Department of Physiology and Biophysics, Weill Medical
College of Cornell University, New York, New York 10021
A mathematical model of the rat
collecting duct (CD) has been developed by concatenating previously
published models of cortical (Weinstein AM. Am J Physiol
Renal Physiol 280: F1072-F1092, 2001); outer medullary
(Weinstein AM. Am J Physiol Renal Physiol 279: F24-F45, 2000); and inner medullary segments (Weinstein AM.
Am J Physiol Renal Physiol 274: F841-F855, 1998).
Starting with end-distal tubular flow rate and composition, plus
interstitial solute profiles, the model predicts urinary solute flows,
including the buffer concentrations required to assess net acid
excretion. In the model CD, the interstitial corticomedullary osmotic
gradient provides the basis for the flow effect on the transport of
several solutes. For substances that have an interstitial accumulation
and that can have diffusive secretion (e.g., urea and
NH
), enhanced luminal flow increases excretion by
decreasing luminal accumulation. For substances that are reabsorbed
(e.g., K+ and HCO
), and for which
luminal accumulation can enhance reabsorption, increasing luminal flow
again increases excretion by decreasing luminal solute concentration.
In model calculations, flow-dependent increases in
HCO
and NH
approximately balance,
so net acid excretion is little changed by flow, albeit at a higher
urinary pH. The model identifies delivery flow rate to the CD as a
potent determinant of urinary pH, with high flows blunting maximal
acidification. At even modestly high flows (9 nl · min
1 · tubule
1,
with 6% of filtered Na+ entering the CD), the model cannot
achieve a urinary pH <5.5 unless the delivered HCO
concentration is extremely low (<2 mM). Nevertheless, simulation of
Na2SO4 diuresis does yield both an increase in
net acid excretion and a decrease in urinary HCO
(i.e., a decrease in pH) despite the increase in urinary flow. This
model should provide a tool for examining hypotheses regarding
transport defects underlying distal renal tubular acidosis.
 |
INTRODUCTION |
STARTING WITH
DISTAL DELIVERY of tubular fluid, final urinary composition is
determined by transport along the entire collecting duct (CD).
Nevertheless, information regarding CD physiology has accrued largely
from segmental studies: isolated tubule experiments for cortical and
outer medullary segments and mainly studies in vivo (micropuncture and
microcatheterization) of the inner medullary segment. From the few
micropuncture experiments that have compared late distal delivery with
final urine, we know that under "control conditions" the CD
reabsorbs Na+ (12, 19), HCO
(17), and to a lesser extent, K+
(18). Although one group found relatively little change in the tubular flow of titratable acid (16) or
NH
(15), substantial addition of
NH
had been found in microcatheterization experiments
(14, 23). With respect to flow effects, urinary
Na+ excretion increases sharply above a threshold value of
distal delivery (25). K+ excretion is enhanced
with maneuvers that increase urinary flow, such as saline or osmotic
diuresis (19), and distal microinjection experiments have
demonstrated flow-enhanced CD K+ reabsorption in
Na+-deprived rats (9). There appears to be no
information about the effect of flow on the components of net acid
excretion, although there is indirect evidence (urinary
PCO2) that enhanced CD Na+ delivery
(via furosemide) increases CD proton secretion (10).
Mathematical models of CD function have been limited in number,
and, as in experimental investigation, have been developed segmentally.
The earliest models were those of rabbit cortical CD (CCD), first as
tubule lumen (24) and then with cellular compartments
lining the lumen (26, 27). More recently, models of all
segments of rat CD have been developed: inner medulla (IMCD) (29,
30), outer medulla (OMCD) (31), and CCD
(32). The rat models have included sufficient detail for
simulation of acid/base transport, and the model calculations were done
using luminal and peritubular conditions thought likely to be
encountered in vivo. The inner medullary model included the coalescing
structure of the IMCD, so input to this model was total inner medullary delivery and output was urinary excretion for a single kidney. What has
not yet been done, and is the subject of this work, is an examination
of the three CD segments in series in a peritubular environment
presumed similar to conditions in vivo. This is intended to yield a
sense of proportion among the segments with respect to their relative
roles in modulating CD electrolyte flows. With this configuration, the
simulations to be considered are the effects of input variation,
specifically water or saline or osmotic diuresis, on final urinary
composition and net acid excretion. One important limitation of these
simulations is that secondary changes in the peritubular environment,
which may derive from altered CD transport, remain outside the scope of
the model calculations.
 |
MODEL CD |
The model CD is depicted in Fig. 1,
which shows the series configuration and the coalescing of IMCD
tubules. Within the IMCD, both luminal cross-sectional and epithelial
transport areas decrease from the outer-inner medullary junction (OIMJ)
toward the papillary tip. Starting with 7,200 tubules at the base of
the IMCD, a series of 6 mergings of pairs of ducts reduces the final
number of papillary collecting ducts by 1/64 to 113. 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
|
(1)
|
Thus with unbranched CCD and OMCD, total delivery is distributed
among 7,200 collecting ducts. CD delivery of 54 µl/min [~10% of
the single-kidney glomerular filtration rate (GFR)] corresponds to an
individual tubule flow of 7.5 nl/min. In each segmental model,
epithelial compartments line the tubule lumen, and the entire CD is
surrounded by a peritubular solution of specified composition.

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Fig. 1.
Schematic representation of rat collecting duct (CD),
including cortical (CCD), outer medullary (OMCD), and inner medullary
(IMCD) segments. CCD and OMCD segments consist of 7,200 unbranched
tubules; IMCD segments coalesce 6 times to yield ~113 papillary CDs.
JMI, JME,
JIE, JIS, and
JES: intraepithelial flux of any of the luminal
cell membranes, tight junctions, lateral cell membranes, basal cell
membranes, or interspace basement membranes, respectively; LIS, lateral
intercellular space; FM, axial flow of a solute.
|
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The cellular transport pathways of the CCD, OMCD, and IMCD are depicted
in Fig. 2. Peritubular concentrations are
constant along the CCD but vary linearly with depth along the OMCD and IMCD. Luminal and epithelial concentrations vary axially as a consequence of transport. 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. Within each compartment, the concentration of species i
is designated C
(i), where
is the
lumen (M), interspace (E), cell (I), or peritubular solution (S). (In
the CCD, 3 cellular compartments are distinguished, corresponding to principal and intercalated cells.) Along the tubule lumen, axial
flows of solute are designated FM(i)
(mmol/s). Intraepithelial flux of volume or solute i
across membrane 
is denoted J
(v) or
J
(i), where 
may refer to
tight junction (ME), interspace basement membrane (ES), any of the
luminal cell membranes (MI), lateral cell membranes (IE), or basal cell
membranes (IS). Volume flux across the model membranes is represented
as
|
(2)
|
where Lp
is membrane water
permeability, P
and 
are
hydrostatic and oncotic pressures, RT is the product of gas
constant and temperature, and 

is the osmotic reflection coefficient of solute i (unity for all cell
membranes and 0 for interspace basement membranes). Solute transport 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
|
(3)
|
In this equation, the first term is the Goldman relationship for
ionic fluxes, where h
(i) is a
solute's permeability, and
is a normalized electrical potential difference, with
zi the valence of i, F the
Faraday, RT the product of gas constant and temperature, and


the potential difference between compartments
and
.

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Fig. 2.
Cellular transport pathways of CCD, OMCD, and IMCD, with luminal
membranes facing left. Ion channels are specified by Goldman
equations, coupled transporters by linear nonequilibrium thermodynamic
formalism (with the exception of a kinetic model of -intercalated
cell Cl /HCO exchanger in CCD and
OMCD), and each ATPase ion pump by its own functional representation.
Within each cell, carbonic anhydrase is present, but there is none
within the lumen.
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For each segment model, the equations of mass conservation with
multiple reacting solutes have had a consistent formulation. First, an
expression for the generation of each species within each model
compartment is defined. Although these have been presented previously
with the inclusion of time-dependent terms (to represent accumulation),
for the purpose of this model only the steady-state equations will be
considered. Within a cell, the generation of volume,
sI(v), or of solute i,
[sI(i)], is equal to its net export
|
(4)
|
and for the interspace
|
(5)
|
Within the tubule lumen, mass generation is appreciated as an
increase in axial flux or as transport into the epithelium
|
(6)
|
With this notation, the equations of mass conservation for volume
and for the nonreacting species (Na+, K+,
Cl
, and urea) are written
|
(7)
|
where
= I, E, or M. For the phosphate and for the ammonia
buffer pairs, there is conservation of total buffer
|
(8)
|
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(9)
|
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
|
(10)
|
for
= I or E, and V
is the volume of the
compartment (cm3/cm). For the tubule lumen
|
(11)
|
In each compartment (
= I, E, or M), conservation of total
CO2 is expressed
|
(12)
|
Corresponding to conservation of protons is the equation for
conservation of charge for all the buffer reactions
|
(13)
|
where zi is the valence of species
i. In this model, conservation of charge for the buffer
reactions (Eq. 13) may be rewritten
so by virtue of total phosphate conservation (Eq. 8)
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
= I, E, or
M, is the equation for electroneutrality
|
(14)
|
In the case of HCO
diuresis, with high
HCO
concentrations in an alkaline urine, there has
been concern about the possible importance of luminal
CO
concentration, particularly as a source for
delayed CO2 generation via the reaction
Within the scope of the present model, this concern can be
addressed by including an additional luminal variable
CM(CO
) plus an additional equation for
its chemical equilibrium
|
(15)
|
It will be assumed that there is negligible transepithelial flux
of CO
, so that its luminal generation (Eq. 6) is represented as
|
(16)
|
Accordingly, to accommodate the reactivity of
CO
, Eq. 12 for luminal total
CO2 conservation must be modified
|
(17)
|
and Eq. 13 for charge conservation of the buffer
reactions also takes on a single additional term
|
(18)
|
These modifications will not be included in the calculations in
this paper but will be examined in the consideration of
HCO
diuresis in the companion manuscript
(33).
 |
MODEL PARAMETERS |
With two exceptions, all of the model geometric and transport
parameters are identical to those that were selected for the segmental
models (29, 31, 32). The changes are the principal cell
permeabilities to CO2 within the CCD and IMCD. In the CCD, OMCD, and IMCD models, the overall epithelial permeabilities to CO2 in the published models had been (×10
2
cm/s) 9.4, 0.92, and 4.5, respectively. Although there are no measurements of CD CO2 permeability, these may be compared
with the measured value for proximal tubule, 1.3 × 10
2, which is about one-half that which could be ascribed
to free diffusion of CO2 through a comparable thickness of
water (22). Thus to bring CCD and IMCD permeabilities into
a realistic range, the unit permeabilities of both luminal and
peritubular cell membranes (which had been equal) have each been
reduced by a factor of 6.5. In the calculations that follow, however,
this change is of little consequence, in the sense that there are still
no appreciable PCO2 differences that develop
across any tubule segment.
Table 1 contains the baseline conditions
assumed for entering luminal fluid and the peritubular interstitium.
Peritubular conditions are identical to those previously chosen for the
model segments. Cortical concentrations pertain to the 2-mm CCD and to
the OMCD at the corticomedullary junction (CMJ). Within the OMCD, from
the CMJ to the OIMJ, there is a doubling of interstitial NaCl and KCl,
a 50% increase in total phosphate, a 9-fold increase in ammonia, and a
4-fold increase in urea. All of these concentrations increase linearly
with distance along this 2-mm segment. Of note, the OIMJ urea
concentration (20 mM) is identical to that chosen for the OMCD model,
but this had been revised down from the concentration used previously
in the IMCD model (200 mM); the final papillary urea concentration (500 mM) is identical. Within the 5-mm IMCD, from OIMJ to tip, the only
other peritubular concentration that varies is that of KCl, which
increases linearly from 10 to 20 mM.
With respect to delivered volume and solute loads to the CD, some
ambiguity derives from the merging of ~36,000 connecting segments
within arcades to form ~7,200 CCDs. Because connecting segments have
water reabsorption and coalesce, there is uncertainty in identifying
volume flows measured by micropuncture with the CCD entering flow.
However, there should be greater reliability in measurements of
fractional solute deliveries (X/Inulin TF/P). In the model, the
delivered load of NaCl, ~5% of filtered, is perhaps higher than that
found in the hydropenic rat but easily observed with mild volume
expansion (11, 12). The delivered load of KCl, ~50% of
filtered, may also be higher than expected for control but well within
the range of mild volume expansion (18, 19). The rationale
for selecting generous delivered loads of both Na+ and
K+ is to avoid circumstances in which tubule flows of these
cations may be rate limiting to proton secretion. The delivery of
HCO
to the CCD, 2.8%, is nearly identical to that
observed in control rats (Table 11 in Ref. 17). The total phosphate
delivery is 0.42 µmol/min, at a pH close to the phosphate
pK, so that this corresponds to a titratable acid (TA)
delivery of 0.13 µmol/min; on a per tubule basis, this is 58 and 18 pmol/min for total phosphate and TA, respectively. These numbers may be
compared with micropuncture determinations of late distal TA delivery
[15 pmol/min (16)], acid phosphate delivery [14
pmol/min (4)], or total phosphate delivery in the
presence of ADH [20 pmol/min (8)]. They may also be
compared with single-kidney total phosphate excretion [0.55 µmol/min
(16)] or IMCD total phosphate delivery [0.23-0.43 µmol/min (14)]. The ammonium delivery (0.11 µmol/min)
corresponds to a tubular flow of 15 pmol/min, with micropuncture
reports of late distal NH
delivery of 18 (15) or 3.1 pmol/min (4). This model's urea
delivery is close to that which has been estimated for rats in an
extensive review of available data (20).
 |
MODEL CALCULATIONS |
In all of the model calculations, the 2-mm CCD has been
discretized into 80 segments for numerical integration, using a
second-order centered scheme; the 2-mm OMCD also comprises 80 segments,
but the integration is first-order backward; and the 5-mm IMCD uses a
chop of 500, with a backward scheme. The solutions of the model equations for the control conditions of Table 1 are displayed in Figs.
3 and 4,
and these constitute the standard tableau for presentation of model
results. Table 2 contains a summary of the overall urinary flow and composition for this and several additional input conditions. Columns 1 and 3 in
Table 2 display the delivery and excretion of the indicated species,
and column 2 ("flux") is their difference. In the case
of nonreacting species (water, urea, Na+, and
K+), Eq. 7 guarantees that this flux is truly
the integrated flux over the full CD. For reacting species (e.g.,
HCO
), the flux is only apparent and may bear little
relation to local fluxes. The additional simulations in Table 2 include
high interstitial K+, in which OIMJ K+
concentration has been increased from 10 to 15 mM and papillary K+ from 20 to 45 mM; absent H-K-ATPase along the whole CD
but with a uniform increase of H+-ATPase density 4-fold
over control to match control proton secretion; a low-flow condition,
in which entering volume flow has been decreased from 54 to 42 µl/min
(from 7.5 to 5.8 nl · min
1 · tubule
1);
high Na+ load, in which entering NaCl concentration has
been increased by 40 mM; no ADH, corresponding to a decrease in luminal
membrane water permeability by a factor of 30 along the whole CD
length; and sulfate infusion, in which 50 mM entering luminal
Cl
has been replaced by a 25 mM impermeant anion.

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Fig. 3.
Electrolyte transport along the model CD under control conditions
(Table 1). Left: luminal potential difference (PD; mV) and
the luminal concentrations of Na+, K+,
Cl , and urea (mM; indicated by brackets).
Right: volume flow within the aggregate of all tubule
segments (µl/min) as well as the axial solute flows (µmol/min)
within the entire CD. The abcissa is distance along the CD, with
x = 0 being the initial cortical point and CCD, OMCD,
and IMCD accounting for 2, 2, and 5 mm of CD length, respectively.
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Fig. 4.
Acid-base transport along the model CD under control conditions,
corresponding to the electrolyte profiles in Fig. 3. Left:
luminal pH and the concentrations of HCO , titratable
acid (TA), and NH (mM). Right: flows
within the aggregate of all CD tubule segments of
HCO , TA, and NH (µmol/min),
along with their sum-to-net acid flow (TA + NH HCO ).
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Figure 3 displays the luminal potential difference, the volume flow
rate, and the concentrations and flows of the nonreacting solute
species. This is a tubule in an antidiuretic kidney, and fluid
reabsorption within the CCD, OMCD, and IMCD is ~32, 35, and 22% of
the flow entering the CCD, respectively. For each segment, with
variable volume flow and variable luminal cross section, the tubule
transit time from x0 to
x1,
(x0,x1), must be
computed as an integral using local volume flow and luminal area
|
(19)
|
For the CCD, OMCD, and IMCD, the predicted times are 10.4, 22.8, and 32.4 s, respectively, for a total CD time of 65.6 s (Table 3). This overall transit time is
comparable to observations of a delay of 1-2 min for the urinary
appearance of radioactive inulin, after injection into the distal
convoluted tubule (5, 9). A small decrease in entering
flow (from 54 to 42 µl/min) produces a substantial increase in
transit time (to 107 s) principally due to a delay within the
IMCD. Sharp reductions of transit time follow natriuresis or water
diuresis (Table 3).
Under control conditions, approximately two-thirds of the delivered
Na+ is reabsorbed. With reference to Fig. 3, there is
virtually no Na+ flux in either the CCD or OMCD, although
the concentrations in each segment increase due to water abstraction.
The CCD of this model is a Na+-reabsorbing segment, with
transport rates comparable to those reported in vitro for tubules
exposed to both mineralocorticoid and ADH (32).
Nevertheless, the reabsorptive flux is still relatively small, and
paracellular backflux further reduces this. The only Na+
flux in the OMCD is paracellular backflux, but this secretory contribution is also small. Na+ delivery to the IMCD in
this model is ~3.8 µmol/min, or ~5% of filtered Na+
for a kidney with a GFR of 0.5 ml/min. This is higher than the delivered load of 2.6 µmol/min used in developing the IMCD model (29). In comparison with that segmental model, the luminal
Na+ concentration here is 90% higher (210 mM compared with
110 mM), but the luminal volume flow is 25% lower, yielding a 50%
greater Na+ delivery in this CD. In both models, IMCD
transporters are identical. In this CD, about two-thirds of the
delivered Na+ load is reabsorbed, whereas in the segmental
model fractional reabsorption was 75%.
Under control conditions, ~80% of delivered K+ is
reabsorbed in the model CD. There is relatively little flux of
K+ within CCD, because the luminal K+
concentration is close to the limiting gradient for secretory K+ flux (32). However, within the OMCD and
IMCD the model predicts substantial reabsorption, due in part to
transcellular uptake by luminal H-K-ATPase but most importantly due to
concentration of luminal K+ by water abstraction and
diffusive backflux across the tight junctions and, within the IMCD,
luminal cation channels. In the absence of luminal membrane H-K-ATPase,
60% of delivered K+ is still reabsorbed (Table 2). The
gradient-driven K+ flux may be modulated by peritubular
conditions so that when the mean interstitial K+
concentration is doubled in the IMCD, fractional K+
excretion increases by 50%. Even more dramatic changes in fractional excretion are achieved by maneuvers that change luminal flow: with low
flow, nearly all of the entering K+ is reabsorbed, and,
with natriuresis or diuresis, fractional K+ excretion is
enhanced (Table 2). Predictably, flow-dependent excretion is also
evident with urea, which under control conditions shows only trivial
net flux.
Acid-base transport by this CD is displayed in the tableau in Fig. 4,
which shows luminal pH, the concentrations of HCO
, TA, and NH
, and their associated flows. The TA
concentration is estimated as the base required to titrate total
luminal phosphate to a pH of 7.40. Net acid excretion is the sum of TA
plus NH
flow, less any flow of
HCO
. Overall, the model CD contributes 0.57 µmol/min to net acid secretion, of which 0.35 is attributed to
HCO
reabsorption, 0.13 to NH
secretion, and 0.09 to HPO
titration. In the final
urine, net acid excretion is 0.44 µmol/min, split nearly equally
between TA and NH
excretion (Table 2). Within the CD
(Fig. 4), the bulk of proton secretion and HCO
reabsorption occurs in the OMCD and early IMCD. The OMCD is also the
site for most of the NH
secretion, because high
luminal NH
concentrations blunt further secretion within the IMCD. Indeed, with a low volume flow into the CD, not only
is NH
delivery decreased by 27% but also the luminal
concentration of NH
decreases secretory flux by 31%,
so overall NH
excretion is down 30%. With
natriuresis or diuresis, excretion of NH
is enhanced
dramatically (Table 2).
Figures 5 and
6 display the CD tableau for the
simulated sulfate infusion, in which 50 mM entering luminal
Cl
has been replaced by a 25 mM impermeant ion. The
control conditions are reproduced as dotted curves. In Fig. 5, the
effects of the Cl
replacement are straightforward: a
sharp decrease in luminal Cl
, luminal hyperpolarization
that reaches
25 mV within the IMCD, and secondary retention of
Na+ and K+. There is a 50% increase in luminal
flow, with a smaller fractional increase in urea excretion. The impact
on acid-base transport is an increase in all components of net acid
excretion, which is displayed in Fig. 6. With respect to luminal
HCO
, there is a sharp reduction in concentration
within the IMCD (to <1 mM) and a concomitant reduction in
HCO
flow. Thus along the IMCD, sulfate produces a
greater decline in luminal pH than in control, with the final urinary
pH reaching 5.48 (compared with 6.14). With sulfate, however, both
luminal TA and NH
concentrations are less than in
control, due to the osmotic effect of the impermeant, but the excretory
flow of each is increased. Thus with sulfate infusion, net acid
excretion by this CD is increased by 23% compared with control (Table
2).

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Fig. 5.
Electrolyte transport along the model CD when 25 mM luminal
Na2SO4 has replaced 50 mM NaCl.
Left: luminal PD and solute concentrations.
Right: volume and solute flows for a single kidney. The
dotted curves correspond to the model solution for control conditions
(Fig. 3).
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Fig. 6.
Acid-base transport along the model CD during
Na2SO4 diuresis, corresponding to the
electrolyte profiles in Fig. 5. Left: pH and the components
of net acid excretion. Right: associated flows for a single
kidney. The dotted curves correspond to control conditions (Fig. 4).
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The impact of flow on CD transport can be considered systematically in
a series of simulations in which luminal entry is varied, without
changing luminal composition from control (Table 1). These are
displayed in Fig. 7, in which the abcissa
is the flow to all 7,200 CDs, from 36 to 180 µl/min (control 54 µl/min), and the curves are obtained by solving the CD model at
25 flows. The panels show final urinary output and solute excretion for
nonreactive species and for the important acid-base constituents. For
each solute, CD delivery changes with entering flow and is also
plotted. Thus in the Na+ and K+ panels, the
nearly parallel delivery and excretion curves indicate that above the
very lowest input rates, transport of these species is independent of
flow. This derives from the fact that the delivered Na+ is
well above the apparent Km for Na+
reabsorption, and this luminal concentration increases further as a
result of water abstraction. The second pattern is that of urea and
NH
, and to some extent TA, in which excretion exceeds
delivery and in which development of high luminal concentrations blunts
secretion. For these solutes, an increase in flow above control
enhances secretion. The third pattern is that of
HCO
, in which excretion is less than delivery and
development of high luminal concentration enhances reabsorption. In
this case, increase in flow above control blunts reabsorption. This is
even more apparent in the very high flows corresponding to absent ADH
(Table 2), in which CD HCO
reabsorption falls to
0.20 from 0.35 µmol/min in control. In the face of flow-enhanced
NH
secretion plus flow-diminished
HCO
reabsorption, the overall effect of flow on net
acid excretion is negligible (Fig. 7). This stability of net acid
excretion with varying flow is also seen in the water diuresis and the
natriuresis simulations in Table 2. It should also be noted that only
under delivery flow less than the control rate can nearly all of the
delivered HCO
be reclaimed. In particular, this means that testing maximal urinary acidification (i.e., minimal urinary
pH) for this CD requires sharp limitations on urinary flow.

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Fig. 7.
Collecting duct excretion as a function of entering flow. With
entering fluid composition that for control (Table 1), the initial CCD
flow rate has been varied from 36 to 180 µl/min, and this appears as
the abcissa. (Corresponding tubule flows range from 5 to 25 nl/min.)
Corresponding to each of the 25 abcissa points is a solution of the
full CD model. The panels display the single-kidney urinary output
(µl/min) and the delivery (x = 0) and excretion
(x = 9 mm) of Na+, K+, urea,
and the components of net acid flow.
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In simulations in which entering volume flow is fixed at the control
rate, but NaCl concentration is varied, luminal flows are changed but
with a constancy of delivered load for most solutes. These are
displayed in Fig. 8, in which entering
Na+ concentration is varied from 24 to 145 mM (control 70 mM). With increasing Na+ load and natriuresis, excretion of
urea and NH
is again enhanced with increasing luminal
flow. The pattern for HCO
is also similar to that
shown in Fig. 7, perhaps more clearly here, with increased luminal flow blunting the increase in luminal HCO
concentration
and thus decreasing reabsorption. Once again, only at low rates of CD
Na+ delivery can HCO
be fully
reabsorbed, suggesting that it may be misleading to test maximal
acidification under higher flows. Overall, the flow effects for
NH
and HCO
approximately cancel
each other so that except when Na+ excretion vanishes, net
acid excretion is independent of Na+ load to the CD. Of
note, the effect of Na+ load on K+ excretion is
similar to the pattern seen for HCO
, with decreased
K+ reabsorption as the consequence of diminished luminal
K+ concentration. In all of these model calculations, the
delivered load of K+ was chosen to be ample, so as to avoid
luminal K+ concentration becoming rate limiting for acid
excretion via the H-K-ATPase. The sensitivity of K+
reabsorption to the delivered load of K+ is examined in
Fig. 9, where the entering concentration
of K+ on the abcissa is varied from 12 to 60 mM (control 24 mM) by addition of KCl. At all levels of delivery, CD K+
reabsorption is substantial, nearly complete at the lowest loads and
about two-thirds at the highest delivery. The increase in absolute
K+ reabsorption reflects the development of lumen-to-blood
K+ concentration gradients with water abstraction. With
increasing KCl, there is diuresis, and with this increase in urinary
flow, the expected increases in urea, NH
, and
HCO
excretion are all evident. Again, the changes in
NH
and HCO
nearly cancel, so there
is no significant change in net acid excretion.

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Fig. 8.
Collecting duct excretion as a function of entering Na+
concentration. With entering flow rate that for control (54 µl/min),
initial CCD Na+ concentration has been varied by varying
luminal NaCl, and this appears as the abcissa. Corresponding to each of
25 abcissa points is a solution of the full CD model. The panels
display the single-kidney urinary output (µl/min) and the delivery
(x = 0) and excretion (x = 9 mm) of
Na+, K+, urea, and the components of net acid
flow.
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Fig. 9.
Collecting duct excretion as a function of entering K+
concentration. With entering flow rate that for control (54 µl/min),
initial CCD K+ concentration has been varied by varying
luminal KCl, and this appears as the abcissa. Corresponding to each of
25 abcissa points is a solution of the full CD model. The panels
display the single-kidney urinary output (µl/min) and the delivery
(x = 0) and excretion (x = 9 mm) of
Na+, K+, urea, and the components of net acid
flow.
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The observations of excretion as a function of load suggest that
maneuvers which increase urinary flow rate within the CD will produce
bicarbonaturia and thus increase urinary pH. Conversely, generation of
an acidic urine in the face of increased flow should require a
reduction in the concentration of delivered HCO
. This is explored systematically in the top panel of Fig.
10, in which the abcissa shows a range
of CD delivery rates from 30 to 70 µl/min and the ordinate shows a
range of HCO
concentrations from 2 to 7 mM. Division
of the abcissa into 25 subunits (26 entering flows) and division of the
ordinate into 25 subunits (26 HCO
concentrations) define a grid of 676 CD model calculations, in which the entering flow
is specified and the entering HCO
concentration is
varied by Cl
-for-HCO
substitution
(along with appropriate adjustment of other buffer concentrations).
Each calculation determines a urinary pH associated with the
appropriate grid point. What is plotted are level curves corresponding
to urinary pH of 4.25, 4.50, 5.50, 6.00, and 6.25. These curves are
exact for each of the entering flows, and the entering
HCO
concentrations have been determined by linear
interpolation. From Fig. 10, it appears that to achieve a urinary pH of
5.5 at a delivered flow of either 48 or 60 µl/min, one requires
entering HCO
to be 6 or 3 mM, or, equivalently, an
entering pH of 6.7 or 6.4. Beyond an entering flow of 66 µl/min, it
seems impossible to achieve a sufficiently low entering
HCO
. One should note that if only the delivered load
of HCO
determined the total HCO
reabsorption, then lines of constant urinary pH would be true
hyperbolas in the top panel of Fig. 10. To examine this, the
same pH data are replotted in the bottom panel, in which the
abcissa is the logarithm of entering flow and the ordinate is the
logarithm of the entering HCO
. The four curves
corresponding to urinary pH 4.5 and above are all linear in this
log-log plot, and their slopes have been indicated. If only delivered
HCO
load were the determinant of urinary pH, then
all of the slopes would be unity; however, what is found is
progressively steeper slopes with more acidic urine. This implies that
small increases in flow require larger fractional decreases in
delivered HCO
concentration to achieve the same
urinary pH and that these fractional changes get quite a bit larger for
the most acidic pH.

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Fig. 10.
Relationship between entering flow rate and entering
HCO concentration in determining final urinary pH.
Top: the abcissa is entering flow to the entire CD
(µl/min), and the ordinate is HCO (mM).
Corresponding to a grid determined by 26 abcissa points and 26 ordinate
points are solutions of the CD model for all 676 grid points. What is
plotted in the figure are level curves corresponding to loci of
constant urinary pH, from 4.25 to 6.25. Bottom: the same
data as in the top panel replotted on a log-log scale.
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DISCUSSION |
This has been the first examination of a model CD comprising
previously developed segmental models, to see how the parts might function together in vivo. Because this model represents all CD tubules
for a single rat kidney, the output can be identified with
single-kidney urinary flow. The inputs, namely, distal delivery and
interstitial composition, are an obligatory source of uncertainty, in
view of the uncertainty of their estimation in vivo. The first step was
to consider CD function under control, antidiuretic conditions. In this
model, CD Na+ delivery was 5% of filtered load, with
one-third of that reaching the final urine. Thus Na+
excretion was ample and could not be considered rate limiting for other
solute fluxes. One salient finding in this CD was that K+
reabsorption was substantial, with CD delivery one-half of filtered load and urinary excretion only one-fifth of entering K+
(or 10% of filtered load). This may be compared with the observation of Malnic et al. (18) (Fig. 3), which shows distal
K+ delivery from cortical nephrons as being close to
one-third of filtered load and final excretion about one-half of
delivery. Significant K+ secretion between the last
accessible distal convoluted tubule segment and the start of the CD
could provide a means for reconciling the difference between the
fractional K+ reabsorption observed and that predicted by
this model. A similar effect could also be attributed to deep nephrons
if they contributed a higher K+ load to the CD.
Alternatively, if this model substantially overestimated luminal
H-K-ATPase activity, underestimated interstitial K+
concentrations, or overestimated CD K+ permeability, the
predicted fractional K+ excretion would be higher (Table
2).
In discussing their data, Malnic et al. (18) noted that
specific transporters for CD K+ uptake were not required to
rationalize a reabsorptive flux. They reasoned that CD water
abstraction would concentrate luminal K+ and thus create a
favorable gradient for diffusion from lumen to blood. Indeed, in
subsequent work Malnic et al. (19) found that both CD
K+ delivery and urinary K+ excretion were
critically dependent on urinary Na+ excretion, itself
manipulated by changing extracellular volume or administration of a
diuretic. They documented that K+ reabsorption could be
nearly complete under conditions of low Na+ excretion. The
simulations here certainly support the proposed importance of osmotic
modulation of diffusive reabsorption as a mediator for flow-dependent
K+ excretion. This is best seen in Fig. 8, in which
K+ delivery is constant and enhanced K+
excretion accompanies increased Na+ delivery. With a pure
increase in K+ delivery (Fig. 9), fractional K+
reabsorption again decreases, presumably due to a flow effect. In
essence, a doubling of entering K+ concentration does not
propagate to a doubling of luminal K+ all along the CD, due
to enhanced fluid flow. The simulation of Fig. 7, in which luminal flow
is varied in the absence of a compositional change, incorporates both
natriuresis and kaliuresis, with an even more prominent increase in
fractional K+ excretion. The curves in Fig. 7 give the
appearance of fixed absolute K+ reabsorption, but that is
just fortuitous. In this model, only 30% of reabsorptive
K+ flux is attributable to the H-K-ATPase (Table 2), and
thus only a small component could qualify as fixed.
Under the control conditions of the model CD, about two-thirds of
proton secretion is devoted to reclaiming delivered
HCO
and the remainder to increasing the axial flow
of TA and NH
. Net acid excretion by this CD is 0.44 µmol/min, split evenly between TA and NH
. Several
studies are available with complete data for acid excretion by rat
kidney, and these have been summarized in Table
4. This table illustrates the variability among the data, deriving in part from differences in renal size (as
seen in the range of GFR). Urinary acidification by this model is
perhaps closest to the data from Sabatini et al. (21),
with agreement for final urinary pH and net acid excretion, and this is
also the study for which the model GFR is also the best match. With
infusion of Na2SO4, Sabatini et al. found a
doubling of Na+ excretion, a near tripling of
K+ excretion, a decrease in urinary pH by 0.66 unit, and an
increase of 0.20 µmol/min in net acid excretion. This may be compared
with the model doubling of urinary Na+ and K+,
a decrease in urinary pH by 0.64 unit, and an increase in net acid
excretion by 0.10 µmol/min. As observed by Sabatini et al., Na2SO4 infusion increased both TA and
NH
excretion by the model CD (Table 2). Failure of
the model to match the net acid excretion noted by Sabatini et al. may
simply reflect Na2SO4-enhanced acidification in
tubule segments proximal to the CD.
In the model simulations, osmotic effects on solute reabsorption are
not limited to K+. In general, for substances that have an
interstitial accumulation and can have diffusive secretion, enhanced
luminal flow increases excretion by decreasing luminal accumulation.
This applies to urea and NH
fluxes. For substances that are reabsorbed, and for which luminal accumulation can enhance reabsorption, increasing luminal flow again increases excretion by
decreasing luminal solute concentration. This applies to
HCO
as well as to K+. In the simulations
considered here, flow-dependent increases in HCO
and
NH
approximately balanced, so net acid excretion was
little changed by flow, albeit at a higher urinary pH. Indeed, the
model identified CD delivery flow rate as a very potent determinant of
urinary pH. At even modestly high flows, it seems almost impossible to achieve a urinary pH <5.5 unless the delivered HCO
concentration were extremely low (Fig. 10). Consistent with the predictions of this model, it was an early observation that furosemide administration acutely produced bicarbonaturia in direct relation to
the extent of natriuresis (1). In humans, this direct
relationship between urinary flow after furosemide and urinary pH has
also been observed (Fig. 2 in Ref. 3). This is important in
understanding the use of furosemide as a provocative test in clinical
settings to assess minimal urinary pH for a patient with a suspected
acidification defect. In this test, the urine initially alkalinizes
during the first 2 h after a discrete furosemide dose, and then
minimal urinary pH is observed in hours 3 and 4,
when urinary flow is lowest, presumably when the kidney has become
Na+ avid (3).
In summary, this model of the full CD of the rat displays relatively
minor transport within the CCD (despite tubule fluxes at the high end
of experimental observation), important reabsorption of K+
and HCO
and secretion of NH
within
the OMCD, and substantial reabsorption of Na+ and
Cl
within the IMCD. Beyond the individual segmental
models, what this model offers is the ability to provide estimates of
delivery to the distal segments based on best estimates of transport by inaccessible structures. In this model, water abstraction along the CD
engenders luminal concentrations much higher than those used in vitro,
and in some cases, different from what had been used in prior segmental
models. This corticomedullary osmotic gradient provides the basis for
the effect of flow on the transport of several solutes, including
K+ and the components of net acid excretion. Compared with
micropuncture studies, the model CD has greater K+
reabsorption, although the flow dependence of K+ excretion
is similar to observations. The higher estimates for K+
reabsorption could be due to an experimental underestimate of CD
K+ delivery or lower tubule K+ permeability in
vivo than in vitro, but only in part to a high estimate for rat CD
H-K-ATPase activity. Model estimates for CD acid excretion predict
strong flow dependence of urinary pH; failure to achieve a minimal
urinary pH derives from rapid CD transit that comes with diuresis. Flow
dependence of acid-base transport within the CD has yet to be examined
experimentally and would provide a critical test of these model predictions.
This investigation was supported by Public Health Service Grant
1-R01-DK-29857 from the National Institute of Arthritis, Diabetes, and
Digestive and Kidney Diseases.
Address for reprint requests and other correspondence:
A. M. Weinstein, Dept. of Physiology and Biophysics, Weill
Medical College of Cornell University, 1300 York Ave., New York, NY
10021 (E-mail:
alan{at}nephron.med.cornell.edu).
The costs of publication of this
article were defrayed in part by the
payment of page charges. The article
must therefore be hereby marked
"advertisement"
in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.