ there is modulation of luminal membrane
Na+ conductance. In both cases, changes in membrane
potentials act to leave the limiting K+ gradient relatively
unaffected.

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Fig. 10.
Luminal equilibrium K+ concentrations. For
the simulations considered previously, luminal KCl is adjusted until
net transepithelial K+ flux is 0. Top: varied
principal cell luminal membrane Na+ permeability (as in
Fig. 6). Middle: varied principal cell K+
permeabilities (Fig. 8). Bottom: varied principal cell
luminal NaCl cotransport (NCC; left) (Fig. 7) and
examination of a range of luminal NaCl (right) (Fig. 4).
|
|
When the CCD epithelium is configured as a tubule, it permits
predictions about its function in vivo. Figure 2A displays
estimates of conditions and fluxes in the early collecting duct.
Conditions are taken to resemble those in late distal tubule:
Na+ = 35.0, K+ = 12.0, Cl
= 36.2 (5, 11), and
HCO
= 7.0 mM (8, 22). For an axial
volume flow of 7.5 nl · min
1 · tubule
1 (54 µl/min for all 7,200 tubules) and a single-kidney glomerular filtration rate of 500 µl/min, these concentrations correspond to
~2.5, 30, 3, and 3.5% of filtered loads of Na+,
K+, Cl
, HCO
, respectively.
The total phosphate, 3.9 mM, is a lower concentration than reported
(11), but for a plasma concentration of 2.5 mM
(22) it represents a delivery of 16% of filtered load
and, from that perspective, is consistent with observations
(11). The urea concentration, 30 mM, corresponds to
~60% of filtered load and, although a low concentration, is compatible with determinations of urea flow at this point
(30). The ammonia concentration, 2 mM, sits between
measured values for late distal tubule fluid in the rat (21,
38). With these values as input, the concentrations and flows
have been determined for a 2.0-mm tubule segment, whose peritubular
solution is shown in Table 2. Figure 11
shows the tubule fluid osmolality and solute concentrations
(left) and axial volume and solute flows (right). It is apparent that over the first 20% of the tubule length, about half of the tubule water is reabsorbed in the transition to
isotonicity, and consequently luminal solute concentrations double.
Moreover, along the length of the tubule, ~20% of the delivered NaCl
is reabsorbed, resulting in additional water loss and still further increases in luminal K+ concentration. Although
K+ is initially secreted (
15.2
pmol · mm
1 · min
1, Fig.
2A), luminal K+ concentration rises above its
limiting value, so that by the tubule exit, K+ flux is
reabsorptive (6.1 pmol · mm
1 · min
1). The
end-tubule concentrations and fluxes have been indicated in Fig.
2B. By this point in the tubule, luminal Na+ and
K+ have increased to 77 and 31 mM, respectively, and
luminal PD has hyperpolarized slightly to
29 mV. Overall CCD
K+ flux is small, with reabsorption ~6% of delivered
load. Osmotic concentration of luminal solutes also enhances urea and
ammonia reabsorption (9 and 38% of delivered load, respectively). The whole-tubule economy of solute fluxes has been summarized in Table 5.

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Fig. 11.
Transport along the CCD under baseline conditions (as in
Fig. 2). Left: luminal concentrations as a function of
tubule length. Right: volume and solute flows.
|
|
The observations in Fig. 11 are that luminal K+
concentration can be driven above its limiting value, by either pure
water abstraction or isosmotic NaCl reabsorption. Either of these
features can be enhanced by changing luminal conditions. In Fig.
12, entering flow and concentrations
are identical to those in Fig. 11, with the exception that entering KCl
concentration has been doubled to 24 mM. Now, osmotic abstraction of
water rapidly increases luminal K+ concentration to 45 mM,
nearly twice the limiting K+ gradient, and under these
circumstances there is an increase in both absolute and fractional
K+ reabsorption (~25% of delivered load). What should
also be noted is that although the luminal K+ concentration
is much higher than can be sustained by K+ secretion, the
relaxation of luminal K+ toward the limiting value is slow,
due to the relatively low K+ permeability of this segment.
The end-luminal K+ concentration is still more than
eightfold greater than the peritubular concentration. The impact of
isosmotic NaCl reabsorption is emphasized in Fig.
13, wherein the initial conditions are
those in Fig. 11, with the exception that the entering flow is 20% of
baseline. With such a sluggish flow, NaCl reabsorption is ~80% of
delivered load and occurs more rapidly than K+
reabsorption. This produces a sustained increase in luminal
K+ concentration, to 33 mM by tubule end.

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Fig. 12.
Transport along the CCD when inlet luminal
K+ has been increased from 12 to 24 mM (KCl addition).
Conditions are otherwise as in Fig. 2.
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Fig. 13.
Transport along the CCD when inlet luminal flow has been
decreased from 7.5 to 1.5 nl/min. Conditions are otherwise as in
Fig. 2.
|
|
It should not be concluded from the foregoing figures that this model
CCD cannot be a site of net K+ secretion. In Fig.
14, the luminal membrane
Na+ permeability and the density of peritubular Na-K-ATPase
have both been increased by 50% (as in Fig. 6, right). In
the CCD epithelial model, using perfusion bath conditions, this
parameter change yields Na+ reabsorption and K+
secretion of 94.1 and
43.8
pmol · mm
1 · min
1,
respectively, comparable to the maximal fluxes observed in (ADH- and
DOC-treated) rat tubules in vitro (40). Figure 14 contains the model prediction of such a tubule in vivo, using the perfusion conditions of Fig. 11. In this case, CCD Na+ reabsorption
increases from 19 to 32%, the tubule hyperpolarizes to
35 mV, and
the K+ reabsorption at baseline now becomes a net
K+ secretion equal to 15% of delivered load (Table 5).
With these parameters, the limiting K+ concentration is
~35 mM, and the final tubular K+ concentration is nearly
40 mM. Considerations of osmotic water flux suggest that, by blunting
the ADH effect, luminal hypotonicity might also render the CCD a site
for K+ secretion. In the calculations of Fig.
15, the luminal
Pf is set at 10% of its baseline value, and
luminal osmolality reaches only 240 mosmol/kgH2O. Along
most of the tubule length, the luminal K+ concentration
remains less than its limiting value, and, until the very end of the
tubule, the K+ flux remains secretory. Thus, instead of the
6% overall K+ reabsorption at baseline, low tubule water
permeability results in a 14% increase in K+ delivery to
the medullary collecting duct. Furthermore, what had been 38%
NH
reabsorption by the antidiuretic CCD becomes a net
secretion of 3% (Table 5). Finally, as had been discerned
experimentally, the greatest K+ secretion can be achieved
when luminal Cl
is replaced by an impermeant anion. This
is illustrated in Fig. 16, in which 30 mM of an impermeant have been substituted for luminal Cl
.
In this tubule, distal delivery of K+ is now 28% greater
than the entering load. Compared with control, both luminal
hyperpolarization and decreased volume reabsorption combine to render
conditions favorable for sustained K+ secretion.

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Fig. 14.
Transport along the CCD when epithelial Na+
reabsorption has been increased. Principal cell luminal membrane
Na+ permeability and peritubular Na-K-ATPase density have
both been increased by 50%. Conditions are otherwise as in Fig. 2.
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Fig. 15.
Transport along the CCD when principal cell luminal
membrane water permeability has been reduced to 10% of control.
Conditions are otherwise as in Fig. 2.
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Fig. 16.
Transport along the CCD when 30 mM of luminal
Cl have been replaced by an impermeant anion. Conditions
are otherwise as in Fig. 2.
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|
 |
DISCUSSION |
In the assessment of disorders of potassium excretion in humans,
it has become customary to calculate the transtubular K+
gradient (TTKG)
|
(29)
|
Equation 29 is intended to estimate the tubule fluid
K+ concentration at a point at which it was last isotonic
to plasma, namely, the CCD. The formula assumes that there is
negligible transport of solute in the medullary collecting
duct, so that K+ concentration changes from cortex to final
urine derive only from water abstraction in the medullary collecting
duct. (This is an important limitation to the ultimate significance of
the TTKG. If, for example, half of the sodium and urea delivered to the
medullary collecting duct were to be reabsorbed there, then the
computed TTKG would overestimate the CCD K+ concentration
by approximately a factor of 2.) Nevertheless, in the antidiuretic
human kidney, the TTKG does increase with K+ loading or
administration of mineralocorticoid (13). It is tempting,
therefore, to try to interpret the TTKG with reference to ion transport
by CCD and its modulation by mineralocorticoid. In the present work, a
model of the CCD was developed with parameters assigned to simulate
observed fluxes and permeabilities of rat tubules in vitro under
standard perfusion conditions. In these experiments, CCD K+
secretion is responsive to both ADH and mineralocorticoid, and these
observations focus attention on the limiting luminal K+
concentration that can be established by tubular secretion. The principal finding of this modeling effort is that, in the antidiuretic CCD in vivo, water abstraction may be a critical factor in determining the TTKG, in the sense that the TTKG may be well above the limiting K+ gradient.
This conclusion regarding the magnitude of the limiting K+
gradient is a quantitative one and contingent on the accuracy of the
model parameters, specifically the transport rates and certain key
permeabilities. With respect to overall transport rates, Table 6 provides a summary of observations of
CCD fluxes from rabbit and rat under similar perfusion conditions
(luminal Na,+ 145 mM; K+ , 5 mM) with or
without additional ADH and mineralocorticoid. The baseline
parameter set selected for the model calculations yielded net
Na+ reabsorption and K+ secretion of 68.6 and
29.8 pmol · mm
1 · min
1,
respectively, and a transepithelial PD of
18.3 mV, consistent with
perfusions in which both hormone effects are present. The measured
Na+ flux largely determines the choice of luminal membrane
Na+ permeability of the principal cell. Given this
permeability, measurements of the K+ transference number
for the luminal membrane (44) provide an estimate of
luminal K+ permeability. Finally, the measurement of the
fractional apical resistance (together with the observation that, in
the rat, the peritubular membrane is nearly entirely K+
selective) determines peritubular K+ permeability. From
these considerations, it appears that the overall permeability of the
transcellular K+ pathway is likely to be severalfold
greater than that across the tight junction. Because of the high
cytosolic K+ concentration, the difference in
K+ conductance between paracellular and transcellular
pathways is higher still in this model, ~10-fold greater. It must be
acknowledged that in the above argument, experimental determination of
tight junctional K+ permeability is not available, so that
the measurement of tight junctional conductance was used as a surrogate
(with the assumption that K+ permeability is 20% greater
than that of Na+).
From the parameters selected here, the prediction for the limiting
K+ concentration was 23 mM (Fig. 10). This may be compared
with micropuncture determinations of late distal fluid-to-plasma
K+ concentration ratio of 3.8 in rats on a
low-Na+ diet and treated with DOCA (29). In a
microperfusion study, the limiting K+ concentration of rat
DCT has been estimated to be 13-15 mM by extrapolating between
secretory (10 mM) and reabsorptive (25 mM) luminal K+
concentrations (18). Perhaps a more direct determination
of the limiting K+ concentration is the "stationary
K+ concentration" observed in a split drop, 30.6 mM under
control conditions (2). There do not appear to be
measurements of limiting K+ gradients in perfused rat CCD,
although, in a study of rabbit CCD, with prolonged contact time the
tubules could raise luminal K+ concentration to >100 mM
(19). An approximate analysis of the limiting luminal
K+ potential indicated that it is roughly the ratio of
principal cell peritubular uptake of K+ relative to the
peritubular membrane K+ conductance and is influenced
little by luminal membrane K+ conductance. Although it was
possible to achieve high limiting K+ concentrations by
decreasing peritubular membrane K+ permeability,
electrophysiology of rabbit CCD has indicated that peritubular
K+ conductance actually increases under the influence of
mineralocorticoid (39). Thus the effect of aldosterone to
enhance the limiting K+ concentration is likely to be due
to enhanced peritubular K+ uptake plus luminal
hyperpolarization, rather than decreasing K+ backleak.
Once DCT fluid reaches the water-permeable segment of the nephron,
osmotic equilibration produces a near doubling of solute concentrations. If tubule fluid K+ is less than its
limiting concentration, it can increase rapidly toward the limiting
gradient (Fig. 11), and if tubule fluid K+ is at its
limiting concentration at the start of CCD, it can increase well above
the limiting gradient (Fig. 12). In the first case, K+
secretion by CCD is shut off, and in the second case K+
transport by CCD becomes reabsorptive. This osmotic equilibration is
contingent on ADH-mediated hypotonic reabsorption. In the absence of
ADH, the persistence of luminal hypotonicity maintains a favorable secretory gradient for K+ along the length of CCD (see Fig.
15). These calculations highlight the necessity for ADH-modulated
K+ transport to maintain K+ balance during
transitions between antidiuresis and diuresis. They also underscore the
energetic advantage of this organization of transport along the
nephron, wherein the bulk of K+ secretion is completed
before water abstraction (28, 29).
After osmotic equilibration with hypotonic reabsorption, there may be
further isotonic reabsorption of luminal fluid due to NaCl transport by
CCD. In this case, the limiting K+ concentration is no
longer the zero-flux concentration used to obtain Eqs. 27 and 28 but rather some higher concentration that will
support reabsorptive K+ flux at a rate comparable to the
volume reabsorption, Jv. If one ignores
paracellular and intercalated cell fluxes, then the limiting
condition satisfies
|
(30)
|
where LMS is the series equivalent
coefficient specified in Eq. 26. Setting C
to be the limiting concentration that satisfies the zero-flux in
Eq. 27 and denoting
one obtains
|
(31)
|
in which it has been assumed that luminal PD is little changed by
the isotonic NaCl reabsorption. Defining an equivalent K+
permeability by
Eq. 31 may be rewritten
|
(32)
|
where the approximation in Eq. 31 requires that
Jv/hMS be small. For the
baseline model parameters, Table 4 shows the equivalent permeability,
hMS, and in view of the end-tubule volume
reabsorption, Jv = 6 × 10
6 cm3/cm2, the reabsorptive
volume flow is sufficiently small to apply Eq. 31.
The estimated increment in limiting K+ concentration (Table
4) is 15-30% above the zero-flux limiting concentration. From these considerations, the increase in the limiting K+
gradient due to isotonic NaCl transport should be a small contribution to the TTKG. This is not to say that the overall K+
reabsorption by CCD must be small. With reference to Fig. 13, in which
the impact of slow axial flow is considered, when there was
reabsorption of 75% of the delivered Na+ there was
reabsorption of 60% of delivered K+ and an approach of
luminal K+ concentration to the estimated limiting
gradient. In one set of observations in normal rats, the TTKG was found
to be 10, and the estimated CCD K+ concentration was
back-calculated to be 40 mM (Table 8 in Ref. 72). This
most closely corresponds to the situation in Fig. 12, in which the
luminal K+ concentration delivered to CCD was 24 mM, which
approximately doubled during osmotic equilibration. In this simulation,
CCD K+ reabsorption was ~20% of delivered load.
A critical issue for model development was estimating the relative
contributions of K+ secretion and Cl
reabsorption to maintaining overall electroneutral solute transport by
CCD. The fraction of Na+ reabsorption that was balanced by
K+ secretion was selected to be ~50%, according to the
observations of Schafer and Troutman (40). What
had not been anticipated in construction of the model was that, once
this ratio had been selected for the baseline case, it persisted over
the whole range of Na+ reabsorption rates engendered by
varying luminal Na+ permeability (Fig. 7). In contrast,
increasing delivered NaCl load resulted in increased NaCl reabsorption
with little change in K+ flux. Most of the Cl
reabsorption by this model tubule was paracellular. Transport assigned
to the
-cell was assumed to be small (17%). Supporting this
assumption is the observation that when luminal Cl
was
removed from perfused (control) rat CCD, there was no significant change in HCO
reabsorption (17). A
small fraction of Cl
reabsorption (10%) was assigned to
coupled transport across the principal cell via an NaCl cotransporter
(57). It was found that even a small peritubular
Cl
conductance was sufficient to reduce cytosolic
Cl
to levels compatible with experimental observation
(43). As long as changes in the rate of luminal membrane
NaCl cotransport and peritubular membrane Cl
permeability
varied in parallel, there would be little discernible change in
cytosolic Cl
concentrations. When Na+ fluxes
were varied by increasing luminal NaCl cotransport, there was no change
in K+ secretion, despite the increased flux through the
Na-K-ATPase. (The peritubular Cl
current depolarized the
epithelium, so that despite increased peritubular uptake of
K+, secretion was not enhanced.) Application of bradykinin,
which can alter CCD Cl
secretion without changing
transepithelial PD, is difficult to understand in the absence of a
transcellular electroneutral mechanism for Cl
reabsorption (58). In simulations not shown, the impact of decreasing
-cell H+-ATPase density on CCD K+
secretion was negligible, so that these calculations do not offer insight into the K+ wasting of distal renal tubule acidosis.
Issues of CCD acid-base transport have not been systematically explored
in this initial presentation of the model. In the normal rat, the CCD
is a net proton-secreting segment, but the rate of luminal
acidification appears to be lower than either DCT or medullary
collecting duct. Length-specific proton secretory rate in late DCT is
perhaps comparable to that of CCD, whereas early DCT may be threefold
greater (62). In the calculations of this model, osmotic
water loss doubles luminal HCO
concentration in the
early portion of the tubule, and there is a relatively stable
concentration along the remainder of the segment. In the reference
condition, ~16% of the delivered HCO
was
reabsorbed, but slow flow could enhance this substantially. A subtle,
but perhaps important observation is that in antidiuresis there is also
an abrupt increase in luminal ammonia concentrations, with brisk
reabsorption of significant amounts of base (NH3). Quantitatively, this base reabsorption is a significant fraction (2/3)
of the net HCO
reabsorption. When this base exit
from the lumen was eliminated in diuresis (Fig. 15 and Table 5), CCD
HCO
reabsorption also virtually ceased. This
mechanism for HCO
reclamation seems not to
have been recognized previously. However, it should not be specific to
some special feature of this model. Rather, substantial NH3
reabsorption by the CCD follows directly from the delivery of ammonia
at a concentration an order of magnitude higher than in cortical blood,
further concentration of ammonia and HCO
by water
abstraction, and a high CCD permeability to NH3. Although
overall collecting duct transport of ammonia has been found to be small
(21), prior models of the medullary collecting duct have
predicted significant ammonia addition (69, 70). Thus
sequential ammonia reabsorption and secretion along the collecting duct
would not be incompatible with the overall micropuncture accounting.
In sum, with the emphasis of the present work on K+
excretion, CCD model parameters have been selected to represent the rat kidney under the influence of both ADH and aldosterone. The model was
used to identify determinants of the transtubular K+
gradient. From the perspective of an epithelial model, the zero-flux equilibrium is the natural calculation, and in this the luminal concentration derives from the magnitude of principal cell peritubular K+ uptake relative to peritubular membrane K+
permeability. When the model is configured as a tubule and there is
isosmotic NaCl reabsorption, the condition of luminal K+
equilibrium requires that overall K+ flux be reabsorptive
and keep pace with the reabsorptive volume flux. This provides a small
increment to the zero-flux equilibrium concentration. The most striking
effect, however, in the simulation of conditions in vivo comes with
osmotic equilibration of luminal fluid. With this, there is a doubling
of the initial K+ concentration, which, depending on
delivered load, may be substantially greater than the equilibrium
value. This implies that the CCD could be a site for K+
reabsorption, although the relatively low permeability ensures that
luminal K+ concentration declines slowly. Thus assessment
of the TTKG from final urine composition may yield a value well above
that which can be rationalized from transport studies in isolated
perfused tubules.
 |
ACKNOWLEDGEMENTS |
This investigation was supported by Public Health Service Grant
1-R01-DK-29857 from the National Institute of Diabetes and Digestive
and Kidney Diseases.
 |
FOOTNOTES |
Address for reprint requests and other correspondence: A. M. Weinstein, Dept. of Physiology and Biophysics, Weill Medical College
of Cornell University, New York, NY 10021.
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 9 October 2000; accepted in final form 12 February 2001.
 |
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