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) is used to examine the effect of delivered load of
bicarbonate and nonbicarbonate buffer on urinary acidification.
Increasing the delivered load of HCO
produces
bicarbonaturia, and, with luminal carbonic anhydrase absent, induces a
disequilibrium luminal pH and a postequilibration increase in urinary
PCO2. At baseline flows, this disequilibrium
disappears when luminal carbonic anhydrase rate coefficients reach 1%
of full catalysis. The magnitude of the equilibration
PCO2 depends on the product of urinary acid phosphate concentration and the disequilibrium pH. Thus, although increasing phosphate delivery to the CD decreases the disequilibrium pH, the increase in urinary phosphate concentration yields an overall
increase in postequilibration PCO2. In
simulations of experimental HCO
loading in the rat, model predictions of urinary PCO2 exceed the
measured PCO2 of bladder urine. In part, the
higher model predictions for urinary PCO2 may
reflect higher urinary flow rates and lower urinary phosphate concentrations in the experimental preparations. However, when simulation of CD function during HCO
loading
acknowledges the high ambient renal medullary
PCO2 (5), the predicted urinary
PCO2 of the model CD is yet that much greater. This discrepancy cannot be resolved within the model but requires additional experimental data, namely, concomitant determination of
urinary buffer concentrations within the tubule fluid sampled for
PCO2 and pH. This model should provide a means
for simulating formal testing of urinary acidification and thus for
examining hypotheses regarding transport defects underlying distal
renal tubular acidosis.
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INTRODUCTION |
ESTABLISHING THE MINIMAL
URINARY pH during acid challenge and low renal
HCO
delivery is perhaps the most widely used test of
urinary acidification. Nevertheless, there has been considerable
interest in kidney function during buffer excess, particularly
HCO
diuresis, as an indicator of collecting duct
(CD) acidification. The basic observation is that the
PCO2 of an alkaline urine may be several-fold higher than that in arterial blood (16), and this increase
in PCO2 is enhanced when urinary phosphate
concentrations are high (12, 22). Fundamentally, the
PCO2 elevation derives from titration of
urinary HCO
and a delay in dehydration of the
resulting H2CO3 until a point along the nephron
(or beyond) from which CO2 is not readily absorbed
(3). The critical experiment was intravenous
administration of carbonic anhydrase and the observation that the
presence of this enzyme within the alkaline urine obliterated the
PCO2 elevation (15). Subsequent
studies have demonstrated that the elevation of urinary
PCO2 varies directly with urinary HCO
concentration and thus depends in part on an
intact mechanism for water abstraction from the CD (2, 17,
21). Clinical interest in urinary PCO2
derives from the observation that patients with primary defects in
urinary acidification (10), as well as some with acquired
defects (reviewed in Refs. 1 and 3), are also deficient in
the capacity to raise urinary PCO2.
Sampling of CD urine, by microcatheterization (9) or
by micropuncture (5), has shown that the elevation of
PCO2 in bicarbonate diuresis is an intrarenal
phenomenon. Looked at more closely, there are two classes of factors at
work to elevate urinary PCO2. The first class
comprises those local to the collecting duct lumen, which lead to
formation of H2CO3 or enhance the delay in its
dehydration: 1) luminal proton secretion to establish an
acid disequilibrium pH (5), 2) effect of
phosphate buffer to decrease H2CO3
concentration and thus (via mass action) slow dehydration, and possibly
3) osmotic concentration of HCO
with
resulting generation of CO
plus
H2CO3 (13). The second class of
factors relates to diffusion trapping of CO2 within the
renal medulla. It has been observed that vasa recta PCO2 is comparable to that in the CD
(5) and that after administration of carbonic anhydrase to
rats, it still takes 1 h for urinary PCO2
to equilibrate with blood PCO2
(8). This means that increases in urinary
PCO2 must be understood in part as a
consequence of a countercurrent mechanism, presumably with local CD
events as the single effect. The CD model in the companion paper
(24) has sufficient detail for making predictions
regarding the luminal factors that increase urinary
PCO2. In particular, the model incorporates the
effects of proton secretion, water abstraction, and tubular flow rate
to yield a prediction for final urinary PCO2.
It permits estimation of the extent to which disequilibrium conditions
established within cortical and outer medullary segments can persist
through the longer inner medullary collecting duct (IMCD). The model
affords predictions of the impact of CD buffer delivery on measures of urinary acidification. Ultimately, the model may give a sense of the
reliability of inferences made about the health of the CD from an
assessment of urinary PCO2.
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MODEL FORMULATION |
The model of this investigation is identical to that of the
companion paper (24), with only variation in CD delivery
or carbonic anhydrase activity. In this study, there are supplemental calculations of equilibrium conditions for the luminal fluid at all
points along the CD. This includes the concentrations of
H2CO3, HCO
, CO2,
and possibly CO
, along with concentrations of
phosphate and ammonia species. These equilibrium calculations are
undertaken once the full CD model results are obtained along the whole
CD. Qualitatively, the equilibration process is illustrated in Fig.
1, in which the three buffers, total
CO2, phosphate, and ammonia, are depicted as discrete
components. An acid disequilibrium pH (an
H2CO3/CO2 ratio greater than the equilibrium ratio of the dehydration reaction) may be the result of
either proton addition or water abstraction (with fixed
PCO2). Within the bladder, or if tubule fluid
is removed within a closed pipette, the
H2CO3/CO2 ratio moves to
equilibrium. As H2CO3 is dehydrated to
CO2, pH alkalinizes and, with alkalinization, protons from
phosphate and ammonia titrate HCO
to generate new
CO2. Throughout this process, there is conservation of
total buffer for each component.

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Fig. 1.
Schematic for the calculation of disequilibrium pH and
postequilibration PCO2 of collecting duct
(CD) luminal fluid, with the 3 buffers, total CO2,
phosphate, and ammonia depicted as discrete components. An acid
disequilibrium pH (increase in the
H2CO3/CO2 ratio) is obtained by
either proton addition or water abstraction with fixed
PCO2. In the equilibration phase, there is
conservation of total buffer for each component, while protons from
phosphate and ammonia titrate HCO to generate new
CO2.
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To quantify the equilibration process, there are eight reactive species
with eight unknown luminal concentrations
Solution of the CD model provides the total buffer concentrations
at any point, and these totals are unchanged during equilibration
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(1)
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Denote the equilibrium constant for titration of
HCO
, pKb = 3.57, and for titration of CO
, pKd = 10.1 (13); and
corresponding to phosphate and ammonia are
pKp = 6.80 and
pKn = 9.15. Then, the ratios of base and
acid moieties of each buffer pair may be expressed as
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(2)
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This supplies an additional four equations with one new variable,
luminal pH. Corresponding to the chemical reaction
is the equilibrium condition that the
H2CO3 concentration is proportional to the
dissolved CO2
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(3)
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In the calculations below, kh = 0.145 s
1 and kd = 49.6 s
1, with an equilibrium ratio
kh/kd = 2.92 × 10
3.
From the solution of the CD model, one obtains all of the local
solute concentrations. For any perturbation of the luminal pH, the
three total buffer conservation relationships, plus the four buffer
pair equilibria, plus the dissolved H2CO3
equilibrium together provide the eight linear equations that yield the
new solute concentrations. Corresponding to a closed system, one then seeks an equilibrium value of the luminal pH that corresponds to proton
conservation, or equivalently, conservation of charge in the buffer
reactions. Denote by superscript o the (disequilibrium) values of the
model variables computed by the CD model. Then, charge conservation may
be expressed as
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(4)
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For an initial guess at the disequilibrium pHM,
Eq. 4 is evaluated to determine the error in charge
conservation, and then Newton iterations are used to refine the guess
to a solution. In the calculations below, the solution is also
displayed for an "open" system, in which there is no conservation
of total CO2 but rather a specified
PCO2. In this case, the dissolved
CO2 concentration is no longer a variable, and the equation
for conservation of total CO2 is eliminated.
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MODEL CALCULATIONS |
Bicarbonate loading can be simulated either with addition of
NaHCO3 or with HCO
-for-Cl
substitution in the entering luminal fluid. The latter is simpler in
that it avoids a superimposed natriuresis, augmenting what is already a
generous CD Na+ delivery. The solid curves in Figs.
2-4 display the results of a simulation in which 25 mM of luminal
Cl
has been replaced by HCO
, so that entering HCO
concentration is 32 mM. Figure 2
is a tableau of nonreacting species, with luminal PD and solute
concentrations on the left, and volume and solute flows for
the ensemble of all collecting ducts on the right. The
dotted curves are the solution of the model equations for the control simulation, in which luminal HCO
is 7 mM. The
abcissa for each panel is distance along the 9-mm CD. In this
simulation, the transit times for the cortical CD (CCD), outer
medullary CD (OMCD), and IMCD are 10.5, 23.0, and 31.6 s, respectively. Compared with control, the most obvious change is the
decrease in luminal Cl
at CD entry, which propagates
through the CCD and OMCD, and is amplified in the IMCD. There is a
small luminal hyperpolarization, but little difference in volume or
urea flows. Both Na+ and K+ concentrations
appear to be increased over control, and with reference to Table
1, the increase in urinary excretion
rates are, respectively, 20 and 50%. Figure 2 suggests that the
increase in K+ excretion derives largely from decreased
reabsorption within the IMCD. Examination of the detailed model output
provides the rationalization. At the midpoint of the IMCD under control
conditions, luminal and cytosolic HCO
concentrations are 6.3 and 18 mM, respectively, and cellular proton secretion is 0.75 nmol · s
1 · cm
2,
with 90% through the H-K-ATPase. During HCO
loading, luminal and cytosolic HCO
concentrations
are 80 and 24 mM, respectively, with proton flux increased to 0.84 nmol · s
1 · cm
2
but only 48% via the H-K-ATPase. These differences reflect the different pH sensitivities of the two luminal proton pumps
(23).

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Fig. 2.
Electrolyte transport along the model CD with 25 mM
HCO for Cl replacement (solid curves)
and under control conditions (dotted curves). Left: luminal
potential difference (PD; mV) and the luminal concentrations of
Na+, K+, Cl , and urea (mM).
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 the initial cortical point, and cortical (CCD),
outer medullary (OMCD), and inner medullary CD (IMCD) accounting for 2, 2, and 5 mm of CD length, respectively.
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In Fig. 3, luminal pH is displayed along with the components of net
acid excretion in both HCO
loading and control. With
HCO
delivery 4.5-fold greater than control,
HCO
reabsorption increased by more than a factor of
3 (Table 1). Despite increased reabsorption, urinary
HCO
concentration rose with HCO
loading, and this high urinary HCO
is similar to
values obtained in experimental studies in rats (e.g., Ref. 5). Figure 3 also shows that virtually all of the change
in net acid flow derives from the change in HCO
flow, with only trivial changes in titratable acid (TA) or
NH
along the CD. Despite the increase in luminal
HCO
, the pH along the CD actually decreases to 7.06, due to the development of a disequilibrium pH. The impact of delayed
dehydration of H2CO3 is explored in more detail
in Fig. 4 and Table 2. In the top left of Fig. 4, the net flux through the dehydration reaction is
plotted at each point along the CD. A maximal value occurs within the
OMCD, ~0.2
µmol · min
1 · mm
1
for all 7,200 tubules, or 28 pmol · mm
1 · min
1 · tubule
1.
This may be related to the local proton secretory rate of ~0.6 nmol · s
1 · cm
2,
equivalently 34 pmol · mm
1 · min
1
for an OMCD of 30 µm diameter. In the bottom left panel of
Fig. 4, the curves in the top left panel have been
integrated numerically to yield the net generation of CO2
along the CD. The end-luminal value, ~0.9 µmol/min, is slightly
less than net CD HCO
reabsorption, 1.15 µmol/min
(Table 2), reflecting the presence of other pathways for direct
HCO
reabsorption, which do not involve titration to
CO2.

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Fig. 3.
Acid-base transport along the model CD with HCO
loading as in Fig. 2 (solid curves) and under control conditions
(dotted curves). 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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Fig. 4.
CO2 generation along the CD with HCO
loading as in Fig. 2 (solid curves) and under control conditions
(dotted curves). Top left: rate of CO2 formation
via the dehydration of luminal H2CO3.
Bottom left: integral of this rate along the CD. Top
right: luminal disequilibrium pH on the assumption of either a
perfectly closed sampling system or else complete equilibration with an
ambient PCO2 of 50 mmHg. Bottom
right: equilibrium PCO2 of luminal fluid,
with and without scaling for local osmolality.
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The top right panel in Fig. 4 displays the disequilibrium pH
along the CD, on the assumption of either a fully closed system or a
fully open system with equilibration to a PCO2
of 50 mmHg. For the closed system, the maximal disequilibrium pH is
0.26, which occurs early in the OMCD; end-urinary disequilibrium pH is
0.17. The interpretation of these disequilibrium values is not
totally straightforward. As indicated above, the proton secretory rates
within the OMCD and IMCD are ~0.6 and 0.8 nmol · s
1 · cm
2,
apparently at odds with the relative magnitudes of the disequilibrium pH. What must also be considered, however, is that in the transition from the OMCD to IMCD, the concentration of urinary phosphate has
increased several-fold (Fig. 3), so that the decline in disequilibrium pH must be referred to this increase in urinary buffer. The effect of
phosphate delivery will be given more attention in calculations below.
The bottom right panel in Fig. 4 shows the equilibrium PCO2 for the closed system, which rises
monotonically along the CD to 200 mmHg by tubule end. The components of
end-luminal equilibration (corresponding to each of the variables in
Eqs. 1-4) are displayed in Table 2, and the
top section of the table shows the numerical results
corresponding to the data in Fig. 4. The end-luminal fluid shows pH of 7.064, HCO
of 85.9 mM, and
CO2 = 1.6 mM. With equilibration, there is titration
of 4.45 mM HCO
by acid phosphate to form a nearly
equal amount of dissolved CO2; this corresponds to an
increase in PCO2 of 148 mmHg. The contribution
from the ammonia buffer is negligible. The calculations in Figs.
2-4 have been repeated with the inclusion of luminal
CO
(with the model formulation indicated in the
companion paper) (24). The results of these calculations
are displayed in the middle section of Table 2, and there
are no significant differences between the two models, in terms of
either disequilibrium pH or PCO2. The
contribution of CO
remains trivial, essentially due
to the high PCO2 of the closed system. Finally,
the bottom right panel in Fig. 4 includes an attempt to
factor out the osmotic impact on the equilibrium
PCO2, in the curve that has been scaled for
osmolality. The rationale for this curve is that even if there were 0 proton secretion along the CD, water abstraction would elevate all CD
solutes, with the exception of dissolved CO2. Whether this
scaling to an isotonic urine might provide additional sensitivity to
detect proton secretory defects remains to be determined.
To generate Figs. 5 and
6, the delivered
HCO
concentration has been varied over the range
5-45 mM (in steps of 2 mM) via
Cl
-for-HCO
substitution. Figure 5
shows the urinary output, along with the delivery and excretion of the important solutes. In this range of delivered HCO
, no diuresis is produced, nor is there significant change in flows of
urea or Na+. With respect to K+, the effect of
high HCO
is a near doubling of K+
excretion, compared with that at the most acidic urine, again due to
the shift in proton flux from the H-K-ATPase to the
H+-ATPase. With increasing HCO
delivery, there is an increase in both excretion and in the difference between delivery and excretion (i.e., an increase in CD HCO
reabsorption). The increase in HCO
is acompanied by
a sharp decline in urinary NH
, reflecting the
increased concentration of NH3 in progressively alkaline
urine. The urinary CO2 data for these simulations have been
summarized in Fig. 6, in which the bottom panel documents the increase in HCO
consumption. The panels show
urinary pH increasing from 5.82 to 7.21, urinary HCO
concentration increasing from 1.9 to 128 mM, and the disequilibrium pH
for a closed system increasing from
0.05 to
0.21, all progressively over the range of delivered HCO
. Nevertheless, the
equilibrium PCO2 clearly plateaus. Indeed,
above a delivered concentration of 25 mM, the equilibrium
PCO2 values are all within 2% and show a
maximum value at the delivered concentration, 33 mM. As indicated with
the analysis of Table 2, the magnitude of the disequilibrium
CO2 reflects both the concentration of acid phosphate plus
the magnitude of disequilibrium pH. With progressively alkaline urine,
the H2PO4 concentration is decreasing while the disequilibrium pH is increasing, setting the stage for nearly constant
CO2 generation. Repetition of these calculations with luminal NaHCO3 addition (rather than Cl
substitution) adds relatively little to this discussion. The results
are those expected from the preceding considerations of natriuresis
(24): an increase in urinary flow and in the excretion of
urea, K+, NH
, and HCO
. The CO2 data for these simulations look similar to the
curves in Fig. 6, including the plateau of the equilibrium
CO2.

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Fig. 5.
CD
excretion as a function of entering HCO during
HCO -for- Cl replacement. Entering
HCO concentration has been varied from 5 to 45 mM,
and corresponding to each of 21 abcissa points is a solution of the
full CD model. The panels display the single-kidney urine 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. 6.
Urinary acid-base parameters as a function of entering
HCO during HCO -for-
Cl replacement as in Fig. 5. Left: end-luminal
pH and HCO before equilibration, respectively.
Top right: end-luminal disequilibrium pH for either a
perfectly closed system or one completely equilibrated with an ambient
PCO2 of 50 mmHg. Bottom right:
equilibrium PCO2 of luminal fluid with and
without scaling for urinary osmolality. Bottom: total
CO2 generation along the CD.
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The effect of isolated variation in phosphate delivery to the CD is
examined in Figs. 7 and
8, in which the abcissa is luminal total phosphate concentration over the range 0.8-24.8 mM. In these simulations, entering HCO
concentration = 7 mM,
as in control conditions, and the phosphate is varied as a substitution
for Cl
. The excretion profiles in Fig. 7 indicate that
changing phosphate delivery has only a small effect on luminal flow and
on excretion of urea or Na+. There is a 60% increase in
K+ excretion over the full range of phosphate delivery, but
the absolute rate of K+ excretion remains small. The major
impact of phosphate delivery is the increase in net acid excretion due
to the increase in TA. With the additional urinary buffer,
HCO
reabsorption is blunted, and urinary excretion
increases. By itself, the flow increase would increase
NH
excretion, and, by itself, the increase in urinary
HCO
would decrease NH
excretion.
Thus the net effect of phosphate delivery is to leave
NH
excretion unchanged. The urinary CO2
data for these calculations are summarized in Fig. 8. The
bottom and left panels indicate in more detail
the effect of increasing phosphate delivery to decrease
HCO
titration, increase its end-luminal
concentration, and alkalinize the urine. The top right panel
displays the disequilibrium pH of the final urine for a closed system,
and one equilibrated to the ambient PCO2.
Figure 8 shows that above the lowest delivery rates, increasing
phosphate buffer decreases the disequilibrium pH. At the very lowest
phosphate delivery, urinary HCO
is exhausted, so the
disequilibrium pH also collapses. Overall, the effect of increasing
phosphate delivery is to increase the disequilibrium PCO2. This occurs despite the fact that
progressive urinary alkalinization acts to decrease the relative
abundance of acid phosphate; also, the decline in disequilibrium pH
diminishes the fraction of this acid phosphate that titrates
HCO
. The effect of phosphate delivery in an
alkaline urine is shown in Table 1, in the simulation in which entering
HCO
and entering phosphate are 32 mM, both as
Cl
substitutions. In this case, there is little impact on
HCO
reabsorption, and only an increase in net acid
excretion with the increase in TA. By replacing much of the available
Cl
, the additional phosphate does act as a
nonreabsorbable ion and enhances urinary flow plus excretion of urea,
Na+, and K+.

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Fig. 7.
CD
excretion as a function of entering phosphate during phosphate-for-
Cl replacement. Entering HCO
concentration is 7 mM, and entering total phosphate is varied from 0.8 to 24.8 mM. Corresponding to each of 25 abcissa points is a solution of
the full CD model. Shown are single-kidney urine 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. 8.
Urinary acid-base parameters as a function of entering phosphate
during phosphate-for- Cl replacement as in Fig. 7.
Left: end-luminal pH and HCO before
equilibration, respectively. Top right: end-luminal
disequilibrium pH for either a perfectly closed system or one
completely equilibrated with an ambient PCO2 of
50 mmHg. Bottom right: equilibrium
PCO2 of luminal fluid with and without scaling
for urinary osmolality. Bottom: total CO2
generation along the CD.
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The interplay of luminal buffers in establishing the disequilibrium pH
and PCO2 of the final urine is examined
systematically in Fig. 9. The abcissa of
each panel is a range of total phosphate concentrations in delivered
fluid, from 0.8 to 24.8 mM, and the ordinate of each panel is a range
of HCO
concentrations in delivered fluid, from 5 to
45 mM. Division of the abcissa into 24 subunits (25 phosphate
concentrations) and division of the ordinate into 20 subunits (21 HCO
concentrations) defines a grid of 525 CD model
calculations. In each calculation, constancy of entering anions is
achieved by adjusting the Cl
concentration, and thus a
final urinary PCO2 and disequilibrium pH is
determined for each grid point. The top panel shows level curves of constant PCO2 on the
PO4-HCO
grid, determined by linear
interpolation among the grid points. It is clear that the highest
PCO2 values are obtained when both buffers are
increased nearly proportionally and that when only one buffer is
increased, a plateau is quickly reached. For each of these
calculations, the interstitial PCO2 is 50 mmHg,
so that the urine
blood PCO2 difference
is just that fixed offset. The bottom panel displays the
level curves for disequilibrium pH. It is apparent that at each
entering HCO
level, increasing the delivered
phosphate depresses the disequilibrium pH. Conversely, at each entering
phosphate concentration, increasing the delivered
HCO
increases the disequilibrium pH, and this effect
is most pronounced at the lower phosphate concentrations. Finally, in
regions of buffer abundance, the level curves are linear, suggesting
that the disequilibrium pH depends on the concentration ratio of
HCO
and phosphate.

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Fig. 9.
Relationship between entering phosphate and entering
HCO concentrations on determining final urinary
disequilibrium pH and PCO2. For each panel, the
abcissa is entering phosphate, and the ordinate is
HCO (mM). Corresponding to a grid determined by 25 abcissa points and 21 ordinate points are solutions of the CD model for
all 525 grid points. Top: plot with level curves
corresponding to loci of constant urinary (end-luminal)
postequilibration PCO2 from 80 to 240 mmHg.
Bottom: loci of constant end-luminal disequilibrium pH.
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Experimental observation in the rat requires that the rate coefficients
for the dehydration/hydration of H2CO3/
CO2 within the CD lumen be taken at values consistent with
complete absence of carbonic anhydrase. To assess the sensitivity of
the model results to this assumption, the calculations in Figs.
2-4 (entering HCO
= 32 mM) have been repeated
over a range of rate coefficients, from complete absence to full
catalysis (10,000-fold increase). With a variation in catalysis, the
changes in urinary flow or excretion of urea, Na+, and
K+ are all minor. Table 1 contains a summary of these
values for full catalysis, which may be compared with the 0-catalysis
condition. Figure 10 displays the
CO2 data for these simulations, in which the abcissa is the
log (base 10) of the ratio of the rate coefficients to the uncatalyzed
value. It is clear that as the rate coefficients increase, the
disequilibrium pH falls to 0, and PCO2
approaches the ambient value. As catalysis increases, CO2
titration increases by ~60%, from 0.93 to 1.47 µmol/min, which may
be compared with the delivered HCO
load, 1.73 µmol/min. From the perspective of HCO
excretion, full catalysis decreases this by two-thirds (Table 1). Thus Fig. 10
shows a decline in urinary HCO
concentration from 85 to 37 mM over the range of catalysis. Nevertheless, the urinary pH is
predicted to increase from 7.06 to 7.46 due to the fall in the luminal
concentration of H2CO3 (not shown). What Fig. 10 offers, beyond the data of Table 1, is the demonstration that virtually all of the changes in urinary composition with catalysis are
complete when the rate coefficients are just 1% of full catalysis, and
most of this occurs within 0.1%. The urinary flow rates of these
simulations are relatively low (0.16 ml/day), consistent with the
assumed antidiuretic conditions. More rapid urinary flows would be
expected to require higher rate coefficients and thus shift the curves
of Fig. 10 to the right.

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Fig. 10.
Urinary acid-base parameters as a function of the rate coefficient
for dehydration of urinary H2CO3. The delivered
solution is the high-HCO solution used in the
calculations of Figs. 2-4. Both luminal rate coefficients, for
H2CO3 dehydration and CO2
hydration, have been scaled by the same factor, from 1 (for no
catalysis) up to 10,000 (for full carbonic anhydrase activity; CA), and
the logarithm of this scaling constitutes the abcissa. The tableau of
panels are those in Figs. 6 and 8.
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Finally, in view of the observation that during HCO
loading the ambient medullary PCO2 is well above that in arterial blood (5), the model CD should be
examined in those circumstances. In the calculations in Figs.
11 and
12, we set the following interstitial
PCO2: ambient PCO2 in
the cortex remains at 50 mmHg but increases linearly through the outer
medulla to 100 mmHg at the outer-inner medullary junction, then further increases linearly in the IMCD to 200 mmHg at the papillary tip. The
impact on urinary flow rate and urea and Na+ excretion is
relatively minor and is indicated in Table 1. The effect on
K+ is more significant, with an 18% increase in
reabsorption that translates into a 46% decrease in K+
excretion. This is due to increased IMCD proton secretion in response
to cytosolic acidosis, specifically an increase in flux through the
H-K-ATPase. With the higher PCO2, IMCD total
and H-K-ATPase proton secretion are ~1.20 and 0.85 nmol · s
1 · cm
2,
respectively, which may be compared with 0.84 and 0.40 nmol · s
1 · cm
2
under the conditions in Figs. 2-4. Flux through the model
H+-ATPase is sensitive only to pH differences across the
luminal membrane and is thus unchanged by the change in
PCO2, whereas the model H-K-ATPase is
preferentially activated by changes in cytosolic pH (23).
Figure 11 displays the tableau of acid-base fluxes along the CD in the
high PCO2 condition (solid curves), with the
lower PCO2 calculations (Fig. 3) reproduced as
dotted curves. Under high PCO2, there is a
depression in luminal pH and enhanced HCO
reabsorption and TA generation. The effect on net acid excretion is
blunted by the decrease in NH
trapping. In Fig. 12,
the enhanced generation of CO2, beginning in the OMCD, is
apparent. It is also clear from the bottom right panel that
the equilibrium PCO2 of the closed system is
quite a bit higher with the high ambient PCO2,
in fact nearly additive with the PCO2 increase.
However, the top right panel in Fig. 12 shows that that the
disequilibrium pH for the closed system is similar to that with the
lower ambient PCO2, and this occurs despite the
fact that luminal proton secretion is 50% higher with the high
PCO2. The explanation can be gleaned from the
data in the bottom section of Table 2. At the higher CO2 concentration, the equilibrium
H2CO3 concentration is also increased. Because
comparable disequilibrium pH corresponds to the same ratio of
H2CO3 concentrations (disequilibrium to
equilibrium), a higher ambient PCO2 will
necessitate a higher disequilibrium H2CO3. This
will drive H2CO3 dehydration more rapidly and
thus require a higher rate of luminal proton secretion to generate this disequilibrium pH. Thus in a truly closed system, the
predicted equilibrium PCO2 is quite a bit
higher than any value measured in urine or by micropuncture.

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Fig. 11.
Acid-base transport along the model CD with HCO
loading and high ambient PCO2. Cortical
PCO2 remains at 50 mmHg but increases linearly
through the OMCD to reach 100 mmHg by the outer-inner medullary
junction (OIMJ), and there is a further linear increase to 200 mmHg by
the papillary tip (solid curves). The abcissa is length along the CD,
and the tableau of panels is that in Fig. 3. Dotted curves correspond
to the uniform PCO2 (50 mmHg) used in the
calculations of Fig. 3.
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Fig. 12.
CO2 generation along the CD with
HCO loading and high ambient
PCO2 (Fig. 11). The abcissa is length along the
CD, and the tableau of panels is that in Fig. 4. Dotted curves
correspond to the uniform PCO2 (50 mmHg) used
in the calculations of Fig. 4.
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DISCUSSION |
This has been the first effort to explore determinants of urinary
PCO2 using a mathematical model of the CD. The
first model of disequilibrium conditions developing along a kidney
tubule was that of Star et al. (20). This was a simpler
CCD model, without cellular structure or nonreactive solutes, but one
that included the important buffers and sufficient kinetics to predict the axial evolution of a disequilibrium pH. In their simulations of
experiments in vitro, no attention was directed to osmotic effects or
the generation of a disequilibrium PCO2. The
paper of Berliner and DuBose (3), however, focused
squarely on both disequilibrium pH and CO2 generation in
HCO
-rich urine. The calculations in that work were
from the perspective of solution chemistry in a well-mixed beaker, but
they were sufficient to identify the critical kinetic factors. The key
step to developing the disequilibrium pH is luminal proton addition,
with HCO
titration. Berliner and DuBose emphasized
that the dehydration coefficient for H2CO3 is
relatively large and that in the absence of nonbicarbonate buffer, the
disequilibrium pH would be substantial and dehydration would be rapid.
With high phosphate concentrations, accumulation of
H2CO3 is diminished, and thus, from mass action considerations, equilibration is slowed. They also examined the effect
of water abstraction (with fixed ambient PCO2)
to concentrate urinary HCO
and
H2CO3 and, consequently, generate an acid
disequilibrium. Again in this circumstance, substantial phosphate
buffering would delay dehydration of H2CO3.
Berliner and DuBose considered CO
generation as a
proton source, recognizing that this reaction would be immediate and,
compared with phosphate buffering, relatively minor.
With respect to assessing whether kinetic coefficients are large or
small, or whether processes are rapid or slow, the relevant frame of
reference is the timing of events within the collecting duct. In
particular, this means considering the rate of
H2CO3 dehydration in relation to the rate of
proton secretion and in relation to the rate of water abstraction, and
within the time period of fluid transit through the CD. Because this is
sufficiently complex, it is difficult to do outside the framework of a
CD model. Nevertheless, some crude estimates can be made. The first
important rate is that of dehydration of H2CO3.
With regard to Table 2, 10 µM might be considered a representative
disequilibrium H2CO3 concentration, so that for
a dehydration rate coefficient of 50 s
1, one would expect
a CO2 generation rate of 0.5 mM/s. The proton secretory
rate for the model IMCD with HCO
loading is ~1
nmol · s
1 · cm
2.
For a cylindrical tubule of 30-µm diameter, this rate is ~1 pmol · s
1 · mm
1,
and the tubule volume is 0.7 nl/mm, so this rate of proton secretion corresponds to a titration of HCO
of 1.4 mM/s.
Finally, over the length of the inner medulla, water abstraction
produces a near doubling of HCO
. By itself, this
would yield a disequilibrium pH of 0.3. If one started with 45 mM total
phosphate at a pH of 7.1 (HPO
/H2PO
= 30:15),
then a shift of 0.3 pH units to 7.4 corresponds to deprotonation of 6 mM H2PO
(HPO
/H2PO
= 36:9).
In this model, the tubular fluid spends ~30 s in the IMCD, so that
the corresponding rate of HCO
titration is 0.2 mM/s.
From these considerations, the proton secretory rate is clearly the
dominant factor in generating and maintaining the disequilibrium.
Indeed, with regard to Fig. 4, the disequilibrium PCO2 increases along the IMCD, consistent with
a rate of HCO
titration greater than
H2CO3 dehydration. The osmotic effect is small
but nontrivial.
Despite the complexity of the system, certain qualitative aspects of
the buffer effects on disequilibrium pH and
PCO2 emerge (Fig. 9). With increasing phosphate
buffer, the disequilibrium pH is blunted, and with increasing
HCO
it is enhanced. Although the disequilibrium pH
predicted by this model is less than that observed in the rat IMCD
(5), Fig. 9 indicates that in this pH neighborhood small
decreases in entering phosphate, combined with an increase in delivered
HCO
concentration, could reproduce the experimental
finding. Furthermore, with reference to Figs. 4 and 6, the
disequilibrium pH determination is clearly sensitive to small losses of
dissolved CO2 during the experimental protocol. The
disequilibrium PCO2 depends on the product of
end-luminal H2PO
concentration and the
disequilibrium pH. Increasing either buffer alone had a smaller impact
on the disequilibrium PCO2 than when both
buffers increase in proportion. In this model, there was no significant impact of the ampholyte property of HCO
, namely,
generation of CO
and H2CO3. In large measure, this derived from the fact that with end-luminal equilibration in a closed system, the resulting
PCO2 increase kept the tubule fluid relatively
acid, with virtually no generation of CO
. Even when
the system was open and the urine alkalinized, there was a
concomitantly larger contribution (to HCO
titration)
from the phosphate buffer, so that the
CO
effect remained insignificant (Table 2).
The most disturbing feature of these model simulations is the predicted
magnitude of the equilibrium PCO2 within a
truly closed system. In the calculations of Figs. 2-4, the
predicted urinary HCO
concentration was 86 mM, total phosphate was 53 mM, and equilibrium pH and
PCO2 were 7.23 and 202 mmHg, respectively. For
comparison, Table 3 includes data from a
number of HCO
-loading studies in rats. For all of
these studies, the emphasis had been on urinary PCO2, so that only two (18, 19)
had included measurements of urinary phosphate concentrations. Although
some studies did not record urinary flow, all indicated the
steady-state infusion rate of their HCO
solution,
and this should be a good approximation of urinary flow. While the
model CD generally matches the urinary HCO
concentrations, it is clear that the model equilibrium pH is lower and
the equilibrium PCO2 is higher than measured
values. To some extent, this may be explained by the fact that the
model has assumed a moderately antidiuretic rat kidney, with a final
urinary osmolality of 945 mosmol/kgH2O and thus a higher
concentration of urinary phosphate. Indeed, Arruda et al. (Fig. 2 in
Ref. 2) demonstrated an inverse relationship between urinary flow rate
and urinary PCO2. Consistent with those
findings are the observations of DuBose et al. (5, 6), who
worked with much lower infusion rates and found much higher urinary
PCO2. Nevertheless, when the data of Arruda et
al. (2) are examined in the range of urinary flow rates
comparable to those of the model CD (~1.4% of glomerular filtration
rate), they found urinary PCO2 to be only ~50
mmHg higher than plasma PCO2 (35 mmHg). The
model CD can be used to estimate the impact of increasing urinary flow.
This has been done in Table 3 by increasing the delivery of the
HCO
-rich perfusate by a factor of three. This leads
to a 10-fold increase in urinary flow, a 60% decrease in urinary
phosphate, and a 30% decrease in the equilibrium
PCO2. The urinary flow and buffer concentrations are now closer to some of the published studies, but
urinary PCO2 remains twofold greater.
The model estimate of equilibrium PCO2 is
considerably more discrepant with data if one tries to accommodate the
observation that ambient medullary PCO2 is
higher than arterial blood PCO2 and is in fact
relatively close to the values reported for bladder urine (5,
9). When ambient PCO2 is increased in
the model CD (Figs. 11 and 12), the predicted disequilibrium
PCO2 is nearly additive with the increment.
This places the predicted urinary PCO2 well
beyond reported values. Indeed, in the study by DuBose (5), the difference in PCO2
between IMCD fluid and bladder urine during HCO
infusion was only 14 mmHg. Concern regarding CO2
escape from the renal collecting system does not appear to have
received extensive experimental investigation. Kennedy et al.
(12) acknowledged that they had undertaken a comparison of
ureteral and bladder urinary CO2 but indicated only that
renal pelvic urine showed "the same general relationship" to
arterial PCO2 as did bladder samples.
Estimating the increase in ambient medullary
PCO2 with HCO
loading is
beyond the scope of this model, although this study can offer something
to the discussion of the "single effect" of medullary
CO2 generation. With regard to Fig. 4, it is clear that the
preponderance of CO2 generation occurs in the OMCD or near the outer-inner medullary junction. This CO2 will diffuse
into descending vasa recta, be carried toward the papilla, and become trapped within inner medulla. A simple analytic model of vasa recta
flow predicts that OMCD CO2 generation can yield a
substantial increase in papillary PCO2 even in
the absence of IMCD CO2 generation.
In summary, this model has provided a means for simulating CD function
during HCO
loading. In the rat, with absent luminal
carbonic anhydrase, the model yields predictions for the luminal
disequilibrium pH and the equilibrium increment in urinary
PCO2. From the rate of luminal proton secretion and the transit time of tubular fluid within the CD, the kinetic coefficient for the dehydration of H2CO3 must
be within 1% of its uncatalyzed value to obtain significant end-tubule
disequilibrium. The concentrations of urinary HCO
and phosphate are critical to the magnitude of disequilibrium pH and
equilibrium PCO2, so the antidiuretic state of
the animal is a major factor in these calculations. The disequilibrium
pH increases with either increased HCO
or decreased
phosphate; the increment in equilibrium PCO2
increases with increases in either species. A prediction of this model
is that in a truly closed system, the disequilibrium pH should be slightly lower than the reported value and that the urinary equilibrium PCO2 for HCO
-loaded rats
should be substantially higher than reported values. Certainly, a
CO2 "leak" during experimental determinations would be
compatible with both discrepancies, but in view of the experimental
care exercised, other explanations may need to be sought. Specifically,
the first item of investigation should be concomitant determination of
urinary buffer concentrations within the tubular fluid sampled for
PCO2 and pH. This model should provide a means
for exploring the impact of proposed transport defects during formal
testing of urinary acidification.
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.