A mathematical model of the rabbit cortical collecting
tubule
JON STRIETER, JOHN L. STEPHENSON, GERHARD GIEBISCH, AND ALAN M. WEINSTEIN Department of Physiology and Biophysics, Cornell University Medical College, New York, New York 10021; and Department of Physiology, Yale University School of Medicine, New Haven, Connecticut 06510 Strieter, Jon, John L. Stephenson, Gerhard Giebisch, and Alan M. Weinstein. A mathematical model of the rabbit cortical collecting tubule. Am. J. Physiol. 263 (Rend Fluid ELectrolyte Physiol. 32): F1063-F1075, I992.-The epithelium of the cortical collecting tubule of the rabbit is represented as four well-stirred compliant compartments corresponding to principal cell, cy- and P-intercalated cells, and lateral interspace. Model variables include the concentrations of Na, K, Cl, and HCOs, pH, cell volume, and electrical potential. The model equations specify mass conservation and chemical equilibrium for buffer reactions. Ionic conductance is represented by the Goldman constant-field equation. For the intercalated cells, phenomenological expressions describing the proton pumps are structured to agree with data of 0. S. Andersen, J. E. N. Silveira, and P. R. Steinmetz (J. Gen. Physiol. 86: 215-234, 1985) in the turtle bladder. Coupled transport via Na/H and C1/HC03 exchangers is represented according to the formalism of linear nonequilibrium thermodynamics. To construct the tubule model, the flat epithelium is wrapped into a cylinder, creating a luminal compartment. Luminal variables include volume flow, hydrostatic pressure, electrical potential, and ionic concentrations. A specific aim of this investigation was to simulate the capability of the epithelium to maintain Na reabsorption in the presence of low luminal salt concentration. In this regard, critical features of the model include tight junctional conductance and the apical Na permeability of the principal cell. In particular, we examine a principal cell apical Na permeability inversely dependent on luminal and intracellular Na concentrations (M. M. Civan and R. J. Bookman. J. Membr. BioL. 65: 63-80, 1982). This concentration-dependent permeability together with a low junctional conductance produces three results congruent with experimental data: 1) dilution of luminal Na and maintenance of reabsorptive Na transport despite a steep transtubular gradient, 2) a relatively constant level of K secretion over a wide range of luminal Na concentrations, and 3) a relatively constant transepithelial potential over this range of luminal Na.
that these cells may also play a role in K reabsorption (60) during states of K depletion. As the intercalated cells constitute almost 30% of the population (27), major variations in the electrophysiology of the CCT in general reflect the transport characteristics of a heterogeneous cell population, along with the conductive properties of the tight junction. Experimentally, the in vitro microperfusion of isolated collecting tubules has helped to elucidate the electrophysiology and transport characteristics of this segment. Several workers have studied transepithelial transport and potential as a function of Na and K composition in lumen and bath. Demonstrating the tubule’s ability to transport against steep Na gradients, Grantham et al. (13) showed that the rabbit CCT can maintain an initial luminal Na concentration of 8 mM. Offering evidence for the interdependence of K secretion and Na reabsorption, they also found that K secretion fell progressively as luminal Na was lowered below 30 mM. Stokes (44) measured transepithelial Na and K flux, luminal potential, and final tubule concentration while varying initial luminal Na concentration. Similar to the earlier findings, the tubule continued to reabsorb Na despite luminal Na concentrations as low as 8 mM. Furthermore, although Na transport clearly decreased with reduction in luminal Na, there were no significant changes in transepithelial K secretion nor in transepithelial potential. In rat distal tubule, Good and Wright (12) also found that variations in luminal Na concentration between 20 and 100 mM did not affect distal K secretion. In their study, reduction of early distal Na to 15 mM caused reversal in the direction of net Na transport. principal cell; sodium reabsorption; tight junction; sodium perThe principal cell has been modeled previously in the meability setting of frog skin, toad skin, and toad urinary bladder (6, 15, 20, 47). Hviid Larsen (15) examined the steadyTHECORTICALCOLLECTINGTUBULE (CCT)iscapableof state effects of changes in mucosal Na concentration maintaining steep transepithelial Na gradients, required for the conservation of salt and water, and is also the and apical Na permeability on short-circuit current and major site of K secretion along the nephron (13, 23, 32). cell volume in a model of toad skin. Although shortThe epithelium is composed of at least three cell types, circuit current varied, maintaining a constant ratio of each subserving a unique function. The principal cell apical Na conductance to basolateral K conductance re(27) functions as the prototypical salt-transporting cell duced fluctuations in cell volume. It was also concluded of “tight” epithelia. Characterized by apical Na and K that an inverse relationship between mucosal Na concentration and apical Na permeability allowed the cells conductances and a basolateral sodium-potassium-adenosinetriphosphatase (Na-K-ATPase), it absorbs Na and to function in low outer Na concentrations while presecretes K (13). The minority cell population consists of venting them from swelling on sudden increases in outer Na concentration. Lew et al. (20) modeled the ability of two types of intercalated cells (39). The less frequent a-intercalated cell secretes protons via an apical H- frog skin to transport Na against a steep gradient. It was ATPase while the ,&intercalated cell participates in bi- found that Na reabsorption continued despite Na concarbonate secretion and Cl reabsorption via an apical centrations below 1 mM, provided that the tight juncC1/HC03 exchanger (40, 42, 52). It has been proposed tional Na permeability and the ratio of basolateral to 0363-6127/92
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0 1992 The
American
Physiological
Society
Fl063
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F1064
A MATHEMATICAL
apical Na permeability were sufficiently low for a given pump rate. In their model, Civan and Bookman (6) varied junctional Na permeability to investigate the role of the junctions in regulating net Na transport in tight epithelia. They also extended the expression for apical Na permeability to include inverse dependence on intracellular Na concentration as well as mucosal Na concentration. Although low junctional permeability alone eliminated net Na secretion when mucosal Na was reduced to 1.8 mM, addition of the concentration-dependent apical Na permeability permitted positive Na reabsorption. In the latest model of toad skin, Hviid Larsen (16) extended his original principal cell model to consider a single type of intercalated cell (mitochondriarich cell). Although this model did not consider acidbase transport, it included a voltage-gated apical Cl permeability in the intercalated cell and was used to investigate current-voltage (I-V) relations as well as time-dependent voltage-clamp experiments, including application of ouabain and amiloride. To our knowledge, the CCT has been previously modeled only by Star et al. (41) In the present work, we construct a model of rabbit CCT epithelium with three cell types and acid-base relations. For each of the component epithelial membranes, ionic conductances are formulated according to the Goldman constant-field equation. While the basolateral Na/K pumps are represented kinetically, phenomenological expressions describing the intercalated cell proton pumps are structured to agree with the data of Andersen et al. (1) in the turtle bladder. Coupled transport via Na/H and C1/HCOs exchangers is represented according to the formalism of linear nonequilibrium thermodynamics. To predict the overall transport by this nephron segment, this epithelial model is subsequently configured as a tubule. Model parameters have been selected from data describing the rabbit CCT. Simulations of low luminal Na concentration depend critically on tight junctional and principal cell apical membrane permeabilities. As experimental measurements of junctional conductance vary from 1.1 to 7.0 mS/cm2 (18, 28, 33, 53), we explore two sets of junctional permeabilities. Model simulations with a high-conductance junction and a low-conductance junction show distinct differences with regard to axial concentration and potential profiles as well as transepithelial fluxes. Simulations with a low-conductance junction and an apical Na permeability that increases when luminal and intracellular Na concentrations decrease (6) produce results most comparable to experimental observation. MODEL
EQUATIONS
The epithelium of the CCT is represented as compliant cellular and paracellular compartments, bounded by wellstirred mucosal and serosal bathing media (Fig. 1). The system is thus composed of six compartments: mucosal solution (m), serosal solution (s), principal cellular compartment (p), a-intercalated cell compartment (a), ,&intercalated cell compartment (b), and a common lateral intercellular space (e). For each compartment, i, we con-
MODEL
OF CCT
Na
I I
1
K 9
7+ 4
Na K Na
PRINCIPAL
H
(PI F
I
K 1.
-TMUCOSAL
Cl
(ml
SEROSAL (s)
INTERCALATED (a)
Cl HC03
BETAINTERCALATED (b)
H
f
Cl
Fig. 1. Schematic representation of cortical collecting tubule (CCT) epithelium with principal cell (p), a-intercalated cell (a), and P-intercalated cell (b). Although all membranes contain nonzero permeabilities for all 5 species, only the major conductances and transporters are shown.
sider the hydrostatic pressure (Pi), the electrical potential (@), and the electrolyte concentrations (Ck) where k = 1, 2, 3, 4, 5 denotes Na+, K+, Cl-, HCO,, and H+, respectively. The cells also contain impermeant species with osmotic activity (C f,,). Within each cell, buffering is approximated as a single buffer pair with a concentration of unprotonated buffering sites (C b,,) and protonated buffering sites (C hBuf). All activity coefficients are assumed to be unity. The volumes of the principal cell, a-intercalated cell, ,&intercalated cell, and interspace are denoted by VP, Va, Vb, and Ve, respectively. Hydrostatic pressure within the cells is set equal to luminal pressure, although interspace pressure is a variable. The cellular and interspace volume, electrical potential, and ionic concentration constitute the 28 system variables. For simulation of opencircuited experiments, the transepithelial potential is an additional variable. Otherwise, luminal and peritubular compositions are specified as boundary conditions, including epithelial Pco~. There are 11 membranes in the system, superscripted according to the two compartments, i and j, which they separate. Across each membrane, ij, there is a volume flow from compartment i to j, J$ and for each species, k, there is a solute flux, Ji. (For either volume or solute flux, Jij
=
_
Jji )
To formulate the equations of mass conservation, we define the virtual generation (or source) of each species as the sum of net transmural flux and local concentration
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A MATHEMATICAL
change (54). Equations 1 and 2 describe the generation volume (s:) or solute (sk) in compartment i dV s; = Jt” + J:” + Jt” + -, dt
MODEL
of
i
i = p, a, b
dV st = J; + J;” + J; + J;b + J;” + -rrt’ d(ViC;)
s;=J~+J~+JK”+~,
i = p, a, b
where zk is the valence of species k, and zImp is the valence of the impermeant. In all, mass conservation and electroneutrality give a system of 28 equations for the epithelial model. To represent distension and shrinkage of the interspace corresponding to fluctuations in pressure, compliance relations have been included. Tight-junctional area is assumed constant, and the area of the interspace basement membrane (A”“) varies linearly with transmural pressure
(f
A es= A?[1 + ues(Pe - PP)]
d(VeC;)
s;, = Jr + J;p + J;;” + Je$, + J;” + dt
This definition has been particularly useful in coding the numerical solution. For volume and for the nonreacting species, the mass conservation equations are stated simply
s; = 0 k = 1, 2,3
0)
Sk=
(3)
&(C&,f
+ qjB”f)l = 0,
There is
i = p, a, b
(5)
where HBuf subscript indicates pro tonated buffer. Alternatively, we may write in compact notation.
i = p, a, b
SBuf + S4-IBuf= 0,
Additionally, the net creation of protons within the cells must be equal to the new buffer created in the form of bicarbonate or impermeant base
G-I= sko, + &f,
i = p, a, b
(7)
species also obey pH equilibrium
C’kco, pHi = -1ogCh = 6.1 + logo o3 . pc0 .
,
i = p, a, b, e
- 1ogc ;I = pK,,,
+ logr
C’buf HBuf
Electroneutrality is preserved interspace according to
i = p, a, b
(g)
’
within
zlzc; + ZlmpC~mp- cg,, = 0,
=
w,mq9 dx
k=l
(13)
(15)
(16)
k = 1, 2, 3
0,
(17)
The reacting species are again considered separately. Both divalent phosphate ion HPOfand monovalent phosphate ion H,PO, are present in the lumen only, functioning as impermeant buffer. There is conservation of this total buffer within the lumen (18)
GPO, + S;t;,PO,= 0 Again, interconversion librium is obeyed
of species is allowed and pH equi-
sg=s
(10)
ZkC”, = 0
area) is the sum
+ Jr” + Jr” + Jr” + Jzb
s:” = 0
k=l
;
(12)
Luminal mass conservation relations for volume and the nonreacting species may then be written
the cells and the
i = p, a, b
the and
To construct the tubule model, the flat epithelium is wrapped into a cylinder of cross-sectional area Am and length 1 surrounded by a well-stirred peritubular bath. Luminal variables include the axial volume flow (F,“), hydrostatic pressure, electrical potential, and ionic concentrations, and an additional set of luminal equations must be formulated. For the lumen, we redefine the virtual generation of each species as the sum of the local increment to axial flux plus transmural flux
Sk”=
The PCO~ is assumed constant throughout the epithelium, lumen, and bath. For the cells, a similar relation is assumed for the impermeant buffer
+ ue(Pe - Pm)]
L = VP + Va + Vb -I- Ve
(8)
2
Additionally, of interspace
Epithelial volume L (per cm2 of epithelial of cell volumes and interspace volume
s;(X)
G-I= sFEo,
;
V” = V;[l
(6)
while for the interspace
The reacting
where yes is a compliance coefficient. interspace volume varies as a function luminal pressures
(4)
The reacting species are considered separately. conservation of total buffer within the cells
F1065
OF CCT
f;co
3 +
%PO
PKJ-JPO,
+
l”gF
W)
4
C EPO, PHm
=
(20)
WO4
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F1066
A MATHEMATICAL
Electroneutrality
MODEL
is preserved within the lumen i
‘12
c it---
2c HmPO, - C &PO,
=
the pump flux for K
0
The luminal equations are completed with the relation for Poiseuille flow
-
(27)
The variable affinities of the Na and K binding sites on the intracellular and serosal surfaces are represented according to Garay and Garrahan (11)
SW (Am)2
FF
=
pyv,
i
&iA($
+
a$j&Cf,
(23)
V
(
K
(22)
’
where Q is the tubule fluid viscosity. The luminal and epithelial relations together constitute a system of 36 equations. The model equations are completed with the specification of the laws of membrane transport. The volume flow across membrane ij is written .. f ij J =- ‘;T” w AP i
= - _2 J act J act K 3 Na
(21)
k=l
dP -g+-
OF CCT
k=l
1
where AP’j and ACf are pressure and concentration differences, respectively, across membrane ij, Pf is the water permeability, UY is the reflection coefficient of species k, RT is the product of gas constant and absolute temperature, and V, is the partial molar volume of water. Solute flux is the sum of convective, electrodiffusive, and active transport. We have used the Goldman constantfield equation to describe electrodiffusive ionic fluxes. In this representation, solute flux is given by
of species
(ck7KK)2
(26)
i
.
[y=LiaH[-;
(29)
1
-;]
[~I
(30)
where ij is ps or pe for the basal or the lateral membrane of the principal cell, respectively, LgaH is Na/H exchange coefficient, and the electrochemical potential is given by A$ = RTAlnCf
+ zkFAqij
(31)
Similarly, we have described the basolateral Cl/HC03 exchanger in the a-intercalated cell and the apical Cl/ HC03 exchanger in the ,&intercalated cell by
;,z,l
=L&HCO,
[
-:
-:]
[
(32)
;fo3]
where ij is as or ae for the a-cell and ij is bm for the P-cell. The apical proton pump in the a-cell and the basolatera1 proton pump in the ,&cell were described using a phenomenological expression that simulates the proton flux data of Andersen et al. (1) in turtle bladder = - L J ma H ii
1
1 1 + exp[tGga - &;,,)I
JbI--f= LbH
(34)
where LH is the maximum proton flux, ATill is the electrochemical potential difference at which the flux is half of its maximum value, and 4 defines the steepness of the function. Figure 2 shows the shape of this function for [ = 0.4 mmol/J CHOICE
J Et = (Jgkt)max ( c~a2kNa)3
C 1 +m ha
Chaillet et al. (3) have offered evidence of Na/H exchange in the basolateral membrane of the principal cell. Although kinetic models of this exchanger have been presented in proximal tubule (51)) the formalism of nonequilibrium thermodynamics [as in Weinstein (55)] is used here
(2%
Pi is the permeability of membrane ij for species k, F is Faraday’s constant, A#G is the electrical potential difference across the membrane, and J?, is the active transport term. To represent the Na/K pump in the basolateral membrane of the principal and intercalated cells, we have used the kinetic description of Hoffman and Tosteson (14). With three identical noncooperative sites for Na and 2 for K, the pump flux for Na is written
(28)
Oe2
K K=o.l
[
where Ct is the mean membrane concentration k, i.e., - .. c-c AC ij c; - cj, C z Aln[i ln(C~/CjJ
Na =
OF
and A$,,
PARAMETERS
= 2.1 J/mmol. AND
NUMERICAL
SOLUTION
A complete list of model parameters appears in Tables these parameters have been selected from the experimental literature of the rabbit CCT. Where direct experimental determinations are 1 and 2.- Where available,
where & and & are the affinity constants for sodium and potassium, respectively. A stoichiometry of 3:2 gives
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A MATHEMATICAL
0.0
-0.5
-1.0
-1.5
r 1
I
-200
I
1
-100
I
I
0
I
1
I
1
100
JIma ow
Fig. 2. Graph of intercalated cell proton pump flux (JH) as a function of transmembrane potential difference ($, mV), with [ = 0.4 mmol/J and W/2 = 2.1 J/mmol (see Eqs. 33 and 34); ma indicates mucosal to a-intercalated cell compartment.
lacking (e.g., especially those parameters relating to the intercalated cells), they have been chosen so as to achieve concentrations, potentials, and fluxes compatible with available micropuncture and microperfusion data. The principal cell apical and basolateral water permeabilities agree with those reported by Strange and Spring (46). The paracellular pathway has traditionally been considered a physiologically insignificant route for water transport in tight epithelia. Because experimental determinations of tight-junctional water permeability are lacking, we have set this parameter equal to one-tenth of its value for the unstimulated apical membrane of the principal cell. The interspace basement membrane solute permeabilities were chosen sufficiently large to render agreement within 1.0 mM of peritubular and interspace ionic concentrations under all conditions. With these small concentration differences between interspace and bath, and in view of the lack of experimental evidence for variability of interspace composition with bounding cell type, we have considered only a single “average” lateral inter-
MODEL
F1067
OF CCT
space compartment in these models. Reflection coefficients have been taken as unity for all membranes, except the interspace basement membrane where they are zero. Electrophysiological investigation of the principal cell with regard to Na and K transport has been extensive. Given the principal cell concentrations of 15 mM Na and 140 mM K (26,34), apical membrane conductances of 1.5 for Na and 5.0 mS/cm2 for K, and transmembrane potential of 70 mV (25, 28, 33), the apical membrane permeabilities for Na and K were back-calculated with the Goldman equation. In cases where there has been no experimental evidence of a significant membrane conductance for a particular ion, its non-area-adjusted permeability has been set to 1.0 x 10vg cm/s. With respect to the basolateral membrane of the principal cell, Sansom and O’Neil (35) have determined transport numbers of 0.2 and 0.8 for K and Cl, respectively. Given these values, a basolateral K conductance of 2 mS/cm2 (33), a basolatera1 potential of -75 mV, and an intracellular K concentration as stated above, the permeabilities were calculated. For the junctional K permeability we have used the estimate of 1.0 x 10s5 cm/s reported by Stoner et al. (45). Tight-junctional ionic permeabilities for Na and Cl were calculated from the total conductance of 6.6 mS/cm2 of Muto et al. (25) and the Na/Cl permeability ratio of 0.75 of Warden et al. (53). Experimental measurements of junctional conductance, however, range from 1.1 to 7 mS/ cm2 (18, 28, 33, 53). As paracellular backflux contributes in an important way to transepithelial Na transport, the issue of high vs. low junctional conductance is explored. In contrast to the principal cell, experimental characterization of the intercalated cells has been limited. O’Neil and Hayhurst (27) have measured the cell populations per millimeter of tubule length (317 principal cells/mm; 127 intercalated cells/mm) while Schuster et al. (39) have measured the number of a-cells (29%) and
Table 1. Parameter values for interspace and principal cell Dimensions
A”” = 0.01 [l + 0.13 (P” - P”)] Ve = 6 x lo+ [l + 0.8 (P” - Pm)]
Interspace basement membrane area (A”“), cm2/cm2 epithelium Interspace volume (V”), cm3/cm2 epithelium Membrane Tight junction
Area, cm2/cm2 epithelium Pf, cm3. s-l . cm epitheliums2
3.0
0.001 x lo-*
Interspace basement membrane
Properties Apical membrane
Basolateral membrane
1.46
3.4
x 10-l
3.0
x 1o-3
12.25 8.1
x 1o-2
CT
Na 1.0 0.0 1.0 K 1.0 0.0 1.0 Cl 1.0 0.0 1.0 HCO, 1.0 0.0 1.0 H 1.0 0.0 1.0 Pk, cm3 s-l cm epitheliume2 Na 6.0 x lo+ 1.1 x lo-” 3.4 x lo+ K 1.0 x 1o-5 1.1 x 1o-3 5.6 x 1O-5 Cl 8.0 x lo+ 1.1 x 1o-3 1.5 x 1o-g HCO, 1.0 x 1o-g 1.1 x 1o-3 1.5 x 1o-g H 1.0 x 1o-g 1.1 x lo-” 1.5 x 1o-g LkaIH, mmo12. J-l. s-l cm epitheliums2 Pf, water permeability; CT,reflection coefficient; Pk, Goldman permeability; LkaIH, Na/H exchange coefficient. See Eqs. 11 and panying text in MODEL EQUATIONS for further descriptions of dimensions. l
1.0 1.0 1.0 1.0 1.0
l
l
1.2 x 1o-8
2.5 x 1o-5 9.8 1.2 1.2 1.8
x x x x
1o-5 1o-8 1o-8 lo-lo
12 and accom-
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F1068
A MATHEMATICAL
MODEL
OF CCT
Table 2. Parameter values for intercalated cells Membrane
Area, cm2/cm2 epithelium Pf, cm3. s-l. cm epitheliumm2 fJ Na K Cl HC03 H Pk, cm3. s-l. cm epitheliumB2 Na K Cl HC03 H LCI/HCo3, mmo12 J-l s-l. cm epitheliumm2 cu-IC compartment p-IC compartment LH, mmol . s-l cm epitheliumA2 a(-IC compartment p-IC compartment LCIIHCo3, C1/HC03 exchange coefficient; LH, hydrogen l
Properties
a-Apical
wBasolatera1
,&Apical
P-Basolateral
0.084 2.1 x 1o-4
0.742 4.6 x 1O-3
1.82 5.2 x 1o-4
0.206 1.2 x 1o-2
1.0 1.0 1.0 1.0 1.0 8.4 8.4 8.4 8.4 8.4
x x x x x
1.0 1.0 1.0 1.0 1.0
lo-l1 10-l’ lo-l1 lo-l1 lo-l1
7.4 2.2 7.4 7.4 7.4
x x x x x
lo-lo 1o-g lo+ lo-lo lo-lo
1.0 1.0 1.0 1.0 1.0 2.1 2.1 2.1 2.1 2.1
x x x x x
lo-lo lo-‘O lo-lo lo-lo lo-lo
1.0 1.0 1.0 1.0 1.0 1.8 5.5 1.8 1.8 1.8
x x x x x
1O-g 1o-g 1O-5 1O-g 1O-g
l
3.0 x 1o-8 6.0 x 1O-8
l
2.0 x 1o-7 3.6 x 1O-7
pump coefficient; IC, intercalated cell.
P-cells (71%) as a percentage of all intercalated cells. Together with apical and basolateral membrane areas per cell provided by Strange and Spring (46), cell membrane areas per square centimeter of epithelium may be calculated, assuming a tubule diameter of 25 pm (58). These values have then been used to furnish distinct CY-and P-cell water permeabilities based on the data of Strange and Spring (46). For lack of experimental data, the nonarea-adjusted basolateral Cl permeability for both types of intercalated cells was set approximately equal to that of the principal cell. Again, pump and exchange coefficients were chosen in an effort to simulate experimental values of both intracellular pH and total bicarbonate flux (22,
37)
In a preliminary simulation of low luminal NaCl concentration, it became clear that the P-cell was unable to withstand such an environment without threatening increases in cell volume. Decreases in luminal Cl concentration caused an increase in flux through the apical Cl/ HCOs exchanger. Consequent entry of HCOs with an accompanying cation (K) resulted in a significant rise in intracellular pH and volume. To provide volume homeostasis under these conditions, the parameter describing the steepness of the expression for proton pump flux was increased in the ,&cell. Thus, [ for the ,&cell is equal to 4.0, whereas [ for the a-cell is equal to 0.4. This greater sensitivity of the proton pump to changes in intracellular pH resulted in increased proton extrusion and elimination of large volume fluctuations. The numerical methods used here have been used previously in models of the urinary concentrating mechanism (43) as well as in epithelial models (56). In the solution of steady-state problems, the spatial derivatives of the luminal variables were represented by finite differences centered in space. Once the inlet flow rate and solute concentrations are given, a solution of the 28 epithelial equations (Eqs. l-l 0) completes the specification
of the initial data. The contributions of the individual cell types to total transmural flux was weighted according to their numbers within the tubule epithelium (71.4% principal cells; 8.3% a-cells; 20.3% ,&cells). The 36 nonlinear finite difference equations (Eqs. l-10, 14-22) are then solved iteratively along successive points of the tubule by use of Newton’s method. Typically, a spatial chop of 20 mesh points was employed (200 pm spacing). The model was coded in FORTRAN, and calculations were performed on a Hewlett-Packard HP9000/835S. RESULTS
Table 3 displays the solution to the open-circuited epithelium between mucosal and serosal isotonic Ringer solutions. The model’s transepithelial potential of -8 mV is generated by active Na reabsorption through the principal cell driven by the basolateral Na/K pump and is especially sensitive to the magnitude of the apical Na conductance. In particular, as the ratio of apical Na to apical K conductance increases, the lumen hyperpolarizes. Although experimental measurements of transepithelial electrical potential in the mammalian CCT range from +7 to -117 (2, 19), the degree of mineralocorticoid stimulation for the animals examined is not always specified. In the model, principal and intercalated cell basolateral potentials of -80 and -36 mV, respectively, are concordant with experimental observations (18, 25). Principal cell Na and K concentrations are maintained by the basolateral Na/K pump and agree with experimental reports (26, 34). Similar concentrations are maintained within the intercalated cells although there is negligible flux through the pump in these cells compared with the principal cells. Indeed, recent evidence shows markedly less effect of ouabain on intercalated cell concentrations than on principal cell concentrations (36). Principal cell Cl concentration in the model is at an equilibrium value of 6 mM, whereas intercalated cell Cl concentrations are
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A MATHEMATICAL
Table 3. Solution
MODEL
F1069
OF CCT
of model equations for open-circuited epithelium between equal Ringer solutions Compartment
V, rl/, P, G,
X10e4
cm3/cm2
0.08 0.0 0.4
-8.0 0.0
mv mmHg mM Na K Cl
140.0 5.0 120.0 25.0
HCO3
140.9 4.4 120.4 24.9
Imp
7.32
PH
7.32
4.05 -80.2 0.0
0.31 -36.2 0.0
0.71 -36.3 0.0
15.8 140.6 6.0 25.0 103.1 7.32
15.6 140.9 37.6 24.9 71.5 7.32
15.7 141.5 36.5 24.9 72.0 7.32
0.0 0.0 140.0 5.0 120.0 25.0 7.32
Membrane Total me
g, mS/cm2 Na K Cl HCOtj H Total Jh, pmol . s-l. Na K Cl HCO3 H
3.05 0.17 3.47 0.00 0.00 6.69
mP
1.49 4.71 0.00 0.00 0.00 6.20
PS + Pe
0.01 1.75 6.69 0.00 0.00 8.45
ma
as + ae
0.00 0.00 0.00 0.00 0.00 0.00
mb
0.00 0.00 1.65 0.00 0.00 1.65
bs + be
0.00 0.00 0.00 0.00 0.00 0.00
0.00 0.00 3.97 0.00 0.00 3.98
crnp2
1,351. -258. 0. -8. -711. 0. 285. 0. 0. 0. 0. 0. 0. 0. -89. V, volume; #, electrical potential; P, hydrostatic pressure; Cj
tubule
JON STRIETER, JOHN L. STEPHENSON, GERHARD GIEBISCH, AND ALAN M. WEINSTEIN Department of Physiology and Biophysics, Cornell University Medical College, New York, New York 10021; and Department of Physiology, Yale University School of Medicine, New Haven, Connecticut 06510 Strieter, Jon, John L. Stephenson, Gerhard Giebisch, and Alan M. Weinstein. A mathematical model of the rabbit cortical collecting tubule. Am. J. Physiol. 263 (Rend Fluid ELectrolyte Physiol. 32): F1063-F1075, I992.-The epithelium of the cortical collecting tubule of the rabbit is represented as four well-stirred compliant compartments corresponding to principal cell, cy- and P-intercalated cells, and lateral interspace. Model variables include the concentrations of Na, K, Cl, and HCOs, pH, cell volume, and electrical potential. The model equations specify mass conservation and chemical equilibrium for buffer reactions. Ionic conductance is represented by the Goldman constant-field equation. For the intercalated cells, phenomenological expressions describing the proton pumps are structured to agree with data of 0. S. Andersen, J. E. N. Silveira, and P. R. Steinmetz (J. Gen. Physiol. 86: 215-234, 1985) in the turtle bladder. Coupled transport via Na/H and C1/HC03 exchangers is represented according to the formalism of linear nonequilibrium thermodynamics. To construct the tubule model, the flat epithelium is wrapped into a cylinder, creating a luminal compartment. Luminal variables include volume flow, hydrostatic pressure, electrical potential, and ionic concentrations. A specific aim of this investigation was to simulate the capability of the epithelium to maintain Na reabsorption in the presence of low luminal salt concentration. In this regard, critical features of the model include tight junctional conductance and the apical Na permeability of the principal cell. In particular, we examine a principal cell apical Na permeability inversely dependent on luminal and intracellular Na concentrations (M. M. Civan and R. J. Bookman. J. Membr. BioL. 65: 63-80, 1982). This concentration-dependent permeability together with a low junctional conductance produces three results congruent with experimental data: 1) dilution of luminal Na and maintenance of reabsorptive Na transport despite a steep transtubular gradient, 2) a relatively constant level of K secretion over a wide range of luminal Na concentrations, and 3) a relatively constant transepithelial potential over this range of luminal Na.
that these cells may also play a role in K reabsorption (60) during states of K depletion. As the intercalated cells constitute almost 30% of the population (27), major variations in the electrophysiology of the CCT in general reflect the transport characteristics of a heterogeneous cell population, along with the conductive properties of the tight junction. Experimentally, the in vitro microperfusion of isolated collecting tubules has helped to elucidate the electrophysiology and transport characteristics of this segment. Several workers have studied transepithelial transport and potential as a function of Na and K composition in lumen and bath. Demonstrating the tubule’s ability to transport against steep Na gradients, Grantham et al. (13) showed that the rabbit CCT can maintain an initial luminal Na concentration of 8 mM. Offering evidence for the interdependence of K secretion and Na reabsorption, they also found that K secretion fell progressively as luminal Na was lowered below 30 mM. Stokes (44) measured transepithelial Na and K flux, luminal potential, and final tubule concentration while varying initial luminal Na concentration. Similar to the earlier findings, the tubule continued to reabsorb Na despite luminal Na concentrations as low as 8 mM. Furthermore, although Na transport clearly decreased with reduction in luminal Na, there were no significant changes in transepithelial K secretion nor in transepithelial potential. In rat distal tubule, Good and Wright (12) also found that variations in luminal Na concentration between 20 and 100 mM did not affect distal K secretion. In their study, reduction of early distal Na to 15 mM caused reversal in the direction of net Na transport. principal cell; sodium reabsorption; tight junction; sodium perThe principal cell has been modeled previously in the meability setting of frog skin, toad skin, and toad urinary bladder (6, 15, 20, 47). Hviid Larsen (15) examined the steadyTHECORTICALCOLLECTINGTUBULE (CCT)iscapableof state effects of changes in mucosal Na concentration maintaining steep transepithelial Na gradients, required for the conservation of salt and water, and is also the and apical Na permeability on short-circuit current and major site of K secretion along the nephron (13, 23, 32). cell volume in a model of toad skin. Although shortThe epithelium is composed of at least three cell types, circuit current varied, maintaining a constant ratio of each subserving a unique function. The principal cell apical Na conductance to basolateral K conductance re(27) functions as the prototypical salt-transporting cell duced fluctuations in cell volume. It was also concluded of “tight” epithelia. Characterized by apical Na and K that an inverse relationship between mucosal Na concentration and apical Na permeability allowed the cells conductances and a basolateral sodium-potassium-adenosinetriphosphatase (Na-K-ATPase), it absorbs Na and to function in low outer Na concentrations while presecretes K (13). The minority cell population consists of venting them from swelling on sudden increases in outer Na concentration. Lew et al. (20) modeled the ability of two types of intercalated cells (39). The less frequent a-intercalated cell secretes protons via an apical H- frog skin to transport Na against a steep gradient. It was ATPase while the ,&intercalated cell participates in bi- found that Na reabsorption continued despite Na concarbonate secretion and Cl reabsorption via an apical centrations below 1 mM, provided that the tight juncC1/HC03 exchanger (40, 42, 52). It has been proposed tional Na permeability and the ratio of basolateral to 0363-6127/92
$2.00
Copyright
0 1992 The
American
Physiological
Society
Fl063
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F1064
A MATHEMATICAL
apical Na permeability were sufficiently low for a given pump rate. In their model, Civan and Bookman (6) varied junctional Na permeability to investigate the role of the junctions in regulating net Na transport in tight epithelia. They also extended the expression for apical Na permeability to include inverse dependence on intracellular Na concentration as well as mucosal Na concentration. Although low junctional permeability alone eliminated net Na secretion when mucosal Na was reduced to 1.8 mM, addition of the concentration-dependent apical Na permeability permitted positive Na reabsorption. In the latest model of toad skin, Hviid Larsen (16) extended his original principal cell model to consider a single type of intercalated cell (mitochondriarich cell). Although this model did not consider acidbase transport, it included a voltage-gated apical Cl permeability in the intercalated cell and was used to investigate current-voltage (I-V) relations as well as time-dependent voltage-clamp experiments, including application of ouabain and amiloride. To our knowledge, the CCT has been previously modeled only by Star et al. (41) In the present work, we construct a model of rabbit CCT epithelium with three cell types and acid-base relations. For each of the component epithelial membranes, ionic conductances are formulated according to the Goldman constant-field equation. While the basolateral Na/K pumps are represented kinetically, phenomenological expressions describing the intercalated cell proton pumps are structured to agree with the data of Andersen et al. (1) in the turtle bladder. Coupled transport via Na/H and C1/HCOs exchangers is represented according to the formalism of linear nonequilibrium thermodynamics. To predict the overall transport by this nephron segment, this epithelial model is subsequently configured as a tubule. Model parameters have been selected from data describing the rabbit CCT. Simulations of low luminal Na concentration depend critically on tight junctional and principal cell apical membrane permeabilities. As experimental measurements of junctional conductance vary from 1.1 to 7.0 mS/cm2 (18, 28, 33, 53), we explore two sets of junctional permeabilities. Model simulations with a high-conductance junction and a low-conductance junction show distinct differences with regard to axial concentration and potential profiles as well as transepithelial fluxes. Simulations with a low-conductance junction and an apical Na permeability that increases when luminal and intracellular Na concentrations decrease (6) produce results most comparable to experimental observation. MODEL
EQUATIONS
The epithelium of the CCT is represented as compliant cellular and paracellular compartments, bounded by wellstirred mucosal and serosal bathing media (Fig. 1). The system is thus composed of six compartments: mucosal solution (m), serosal solution (s), principal cellular compartment (p), a-intercalated cell compartment (a), ,&intercalated cell compartment (b), and a common lateral intercellular space (e). For each compartment, i, we con-
MODEL
OF CCT
Na
I I
1
K 9
7+ 4
Na K Na
PRINCIPAL
H
(PI F
I
K 1.
-TMUCOSAL
Cl
(ml
SEROSAL (s)
INTERCALATED (a)
Cl HC03
BETAINTERCALATED (b)
H
f
Cl
Fig. 1. Schematic representation of cortical collecting tubule (CCT) epithelium with principal cell (p), a-intercalated cell (a), and P-intercalated cell (b). Although all membranes contain nonzero permeabilities for all 5 species, only the major conductances and transporters are shown.
sider the hydrostatic pressure (Pi), the electrical potential (@), and the electrolyte concentrations (Ck) where k = 1, 2, 3, 4, 5 denotes Na+, K+, Cl-, HCO,, and H+, respectively. The cells also contain impermeant species with osmotic activity (C f,,). Within each cell, buffering is approximated as a single buffer pair with a concentration of unprotonated buffering sites (C b,,) and protonated buffering sites (C hBuf). All activity coefficients are assumed to be unity. The volumes of the principal cell, a-intercalated cell, ,&intercalated cell, and interspace are denoted by VP, Va, Vb, and Ve, respectively. Hydrostatic pressure within the cells is set equal to luminal pressure, although interspace pressure is a variable. The cellular and interspace volume, electrical potential, and ionic concentration constitute the 28 system variables. For simulation of opencircuited experiments, the transepithelial potential is an additional variable. Otherwise, luminal and peritubular compositions are specified as boundary conditions, including epithelial Pco~. There are 11 membranes in the system, superscripted according to the two compartments, i and j, which they separate. Across each membrane, ij, there is a volume flow from compartment i to j, J$ and for each species, k, there is a solute flux, Ji. (For either volume or solute flux, Jij
=
_
Jji )
To formulate the equations of mass conservation, we define the virtual generation (or source) of each species as the sum of net transmural flux and local concentration
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A MATHEMATICAL
change (54). Equations 1 and 2 describe the generation volume (s:) or solute (sk) in compartment i dV s; = Jt” + J:” + Jt” + -, dt
MODEL
of
i
i = p, a, b
dV st = J; + J;” + J; + J;b + J;” + -rrt’ d(ViC;)
s;=J~+J~+JK”+~,
i = p, a, b
where zk is the valence of species k, and zImp is the valence of the impermeant. In all, mass conservation and electroneutrality give a system of 28 equations for the epithelial model. To represent distension and shrinkage of the interspace corresponding to fluctuations in pressure, compliance relations have been included. Tight-junctional area is assumed constant, and the area of the interspace basement membrane (A”“) varies linearly with transmural pressure
(f
A es= A?[1 + ues(Pe - PP)]
d(VeC;)
s;, = Jr + J;p + J;;” + Je$, + J;” + dt
This definition has been particularly useful in coding the numerical solution. For volume and for the nonreacting species, the mass conservation equations are stated simply
s; = 0 k = 1, 2,3
0)
Sk=
(3)
&(C&,f
+ qjB”f)l = 0,
There is
i = p, a, b
(5)
where HBuf subscript indicates pro tonated buffer. Alternatively, we may write in compact notation.
i = p, a, b
SBuf + S4-IBuf= 0,
Additionally, the net creation of protons within the cells must be equal to the new buffer created in the form of bicarbonate or impermeant base
G-I= sko, + &f,
i = p, a, b
(7)
species also obey pH equilibrium
C’kco, pHi = -1ogCh = 6.1 + logo o3 . pc0 .
,
i = p, a, b, e
- 1ogc ;I = pK,,,
+ logr
C’buf HBuf
Electroneutrality is preserved interspace according to
i = p, a, b
(g)
’
within
zlzc; + ZlmpC~mp- cg,, = 0,
=
w,mq9 dx
k=l
(13)
(15)
(16)
k = 1, 2, 3
0,
(17)
The reacting species are again considered separately. Both divalent phosphate ion HPOfand monovalent phosphate ion H,PO, are present in the lumen only, functioning as impermeant buffer. There is conservation of this total buffer within the lumen (18)
GPO, + S;t;,PO,= 0 Again, interconversion librium is obeyed
of species is allowed and pH equi-
sg=s
(10)
ZkC”, = 0
area) is the sum
+ Jr” + Jr” + Jr” + Jzb
s:” = 0
k=l
;
(12)
Luminal mass conservation relations for volume and the nonreacting species may then be written
the cells and the
i = p, a, b
the and
To construct the tubule model, the flat epithelium is wrapped into a cylinder of cross-sectional area Am and length 1 surrounded by a well-stirred peritubular bath. Luminal variables include the axial volume flow (F,“), hydrostatic pressure, electrical potential, and ionic concentrations, and an additional set of luminal equations must be formulated. For the lumen, we redefine the virtual generation of each species as the sum of the local increment to axial flux plus transmural flux
Sk”=
The PCO~ is assumed constant throughout the epithelium, lumen, and bath. For the cells, a similar relation is assumed for the impermeant buffer
+ ue(Pe - Pm)]
L = VP + Va + Vb -I- Ve
(8)
2
Additionally, of interspace
Epithelial volume L (per cm2 of epithelial of cell volumes and interspace volume
s;(X)
G-I= sFEo,
;
V” = V;[l
(6)
while for the interspace
The reacting
where yes is a compliance coefficient. interspace volume varies as a function luminal pressures
(4)
The reacting species are considered separately. conservation of total buffer within the cells
F1065
OF CCT
f;co
3 +
%PO
PKJ-JPO,
+
l”gF
W)
4
C EPO, PHm
=
(20)
WO4
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F1066
A MATHEMATICAL
Electroneutrality
MODEL
is preserved within the lumen i
‘12
c it---
2c HmPO, - C &PO,
=
the pump flux for K
0
The luminal equations are completed with the relation for Poiseuille flow
-
(27)
The variable affinities of the Na and K binding sites on the intracellular and serosal surfaces are represented according to Garay and Garrahan (11)
SW (Am)2
FF
=
pyv,
i
&iA($
+
a$j&Cf,
(23)
V
(
K
(22)
’
where Q is the tubule fluid viscosity. The luminal and epithelial relations together constitute a system of 36 equations. The model equations are completed with the specification of the laws of membrane transport. The volume flow across membrane ij is written .. f ij J =- ‘;T” w AP i
= - _2 J act J act K 3 Na
(21)
k=l
dP -g+-
OF CCT
k=l
1
where AP’j and ACf are pressure and concentration differences, respectively, across membrane ij, Pf is the water permeability, UY is the reflection coefficient of species k, RT is the product of gas constant and absolute temperature, and V, is the partial molar volume of water. Solute flux is the sum of convective, electrodiffusive, and active transport. We have used the Goldman constantfield equation to describe electrodiffusive ionic fluxes. In this representation, solute flux is given by
of species
(ck7KK)2
(26)
i
.
[y=LiaH[-;
(29)
1
-;]
[~I
(30)
where ij is ps or pe for the basal or the lateral membrane of the principal cell, respectively, LgaH is Na/H exchange coefficient, and the electrochemical potential is given by A$ = RTAlnCf
+ zkFAqij
(31)
Similarly, we have described the basolateral Cl/HC03 exchanger in the a-intercalated cell and the apical Cl/ HC03 exchanger in the ,&intercalated cell by
;,z,l
=L&HCO,
[
-:
-:]
[
(32)
;fo3]
where ij is as or ae for the a-cell and ij is bm for the P-cell. The apical proton pump in the a-cell and the basolatera1 proton pump in the ,&cell were described using a phenomenological expression that simulates the proton flux data of Andersen et al. (1) in turtle bladder = - L J ma H ii
1
1 1 + exp[tGga - &;,,)I
JbI--f= LbH
(34)
where LH is the maximum proton flux, ATill is the electrochemical potential difference at which the flux is half of its maximum value, and 4 defines the steepness of the function. Figure 2 shows the shape of this function for [ = 0.4 mmol/J CHOICE
J Et = (Jgkt)max ( c~a2kNa)3
C 1 +m ha
Chaillet et al. (3) have offered evidence of Na/H exchange in the basolateral membrane of the principal cell. Although kinetic models of this exchanger have been presented in proximal tubule (51)) the formalism of nonequilibrium thermodynamics [as in Weinstein (55)] is used here
(2%
Pi is the permeability of membrane ij for species k, F is Faraday’s constant, A#G is the electrical potential difference across the membrane, and J?, is the active transport term. To represent the Na/K pump in the basolateral membrane of the principal and intercalated cells, we have used the kinetic description of Hoffman and Tosteson (14). With three identical noncooperative sites for Na and 2 for K, the pump flux for Na is written
(28)
Oe2
K K=o.l
[
where Ct is the mean membrane concentration k, i.e., - .. c-c AC ij c; - cj, C z Aln[i ln(C~/CjJ
Na =
OF
and A$,,
PARAMETERS
= 2.1 J/mmol. AND
NUMERICAL
SOLUTION
A complete list of model parameters appears in Tables these parameters have been selected from the experimental literature of the rabbit CCT. Where direct experimental determinations are 1 and 2.- Where available,
where & and & are the affinity constants for sodium and potassium, respectively. A stoichiometry of 3:2 gives
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A MATHEMATICAL
0.0
-0.5
-1.0
-1.5
r 1
I
-200
I
1
-100
I
I
0
I
1
I
1
100
JIma ow
Fig. 2. Graph of intercalated cell proton pump flux (JH) as a function of transmembrane potential difference ($, mV), with [ = 0.4 mmol/J and W/2 = 2.1 J/mmol (see Eqs. 33 and 34); ma indicates mucosal to a-intercalated cell compartment.
lacking (e.g., especially those parameters relating to the intercalated cells), they have been chosen so as to achieve concentrations, potentials, and fluxes compatible with available micropuncture and microperfusion data. The principal cell apical and basolateral water permeabilities agree with those reported by Strange and Spring (46). The paracellular pathway has traditionally been considered a physiologically insignificant route for water transport in tight epithelia. Because experimental determinations of tight-junctional water permeability are lacking, we have set this parameter equal to one-tenth of its value for the unstimulated apical membrane of the principal cell. The interspace basement membrane solute permeabilities were chosen sufficiently large to render agreement within 1.0 mM of peritubular and interspace ionic concentrations under all conditions. With these small concentration differences between interspace and bath, and in view of the lack of experimental evidence for variability of interspace composition with bounding cell type, we have considered only a single “average” lateral inter-
MODEL
F1067
OF CCT
space compartment in these models. Reflection coefficients have been taken as unity for all membranes, except the interspace basement membrane where they are zero. Electrophysiological investigation of the principal cell with regard to Na and K transport has been extensive. Given the principal cell concentrations of 15 mM Na and 140 mM K (26,34), apical membrane conductances of 1.5 for Na and 5.0 mS/cm2 for K, and transmembrane potential of 70 mV (25, 28, 33), the apical membrane permeabilities for Na and K were back-calculated with the Goldman equation. In cases where there has been no experimental evidence of a significant membrane conductance for a particular ion, its non-area-adjusted permeability has been set to 1.0 x 10vg cm/s. With respect to the basolateral membrane of the principal cell, Sansom and O’Neil (35) have determined transport numbers of 0.2 and 0.8 for K and Cl, respectively. Given these values, a basolateral K conductance of 2 mS/cm2 (33), a basolatera1 potential of -75 mV, and an intracellular K concentration as stated above, the permeabilities were calculated. For the junctional K permeability we have used the estimate of 1.0 x 10s5 cm/s reported by Stoner et al. (45). Tight-junctional ionic permeabilities for Na and Cl were calculated from the total conductance of 6.6 mS/cm2 of Muto et al. (25) and the Na/Cl permeability ratio of 0.75 of Warden et al. (53). Experimental measurements of junctional conductance, however, range from 1.1 to 7 mS/ cm2 (18, 28, 33, 53). As paracellular backflux contributes in an important way to transepithelial Na transport, the issue of high vs. low junctional conductance is explored. In contrast to the principal cell, experimental characterization of the intercalated cells has been limited. O’Neil and Hayhurst (27) have measured the cell populations per millimeter of tubule length (317 principal cells/mm; 127 intercalated cells/mm) while Schuster et al. (39) have measured the number of a-cells (29%) and
Table 1. Parameter values for interspace and principal cell Dimensions
A”” = 0.01 [l + 0.13 (P” - P”)] Ve = 6 x lo+ [l + 0.8 (P” - Pm)]
Interspace basement membrane area (A”“), cm2/cm2 epithelium Interspace volume (V”), cm3/cm2 epithelium Membrane Tight junction
Area, cm2/cm2 epithelium Pf, cm3. s-l . cm epitheliums2
3.0
0.001 x lo-*
Interspace basement membrane
Properties Apical membrane
Basolateral membrane
1.46
3.4
x 10-l
3.0
x 1o-3
12.25 8.1
x 1o-2
CT
Na 1.0 0.0 1.0 K 1.0 0.0 1.0 Cl 1.0 0.0 1.0 HCO, 1.0 0.0 1.0 H 1.0 0.0 1.0 Pk, cm3 s-l cm epitheliume2 Na 6.0 x lo+ 1.1 x lo-” 3.4 x lo+ K 1.0 x 1o-5 1.1 x 1o-3 5.6 x 1O-5 Cl 8.0 x lo+ 1.1 x 1o-3 1.5 x 1o-g HCO, 1.0 x 1o-g 1.1 x 1o-3 1.5 x 1o-g H 1.0 x 1o-g 1.1 x lo-” 1.5 x 1o-g LkaIH, mmo12. J-l. s-l cm epitheliums2 Pf, water permeability; CT,reflection coefficient; Pk, Goldman permeability; LkaIH, Na/H exchange coefficient. See Eqs. 11 and panying text in MODEL EQUATIONS for further descriptions of dimensions. l
1.0 1.0 1.0 1.0 1.0
l
l
1.2 x 1o-8
2.5 x 1o-5 9.8 1.2 1.2 1.8
x x x x
1o-5 1o-8 1o-8 lo-lo
12 and accom-
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F1068
A MATHEMATICAL
MODEL
OF CCT
Table 2. Parameter values for intercalated cells Membrane
Area, cm2/cm2 epithelium Pf, cm3. s-l. cm epitheliumm2 fJ Na K Cl HC03 H Pk, cm3. s-l. cm epitheliumB2 Na K Cl HC03 H LCI/HCo3, mmo12 J-l s-l. cm epitheliumm2 cu-IC compartment p-IC compartment LH, mmol . s-l cm epitheliumA2 a(-IC compartment p-IC compartment LCIIHCo3, C1/HC03 exchange coefficient; LH, hydrogen l
Properties
a-Apical
wBasolatera1
,&Apical
P-Basolateral
0.084 2.1 x 1o-4
0.742 4.6 x 1O-3
1.82 5.2 x 1o-4
0.206 1.2 x 1o-2
1.0 1.0 1.0 1.0 1.0 8.4 8.4 8.4 8.4 8.4
x x x x x
1.0 1.0 1.0 1.0 1.0
lo-l1 10-l’ lo-l1 lo-l1 lo-l1
7.4 2.2 7.4 7.4 7.4
x x x x x
lo-lo 1o-g lo+ lo-lo lo-lo
1.0 1.0 1.0 1.0 1.0 2.1 2.1 2.1 2.1 2.1
x x x x x
lo-lo lo-‘O lo-lo lo-lo lo-lo
1.0 1.0 1.0 1.0 1.0 1.8 5.5 1.8 1.8 1.8
x x x x x
1O-g 1o-g 1O-5 1O-g 1O-g
l
3.0 x 1o-8 6.0 x 1O-8
l
2.0 x 1o-7 3.6 x 1O-7
pump coefficient; IC, intercalated cell.
P-cells (71%) as a percentage of all intercalated cells. Together with apical and basolateral membrane areas per cell provided by Strange and Spring (46), cell membrane areas per square centimeter of epithelium may be calculated, assuming a tubule diameter of 25 pm (58). These values have then been used to furnish distinct CY-and P-cell water permeabilities based on the data of Strange and Spring (46). For lack of experimental data, the nonarea-adjusted basolateral Cl permeability for both types of intercalated cells was set approximately equal to that of the principal cell. Again, pump and exchange coefficients were chosen in an effort to simulate experimental values of both intracellular pH and total bicarbonate flux (22,
37)
In a preliminary simulation of low luminal NaCl concentration, it became clear that the P-cell was unable to withstand such an environment without threatening increases in cell volume. Decreases in luminal Cl concentration caused an increase in flux through the apical Cl/ HCOs exchanger. Consequent entry of HCOs with an accompanying cation (K) resulted in a significant rise in intracellular pH and volume. To provide volume homeostasis under these conditions, the parameter describing the steepness of the expression for proton pump flux was increased in the ,&cell. Thus, [ for the ,&cell is equal to 4.0, whereas [ for the a-cell is equal to 0.4. This greater sensitivity of the proton pump to changes in intracellular pH resulted in increased proton extrusion and elimination of large volume fluctuations. The numerical methods used here have been used previously in models of the urinary concentrating mechanism (43) as well as in epithelial models (56). In the solution of steady-state problems, the spatial derivatives of the luminal variables were represented by finite differences centered in space. Once the inlet flow rate and solute concentrations are given, a solution of the 28 epithelial equations (Eqs. l-l 0) completes the specification
of the initial data. The contributions of the individual cell types to total transmural flux was weighted according to their numbers within the tubule epithelium (71.4% principal cells; 8.3% a-cells; 20.3% ,&cells). The 36 nonlinear finite difference equations (Eqs. l-10, 14-22) are then solved iteratively along successive points of the tubule by use of Newton’s method. Typically, a spatial chop of 20 mesh points was employed (200 pm spacing). The model was coded in FORTRAN, and calculations were performed on a Hewlett-Packard HP9000/835S. RESULTS
Table 3 displays the solution to the open-circuited epithelium between mucosal and serosal isotonic Ringer solutions. The model’s transepithelial potential of -8 mV is generated by active Na reabsorption through the principal cell driven by the basolateral Na/K pump and is especially sensitive to the magnitude of the apical Na conductance. In particular, as the ratio of apical Na to apical K conductance increases, the lumen hyperpolarizes. Although experimental measurements of transepithelial electrical potential in the mammalian CCT range from +7 to -117 (2, 19), the degree of mineralocorticoid stimulation for the animals examined is not always specified. In the model, principal and intercalated cell basolateral potentials of -80 and -36 mV, respectively, are concordant with experimental observations (18, 25). Principal cell Na and K concentrations are maintained by the basolateral Na/K pump and agree with experimental reports (26, 34). Similar concentrations are maintained within the intercalated cells although there is negligible flux through the pump in these cells compared with the principal cells. Indeed, recent evidence shows markedly less effect of ouabain on intercalated cell concentrations than on principal cell concentrations (36). Principal cell Cl concentration in the model is at an equilibrium value of 6 mM, whereas intercalated cell Cl concentrations are
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A MATHEMATICAL
Table 3. Solution
MODEL
F1069
OF CCT
of model equations for open-circuited epithelium between equal Ringer solutions Compartment
V, rl/, P, G,
X10e4
cm3/cm2
0.08 0.0 0.4
-8.0 0.0
mv mmHg mM Na K Cl
140.0 5.0 120.0 25.0
HCO3
140.9 4.4 120.4 24.9
Imp
7.32
PH
7.32
4.05 -80.2 0.0
0.31 -36.2 0.0
0.71 -36.3 0.0
15.8 140.6 6.0 25.0 103.1 7.32
15.6 140.9 37.6 24.9 71.5 7.32
15.7 141.5 36.5 24.9 72.0 7.32
0.0 0.0 140.0 5.0 120.0 25.0 7.32
Membrane Total me
g, mS/cm2 Na K Cl HCOtj H Total Jh, pmol . s-l. Na K Cl HCO3 H
3.05 0.17 3.47 0.00 0.00 6.69
mP
1.49 4.71 0.00 0.00 0.00 6.20
PS + Pe
0.01 1.75 6.69 0.00 0.00 8.45
ma
as + ae
0.00 0.00 0.00 0.00 0.00 0.00
mb
0.00 0.00 1.65 0.00 0.00 1.65
bs + be
0.00 0.00 0.00 0.00 0.00 0.00
0.00 0.00 3.97 0.00 0.00 3.98
crnp2
1,351. -258. 0. -8. -711. 0. 285. 0. 0. 0. 0. 0. 0. 0. -89. V, volume; #, electrical potential; P, hydrostatic pressure; Cj
