A mathematical model of rat collecting duct I. Flow effects on transport and urinary acidification

Alan M. Weinstein

Department of Physiology and Biophysics, Weill Medical College of Cornell University, New York, New York 10021


    ABSTRACT
TOP
ABSTRACT
INTRODUCTION
MODEL CD
MODEL PARAMETERS
MODEL CALCULATIONS
DISCUSSION
REFERENCES

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<UP><SUB>4</SUB><SUP>+</SUP></UP>), enhanced luminal flow increases excretion by decreasing luminal accumulation. For substances that are reabsorbed (e.g., K+ and HCO<UP><SUB>3</SUB><SUP>−</SUP></UP>), 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<UP><SUB>3</SUB><SUP>−</SUP></UP> and NH<UP><SUB>4</SUB><SUP>+</SUP></UP> 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<UP><SUB>3</SUB><SUP>−</SUP></UP> 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<UP><SUB>3</SUB><SUP>−</SUP></UP> (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.

potassium; ammonium; renal acid excretion; distal renal tubular acidosis


    INTRODUCTION
TOP
ABSTRACT
INTRODUCTION
MODEL CD
MODEL PARAMETERS
MODEL CALCULATIONS
DISCUSSION
REFERENCES

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<UP><SUB>3</SUB><SUP>−</SUP></UP> (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<UP><SUB>4</SUB><SUP>+</SUP></UP> (15), substantial addition of NH<UP><SUB>4</SUB><SUP>+</SUP></UP> 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
TOP
ABSTRACT
INTRODUCTION
MODEL CD
MODEL PARAMETERS
MODEL CALCULATIONS
DISCUSSION
REFERENCES

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
<FR><NU>A<SUB>M</SUB>(<IT>x</IT>)</NU><DE><IT>A</IT><SUB>M</SUB>(0)</DE></FR><IT>=</IT><FR><NU><IT>B</IT><SUB>M</SUB>(<IT>x</IT>)</NU><DE><IT>B</IT><SUB>M</SUB>(0)</DE></FR><IT>=</IT>2<SUP>−6<IT>x/L</IT></SUP> (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.

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<UP><SUB>3</SUB><SUP>−</SUP></UP>, CO2, H2CO3, HPO<UP><SUB>4</SUB><SUP>2−</SUP></UP>, H2PO<UP><SUB>4</SUB><SUP>−</SUP></UP>, NH3, NH<UP><SUB>4</SUB><SUP>+</SUP></UP>, 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 Calpha (i), where alpha  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 alpha beta is denoted Jalpha beta (v) or Jalpha beta (i), where alpha beta 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
J<SUB>&agr;&bgr;</SUB>(v)<IT>=Lp<SUB>&agr;&bgr;</SUB></IT>{(<IT>P<SUB>&agr;</SUB>−P<SUB>&bgr;</SUB></IT>)<IT>+</IT>(<IT>&pgr;<SUB>&bgr;</SUB>−&pgr;<SUB>&agr;</SUB></IT>) (2)

<IT>+RT</IT><LIM><OP>∑</OP><LL><IT>i</IT></LL></LIM><IT>&sfgr;<SUB>&agr;&bgr;</SUB></IT>(<IT>i</IT>)[C<SUB><IT>&bgr;</IT></SUB>(<IT>i</IT>)<IT>−</IT>C<SUB><IT>&agr;</IT></SUB>(<IT>i</IT>)]}
where Lpalpha beta is membrane water permeability, Palpha and pi alpha are hydrostatic and oncotic pressures, RT is the product of gas constant and temperature, and sigma alpha beta 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
J<SUB>&agr;&bgr;</SUB>(i)=h<SUB>&agr;&bgr;</SUB>(i)&zgr;<SUB>&agr;&bgr;</SUB>(i)<FENCE><FR><NU>C<SUB><IT>&agr;</IT></SUB>(<IT>i</IT>)<IT>−</IT>C<SUB><IT>&bgr;</IT></SUB>(<IT>i</IT>)<IT>e</IT><SUP><IT>−&zgr;<SUB>&agr;&bgr;</SUB></IT>(<IT>i</IT>)</SUP></NU><DE>1<IT>−e</IT><SUP><IT>−&zgr;<SUB>&agr;&bgr;</SUB></IT>(<IT>i</IT>)</SUP></DE></FR></FENCE> (3)

<IT>+</IT><LIM><OP>∑</OP><LL><IT>j</IT></LL></LIM><IT>L<SUB>&agr;&bgr;</SUB></IT>(<IT>i,j</IT>)[<IT><A><AC>&mgr;</AC><AC>&cjs1171;</AC></A><SUB>&agr;</SUB></IT>(<IT>j</IT>)<IT>−<A><AC>&mgr;</AC><AC>&cjs1171;</AC></A><SUB>&bgr;</SUB></IT>(<IT>j</IT>)]<IT>+J</IT><SUP><IT>act</IT></SUP><SUB><IT>&agr;&bgr;</IT></SUB>(<IT>i</IT>)
In this equation, the first term is the Goldman relationship for ionic fluxes, where halpha beta (i) is a solute's permeability, and
&zgr;<SUB>&agr;&bgr;</SUB>(i)=<FR><NU>z<SUB>i</SUB>F</NU><DE>RT</DE></FR>(&psgr;<SUB>&agr;</SUB>−&psgr;<SUB>&bgr;</SUB>)
is a normalized electrical potential difference, with zi the valence of i, F the Faraday, RT the product of gas constant and temperature, and psi alpha  - psi beta the potential difference between compartments alpha  and beta .


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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 alpha -intercalated cell Cl-/HCO<UP><SUB>3</SUB><SUP>−</SUP></UP> 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.

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
s<SUB>I</SUB>(v)<IT>=J</IT><SUB>IE</SUB>(v)<IT>+J</IT><SUB>IS</SUB>(v)<IT>−J</IT><SUB>MI</SUB>(v) (4)

s<SUB>I</SUB>(<IT>i</IT>)<IT>=J</IT><SUB>IE</SUB>(<IT>i</IT>)<IT>+J</IT><SUB>IS</SUB>(<IT>i</IT>)<IT>−J</IT><SUB>MI</SUB>(<IT>i</IT>)
and for the interspace
s<SUB>E</SUB>(v)<IT>=J</IT><SUB>ES</SUB>(v)<IT>−J</IT><SUB>ME</SUB>(v)<IT>−J</IT><SUB>IE</SUB>(v) (5)

s<SUB>E</SUB>(<IT>i</IT>)<IT>=J</IT><SUB>ES</SUB>(<IT>i</IT>)<IT>−J</IT><SUB>ME</SUB>(<IT>i</IT>)<IT>−J</IT><SUB>IE</SUB>(<IT>i</IT>)
Within the tubule lumen, mass generation is appreciated as an increase in axial flux or as transport into the epithelium
s<SUB>M</SUB>(v)<IT>=</IT><FR><NU>dF<SUB>M</SUB>(v)</NU><DE>d<IT>x</IT></DE></FR><IT>+B</IT><SUB>M</SUB>[<IT>J</IT><SUB>ME</SUB>(v)<IT>+J</IT><SUB>MI</SUB>(v)] (6)

s<SUB>M</SUB>(<IT>i</IT>)<IT>=</IT><FR><NU>dF<SUB>M</SUB>(<IT>i</IT>)</NU><DE>d<IT>x</IT></DE></FR><IT>+B</IT><SUB>M</SUB>[<IT>J</IT><SUB>ME</SUB>(<IT>i</IT>)<IT>+J</IT><SUB>MI</SUB>(<IT>i</IT>)]
With this notation, the equations of mass conservation for volume and for the nonreacting species (Na+, K+, Cl-, and urea) are written
s<SUB>&agr;</SUB>(i)=0 (7)
where alpha  = I, E, or M. For the phosphate and for the ammonia buffer pairs, there is conservation of total buffer
s<SUB>&agr;</SUB>(HPO<SUP>2−</SUP><SUB>4</SUB>)<IT>+s<SUB>&agr;</SUB></IT>(H<SUB>2</SUB>PO<SUP>−</SUP><SUB>4</SUB>)<IT>=</IT>0 (8)

s<SUB>&agr;</SUB>(NH<SUB>3</SUB>)<IT>+s<SUB>&agr;</SUB></IT>(NH<SUP>+</SUP><SUB>4</SUB>)<IT>=</IT>0 (9)
Although peritubular PCO2 will be specified, the CO2 concentrations of the cells, interspace, and lumen are model variables. The relevant reactions are
H<SUP>+</SUP> + HCO<SUP>−</SUP><SUB>3</SUB> ⇄ H<SUB>2</SUB>CO<SUB>3</SUB><LIM><OP><ARROW>⇄</ARROW></OP><LL><IT>k</IT><SUB>d</SUB></LL><UL><IT>k</IT><SUB>h</SUB></UL></LIM>H<SUB>2</SUB>O + CO<SUB>2</SUB>
where dissociation of H2CO3 is rapid and assumed to be at equilibrium. Because HCO<UP><SUB>3</SUB><SUP>−</SUP></UP> and H2CO3 are interconverted, mass conservation requires
s<SUB>&agr;</SUB>(HCO<SUP>−</SUP><SUB>3</SUB>)<IT>+s<SUB>&agr;</SUB></IT>(H<SUB>2</SUB>CO<SUB>3</SUB>) = V<SUB><IT>&agr;</IT></SUB>[<IT>k</IT><SUB>h</SUB>C<SUB><IT>&agr;</IT></SUB>(CO<SUB>2</SUB>)<IT>−k</IT><SUB>d</SUB>C<SUB><IT>&agr;</IT></SUB>(H<SUB>2</SUB>CO<SUB>3</SUB>)] (10)
for alpha  = I or E, and Valpha is the volume of the compartment (cm3/cm). For the tubule lumen
s<SUB>M</SUB>(HCO<SUP>−</SUP><SUB>3</SUB>)<IT>+s</IT><SUB>M</SUB>(H<SUB>2</SUB>CO<SUB>3</SUB>) (11)

<IT>=A</IT><SUB>M</SUB>[<IT>k</IT><SUB>h</SUB>C<SUB>M</SUB>(CO<SUB>2</SUB>)<IT>−k</IT><SUB>d</SUB>C<SUB>M</SUB>(H<SUB>2</SUB>CO<SUB>3</SUB>)]
In each compartment (alpha  = I, E, or M), conservation of total CO2 is expressed
s<SUB>&agr;</SUB>(HCO<SUP>−</SUP><SUB>3</SUB>)<IT>+s<SUB>&agr;</SUB></IT>(H<SUB>2</SUB>CO<SUB>3</SUB>)<IT>+s<SUB>&agr;</SUB></IT>(CO<SUB>2</SUB>)<IT>=</IT>0 (12)
Corresponding to conservation of protons is the equation for conservation of charge for all the buffer reactions
<LIM><OP>∑</OP><LL>i</LL></LIM>z<SUB>i</SUB><IT>s<SUB>&agr;</SUB></IT>(<IT>i</IT>)<IT>=</IT>0 (13)
where zi is the valence of species i. In this model, conservation of charge for the buffer reactions (Eq. 13) may be rewritten
s<SUB>&agr;</SUB>(H<SUP>+</SUP>)<IT>+s<SUB>&agr;</SUB></IT>(NH<SUP>+</SUP><SUB>4</SUB>)<IT>−s<SUB>&agr;</SUB></IT>(HCO<SUP>−</SUP><SUB>3</SUB>)

<IT>−</IT>2<IT> · s<SUB>&agr;</SUB></IT>(HPO<SUP>2−</SUP><SUB>4</SUB>)<IT>−s<SUB>&agr;</SUB></IT>(H<SUB>2</SUB>PO<SUP>−</SUP><SUB>4</SUB>)<IT>=</IT>0
so by virtue of total phosphate conservation (Eq. 8)
s<SUB>&agr;</SUB>(H<SUP>+</SUP>)<IT>+s<SUB>&agr;</SUB></IT>(NH<SUP>+</SUP><SUB>4</SUB>)<IT>−s<SUB>&agr;</SUB></IT>(HCO<SUP>−</SUP><SUB>3</SUB>)<IT>−s<SUB>&agr;</SUB></IT>(HPO<SUP>2−</SUP><SUB>4</SUB>)<IT>=</IT>0
The solute equations are completed with the chemical equilibria of the buffer pairs: HPO<UP><SUB>4</SUB><SUP>2−</SUP></UP>:H2PO<UP><SUB>4</SUB><SUP>−</SUP></UP>, NH3:NH<UP><SUB>4</SUB><SUP>+</SUP></UP>, and HCO<UP><SUB>3</SUB><SUP>−</SUP></UP>:H2CO3. Corresponding to the electrical potentials, psi alpha , for alpha  = I, E, or M, is the equation for electroneutrality
<LIM><OP>∑</OP><LL>i</LL></LIM>z<SUB>i</SUB>C<SUB><IT>&agr;</IT></SUB>(<IT>i</IT>)<IT>=</IT>0 (14)
In the case of HCO<UP><SUB>3</SUB><SUP>−</SUP></UP> diuresis, with high HCO<UP><SUB>3</SUB><SUP>−</SUP></UP> concentrations in an alkaline urine, there has been concern about the possible importance of luminal CO<UP><SUB>3</SUB><SUP>2−</SUP></UP> concentration, particularly as a source for delayed CO2 generation via the reaction
2HCO<SUP>−</SUP><SUB>3</SUB> ⇄ H<SUB>2</SUB>CO<SUB>3</SUB> + CO<SUP>2−</SUP><SUB>3</SUB>
Within the scope of the present model, this concern can be addressed by including an additional luminal variable CM(CO<UP><SUB>3</SUB><SUP>2−</SUP></UP>) plus an additional equation for its chemical equilibrium
pH = 10.1 + log<SUB>10</SUB><FENCE><FR><NU>C<SUB>M</SUB>(CO<SUP>2−</SUP><SUB>3</SUB>)</NU><DE>C<SUB>M</SUB>(HCO<SUP>−</SUP><SUB>3</SUB>)</DE></FR></FENCE> (15)
It will be assumed that there is negligible transepithelial flux of CO<UP><SUB>3</SUB><SUP>2−</SUP></UP>, so that its luminal generation (Eq. 6) is represented as
s<SUB>M</SUB>(CO<SUP>2−</SUP><SUB>3</SUB>)<IT>=</IT><FR><NU>dF<SUB>M</SUB>(CO<SUP>2−</SUP><SUB>3</SUB>)</NU><DE>d<IT>x</IT></DE></FR> (16)
Accordingly, to accommodate the reactivity of CO<UP><SUB>3</SUB><SUP>2−</SUP></UP>, Eq. 12 for luminal total CO2 conservation must be modified
s<SUB>M</SUB>(CO<SUP>2−</SUP><SUB>3</SUB>)<IT>+s</IT><SUB>M</SUB>(HCO<SUP>−</SUP><SUB>3</SUB>)<IT>+s</IT><SUB>M</SUB>(H<SUB>2</SUB>CO<SUB>3</SUB>)<IT>+s</IT><SUB>M</SUB>(CO<SUB>2</SUB>)<IT>=</IT>0 (17)
and Eq. 13 for charge conservation of the buffer reactions also takes on a single additional term
s<SUB>M</SUB>(H<SUP>+</SUP>)<IT>+s</IT><SUB>M</SUB>(NH<SUP>+</SUP><SUB>4</SUB>)<IT>−s</IT><SUB>M</SUB>(HCO<SUP>−</SUP><SUB>3</SUB>) (18)

<IT>−s</IT><SUB>M</SUB>(HPO<SUP>2−</SUP><SUB>4</SUB>)<IT>−</IT>2<IT> · s</IT><SUB>M</SUB>(CO<SUP>2−</SUP><SUB>3</SUB>)<IT>=</IT>0
These modifications will not be included in the calculations in this paper but will be examined in the consideration of HCO<UP><SUB>3</SUB><SUP>−</SUP></UP> diuresis in the companion manuscript (33).


    MODEL PARAMETERS
TOP
ABSTRACT
INTRODUCTION
MODEL CD
MODEL PARAMETERS
MODEL CALCULATIONS
DISCUSSION
REFERENCES

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.

                              
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Table 1.   Luminal and peritubular composition

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<UP><SUB>3</SUB><SUP>−</SUP></UP> 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<UP><SUB>4</SUB><SUP>+</SUP></UP> 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
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ABSTRACT
INTRODUCTION
MODEL CD
MODEL PARAMETERS
MODEL CALCULATIONS
DISCUSSION
REFERENCES

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<UP><SUB>3</SUB><SUP>−</SUP></UP>), 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<UP><SUB>3</SUB><SUP>−</SUP></UP>, titratable acid (TA), and NH<UP><SUB>4</SUB><SUP>+</SUP></UP> (mM). Right: flows within the aggregate of all CD tubule segments of HCO<UP><SUB>3</SUB><SUP>−</SUP></UP>, TA, and NH<UP><SUB>4</SUB><SUP>+</SUP></UP> (µmol/min), along with their sum-to-net acid flow (TA + NH<UP><SUB>4</SUB><SUP>+</SUP></UP> - HCO<UP><SUB>3</SUB><SUP>−</SUP></UP>).


                              
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Table 2.   Collecting duct tubule input flows, reabsorptive fluxes, and excretion

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, tau (x0,x1), must be computed as an integral using local volume flow and luminal area
&tgr;(x<SUB>0</SUB>,x<SUB>1</SUB>)= <LIM><OP>∫</OP><LL>x<SUB>0</SUB></LL><UL>x<SUB>1</SUB></UL></LIM><FR><NU>A<SUB>M</SUB></NU><DE>F<SUB>vM</SUB></DE></FR>d<IT>x</IT> (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).

                              
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Table 3.   Collecting duct transit times

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<UP><SUB>3</SUB><SUP>−</SUP></UP>, TA, and NH<UP><SUB>4</SUB><SUP>+</SUP></UP>, 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<UP><SUB>4</SUB><SUP>+</SUP></UP> flow, less any flow of HCO<UP><SUB>3</SUB><SUP>−</SUP></UP>. Overall, the model CD contributes 0.57 µmol/min to net acid secretion, of which 0.35 is attributed to HCO<UP><SUB>3</SUB><SUP>−</SUP></UP> reabsorption, 0.13 to NH<UP><SUB>4</SUB><SUP>+</SUP></UP> secretion, and 0.09 to HPO<UP><SUB>4</SUB><SUP>2−</SUP></UP> titration. In the final urine, net acid excretion is 0.44 µmol/min, split nearly equally between TA and NH<UP><SUB>4</SUB><SUP>+</SUP></UP> excretion (Table 2). Within the CD (Fig. 4), the bulk of proton secretion and HCO<UP><SUB>3</SUB><SUP>−</SUP></UP> reabsorption occurs in the OMCD and early IMCD. The OMCD is also the site for most of the NH<UP><SUB>4</SUB><SUP>+</SUP></UP> secretion, because high luminal NH<UP><SUB>4</SUB><SUP>+</SUP></UP> concentrations blunt further secretion within the IMCD. Indeed, with a low volume flow into the CD, not only is NH<UP><SUB>4</SUB><SUP>+</SUP></UP> delivery decreased by 27% but also the luminal concentration of NH<UP><SUB>4</SUB><SUP>+</SUP></UP> decreases secretory flux by 31%, so overall NH<UP><SUB>4</SUB><SUP>+</SUP></UP> excretion is down 30%. With natriuresis or diuresis, excretion of NH<UP><SUB>4</SUB><SUP>+</SUP></UP> 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<UP><SUB>3</SUB><SUP>−</SUP></UP>, there is a sharp reduction in concentration within the IMCD (to <1 mM) and a concomitant reduction in HCO<UP><SUB>3</SUB><SUP>−</SUP></UP> 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<UP><SUB>4</SUB><SUP>+</SUP></UP> 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).

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<UP><SUB>4</SUB><SUP>+</SUP></UP>, 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<UP><SUB>3</SUB><SUP>−</SUP></UP>, 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<UP><SUB>3</SUB><SUP>−</SUP></UP> reabsorption falls to 0.20 from 0.35 µmol/min in control. In the face of flow-enhanced NH<UP><SUB>4</SUB><SUP>+</SUP></UP> secretion plus flow-diminished HCO<UP><SUB>3</SUB><SUP>−</SUP></UP> 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<UP><SUB>3</SUB><SUP>−</SUP></UP> 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.

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<UP><SUB>4</SUB><SUP>+</SUP></UP> is again enhanced with increasing luminal flow. The pattern for HCO<UP><SUB>3</SUB><SUP>−</SUP></UP> is also similar to that shown in Fig. 7, perhaps more clearly here, with increased luminal flow blunting the increase in luminal HCO<UP><SUB>3</SUB><SUP>−</SUP></UP> concentration and thus decreasing reabsorption. Once again, only at low rates of CD Na+ delivery can HCO<UP><SUB>3</SUB><SUP>−</SUP></UP> be fully reabsorbed, suggesting that it may be misleading to test maximal acidification under higher flows. Overall, the flow effects for NH<UP><SUB>4</SUB><SUP>+</SUP></UP> and HCO<UP><SUB>3</SUB><SUP>−</SUP></UP> 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<UP><SUB>3</SUB><SUP>−</SUP></UP>, 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<UP><SUB>4</SUB><SUP>+</SUP></UP>, and HCO<UP><SUB>3</SUB><SUP>−</SUP></UP> excretion are all evident. Again, the changes in NH<UP><SUB>4</SUB><SUP>+</SUP></UP> and HCO<UP><SUB>3</SUB><SUP>−</SUP></UP> 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.

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<UP><SUB>3</SUB><SUP>−</SUP></UP>. 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<UP><SUB>3</SUB><SUP>−</SUP></UP> 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<UP><SUB>3</SUB><SUP>−</SUP></UP> concentrations) define a grid of 676 CD model calculations, in which the entering flow is specified and the entering HCO<UP><SUB>3</SUB><SUP>−</SUP></UP> concentration is varied by Cl--for-HCO<UP><SUB>3</SUB><SUP>−</SUP></UP> 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<UP><SUB>3</SUB><SUP>−</SUP></UP> 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<UP><SUB>3</SUB><SUP>−</SUP></UP> 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<UP><SUB>3</SUB><SUP>−</SUP></UP>. One should note that if only the delivered load of HCO<UP><SUB>3</SUB><SUP>−</SUP></UP> determined the total HCO<UP><SUB>3</SUB><SUP>−</SUP></UP> 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<UP><SUB>3</SUB><SUP>−</SUP></UP>. 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<UP><SUB>3</SUB><SUP>−</SUP></UP> 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<UP><SUB>3</SUB><SUP>−</SUP></UP> 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<UP><SUB>3</SUB><SUP>−</SUP></UP> concentration in determining final urinary pH. Top: the abcissa is entering flow to the entire CD (µl/min), and the ordinate is HCO<UP><SUB>3</SUB><SUP>−</SUP></UP> (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.


    DISCUSSION
TOP
ABSTRACT
INTRODUCTION
MODEL CD
MODEL PARAMETERS
MODEL CALCULATIONS
DISCUSSION
REFERENCES

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<UP><SUB>3</SUB><SUP>−</SUP></UP> and the remainder to increasing the axial flow of TA and NH<UP><SUB>4</SUB><SUP>+</SUP></UP>. Net acid excretion by this CD is 0.44 µmol/min, split evenly between TA and NH<UP><SUB>4</SUB><SUP>+</SUP></UP>. 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<UP><SUB>4</SUB><SUP>+</SUP></UP> 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.

                              
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Table 4.   Acid excretion by rat kidney: control and SO4 infusion

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<UP><SUB>4</SUB><SUP>+</SUP></UP> 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<UP><SUB>3</SUB><SUP>−</SUP></UP> as well as to K+. In the simulations considered here, flow-dependent increases in HCO<UP><SUB>3</SUB><SUP>−</SUP></UP> and NH<UP><SUB>4</SUB><SUP>+</SUP></UP> 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<UP><SUB>3</SUB><SUP>−</SUP></UP> 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<UP><SUB>3</SUB><SUP>−</SUP></UP> and secretion of NH<UP><SUB>4</SUB><SUP>+</SUP></UP> 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.


    ACKNOWLEDGEMENTS

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.


    FOOTNOTES

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.

August 6, 2002;10.1152/ajprenal.00162.2002

Received 29 April 2002; accepted in final form 25 July 2002.


    REFERENCES
TOP
ABSTRACT
INTRODUCTION
MODEL CD
MODEL PARAMETERS
MODEL CALCULATIONS
DISCUSSION
REFERENCES

1.   Alguire, PC, Bailie MD, Weaver WJ, Taylor DG, and Hook JB. Differential effects of furosemide and ethacrynic acid on electrolyte excretion in anesthetized dogs. J Pharmcol Exp Ther 190: 515-522, 1974[ISI][Medline].

2.   Bander, SJ, Buerkert JE, Martin D, and Klahr S. Long-term effects of 24-hr unilateral obstruction on renal function in the rat. Kidney Int 28: 614-620, 1985[ISI][Medline].

3.   Batlle, DC. Segmental characterization of defects in collecting tubule acidification. Kidney Int 30: 546-554, 1986[ISI][Medline].

4.   Bengele, HH, McNamara ER, and Alexander EA. Effect of acute thyroparathyroidectomy on nephron acidification. Am J Physiol Renal Fluid Electrolyte Physiol 246: F569-F574, 1984[Abstract/Free Full Text].

5.   Danielson, RA, and Schmidt-Nielsen B. Recirculation of urea analogs from renal collecting ducts of high- and low-protein fed rats. Am J Physiol 223: 130-137, 1972[Free Full Text].

6.   DiTella, PJ, Sodhi B, McCreary J, Arruda JAL, and Kurtzman NA. Mechanism of the metabolic acidosis of selective mineralocorticoid deficiency. Kidney Int 14: 466-477, 1978[ISI][Medline].

7.   DuBose, TD, and Caflisch CR. Effect of selective aldosterone deficiency on acidification in nephron segments of the rat inner medulla. J Clin Invest 82: 1624-1632, 1988[ISI][Medline].

8.   Elalouf, JM, Roinel N, and deRouffignac C. Effects of antidiuretic hormone on electrolyte reabsorption and secretion in distal tubules of rat kidney. Pflügers Arch 401: 167-173, 1984[ISI][Medline].

9.   Fowler, N, Giebisch G, and Whittembury G. Distal tubular tracer microinjection study of renal tubular potassium transport. Am J Physiol 229: 1227-1233, 1975[Abstract/Free Full Text].

10.   Giammarco, RA. Effect of furosemide on collecting-duct hydrogen ion secretion in the rabbit. J Lab Clin Med 97: 390-395, 1981[ISI][Medline].

11.   Giebisch, G, Klose RM, and Windhager EE. Micropuncture study of hypertonic sodium chloride loading in the rat. Am J Physiol 206: 687-693, 1964[Abstract/Free Full Text].

12.   Giebisch, G, and Windhager EE. Renal tubular transfer of sodium, chloride and potassium. Am J Med 36: 643-669, 1964[ISI].

13.   Gouge, TH, and Andriole VT. An experimental model of amphotericin B nephrotoxicity with renal tubular acidosis. J Lab Clin Med 78: 713-724, 1971[ISI][Medline].

14.   Graber, ML, Bengele HH, Mroz E, Lechene C, and Alexander EA. Acute metabolic acidosis augments collecting duct acidification rate in the rat. Am J Physiol Renal Fluid Electrolyte Physiol 241: F669-F676, 1981[Abstract/Free Full Text].

15.   Jaeger, P, Karlmark B, and Giebisch G. Ammonium transport in rat cortical tubule: relationship to potassium metabolism. Am J Physiol Renal Fluid Electrolyte Physiol 245: F593-F600, 1983[Abstract/Free Full Text].

16.   Karlmark, B, Jaeger P, and Giebisch G. Luminal buffer transport in rat cortical tubule: relationship to potassium metabolism. Am J Physiol Renal Fluid Electrolyte Physiol 245: F584-F592, 1983[Abstract/Free Full Text].

17.   Malnic, G, de Mello Aires M, and Giebisch G. Micropuncture study of renal tubular hydrogen ion transport in the rat. Am J Physiol 222: 147-158, 1972[Free Full Text].

18.   Malnic, G, Klose RM, and Giebisch G. Micropuncture study of renal potassium excretion in the rat. Am J Physiol 206: 674-686, 1964[Abstract/Free Full Text].

19.   Malnic, G, Klose RM, and Giebisch G. Micropuncture study of distal tubular potassium and sodium transport in rat nephron. Am J Physiol 211: 529-547, 1966[Free Full Text].

20.   Marsh, DJ, and Knepper MA. Renal handling of urea. In: Handbook of Physiology. Renal Physiology. Bethesda, MD: Am Physiol Soc, 1992, vol. I, sect. 8, chapt. 29, p. 1317-1348.

21.   Sabatini, S, Alla V, Wilson A, Cruz-Soto M, deWhite A, and Kurtzman NA. The effects of chronic papillary necrosis on acid excretion. Pflügers Arch 393: 262-268, 1982[ISI][Medline].

22.   Schwartz, GJ, Weinstein AM, Steele RE, Stephenson JL, and Burg MB. Carbon dioxide permeability of rabbit proximal convoluted tubules. Am J Physiol Renal Fluid Electrolyte Physiol 240: F231-F244, 1981[Abstract/Free Full Text].

23.   Sonnenberg, H, Cheema-Dhadli S, Goldstein MB, Stinebaugh BJ, Wilson DR, and Halperin ML. Ammonia addition into the medullary collecting duct of the rat. Kidney Int 19: 281-287, 1981[ISI][Medline].

24.   Star, RA, Kurtz I, Mejia R, Burg MB, and Knepper MA. Disequilibrium pH and ammonia transport in isolated perfused cortical collecting ducts. Am J Physiol Renal Fluid Electrolyte Physiol 253: F1232-F1242, 1987[Abstract/Free Full Text].

25.   Stein, JH, Osgood RW, Boonjarern S, Cox JW, and Ferris TF. Segmental sodium reabsorption in rats with mild and severe volume depletion. Am J Physiol 227: 351-359, 1974[Free Full Text].

26.   Strieter, J, Stephenson JL, Giebisch GH, and Weinstein AM. A mathematical model of the cortical collecting tubule of the rabbit. Am J Physiol Renal Fluid Electrolyte Physiol 263: F1063-F1075, 1992[Abstract/Free Full Text].

27.   Strieter, J, Weinstein AM, Giebisch GH, and Stephenson JL. Regulation of potassium transport in a mathematical model of the cortical collecting tubule. Am J Physiol Renal Fluid Electrolyte Physiol 263: F1076-F1086, 1992[Abstract/Free Full Text].

28.   Walls, J, Buerkert JE, Purkerson ML, and Klahr S. Nature of the acidfying defect after the relief of unilateral obstruction. Kidney Int 7: 304-316, 1975[ISI][Medline].

29.   Weinstein, AM. A mathematical model of the inner medullary collecting duct of the rat: pathways for Na and K transport. Am J Physiol Renal Physiol 274: F841-F855, 1998[Abstract/Free Full Text].

30.   Weinstein, AM. A mathematical model of the inner medullary collecting duct of the rat: acid/base transport. Am J Physiol Renal Physiol 274: F856-F867, 1998[Abstract/Free Full Text].

31.   Weinstein, AM. A mathematical model of the outer medullary collecting duct of the rat. Am J Physiol Renal Physiol 279: F24-F45, 2000[Abstract/Free Full Text].

32.   Weinstein, AM. A mathematical model of rat cortical collecting duct: determinants of the transtubular potassium gradient. Am J Physiol Renal Physiol 280: F1072-F1092, 2001[Abstract/Free Full Text].

33.   Weinstein, AM. A mathematical model of rat collecting duct. II. Effect of buffer delivery on urinary acidification. Am J Physiol Renal Physiol 283: F1252-F1266, 2002. August 6, 2002; 10.1152/ajprenal.00163.2002.


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