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Department of Physiology and Biophysics, Cornell University Medical College, New York, New York 10021
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ABSTRACT |
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A mathematical model of the inner medullary
collecting duct (IMCD) of the rat has been developed representing
Na+, K+, Cl
,
HCO
3, CO2,
H2CO3, phosphate, ammonia, and urea. Novel
model features include: finite rates of hydration of CO2, a
kinetic representation of the H-K-ATPase within the luminal cell
membrane, cellular osmolytes that are regulated in defense of cell
volume, and the repeated coalescing of IMCD tubule segments to yield
the ducts of Bellini. Model transport is such that when entering
Na+ is 4% of filtered Na+, approximately 75%
of this load is reabsorbed. This requirement renders the area-specific
transport rate for Na+ comparable to that for proximal
tubule. With respect to the luminal membrane, there is experimental
evidence for both NaCl cotransport and an Na+ channel in
parallel. The experimental constraints that transepithelial potential
difference is small and that the fractional apical resistance is
greater than 85% mandate that more than 75% of luminal
Na+ entry be electrically silent. When Na+
delivery is limited, an NaCl cotransporter can be effective at reducing
luminal Na+ concentration to the observed low urinary
values. Given the rate of transcellular Na+ reabsorption,
there is necessarily a high rate of peritubular K+
recycling; also, given the lower bound on luminal membrane
Cl
reabsorption, substantial peritubular
Cl
flux must be present. Thus, if realistic limits on
cell membrane electrical resistance are observed, then this model
predicts a requirement for peritubular electroneutral KCl exit.
epithelial sodium ion transport; sodium chloride cotransport; potassium chloride cotransport
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INTRODUCTION |
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IN THE RAT, the inner medullary collecting duct (IMCD) receives about 5% of the filtered fluid, containing roughly 3.0-4.5% of filtered Na+ and 20% of filtered K+ (3, 43). Final regulation of urinary osmolality, Na+, K+, and pH occurs as some 7,200 tubules merge 5 or 6 times to form 100-200 papillary collecting ducts (24, 32). In vivo studies in the rat have indicated that IMCD Na+ transport rates can be comparable to those of the proximal convoluted tubule (2, 7, 52), although rates in dissected tubules may be substantially lower than those in vivo (34). Although the luminal cell membrane contains a cation channel (29) and a component of Na+ reabsorption is amiloride inhibitable (21, 35, 62), the transepithelial potential difference (PD) of this segment is relatively small (16, 19, 34, 37, 47), and thiazides can inhibit a significant fraction of Na+ transport by the IMCD (35, 61). Under normal circumstances, the IMCD appears to be a site for K+ reabsorption (5, 7), whereas in times of K+ loading, IMCD K+ secretion can be detected (3, 7). There is luminal proton secretion along the IMCD (16, 53), which appears referable in large measure to an H-K-ATPase (14, 23, 31). Furthermore, the rate of acid secretion is modulated by NH3 availability to mediate peritubular base exit (54).
From these considerations, any model of IMCD which seeks to represent Na+ transport must also provide a reasonable picture of K+ and acid/base fluxes. Although most of the important features of these cells have been represented in prior models, several aspects of IMCD have not been modeled previously in epithelial simulations. These include the finite rate of hydration of CO2 within the cell and extracellular compartments, the luminal membrane H-K-ATPase, the large variation in peritubular solute composition over the length of the duct, and the progressive coalescing of the tubules to become the papillary collecting ducts. In this report, the basic structure of the model will be presented, with emphasis on the pathways for Na+ and K+ transport. In the companion study (59a), features of acid/base transport will be examined. In this work, the epithelial model will also be cast as a tubule model to ensure that the computed epithelial fluxes result in realistic axial solute gradients. The calculations to be presented suggest that the magnitude of the transcellular Na+ and K+ fluxes are such that both luminal NaCl cotransport and peritubular KCl cotransport must be important pathways in this epithelium.
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MODEL STRUCTURE |
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The 12 model solutes are Na+, K+,
Cl
, HCO
3,
CO2, H2CO3,
HPO2
4,
H2PO
4, NH3,
NH+4, H+, and urea, as
well as an impermeant species within the cells and possibly within the
lumen. These are the minimal set of solutes which will permit
representation of net acid excretion. Compared with the previous
cortical collecting duct (CCD) model of Strieter et al. (51), the
additional solutes are H2CO3, the two ammonia species, and urea. As in previous work, the model will be composed of
compliant cellular and intercellular compartments lining the tubule
lumen (Fig. 1). 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 ) (in mmol/s).
To formulate the equations of mass conservation with multiple reacting
solutes, it has been convenient to identify the generation of each
species as an intermediate variable. Within a cell or interspace, the generation of i (s
(i )) is
equal to its net export plus its accumulation
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(1) |
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(2) |
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
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(3) |
, and urea)
are written
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(4) |
= E, I, or M. For the phosphate and for the
ammonia buffer pairs, there is conservation of total buffer
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(5) |
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(6) |
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One important feature of this model will be the handling of CO2 and the disequilibrium pH. A medullary interstitial PCO2 profile will be specified, but the CO2 concentrations of the cells, interspace, and lumen are model variables. The relevant reactions are
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3 and
H2CO3 are interconverted, mass conservation
requires
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(7) |
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= I or E, whereas for the tubule lumen
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(8) |
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= I, E, or M), conservation of
total CO2 is expressed as
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(9) |
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(10) |
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(11) |
4:H2PO
4, NH3:NH+4, and
HCO
3:H2CO3.
Corresponding to the electrical potentials, 
, for
= E, I, or M, is the equation for electroneutrality
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(12) |
With respect to volume flow, the approach taken heretofore has been to utilize the volume conservation equations for lumen, interspace, and cell to compute the three unknowns: luminal volume flow, lateral interspace hydrostatic pressure, and cell volume. (Since cell hydrostatic pressure was set equal to luminal pressure, and total cell impermeant content was also fixed, changes in cell volume adjusted the concentration of the impermeant to achieve cell water balance.) In a model of IMCD epithelium, this approach is unsatisfactory, since the large variations in peritubular osmolality would impact unrealistically on cytosolic electrolytes. In vivo, chronic changes in interstitial osmolality leave cytosolic electrolyte composition relatively unperturbed because of compensating changes in organic osmolytes (42). Indeed, changes in cytosolic osmolyte concentrations can occur relatively rapidly in response to ambient conditions (41). Ideally, one would specify the concentrations of organic osmolytes as additional model variables with their own kinetics. However, what has been done here is to restrict the simulations to steady-state problems and 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 cell each 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
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(13) |
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
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(14) |

(i ) is
a solute permeability, and C
(i ) and
C
(i ) are the concentrations of i
in compartments
and
, respectively. Here
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(15) |


is the PD 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
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(16) |
In this model, the Na-K-ATPase within the peritubular membrane is represented by the expression (12)
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(17) |
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(18) |
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(19) |
In this model, both the cross-sectional area and the epithelial transport area decrease as one proceeds from the outer-inner medullary junction (OIMJ) toward the papillary tip. Starting with 7,200 tubules at the base of IMCD, a series of 6 mergings of pairs of ducts reduces the final number of papillary collecting ducts by 1/64 to 113 (24). With each merging, the lateral surface for transport, BM, and the axial cross section for flow, AM, are halved. In the computer code, this is accomplished using a continuous formulation as a function of distance, x, along the IMCD of total length L
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(20) |
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MODEL PARAMETERS |
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In the selection of model parameters, effort was taken to achieve
compatibility with observations in rat IMCD, where both in vivo and in
vitro data are available. A simplification was made in representing the
IMCD cell as a single entity, ignoring differences from base to
papillary tip. Model parameters are indicated in Table
1. The luminal diameter, set to 30 µm, is
comparable to the observation of Sands et al. (39), and cell volume, 8 × 10
4 cm3/cm2, is in the range
reported by Flamion and Spring (9). Morphological data of Rastegar et
al. (33) indicate apical-to-basolateral membrane area ratio of 9.5 and
basolateral-to-tubular basement membrane area of 5.6. For a tubule with
inner and outer radii of 15 and 23 µm, respectively, these
measurements are compatible with an unamplified luminal membrane, whose
area is 0.6 cm2/cm2 relative to the tubule
basement membrane. Total tubule length is 5 mm (61). The volume of the
lateral intercellular space was taken to be ~10% of the epithelial
volume (with a relatively small compliance), a value comparable to that
observed in CCD (60). Carbonic anhydrase (CA) activity diminishes along
the IMCD, from substantial staining at the outer third to negligible at
the papillary tip (30). This is, presumably, the cytosolic enzyme
CA-II, since the membrane bound carbonic anhydrase, CA-IV, does not
appear to be present in IMCD (4, 55). Accordingly, within the lumen and
lateral interspace, the rate constants for hydration and dehydration
are the uncatalyzed values (13); within the cells, the coefficients
have been assumed to be 100-fold larger (i.e., 1% of complete
catalysis). Permeabilities of the tight junction were taken to yield
compatibility with the overall electrical resistance of the epithelium
and a relative anion selectivity (39). 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.
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In Fig. 2, the important cellular transport
pathways have been indicated. On the peritubular membrane is an
Na-K-ATPase, with competition between
NH+4 and K+ for cellular
uptake. The peritubular membrane of these cells contains
Na+/H+ exchange (18, 57) and
Cl
/HCO
3 exchange (27,
48). A peritubular cotransporter for
Na+-K+-2Cl
has also been
identified (17, 36). Also shown within the peritubular membrane is a
cotransporter for HPO2
4, (in parallel
with a small peritubular phosphate leak) so that the cells will have a
nonzero concentration of phosphate, serving as a second internal
buffer; there is, however, no vectorial transport of phosphate. The
peritubular KCl cotransporter has not been identified experimentally
but is a prediction of these model calculations and will be considered
below. The conductance of the peritubular membrane is predominantly
that for K+ (47). With such K+ channels, an
NH+4 permeability may also be
associated which may be less than (20) or comparable (6) to that for K+. In this case, the NH+4
permeability was taken to be 20% that for K+. The
Cl
permeability was assumed small, but not
insignificant, namely 3% of the K+ permeability, and the
HCO
3 permeability was one-half that
for Cl
. At the luminal membrane, the only significant
conductance is that of a nonselective cation channel, with equal
permeabilities to Na+ and K+ (29). The
importance of this channel for luminal Na+ entry, relative
to NaCl cotransport, is considered in the model calculations below.
Proton secretion at the luminal membrane derives at least in part (14),
and perhaps exclusively (23, 31, 58), from an H-K-ATPase. In the
calculations of this study, this is the only luminal proton pump.
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Although not pictured in Fig. 2, membrane permeabilities have been
assigned for CO2, H2CO3,
NH3, urea, and water. The unit membrane permeabilities for
CO2 and H2CO3 are those used for
proximal tubule cell membrane (26) and have been assumed to apply to both luminal and peritubular membranes. The NH3
permeability is derived from the whole epithelial determination of
Flessner et al. (10), by assuming the unit membrane permeabilities of
luminal and peritubular membranes are comparable. (Since the
peritubular membrane area is 10-fold greater than the luminal, it is
the luminal membrane permeability that is measured in the
transepithelial determination.) Sands et al. (38) have found an IMCD
urea permeability 1.7 × 10
4 cm/s and 6.9 × 10
4 cm/s in the absence and presence of ADH.
Accordingly, the tight junction urea permeability was taken as 30% of
the unstimulated epithelial permeability, and the unit membrane urea
permeabilities of luminal and peritubular surfaces were assumed to be
comparable (giving the peritubular membrane a 10-fold greater urea
permeability). In this way, the unit membrane peritubular urea
permeability (9 × 10
5 cm/s) is comparable to that
determined by Star (49). The stimulated luminal membrane urea
permeability was taken as 20-fold greater than its unstimulated value,
and all calculations were performed with the assumption that
antidiuretic hormone (ADH) is present. ADH also produces a threefold
increase in the IMCD water permeability (Pf) from
7 × 10
3 to 19 × 10
3 cm/s (38). As
with urea, the tight junction water permeability was taken as 30% of
the unstimulated epithelial value, and the unit water permeabilities of
luminal and peritubular membranes were assumed comparable. A stimulated
luminal membrane water permeability fivefold greater than the
unstimulated value yielded a realistic overall epithelial water
permeability.
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MODEL CALCULATIONS |
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Table 2 contains the results of a solution
of the model equations for an open-circuited epithelium between bathing
media in which the solute concentrations are suggestive of the OIMJ. The high luminal K+ (7) and low luminal
HCO
3 (8, 15) have been observed in
vivo. The luminal NH+4 concentration is similar to that found from microcatheterization (15, 46). The
peritubular concentrations of K+ (7) and
NH+4 (50) are both high, whereas the
HCO
3 concentration is not very
different from that of the systemic circulation (8). The computed
cellular electrolyte concentrations are similar to those obtained by
Sone et al. (42) using the electron microprobe. In particular, the cell
Na+ is about twice as high, and the cell Cl
is about five times higher than that found by the same workers in the
CCD (1). In vitro, dissected rat IMCD cells have an acid pH (54), and
this is also achieved with this set of model parameters. The solute
concentrations within the lateral interspace largely reflect those of
the peritubular medium, because of the relatively small
diffusion barrier of the basement membrane. Here, one model prediction
that emerges is a significant disequilibrium alkalosis within the
interspace. This is due to the peritubular HCO
3 extrusion, the lack of
membrane-bound CA, and the micromolar concentrations of
H2CO3. The open-circuit PD is
13 mV, a value
close to that first observed (37) but higher than subsequent
determinations.
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Figure 2 displays several of the important fluxes when the bathing
media concentrations are those assigned to the OIMJ. Luminal membrane
Na+ flux is divided between channel-mediated entry (26%)
and NaCl cotransport; peritubular Na+ extrusion is via the
Na-K-ATPase, with small uptake fluxes through Na+/H+ exchange, phosphate entry, and Na-K-2Cl
cotransport. With reference to Table 2, the electrochemical gradient
favoring blood-to-lumen Na+ flux produces a transjunctional
backflux ~39% of the luminal membrane reabsorption. Under the
conditions shown, luminal membrane K+ flux is also
reabsorptive, both via the nonselective cation channel (29%) and the
H-K-ATPase. At the peritubular membrane, there is also cellular uptake
of K+ via the Na-K-ATPase and, to a minor degree, with
Na-K-2Cl cotransport. Peritubular exit of K+ via
channel-mediated flux is relatively small (12%) in comparison to KCl
cotransport. The high luminal K+ concentration yields a
reabsorptive paracellular flux, which is ~11% of the total
transepithelial flux. With respect to Cl
, reabsorption
at the luminal membrane is cotransport with Na+ and at the
peritubular membrane with K+; paracellular
Cl
flux is secretory but small. Plausibility of these
overall epithelial Na+, K+, and
Cl
fluxes requires scrutiny of the performance of this
IMCD when it is configured as system of tubules.
Using the geometry outlined above, the 7,200 tubules of a single kidney
were perfused at 24 µl/min, or ~5% of single-kidney glomerular
filtration rate (GFR) (43). The perfusing solution composition was that
of the OIMJ (Table 2), so that Na+ delivery was 4% and
K+ delivery 50% of estimated filtered loads. The
peritubular solute concentrations at the terminal IMCD were taken to
be identical to those at the OIMJ, with the exceptions of
urea (which increased from 200 to 500 mM) and K+ (which
increased from 10 to 20 mM), so that peritubular osmolality increased
from 800 to 1,120 mosM. Along the IMCD length, peritubular concentrations were assumed to vary linearly. Model equations were
solved using a first-order implicit differencing scheme, since the
centered scheme used previously for proximal tubule calculations would
not converge. For the baseline parameters, 800 spatial steps were
required to reach the point of no further visual change in the solution
curves. Figure 3 displays computed solute
profiles and flows within the IMCD. About 71% of the entering Na+ and 65% of entering K+ has been
reabsorbed, but with progressive water reabsorption there is a 24%
increase in end-tubule Na+ and a 54% increase in
end-tubule K+ concentrations. Proton secretion via the
H-K-ATPase has reduced the luminal
HCO
3 concentration from 5 mM to near
zero, with pH falling nearly 3 units to 4.0. The
NH+4 concentration, initially 10 mM
and increasing to 58 mM, reflects ammonia addition along the tubule
equal to 47% of entering flow; the magnitude of this secretion is
similar to that reported from microcatheterization (46).
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In view of the large transepithelial solute concentration gradients,
accuracy of the fluxes is contingent on accuracy of the overall
epithelial solute permeabilities. Table 3
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 the following composition (mM): 140 Na+, 10 K+, 119 Cl
, 25 HCO
3, 1.5 CO2, 3.9 total
phosphate, 5.0 urea, 1.0 NH+4, and 0.1 impermeant. A series of calculations were performed in which the
luminal solute concentrations were lowered and then raised by 0.1 mM. After subtracting off the small correction due to
changing volume flux, 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 (in
mS/cm2) and when multiplied by
z(i )FC(i )/RT is the ionic
permeability (in cm/s), and these are also included in Table 3. It is
clear that the important conductances are those for Na+ and
Cl
, and to a minor extent K+ and
HCO
3, and that together these confer
a total epithelial conductance of 14.6 mS/cm2 or 68
· cm2. This may be compared with
experimental conductance determinations of 13.7 (47) and 25.4 mS/cm2 (39). The ion permeabilities for Na+ and
Cl
derived from the conductances are 1.15 and 1.85 × 10
5 cm/s and are similar to those obtained by Sands et
al. (39). The permeabilities derived from imposition of ion gradients
are greater (2.32 and 2.29 × 10
5 cm/s) and reflect
the presence of electroneutral, coupled solute pathways. The model
K+ permeability is 3.1 × 10
5 cm/s; for
comparison, a value for rat IMCD K+ permeability of 4.0 × 10
5 has been obtained by Rocha and Kudo (34), using
isotopic bath-to-lumen K+ secretion rates. The model
permeabilities for NH+4, 0.98 × 10
5, and NH3, 0.20 × 10
2
cm/s, are comparable to those obtained by Flessner et al. (10) for rat
IMCD in vitro. The water permeability, Pf = 0.014 cm/s, and the urea permeability, 3.36 × 10
4 cm/s, are
similar to values reported for ADH-treated rat IMCD (34, 38).
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Luminal membrane NaCl cotransport is a prominent feature of this model
IMCD cell; the constraint on the magnitude of this component is
examined in Fig. 4. In these calculations,
luminal membrane NaCl cotransport is varied by varying the coefficients for NaCl cotransport and peritubular KCl cotransport proportionally. At
the same time, the luminal membrane cation conductance (equal for
Na+ and K+) is adjusted so that transcellular
Na+ flux remains constant at 5.6 neq · s
1 · cm
2.
Thus the fraction of luminal flux via cotransport was changed without
changing the total luminal membrane Na+ flux. This was
achieved by incorporating the IMCD model as a subroutine in a Newton
iteration using the luminal cation conductance as the single
independent variable. The boundary conditions for these calculations
are those for the OIMJ, as in Table 2. The bottom of Fig. 4
displays the luminal membrane cation permeability determined from the
Newton iteration. The middle of Fig. 4 is the fractional apical
resistance, defined as the electrical resistance of the luminal
membrane divided by the sum of luminal and peritubular resistances.
Transepithelial and peritubular electrical PD values are plotted in the
top of Fig. 4; the distance between these two curves is the
luminal membrane PD. The arrows on each abcissa of Fig. 4 indicate the
fractional luminal Na+ flux using the baseline parameters
of Table 1, so that for this case the fractional apical resistance is
85%. The electrophysiological study of rat IMCD by Stanton (47) found
a fractional apical resistance of 94 ± 5% (SD) along with a
transepithelial PD of
3 mV. This observation suggests that the NaCl
cotransporter cannot be any less important than its representation by
the baseline model parameters.
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For the same calculations of Fig. 4, Fig. 5 displays the model K+ fluxes. The curves in the top of Fig. 5 indicate that the paracellular K+ is relatively minor in comparison to the total. The bottom of Fig. 5 contains the total luminal membrane K+, along with its two components, that through the cation channel and via the H-K-ATPase. It is apparent that as the channel permeability starts to increase (moving from right to left along the abcissa), the reabsorptive K+ flux through the channel also increases. However, with increasing channel permeability there is progressive luminal membrane depolarization (Fig. 4). Ultimately, the channel K+ flux declines and becomes secretory to the extent that it nullifies the reabsorptive pump flux, and net luminal membrane K+ flux is negligible.
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The impact of luminal Na+ concentration on the IMCD Na+ and K+ fluxes is examined in Fig. 6. In these calculations, two sets of model parameters are considered: calculations in Fig. 6, left, are obtained using the baseline parameters, in which about three-fourths of the luminal membrane Na+ flux is via cotransport; and calculations in Fig. 6, right, are obtained using parameters in which about three-fourths of the luminal membrane Na+ flux is conductive. This parameter set was derived from the calculations of Fig. 4, so that total luminal cell membrane Na+ fluxes are equal under OIMJ bathing conditions. The bathing conditions for these calculations are those of the OIMJ, except for the luminal NaCl concentration, which is shown on the abcissa of Fig. 6. For both Na+ and K+, and with either parameter set, the junctional fluxes are small. In the top of Fig. 6, it may be observed that for both sets of model parameters, the luminal membrane Na+ fluxes are equal at high luminal Na+ (by design), and both go to zero at a luminal Na+ concentration of 2 mM. However, in the midrange of luminal Na+ concentrations, the cotransport parameters provide a luminal membrane flux that is less sensitive to luminal Na+ concentration. As a consequence, the luminal Na+ concentration at which total epithelial Na+ flux falls to zero is near 20 mM when cotransport dominates but over 40 mM when the channel dominates. With respect to K+ fluxes, the salient difference between the two models is the relative independence of Na+ and K+ fluxes, when NaCl cotransport is most important.
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The ability of the model IMCD to reabsorb Na+ against an adverse electrochemical gradient is further examined in Fig. 7. For these calculations, all of the parameter sets generated from the calculations of Fig. 4 are utilized, and the abcissa identifies each set by the fractional luminal NaCl cotransport under OIMJ bathing conditions. For each parameter set, luminal NaCl concentration is varied to bring some Na+ flux to zero, and that concentration is plotted as the ordinate. This was achieved by incorporating the IMCD model as a subroutine in a Newton iteration using the luminal Na+ concentration as the single independent variable. The bottom curve in Fig. 7 demonstrates that this model cell can reabsorb Na+ down to a luminal concentration of ~2 mM and that this equilibrium value is independent of the mechanism of luminal Na+ uptake. The top curve of Fig. 7 signifies the value of luminal NaCl when the cellular reabsorptive flux is just equal to the paracellular backflux; this is the curve most relevant to predicting the equilibrium Na+ concentration of the luminal fluid in vivo. It is apparent that conductive uptake is less effective than the coupled NaCl cotransporter in clearing the lumen of Na+. In part, this is due to the more negative luminal PD, when luminal Na+ uptake is conductive. This is shown in the calculations of the middle curve of Fig. 7, in which transepithelial Na+ is brought to zero while the epithelium is short circuited; the small slope that remains for this middle curve reflects the differences in shape of the curves of luminal membrane Na+ flux as a function of luminal Na+ concentration, displayed in Fig. 6. At its best, this model epithelium can transport down to ~17 mM, although the rats examined by Diezi et al. (7) achieved end-papillary sodium concentrations of 11.4 mM and final urine concentrations of perhaps one-half that value.
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Another prominent feature of this model is the peritubular KCl
cotransporter, which has yet to be identified in this epithelium experimentally. In the model, if one decreases the coefficient for
peritubular KCl cotransport, then cell K+ and
Cl
concentrations increase, and cell NaCl entry is
blunted. The functional impact of the cotransporter can be replaced by
proportional increases in both peritubular K+ and
Cl
conductances. The Cl
pathway is
required because of the substantial luminal Cl
entry,
and the K+ pathway is required by virtue of peritubular
K+ uptake via the Na-K-ATPase. In the calculations of Fig.
8, as one moves along the abcissa from right to left,
the peritubular KCl coefficient is decreased and the two ion
conductances are proportionally increased in a Newton iteration that
seeks to maintain transcellular Na+ flux constant at 5.6 nmol · s
1 · cm
2.
The bounding solutions are those of the OIMJ. The four top
panels of Fig. 8 show that even though only
luminal Na+ flux was controlled, the variations in luminal
Cl
flux, cytosolic K+ and Cl
,
and peritubular PD are all minor. Thus, with respect to net fluxes and
cell composition, the parallel conductance pathways could substitute
for the cotransporter. In the bottom of Fig. 8, however, are
the peritubular conductances for each parameter set. Although the
peritubular conductance for IMCD has not been determined, one might
consider comparison with values obtained for proximal tubule of rat,
0.011 (11), and rabbit, 0.026 mho/cm2 (28). This
consideration has motivated the choice of the baseline parameter at the
rightmost point of these curves. Parenthetically, the fraction of
peritubular KCl cotransport is greater than unity because the
denominator of this fraction is transcellular Cl
flux,
while the numerator also includes reabsorption of the backflux through
the peritubular Na-K-2Cl cotransporter.
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Rocha and Kudo (36) have observed in perfused IMCD that atrial
natriuretic factor (ANF) eliminates net reabsorption of Na+
and Cl
, both by enhanced secretory flux and decreased
reabsorptive flux. Their isotopic flux determinations indicated ANF
increases the bath-to-lumen fluxes of Na+,
Cl
, and K+, and this increase is blocked by
furosemide. The impact of an isolated activation of peritubular
Na-K-2Cl cotransport is examined in Fig. 9.
The model equations are solved over a range of luminal NaCl
concentrations and compare the baseline parameter set (curves labeled
"low flux") with a parameter set in which the peritubular Na-K-2Cl coefficient has been increased 10-fold ("high flux"). The top of Fig. 9 displays the flux of Na+ through
the cotransporter; under OIMJ conditions, the high flux is
approximately sevenfold greater than control. With activation of the
cotransporter, cell Cl
concentrations are ~15 mM
higher, under all luminal NaCl concentrations (Fig. 9, bottom).
Nevertheless, the middle of Fig. 9 shows that the impact of
isolated Na-K-2Cl cotransport on net transepithelial Na+
flux is small. These calculations suggest that ANF action requires impact on an additional Na+ transport step, most likely
luminal entry.
|
| |
DISCUSSION |
|---|
|
|
|---|
This is the first mathematical model of the IMCD to be developed that includes a simulation of the IMCD epithelial cell. It was intended that the scope of this model should include acid secretion by this nephron segment, and this has mandated inclusion of ammonia and phosphate buffers and has enriched the assortment of transport components. Several aspects of this model are novel, and have not been utilized previously in other epithelial models. Specifically, these include the finite rate of hydration of CO2 and the activity of a luminal cell membrane H-K-ATPase. Whereas previous nephron segment models have consisted of a single cylinder surrounded by a uniform bath, the IMCD model required a system of coalescing tubules traversing an interstitium whose composition varies as a function of distance. In this regard, two simplifications were made in the formulation of this model. The first is that cell volume is constant, and balance of luminal and peritubular water fluxes is secured by adjusting the concentration of an impermeant intracellular solute. The importance of this simplification is avoidance of grossly exaggerated swings in the concentration of intracellular electrolytes with changes in peritubular or luminal osmolality. For the moment, it avoids the issue of identifying volume-mediated solute transport steps, as well as identifying the kinetics of organic osmolytes. The trade-off with this simplification is that only steady-state model calculations are meaningful. The second simplification is the assumption of uniform hydrostatic pressure along the IMCD. With six generations of tubule merging, there is a 64-fold reduction in tubule number, despite only a 25% decline in the volume flow from base to tip in antidiuresis. Thus the tubule fluid velocity is expected to increase some 16-fold from base to tip and should be comparable to a proximal tubule flow rate. Under diuretic conditions, the tip flow rate may further increase 10-fold. For a rigid cylinder under Poiseuille flow, the predicted pressure drop at the higher flows would be unrealistic, but rather than consider a system of distensible tubules, the issue has been suppressed for the purpose of this model.
After model geometry, the most important issue to secure is the rate of
Na+ transport. A number of investigators have provided
information on this issue, and their results have been summarized in
Table 4. Perhaps the most direct
information comes from Ullrich and Papavassiliou (52), who
microperfused medullary capillaries of the rat with a Ringer solution
and assessed IMCD Na+ transport rate by shrinking drop
observation. In their preparation, paracellular backleak should have
been negligible, so that their reported flux rate, 4.0 nmol · s
1 · cm
2,
may be identified with transcellular Na+ reabsorption.
Micropuncture studies in young rats (5, 7) have reported their results
as the fraction of filtered Na+ reabsorbed per length of
tubule (0.6% and 1.5%, respectively). To translate these into
absolute flux rates per unit length requires an assumption of the
number of papillary collecting ducts at the level of micropuncture (for
1,000 IMCD, 7 or 17 nmol · s
1 · cm
2,
respectively). Microcatheterization has been undertaken in older rats
(2, 43), and these investigations report Na+ reabsorption
over the whole IMCD as a fraction of filtered load (1.75% and 3.0%,
respectively). To obtain absolute flux rates, one needs an estimate of
the total branching length of the IMCD (using 14,400 mm,
4.6 and 7.9 nmol · s
1 · cm
2,
respectively). Thus, the data obtained in vivo are in general agreement
and yield an IMCD Na+ transport rate comparable to that of
proximal tubule. Transport rates obtained from in vitro IMCD
preparations are substantially lower. In the isolated perfused IMCD of
the rat, Rocha and Kudo found Na+ reabsorption to be ~1.0
nmol · s
1 · cm
2
(34, 35). Suspension of rabbit IMCD has been found to utilize oxygen at
0.65 µmol
O2 · min
1 · g wet
wt
1 (62). Even if all of this were for Na+
transport, it accounts for only 0.3 nmol · s
1 · cm
2
(6 ATP per O2 and 3 Na+ per ATP). More
directly, in this same preparation, ouabain-sensitive K+
uptake was found to be ~25% of tissue K+ per minute
(25), or ~0.7
nmol · s
1 · cm
2.
In a cell culture of rat IMCD, Wall and Koger (56) observed a maximal
Na-K-ATPase activity of 200 nmol · mg
protein
1 · min
1, or 1.5 nmol · s
1 · cm
2.
In the model calculations presented here, the Na+ flux rate
was chosen to be consistent with in vivo data: IMCD reabsorption of
about three-fourths of delivered Na+ load, or ~3% of
estimated filtered Na+ load.
|
With respect to model development, the importance of the absolute
Na+ transport rate derives from the constraint it imposes
on the magnitude of rheogenic transport. The only reported cellular
electrophysiology on IMCD is that of Stanton (47), who found a small
transepithelial electrical PD and a high fractional apical resistance.
Admittedly, these findings could, in part, be an artifact of the in
vitro preparation and its low reabsorptive Na+ flux. In the
calculations of this model, only when channel-mediated luminal membrane
Na+ reabsorption was less than 25% of the total was the
fractional apical resistance greater than 85%. Although one early
determination of transepithelial PD was
11 mV (37), subsequent
investigations have shown it close to zero (16, 19). Indeed, the
transepithelial PD of this model,
12 mV, is high and is attributable
largely to the component of rheogenic Na+ transport. This
suggests that the bulk of Na reabsorption is electrically silent, as
via a thiazide-sensitive NaCl cotransporter, which has been identified
both in vivo (61) and in vitro (35). Unfortunately, there are no
studies available that can confirm the presence of such a cotransporter
within the IMCD, either by in situ hybridization or by
immunohistochemistry. Although amiloride can increase luminal membrane
electrical resistance (47) and decrease perfused tubule Na+
reabsorption (35), along with tubule oxygen consumption (62), there is
no sure way to use these observations to guarantee the importance of
this pathway in vivo. Of note, the model calculations argue only that
electrogenic Na+ transport is a small fraction of the
total; they do not preclude Na+-urea cotransport (22) as a
significant component within the electrogenic fraction. It is of
interest that the model cell could reabsorb Na+ down to 2 mM of luminal Na+, regardless of rheogenic or
electroneutral luminal uptake. However, when the cellular flux is
channel mediated, there is a higher transepithelial PD, greater tight
junctional backflux, and luminal Na+ equilibrium
concentration can become unrealistically high. The discrepancy between
the luminal Na+ concentration that zeroes cell membrane
Na+ flux and that which zeroes epithelial Na+
flux reflects, in part, the Na+ permeability of the tight
junction. Although the model permeabilities have tried to remain
faithful to reported estimates, it is likely that a tighter tight
junction will be necessary to simulate the low urine Na+
concentrations of which the Na+-deprived rat is capable
(7).
Consequent to the high reabsorptive Na+ flux by IMCD is substantial peritubular K+ uptake via the Na-K-ATPase and, of necessity, a peritubular exit step for K+. Were it known, the overall electrical conductance of the peritubular membrane would serve as a constraint on acceptable values for the peritubular K+ permeability. In the absence of this information, if the peritubular resistance of proximal tubule substitutes as a constraint on this parameter, then most of the peritubular K+ exit must be electroneutral, as for example, via a KCl cotransport. A similar problem was confronted in the development of the model of proximal convoluted tubule (59). In that setting, peritubular KCl cotransport was proposed as a resolution to a modeling exigency; the presence of this transporter in proximal tubule was subsequently identified by Sasaki et al. (40). Overall, in control conditions, the IMCD demonstrates little K+ transport (5, 7, 43); in vitro (34), or under Na+ or K+ depletion (7), K+ transport is reabsorptive; whereas in volume expansion (44, 45), or under K+ loading, secretory K+ flux has been observed. In this model, K+ flux was reabsorptive due to the inclusion of a briskly transporting luminal membrane H-K-ATPase. The flux through this transporter was deliberately highlighted in order to demonstrate its capacity to achieve an acid luminal pH. Given the generous delivery of buffer in these model calculations, the degree of K+ reabsorption was obligatory. It should be noted (Fig. 5) that the K+ flux through the luminal membrane cation channel is predicted to be small; with the baseline parameters, it is 25% of the total, but with higher fractional luminal membrane NaCl cotransport, this fraction declines toward zero. Thus, without specific activation of H-K-ATPase, the model would be compatible with substantially smaller K+ reabsorption. Furthermore, these calculations suggest that when the luminal membrane cation channel is activated, the channel K+ flux can become secretory.
The picture of the IMCD which emerges from this modeling effort is that of a briskly transporting epithelium, in some ways comparable to proximal tubule. The coalescing of tubules within the inner medulla diminishes total tubular surface and, most importantly, prevents axial flow from becoming sluggish. Thus, to modify urinary solute excretion, transport rates must be substantial. Assignment of a definitive set of model parameters is close to impossible, because of the profound impact of regulation on this segment. Indeed, Na+ transport may go from strongly reabsorptive to secretory. However, it has been useful to attempt to generate a "baseline" set of parameters with which to try to simulate in vivo transport under antidiuretic conditions. The calculations presented here have suggested that, like the proximal tubule, electroneutral cotransport mechanisms are likely to be most important in the transcellular ion fluxes of IMCD. Specifically, under normal circumstances, luminal membrane NaCl cotransport is likely responsible for at least 75% of Na+ reabsorption, and KCl cotransport appears to be the dominant pathway for K+ flux across the peritubular membrane. In contrast to proximal tubule, flux across the tight junction appears to be small, and a modulating role for the lateral interspace is not apparent. In the companion study (59a), acid secretion and ammonia transport by this model epithelium will be considered in more detail.
| |
ACKNOWLEDGEMENTS |
|---|
This investigation was supported by National Institutes of Arthritis, Diabetes, and Digestive and Kidney Diseases Grant 1-RO1-DK-29857.
| |
FOOTNOTES |
|---|
Address for reprints requests: A. M. Weinstein, Dept. of Physiol. and Biophysics, Cornell University Medical College, 1300 York Ave., New York, NY 10021.
Received 10 October 1997; accepted in final form 4 February 1998.
| |
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C.-L. Chou, M.-J. Yu, E. M. Kassai, R. G. Morris, J. D. Hoffert, S. M. Wall, and M. A. Knepper Roles of basolateral solute uptake via NKCC1 and of myosin II in vasopressin-induced cell swelling in inner medullary collecting duct Am J Physiol Renal Physiol, July 1, 2008; 295(1): F192 - F201. [Abstract] [Full Text] [PDF] |
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C. Xu, S. Rossetti, L. Jiang, P. C. Harris, U. Brown-Glaberman, A. Wandinger-Ness, R. Bacallao, and S. L. Alper Human ADPKD primary cyst epithelial cells with a novel, single codon deletion in the PKD1 gene exhibit defective ciliary polycystin localization and loss of flow-induced Ca2+ signaling Am J Physiol Renal Physiol, March 1, 2007; 292(3): F930 - F945. [Abstract] [Full Text] [PDF] |
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I. H. Quraishi and R. M. Raphael Computational model of vectorial potassium transport by cochlear marginal cells and vestibular dark cells Am J Physiol Cell Physiol, January 1, 2007; 292(1): C591 - C602. [Abstract] [Full Text] [PDF] |
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A. M. Weinstein A mathematical model of rat distal convoluted tubule. I. Cotransporter function in early DCT Am J Physiol Renal Physiol, October 1, 2005; 289(4): F699 - F720. [Abstract] [Full Text] [PDF] |
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A. T. Layton, T. L. Pannabecker, W. H. Dantzler, and H. E. Layton Two modes for concentrating urine in rat inner medulla Am J Physiol Renal Physiol, October 1, 2004; 287(4): F816 - F839. [Abstract] [Full Text] [PDF] |
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A. M. Weinstein Mathematical models of renal fluid and electrolyte transport: acknowledging our uncertainty Am J Physiol Renal Physiol, May 1, 2003; 284(5): F871 - F884. [Abstract] [Full Text] [PDF] |
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A. M. Weinstein A mathematical model of rat collecting duct I. Flow effects on transport and urinary acidification Am J Physiol Renal Physiol, December 1, 2002; 283(6): F1237 - F1251. [Abstract] [Full Text] [PDF] |
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A. M. Weinstein A mathematical model of rat collecting duct III. Paradigms for distal acidification defects Am J Physiol Renal Physiol, December 1, 2002; 283(6): F1267 - F1280. [Abstract] [Full Text] [PDF] |
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J. J. Grantham and D. P. Wallace Return of the secretory kidney Am J Physiol Renal Physiol, January 1, 2002; 282(1): F1 - F9. [Abstract] [Full Text] [PDF] |
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H. Chang and T. Fujita A numerical model of acid-base transport in rat distal tubule Am J Physiol Renal Physiol, August 1, 2001; 281(2): F222 - F243. [Abstract] [Full Text] [PDF] |
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A. M. Weinstein A mathematical model of rat cortical collecting duct: determinants of the transtubular potassium gradient Am J Physiol Renal Physiol, June 1, 2001; 280(6): F1072 - F1092. [Abstract] [Full Text] [PDF] |
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D. Andreasen, B. L. Jensen, P. B. Hansen, T.-H. Kwon, S. Nielsen, and O. Skott The alpha 1G-subunit of a voltage-dependent Ca2+ channel is localized in rat distal nephron and collecting duct Am J Physiol Renal Physiol, December 1, 2000; 279(6): F997 - F1005. [Abstract] [Full Text] [PDF] |
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H. E. Layton, J. M. Davies, G. Casotti, and E. J. Braun Mathematical model of an avian urine concentrating mechanism Am J Physiol Renal Physiol, December 1, 2000; 279(6): F1139 - F1160. [Abstract] [Full Text] [PDF] |
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S. R. Thomas Inner medullary lactate production and accumulation: a vasa recta model Am J Physiol Renal Physiol, September 1, 2000; 279(3): F468 - F481. [Abstract] [Full Text] [PDF] |
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A. M. Weinstein A mathematical model of the outer medullary collecting duct of the rat Am J Physiol Renal Physiol, July 1, 2000; 279(1): F24 - F45. [Abstract] [Full Text] [PDF] |
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A. M. Weinstein A mathematical model of the inner medullary collecting duct of the rat: acid/base transport Am J Physiol Renal Physiol, May 1, 1998; 274(5): F856 - F867. [Abstract] [Full Text] [PDF] |
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