free medium increases Ca influx into the cell, while the Na/Ca exchanger is inactivated by .... Tight junctions join the principal cells at the api- cal membrane ...
Calcium Dynamics and Homeostasis in a Mathematical Model of the Principal Cell of the Cortical Collecting Tubule Y U A N H U A T A N G a n d JOHN L . S T E P H E N S O N From the Department of Physiology and Biophysics, Cornell University Medical College, New York, New York 10021
ABSTRACT Calcium (Ca) dynamics are incorporated into a mathematical model of the principal cell in the cortical collecting tubule developed earlier in Strieter et al. (1992a. Am. J. Physiol. 263:F1063-1075). The Ca components are modeled after the Othmer-Tang model for IP3-sensitive calcium channels (1993, in Experimental and Theoretical Advances in Biological Pattern Formation, 295-319). There are IP3-sensitive Ca channels and ATP-driven pumps on the membrane of the endoplasmic reticulum. Calcium enters the cell passively down its electrochemical gradient. A Ca p u m p and N a / C a exchange in the basolateral membrane are responsible for the extrusion of cytoplasmic calcium. N a / C a exchange can also operate in reverse mode to transport Ca into the cell. Regulatory effects of cytoplasmic Ca on the apical Na channels are modeled after experimental data that indicate apical Na permeability varies inversely with cytoplasmic Ca concentration. Numerical results on changes in intracellular Ca caused by decreasing NaC1 in the bath and the lumen are similar to those from experiments in Bourdeau and Lau (1990. Am. J. Physiol. 258:F1497-1503). This match of simulation and experiment requires the synergistic action of the N a / C a exchanger and the Ca regulated apical Na permeability. In a homogeneous medium, cytoplasmic Ca becomes oscillatory when extracellular Na is severely decreased, as observed in experiments of cultured principal cells (Koster, H., C. van Os, and R. Bindels. 1993. Kidney Int. 43:828-836). This essentially pathological situation arises because the hyperpolarization of membrane potential caused by Nafree medium increases Ca influx into the cell, while the N a / C a exchanger is inactivated by the low extracellular Na and can no longer move Ca out of the cell effectively. The raising of the total amount of intracellular Ca induces oscillatory Ca movement between the cytoplasm and the endoplasmic reticulum. Ca homeostasis is investigated under the condition of severe extracellular Ca variations. As extracellular Ca is decreased, Ca regulation is greatly impaired if Ca does not regulate apical ionic transport. The simulations indicate that the N a / C a exchanger alone has only limited regulatory capacity. The Ca regulated apical sodium or potassium permeability are essential for regulation of cytoplasmic Ca in the principal cell of the cortical collecting tubule.
INTRODUCTION
Cytoplasmic calcium (Ca) is an i m p o r t a n t second messenger in m a n y cell types, including epithelial cells in the n e p h r o n s of m a m m a l i a n kidney. In the last decade, the regulatory effect of Ca has b e e n e x p l o r e d experimentally in m a n y cell types in the n e p h r o n , including the principal cell of the cortical collecting tubule (CCT) (Windhager et al., 1991). Cytoplasmic Ca regulates the apical sodium (Na) and potassium (K) conductances (Chase a n d Al-Awaqti, 1993; Schlatter et al., 1993; Hirsh et al., 1993; Wang et al., 1993). These
Address correspondence to Dr. John Stephenson, Department of Physiology and Biophysics, Cornell University Medical College, New York, NY 10021. 207
changes in conductivity, h e n c e absorption and secretion rates o f the cells, are manifested u n d e r manipulations that change cytoplasmic Ca, including changes in Na concentration in the mucosal or serosal media, horm o n a l application, and the application of a variety of drugs that affect the function of m e m b r a n e proteins. Such a c o m p l e x pattern of interactions warrants a comprehensive model that incorporates individual components into a c o m p l e t e picture. Since calcium in b l o o d and in the l u m e n are in the millimolar range, a n d cytoplasmic Ca is usually below the m i c r o m o l a r range, calcium enters the cytoplasm down a strong electrochemical gradient. T h e r e is some evidence suggesting the existence of apical Ca channels in the principal cell o f CCT, at least in the cultured principal cells (Bindels et al., 1992). In addition, Ca
J. GEN. PHYSIOL.9 The Rockefeller University Press 9 0022-1295/96/02/207/24 $2.00 Volume 107 February 1996 207-230
may enter the cell from other cation channels in the apical membrane. The principal cell needs certain mechanisms to clear this Ca accumulation, which is accomplished by the ATP-dependent Ca pump and the N a / C a exchange located on the basolateral membrane (Gmaj and Murer, 1988; Windhager et al., 1991). The existence of basolateral Ca-ATPase in the principal cell has been demonstrated both enzymatically and immunologically (Doucet and Katz, 1982; Borke et al., 1987). The electrogenic N a / C a exchanger has been shown to exist on the basolateral membrane of connecting tubule cells through manipulations on basolateral Na concentrations (Bourdeau and Lau, 1990; Windhager et al., 1991) and immunocytochemical localization (Bourdeau et al., 1993). The primary store of intracellular Ca is the endoplasmic reticulum (ER) (Costanzo and Windhager, 1992). The Ca channels on the ER are known to be the inositol triphosphate (IP3)-sensitive type (IP3R) (Koster et al., 1993). The ER membrane also contains a Ca-ATPase that transports Ca into the ER (Moore et al., 1974; Parys et al., 1985). These results indicate that the Ca dynamics in the principal cell are similar to those of other cell types where an IP3-sensitive Ca store exists. A mathematical model has been proposed for CCT (Strieter et al., 1992a). It has been applied to study Na and K transport by the epithelium (Strieter et al., 1992b). Ca components are not included in the model. On the other hand, since the first publication of a kinetic model for IP3R by De Young and Keizer (1992), several simpler kinetic models for IPzR have been proposed, including the Othmer-Tang model (Atri et al., 1993; Othmer and Tang, 1993; Bezprozvanny and Ehrlich, 1094). These IP3R models have been extended to study intracellular Ca dynamics in various cell types, including hepatocytes, Xenopus oocytes, pituitary gonatrophs, and pancreatic [3 cells (Keizer and De Young, 1993; Li et al., 1994; Tang and Othmer, 1994). In epithelial cells, intracellular Ca dynamics are usually coupled to transepithelial fluxes and intracellular homeostasis of many ionic species, including Ca itself. To date, no mathematical model has been developed for Ca dynamics in epithelial cell types. In this paper, we expand the model in Strieter et al. (1992a) to include various Ca components as modeled in Othmer and Tang (1993), with the objective of simulating experimental results that are related to cytoplasmic Ca dynamics and its relation to other electrolytes. Here the model for cytoplasmic Ca components is expanded from that in O t h m e r and Tang (1993) to incorporate Ca buffers in the cytoplasm and inside ER. The model simulation reveals that the regulation of cytoplasmic Ca is mediated by the synergistic action of Ca regulated apical Na permeability and the basolateral N a / C a exchange. This communication between the 208
apical and basolateral membranes mediated through cytoplasmic Na and Ca provides a powerful feedback mechanism restoring cytoplasmic Ca toward its equilibrium value. MATERIALS
AND
METHODS
The Model Network and Equations Components of the model. T h e cortical collecting tubule epithelium with the principal cells a n d tight junctions is modeled after (Strieter et al., 1992a). A schematic diagram of the epithelium is shown in Fig. 1. In the center of the figure is the principal cell. To the left of the principal cell is the mucosaI or luminal soludon; and to the right the serosal or bath solution. T h e mem-
mAm0C i a~l
K
Ca
/a~ Ca_ \
Na
3NaZ
C!
IP3R
ER
-3 Na
[
Ca
(I-)p~176 I C ~ a0ump
FIGURE 1. Schematic diagram of the cortical collecting tubule epithelium with the principal cell and the tight junction. The mucosal (/eft) and serosal (right) solutions are separated by the principal cells and the right junctions joining the cells. The apical membrane of the cell separates the cytoplasm from the mucosal medium (also called lumen), and the basolateral m e m b r a n e separates the cytoplasm and the serosal medium (also called bath). Inside the principal cell, the m e m b r a n e of endoplasmic reticulum (ER) separates cytoplasm from the lumen of ER, which serves as an intracellular Ca store. Although all the membranes contain nonzero permeabilifies to all the four ionic species, only the major conductances and transporters are shown. These include Na and K channels in the apical membrane; and K and C1 channels, N a / K and Ca pumps, a Na:K:2CI cotransporter, and a N a / C a exchanger in the basolateral membrane. O n the ER membrane, there are a IP:r sitive Ca channel (IP:~R)for Ca release and a Ca-ATPase for pumping Ca back to the store. Electrical potential across the epithelium also drives leakage current through the tight junction.
A Model of Calcium Dynamics in CCT
b r a n e of the principal cell separating the cytoplasm from the mucosal solution is the apical m e m b r a n e , a n d that separating the cytoplasm a n d serosal solution is the basolateral m e m b r a n e . Thus, the principal cells in CCT separate asymmetric media, with the apical side facing mucosal m e d i u m a n d basolateral side facing serosal medium. Tight j u n c t i o n s j o i n the principal cells at the apical m e m b r a n e , separating the mucosal solution from basolateral solution at cellular boundaries. They provide the diffusion barrier to volume a n d ionic fluxes for the small spaces between the cells. T h e barrier to ionic a n d volume fluxes in the apical a n d basolateral m e m b r a n e s of the principal cells is b r o k e n by specific m e m b r a n e proteins. In the apical m e m b r a n e there are Na a n d K channels, whereas K a n d chloride (C1) channels locate o n the basolateral m e m b r a n e . In addition, there are active transporters located o n the basolateral m e m b r a n e , including N a / K p u m p a n d Ca p u m p , Na:K:2C1 cotransporter, a n d the N a / C a exchanger. Inside the principal cell, the m e m b r a n e of ER separates cytoplasm from the l u m e n of ER, which serves as an intracellular Ca store. O n the ER m e m b r a n e , there are IP3-sensitive Ca channels for Ca release. T h e ER Ca-ATPase is responsible for Ca uptake a n d for p u m p i n g the released Ca back to the store. It is known that there are three distinct cell types in CCT, namely, the principal cell a n d the ct- a n d [3-intercalated cells. The t~-intercalated cell is chiefly involved in p r o t o n secretion, whereas the [3-intercalated cell performs b o t h HCO3 secretion a n d C1 reabsorption. We will only model the principal cells j o i n e d together by fight junctions, since this simplification will not lose any essential c o m p o n e n t s for Ca dynamics in the epithelium (Fig. 1). Thus, the system contains four compartments: the mu-
cosal (m) a n d serosal (s) solutions, cytoplasmic c o m p a r t m e n t (p), a n d the ER (r) of the principal cell. A superscript i stands for one of the m, p, s, r compartments. T h e three m e m b r a n e s in the system, that between the principal cell and the mucosal medium, that between the principal cell a n d the serosal medium, a n d that between the cytoplasm a n d the ER, are d e n o t e d by superscripts ps, pro, a n d pr. T h e m e m b r a n e forming the tight j u n c t i o n is den o t e d as ms (separating mucosal m e d i u m from serosal medium). Superscript ij indicates one of these membranes. In each compartment, we will consider the concentration of Na, K, C1, Ca, a n d an i m p e r m e a n t species, d e n o t e d by C~ . Here the subscript k indicates the electrolyte species; k = l, 2, 3, 4, 5 or Na, K, C1, Ca, IMP interchangeably, corresponding to Na, K, C1, Ca, a n d the i m p e r m e a n t species. In addition to Ca c o m p o n e n t s in the membranes, we also introduced a Na:K:2C1 cotransporter in the basolateral m e m b r a n e (Fig. 1) that is n o t present in the Strieter et al. (1992a) model. T h e introduction of Na:K:2CI cotransporter is to bring the concentrations of major ionic species into the physiologically observed range. The cotransporter increases the intracellular CI concentration to ~ 2 1 mM (Rick, 1993). Without the cotransporter, the intracellular CI will be at its electrochemical equilibrium of ~ 6 mM (Strieter et al., 1992a). To date, there are no published data that support the existence of the Na:K:2C1 cotransporter in the basolateral m e m b r a n e . According to the numerical simulations of the model proposed here, some type of C1 cotransport, not necessarily Na:K:2CI cotransport, must exist in the principal cell (such as a Na:Ct cotransporter). Fluxes across the epithelial membranes. The fluxes across each mem-
Def'mition of Symbols Subscripts k Superscripts i,j
ij
ionic species, 1 (Na); 2 (K); 3(C1); 4 (Ca); 5(IMP) compartment, p (cytoplasmic); m (mucosal); s (serosal); r (ER) m e m b r a n e between c o m p a r t m e n t s i a n d j
I n d e p e n d e n t variables
+ii +p Bi i Ck ~cc V D e p e n d e n t variables CaB i R
P~ &c W Vr
Fluxes pr JCaP ps J CaP J~ j~ct J~
j~o,r
m e m b r a n e potential across m e m b r a n e ij, mV m e m b r a n e potential of the principal cell in culture, mV free Ca buffer concentration in c o m p a r t m e n t i, mM concentration of species k in c o m p a r t m e n t i, mM proportion IP3R in the inhibited state, dimensionless volume of the principal cells, cm 3 per cm 2 epithelium (cm3/cm 2 ep.) electrochemical potential difference of species k across m e m b r a n e ij,J.mmo1-1 buffered Ca in c o m p a r t m e n t i, mM m e a n m e m b r a n e concentration of species k in m e m b r a n e ij, mM proportion of (IP3R) in the bare receptor state, dimensionless proportion IP3R in the IP3-bound state, dimensionless proportion IP3R in the IP3- a n d positive Ca-bound state, dimensionless volume of the cytoplasm, c m 3 / c m 2 ep. volume of the ER, cm3/cm 2 ep. Ca flux across the ER m e m b r a n e by the Ca-ATPase, p,M/s.cm 2 ep. Ca flux across the basolateral m e m b r a n e carried by Ca pump, m m o l / s . c m 2 ep. Ca flux across the ER m e m b r a n e t h r o u g h IP3R,/~M/s'cm 2 ep. K flux carried by the N a / K p u m p across the basolateral m e m b r a n e , m m o l / s . c m 2 ep. ionic flux of k species across m e m b r a n e ij, m m o l / s . c m 2 ep. flux of species k carried by the Na:K:2C1 cotransporter, m m o l / s . c m 2 ep. (continued on next page)
209
TANG AND STEPHENSON
(continuedfrom previouspage) J($.~ ps
JNa/Ca,Ca ps JNa/Ca,Na
JO Constants F R T Kinetic constant bl b_l 62 J iPn~ax.C, ps J max,Na ps
J sc,Na/Ca pr
K M,Ca K PMSca KP~K
K~'~a k+ k-3 Membrane parameters AiJ
c# P+P~ P~r k
Ca leak flux across the ER membrane, i~M/s.cm 2 ep. Na flux of N a / K p u m p across the basolateral membrane, m m o l / s . c m 2 ep. Ca flux carried by N a / C a exchange, m m o l / s . c m '~ ep. Na flux carried by N a / C a exchange, m m o l / s . c m 2 ep. volume flux across m e m b r a n e ij, cm3/s.cm 2 ep. Faraday constant, c o u l o m b s / m m o l universal gas constant, J.mmol-l.K-1 absolute temperature in kelvins, K valence o f species k binding rate of Ca to cytoplasmic buffers, (l~M's)-1 binding rate of Ca to ER buffers, (l~M.s) -1 unbinding rate of Ca from cytoplasmic buffers, s-1 unbinding rate of Ca from ER buffers, s 1 maximal Ca flux across the ER m e m b r a n e by Ca pump, txM/s-cm2 ep. maximal Ca flux across ps m e m b r a n e by Ca pump, m m o l / s . c m 2 ep. maximal rate of N a / K p u m p across ps membrane, m m o l / s . c m 2 ep. scaling Ca flux across ps m e m b r a n e carried by the N a / C a exchanger, m m o l / s . c m 2 ep. dissociation constant ofIP3 (i = I) and positive regulatory Ca (i = 2) to IP3R, s 1 Michaelis-Menten constant for Ca flux across the ER m e m b r a n e by the Ca-ATPase, ~M Michaelis-Menten constant of Ca p u m p on the basolateral membrane, I~M Michaelis-Menten constant for N a / K p u m p across the basolateral membrane, mM MichaelisoMenten constant for N a / K p u m p across the basolateral membrane, mM binding constant of negative regulatory Ca to IP3R, ~M ~ s t unbinding constant of negative regulatory Ca from IPsR, s-i m e m b r a n e area of m e m b r a n e ij, cm2/cm 2 ep. electrical capacitance of m e m b r a n e ij, ~ F / c m 2 hydraulic water permeability across m e m b r a n e ij, c m 3 / s ' m m H g ' c m2 ep. coefficient for Ca conductance from IP3R, 1 / s ' c m 2 ep. leakage coefficient for the ER m e m b r a n e to Ca, 1 / s ' c m 2 ep. Goldman permeability coefficient of species k across m e m b r a n e ij, cm3/s.cm 2 ep. reflection coefficient of species k across m e m b r a n e ij, dimensionless
O t h e r parameters [Bi]T
I KNa,Ca KK,Ca
v,
total Ca buffers in c o m p a r t m e n t i (i = 1 (cytoplasm), 2 (ER)), mM concentration of IP3 in the cytoplasm, I~M constant for the switching point on the regulation of PNP]' by Ca, mM constant for the switching point on the regulation o f P pm by Ca, mM volume ratio o f the ER to the cytoplasm, dimensionless
brane are composed o f two parts, volume fluxes carried by water, 9 9 ij d e n o t e d byJ~ , and lomc fluxes, d e n o t e d byJk 9We will use the convention ji3 = _jji for either volume or solute flux, with the positive direction from c o m p a r t m e n t i to c o m p a r t m e n t j. J~! is d e t e r m i n e d by the osmotic gradient across the membrane ij and the m e m b r a n e permeability to water.
The electrolyte flux, J~ , is composed of a convective term, a passive conducting term, and an active transport term. The convective term and the passive conducting term due to electrochemical gradients across each m e m b r a n e are modeled by the Goldman constant field equation (Goldman, 1943). '
j v + = /+~Rr C Js - C s++
=
~jr~J_ O'k[_t; k
C
,
(1)
"
t
--ij
ij
j ~ = ~ C1 - , ~ ) C~Jv +
~i
P~J Zk F~f)ijCjk Rrc 1
where L',! is its hydraulic water permeability, and r the reflection coefficient of species k. RT is the product of gas constant (R) and absolute temperature (75. We do not have a term for intracellular pressure for the principal cell as in Strieter et al. (1990), which means that the only driving force for volume flux is the osmolality difference. At equilibrium, the osmolalities of different compartments are the same. 210
+
(1
-
a kij
ij ij )CkJv
( -- zkFl~iJ "~
-- t~kexp~'-"'-R - ~ ) (--~FC'~ -
exp~,-~-~--~-~)
J~.~t, if I~Jl>
o
+ Pkij (C3k" -- C ki) + j~+t 9 ,
(2) if
l
Definition
J
2
Area
The electrochemical potential o f each ion is given by p~Pks = R T l n ( C P / C ~ ) +ZkF~b p~,
k = 1,2,3.
For the ATP-driven Ca p u m p on the basolateral membrane, the rate is described by a Michaelis-Menten form with Hill coefficient 2, as given in Haynes and Mandveno (1987). 2 (cL) ps ps (9) JCaP = J . . . . Ca 2 ps 2 (C~Pa) + (KM, Ca ) The N a / C a exchange is m o d e l e d after Campbell et al. (1988) with 3:1 stoichiometry for Navs Ca. The driving forces for the exchange are the chemical concentration gradients of Na and Ca and the m e m b r a n e potential across the ps membrane. Depending on the concentration gradient o f Na and Ca and the membrane potential across the ps membrane, the exchanger can function in two different directions: one is to transport one Ca outward from the cytoplasm to the serosal m e d i u m in exchange for three Na entering the cell; and the other is to transport three Na outward from the cell in exchange for one Ca entering the cell. The Ca flux carried by this exchange is given by
ps = .]sc, Na/Ca
s 3 3 s ( C N~) CPa exp (-F~bps/2RT) - ( CPa ) Cca exp (F~bpS/2RT) 1 + 0.0095 [CcPa (C~qa) 3 + C~a (CNaP) 3] (10) 211
ps
ms
1.46
cm2/cm2 ep. 12.25
0.001
3.0 • 10-3
cm3/s, mmHg. cme ep. 8.1 X 10-2
(8) Lv
ps JNa/Ca,Ca
pm
TANG AND STEPHENSON
crk, k = 1, ..., 4
1.0
1.0
3.0
X
10-4
1.0
cmVs. cm2 ep. Pk NA K C1 Ca
1.36 • 1.12 • 1.5 X 1.8 X
10-6 10-5 10-9 10 -7
1.2 x 1.0 X 1.0 • 1.0 •
10-8 10-s 10-6 10 -9
pmol/s, cm2 ep. Active transporters
JP~ax.Na
2.4
J~ax,ca
1.9 7.0 • 10-e
J~Na/Ca
•
103
mmoF/J, s. cm2 ep.
Na:K:2CI cotransport coefficient L e~
4.0
Michaelis-Menten constant K~.c~
0.075 p~M
X
10 -6
1.2 • 2.0 x 1.6 • 5.0 •
10-6 10-6 10-7 10 -8
35 30
E
25
....•
10
-96mY
-96mY
8
(a)
(b)
- 7 6 m ~
~Q:.....~ -76mY "~
15
4
"
.'"
.....
*'"~176
"" 10 5 -2
0
-5
-4 0
5
10
15
20
25
30
35
40
45
50
0
50
100
CPNalmM)
Cs
50
96mV
40
E 30
-76m~........
2o
-
--r . . . . . . . . . r~ . . . . . . . . . . . . . . . . . . .
-10
250
(mM)
Na
(d)
25
2o
!~
.'" . . " ' " ' "
200
30
.......
~ 10 0
150
~
-96mV -76mV
-5
-9
-8.5
-8
-7.5
-7
-6.5
-6
-5.5
-5
0
0.5
1
1.5
log CPCa (M)
2
2.5
3
3.5
4
CSca (mM)
FIGURE 2. Ca flux across the basolateral membrane carried by the N a / C a exchange as each of the controlling factors (membrane poten" s tial, Na and Ca concentrations in the cytoplasm and in the bath) is individually varied. (a) Varying CNPa only; (b) v a rylng CNa only; (c) varying CPa only; and (d) varying C~:a only. The other controlling factors in each panel are at fixed values given in Table IIL
of the complete volume. The ER serves as the Ca store, and its membrane contains the IP~-sensitive Ca channels. The volume of the ER relative to that of the cytoplasm is assumed to be at a fixed ratio of vr = Vr/VP. AS the cell volume changes, both the volume of the cytoplasm and the volume of the ER change proportionally so that Vr remains a constant. IP3R is a hetero-tetramer that includes at least two different subtypes (Yamamoto-Hino et al., 1994). This channel is up-regulated by cytoplasmic IP3. Cytoplasmic Ca can both potentiate or inhibit the channel opening, depending on its concentration. The time scales of the interaction for the potentiating effect and the inhibitory effect also differs. De Young and Keizer first propose a kinetic scheme for these aspects of channel regulation (De Young and Keizer, 1992). They successfully showed that their model can explain many aspects of Ca dynamics, including excitability, Ca oscillations, and the highly elevated cytoplasmic Ca levels at over-stimulated cases (Keizer and De Young, 1993; Li et al, 1994). Since the publication of the De Young-Keizer model, several simplified models have appeared. Here, we use a four-state model for the IP3R Ca channel proposed by Othmer and Tang (1993). The transitions between the different states occur according to the following sequential scheme.
212
k1
R+I
~
R[,
k_ I
k: R I + C ~--- Rio+,
(11)
k_ 2
k3
R
IC +
+C~(..~_R k3
IC + C - "
Here R denotes the bare receptor for the IP3-sensitive Ca chann e l , / d e n o t e IPs, and C the cytoplasmic Ca. Superscripts + and on C denote the association of Ca to the positive or negative regulatory site on the receptor. Therefore, RI is the IP3R with IP~ bound to it, R~c+ the IP~R with IP s bound to its site and a Ca bound to the positive regulatory site, and R~c+c- the IP3R with all the regulatory sites occupied. R and R~ are the activatable states, Rio+ the activated state, and P~c+c the inhibited state. Among various schemes proposed for the kinetics of IP3R, this scheme is the simplest, i.e., including the least number of parana-
A Model of Calcium Dynamics in CCT
eters a n d c h a n n e l states (De Young a n d Keizer, 1992; Atri et al., 1993; Bezprozvanny a n d Ehrlich, 1994)9 It addresses adequately the effect of IP3 a n d Ca o n the receptor a n d simulates the experimental data well (Tang et al., 1995). Although it is known that the subunit in the c h a n n e l complex interact cooperatively (Watras et al., 1991), its exact nature of cooperativity is still unclear. Previous theoretical a n d numerical studies o n this issue in Tang (1993) are inconclusive. T h e model used here simply assumes that each subunit functions i n d e p e n d e n t l y in response to IPs a n d Ca changes. Although this approach c a n n o t address the regulation of c h a n n e l events at a single receptor level, it is adequate in explaining experimental data for a population of receptors (Bezprozvanny a n d Ehrlich, 1994). Let R, RI, Rm, a n d Rmc d e n o t e the fractions in states R RI, P~c+, a n d Pqc+c-, respectively, a n d I f o r cytoplasmic IP3 concentration. T h e n the governing equations for the c h a n n e l fractions are as follows ( O t h m e r a n d Tang, 1993).
dR --~ = klI. R + k iR1,
dRi d---[ = - ( k-I + k2CPc~ ) RI + kll" R + k_zRio dR1 c dt = -- ( k - 2 + k3C~:a ) RIC + kzCPa " RL + k 3RICC ,
(12)
dRlcc dt = k3CPa " RIC -- k-3RlCC' We assume that the binding of IP 3 a n d the positive regulatory Ca to the IP3R obey fast dynamics so that approximately,
In addition to the Ca c h a n n e l conductance, we assume that there is a leakage of Ca from the ER into the cytoplasm, a n d it is depicted simply by a linear rate driven by the Ca gradient across the ER m e m b r a n e ( O t h m e r a n d Tang, 1993). rp
Jleak
rp = Pleak
r
(Cca - CPa ).
T h e Ca p u m p o n the ER m e m b r a n e is m o d e l e d similarly to the Ca p u m p on the basolateral m e m b r a n e (Gmaj a n d Murer, 1988; De Young a n d Keizer, 1992), i.e.,
pr JcaP
~
pr p )2 J . . . . Ca ( Cca 2 ( p pr )2
Cca)
dRL dt
The c o m b i n e d Ca flux across the m e m b r a n e of rp is jfIipa
= J~eak rp pr +JrhP - J~aP
K2( 1
- RlCC)
K2 ( 1
-
bl CPa+ B p ~__ CaB p,
RICC) (13)
C(Pa ( 1 -- RICC)
CP~ + K 2 (1 + K i l l )
where Ki = k_i/hi, (i = 1,2), are the dissociation constants for each b i n d i n g / u n b i n d i n g step. H e n c e the governing equations for the IP3-sensitive c h a n n e l reduce to:
dRlc c dt
-
-k_3Ricc+
k3 ( CPca ) 2 (1 - Rmc) CcPa + K 2 ( 1 +KI[I).
(14)
This single gating equation for the c h a n n e l dynamics a n d the algebraic equation for the open channel, Rio in Eq. 13 will be the only equations used in the comprehensive model. The Ca conductance across the IPrsensitive c h a n n e l is m o d e l e d as a linear function of the c h a n n e l fraction in the R4c+ state multiplied by the Ca gradient across the ER m e m b r a n e (De Young a n d Keizer, 1992; O t h m e r a n d Tang, 1993). J[P
=
Pcrhp
9
CP
(1 -- R i c e )
( C c a - CcPa ) .
(15)
CcPa + K 2 ( 1 + K , / 1 )
213
(19)
~
CaB r,
b_ 2
KI/I
CCPa + K 2(1 + K , / I ) ' RIC
(18)
In addition to the fluxes across the three distinct membranes, there are internal biochemical processes that modify Ca concentration in the two compartments. Ca in the cytoplasm can b i n d with polyvalent anions, proteins, and fluorescent dyes once they have e n t e r e d the cell. It is also certain that there are Ca binding molecules inside the ER that buffer Ca. For simplicity, we model the various buffers by assuming a single kinetics for the comb i n e d effects from these buffers. Namely, we assume
C(~a+Br
CPa + /s (1 + K, II) RI =
9
b2
O.
By using the fact that R + / ~ + Ric + RIce = 1, we get R=
(17)
+ ( KM. Ca
b_ I dR dt
(16)
T A N G AND STEPHENSON
where B p a n d B r are for the free buffers in the c o m p a r t m e n t s of the cytoplasm a n d the ER. T h e total a m o u n t of buffers in the cytoplasm will be d e n o t e d [BP]T, a n d that in the ER [ B r ] T . Since the total a m o u n t of buffer in each c o m p a r t m e n t is conserved, we have [BP]T = B p + CaBPand [ B r ] T = B r + CaB r. Parameter values for the IP~R kinetics a n d o t h e r c o m p o n e n t s are based on O t h m e r a n d Tang (1993). For Ca buffering in the cytoplasm a n d in the ER, we assume that they are fast processes c o m p a r e d with the physiological processes we study (Robertson et al., 1987; Luo a n d Rudy, 1994). T h e majority of Ca in the cytoplasm a n d in the ER is in the buffered form. T h e r e are n o reports on typical Ca buffer concentrations for epithelial cells. In the axoplasm of Myxicola, typical Ca buffer concentration is estimated to be 100-200 txM in Abercrombie a n d A1-Baldawi (1990), with dissociation constants of 5-10 txM. T h e axoplasm buffer capacity, defined as, buffered Ca] free Ca ) ' nevertheless, can vary from 0-20 for some a n d 100-150 for the others, suggesting considerable axon-axon difference (A1Baldawi a n d Abercrombie, 1995). In the bovine adrenal chromaffin cells, the average Ca binding capacity is 9 -+ 7 for the poorly mobile c o m p o n e n t a n d 31 + 10 for the fixed c o m p o n e n t , with the dissociation constants h i g h e r than 3 IxM (Zhou a n d Neher, 1993; Wagner a n d Keizer, 1994). For the Ca buffers inside the
ER, n o direct m e a s u r e m e n t s o f the buffer c o n c e n t r a t i o n s a n d their dissociation constants are available in the literature. Seco n d a r y evidence suggests the total buffer c o n c e n t r a t i o n to be s o m e w h e r e f r o m h u n d r e d s o f m i c r o m o l a r to millimolar range, while the free Ca c o n c e n t r a t i o n is in the r a n g e 1 - 1 0 0 I~M (personal c o m m u n i c a t i o n s ) (Milner et al., 1992; Kendall et al., 1994). In o u r m o d e l , the c o n c e n t r a t i o n o f Ca buffers in the cytoplasm is a s s u m e d to be 120 I~M a n d that inside the ER is 1.2 mM. T h e dissociation c o n s t a n t Kd for the cytoplasmic buffer is 6.67 I~M a n d for the ER buffer 50 I~M. T h e p a r a m e t e r values for the intracellular Ca fluxes a n d reactions are listed in Table II. The comprehensive model. To d e v e l o p a c o m p r e h e n s i v e m o d e l o f the principal cell-fight j u n c t i o n c o m p l e x u n d e r the o p e n circuit c o n d i t i o n , we assume the serosal potential to be the g r o u n d potential. T h e i n d e p e n d e n t variables for t h e system are the m e m b r a n e potential o f the principal cell (+P0, the potential across the apical m e m b r a n e (~bPm), the cell volume (V), the electrolyte c o n c e n t r a t i o n s (C p , k = 1 . . . . , 4), Ca inside ER (Cc':a), the Ca c h a n n e l state ~ c + c - (Rmc), a n d the free Ca buffer c o n c e n t r a t i o n in b o t h the cytoplasm a n d the ER (BP a n d B~). T h e total n u m b e r o f the system u n k n o w n s is 11. T h e d e p e n d e n t variables i n c l u d e p C IMP, t h e electrical p o t e n t i a l across t h e e p i t h e l i u m r m~ = _ +pro + +p~, v o l u m e o f t h e cytoplasm a n d ER (VP a n d W), channel states o f l P s R o t h e r t h a n RlCC (R, RI, Ric), a n d the b u f f e r e d Ca in the cytoplasm a n d inside the ER ( CaBP a n d CaBr). T h e ionic c o m p o s i t i o n o f the ER a n d the t r a n s p o r t o f ionic species o t h e r than Ca across t h e ER m e m b r a n e are n o t clear experimentally. It is also u n c e r t a i n if t h e r e is a m e m b r a n e potential across t h e ER m e m b r a n e , b u t m o s t likely not, since the ER m e m b r a n e is relatively p e r m e a b l e to ionic species such as Na a n d K. If t h e r e is an ER m e m b r a n e potential, this m e m b r a n e potential will affect the Ca release t h r o u g h the IP~R. In this m o d e l , we d o n o t have the ionic c o n c e n t r a t i o n s for Na, K, CI a n d i m p e r m e a n t species in the ER as i n d e p e n d e n t variables. Instead, we assume that the ER m e m b r a n e is p e r m e a b l e to Na, K, a n d C1, thus t h e r e is n o m e m b r a n e potential across the ER m e m b r a n e . To m a i n t a i n the electroneutrality o f ER, we assume that as Ca is released, the charge carried by it is b a l a n c e d by the secondary c u r r e n t s o f C1 a n d K (or Na). In addition, we assume that as Ca is released, the osmolality o f t h e ER does n o t change. This is a c c o m p l i s h e d by b a l a n c i n g the release o f Ca to t h e release o f C1 a n d the uptake o f K (Na). T h e s e r e q u i r e m e n t s are fulfilled by the c o t r a n s p o r t o f 2Ca (release) to 1 Cl (release) a n d 3 K (uptake). In a d d i t i o n to t h e c h a n n e l gating e q u a t i o n for IPsR (Eq. 14), t h e o t h e r system equations are o b t a i n e d f r o m conservation for the electrical charge (~bP% qbm), the c o m b i n e d volume o f cytoplasm-ER c o m p l e x (V), a n d t h e mass o f each electrolyte (C P~, TABLE
II
r
C PK, C Pl, C P~, a n d C Ca )" Let us assume that the electrical capacitances for t h e two m e m b r a n e s are C qpm a n d C qP~, which are fixed constants d e t e r m i n e d by the surface areas o f the two m e m branes, a n d the electrical capacitance o f tight j u n c t i o n is 0. T h e ionic c u r r e n t s are c o m p u t e d for different apical a n d basolateral m e m b r a n e surface area, AP~ a n d APs, in the unit o f cmz ep. ( s h o r t h a n d for c m 2 9 e p i t h e l i u m ) . T h e actual cell m e m b r a n e is ~10,000 times smaller than this value. T h e specific m e m b r a n e capacity is set at 1 ~ F / c m 2 (Luo a n d Rudy, 1994). Therefore, C~ = Aij (IxF). T h e system e q u a t i o n s are: 4
cPm dcbPm -- -- E ziF(J pm + j~n~) dt k=l 4
C~qs a + ~
dt
dV p. C P a )
d(g
dt
-
- Z ~,vr k=l
= -J7
- ffv m ,
_
.jp;
_jpr~,
_
_j~s
_ j~ ....
d ( VP. c ~ )
dt d ( V p. CPl )
-
dt
-JP~ -JrP~'
KI ks b1 /~2 Jpmrax P~ P[P~k [B0]T
Value
Parameter
Value
0.667 ~M 0.04 (~M - s) -l 0.03 (~M 9s) -1 0.005 (~M . s)-I 1.5 X 10 -z p.M/s 9cm ~ ep. 1.2/s. cm 2 ep. 0.1/s . cm 2 ep. 120 ~M
K2 k_s b_ 1 b_2 KP~M.Ca vr I [BqT
0.126 I~M 0.021 s -~ 0.2 s -1 0.25 s 1 0.1 I~M 0.185 0.1 ~zM 1.2 ~M
214
1.5 v J , ~ ,
r rp
+ 0.5 V J c a ,
(20)
d ( V p" CPa )
%7
dt
-JL m r rp
+ VJ~, a
-- b l g P C p a B p + b _ I V P ( [ B P ] T -
r" C c a )
d(g
-- -- lfr'rrP --JCa
dt d ( g p" B p)
dt
-
d ( V r. B r) dt -
bl v P c P
--
bz wccr~ B r +
BP),
b 2 v r ( [ Br] T -- B r)
a B p + b_ 1 v P ( [ B p] T -- B P ) ,
b2V~Cc'iaBr+ b-2Vr( [Br]T -- Br)'
dRmc 2 ( 1.0 - Rmc) dt = ks(CPa) K2(KII[+I.O)+Cp
-ksRICC
w h e r e VP stands for the volume o f cytoplasm, a n d Vr the volume o f ER, with VP + Vr = V a n d Vr/VP = Yr. T h e r e a d e r s h o u l d refer to Def'mifion o f Symbols for the usage o f symbols in this equation. We follow the data in Silver et al. (1993) a n d in Frindt et al. (1993) to m o d e l the regulation o f CP, o n PNP~~. Because the c h a n n e l o p e n i n g o r closing in r e s p o n s e to CP, increasing or decreasing c a n n o t be fitted by a single e x p o n e n t i a l curve, we c h o o s e a curve o f two exponentials, namely,
Parameter Values for Intracellular Ca Handling Parameter
- jy),
p pm Na I
pm
PNa, 0exp |
-14(CX:a - K Na, Ca)•
p m o exp [ nrN~,
pm
+ 0.1 PNa, 0
--4 024 C - K • 103 " Ca Na, Ca -[- ~u. "l PNa, oPm
if Cca < KNa, Ca otherwise, (21)
pm
ptn
w h e r e PNa,0 is the PNa value given in Table I, a n d Kya,ca = 150 nM. A basal leakage rate without any apical Na c h a n n e l is ass u m e d at 0.1 P0. A l t h o u g h the evidence for Ca inhibition o n P NaPm
A Model of Calcium Dynamics in CCT
as C Pa increases is well established, that a d e c r e a s e in C P~ increases ppm is still controversial. This issue will be a d d r e s s e d in m o r e detail later. T h e d e p e n d e n c e o f P pm o n C cPa is d e p i c t e d in Fig. 3.
Numerical Implementation T h e system e q u a t i o n (20) has 11 i n d e p e n d e n t variables a n d 11 differential equations. T h u s it can b e solved numerically as an initial value p r o b l e m . T h e solute c o n c e n t r a t i o n s in t h e mucosal a n d serosal c o m p a r t m e n t s , C ki (i = m,s), have to b e given over P the time p e r i o d o f simulation, as well as the initial value o f C IMP" To obtain an e q u i l i b r i u m solution o f t h e system, the initial values for the i n d e p e n d e n t variables can b e c h o s e n arbitrarily. Previous m a t h e m a t i c a l m o d e l s for epithelial cells use t h e ass u m p t i o n o f electrical neutrality to d e t e r m i n e the m e m b r a n e potentials (Strieter et al., 1990). This assumption, which is an app r o x i m a t i o n , c o r r e s p o n d s to selecting C ~ = 0 in o u r m o d e l app r o a c h . M o d e l i n g electrical potential equations directly gives rise to a stiff system w h e r e t h e scale o f the e q u a t i o n s varies greatly. In addition, the d i f f e r e n c e s in t h e c o n c e n t r a t i o n s o f intracellular ions are significant. For e x a m p l e , in cytoplasm, Ca is in the p~M range, while Na, C1, a n d K are in the m M range. Special c o m p u tational p r e c a u t i o n is r e q u i r e d to solve this system. To address these issues, we use t h e GEAR drive s u b r o u t i n e dev e l o p e d by H i n d m a r s h (1974), w h i c h is specifically d e s i g n e d for stiff systems. We use a u n i f o r m c o n t r o l l e d e r r o r o f 1.0 • 10 .5 for each discrete equation. Time steps o f adaptive l e n g t h are used such that each call to the s u b r o u t i n e gives a c o n v e r g e d solution. We have r u n o u r p r o g r a m o n SUN a n d Silicon Graphics work stations a n d o n the IBM RISC 6000. For simulating an e x p e r i m e n t o f 20 min, t h e p r o g r a m c o m p l e t e s within two m i n u t e s o n all o f these m a c h i n e s .
inclusion of the Na:K:2CI cotransporter and the Ca regulatory effect on apical Na conductance. According to this model, the principal cells actively absorb Na, while secreting K at a smaller rate. The difference in the Na absorption rate and that of secretion of K gives rise to a net positive current entering the epithelium from the apical m e m b r a n e of the principal cell. This positive current is balanced by a net negative current across the tight junction, carried mainly by currents of Na, K, and C1. Across the tight junction, Na leaks back into the lumen, and CI is absorbed into the serosal medium due to the lumen-negative potential. K is also secreted across the tight junction. The tight junction is quite tight in the sense that the leak Na current is only ~ 2 5 % of that of absorption. T h e Ca absorption rate is ~ 1 1 % of that reported in Bindels et al. (1992). This value is reasonable, since Bindels' data is an aver-
TABLE
III
Equilibrium Solution for Open-circuited Epithelium between Equal Ringer Solutions
~b
Mucosal
Cytoplasm
-37.1
mV -76.1
Serosal
0.0
10-~ cmS/crtY ep. 1.37
V
mM
cL Na K Cl Ca IMP OSM
RESULTS
Steady State Solution Open-circuit epithelium. Table III shows the solution to the open-circuit epithelium in isotonic media with the 10
140.0 5.0 149.0 2.0 0.0 296.0
13.7 140.2 23.7 118.3 296.00
C~a
CaBP
140.0 5.0 149.0 2.0 0.0 296.0 C~:a
CaBr
/xM Ca
93.5 x 10 -B
1.7
34.4
488.8
8-
Jk 6-
Na K C1 Ca
:r t-~
mp
4-
Jk
2-
0
0
012
014
016
018
CPca (gM) FIGURE 3. T h e effect o f C p o n the permeability o f the apical Na channel. This figure is based o n the observed inhibitory effect o f CPa o n the apical Na c h a n n e l r e p o r t e d in Palmer a n d Frindt (1987). See text for m o r e detail. 215
TANG AND STEPHENSON
843.1 -548.6 0.02 1.2 Na/K pump
Na K C1 Ca
874.5 -583.0
ER flux
VrJ~ak
ps
net
ms
pmol/s.cm2 ep. 843.1 - 246.6 -548.6 -14.7 0.02 35.0 1.2 -0.29 Cotransporter Ca pump pmol/s.cm2 ep. -26.0 -26.0 -51.9
Na/Ca exchange -0.06
1.16
0.73
596.5 -563.3 35.0 0.89
pr Vr.J~ V r.Jpump pmol/s.cm2 ep. 0.76 1.50
0.02
age o f the c o n n e c t i n g tubule a n d that o f the cortical collecting tubule. T h e Ca a b s o r p t i o n rate in c o n n e c t ing d u c t is k n o w n to be significantly h i g h e r t h a n that in the C C T (Costanzo a n d W i n d h a g e r , 1992). Closed-circuit epithelium. In the case o f closed-circuit, electrical potential in the l u m e n a n d the b a t h are coupled t o g e t h e r at g r o u n d potential. T h e m a t h e m a t i c a l description o f the closed-circuit differs f r o m the o p e n circuit by having o n e less i n d e p e n d e n t differential equation, since ~)ps = (~pm a n d + . . . . 0. T h e electrical c u r r e n t passing t h r o u g h the tight j u n c t i o n b e c o m e s zero, because o f the parallel s h o r t circuit o f zero resistance. We d e n o t e the m e m b r a n e potential o f the principal cell by d)P. This m e m b r a n e potential is determ i n e d by the c o m b i n e d fluxes o f the cell f r o m the apical a n d basolateral m e m b r a n e , given in the following equation. 4 pdqbp ~" ziF(jpm +jps) (22) Cq dt
-
k= 1
Mucosal
Ca Dynamics in Open-Circuit Epithelium Decreasing NaCl in the serosal medium, ff N a / C a exchange exists in the principal cells, m a n i p u l a t i o n s in extracellular N a c o n c e n t r a t i o n s h o u l d affect C Pa dynamics. B o u r d e a u a n d L a u tested this idea by isosmotically replacing NaC1 in the serosal solution with o t h e r n o n p e r m e a b l e species in the c o n n e c t i n g tubule ( B o u r d e a u a n d Lau, 1990). T h e effect o n C Pa by decreasing NaC1 in the serosal m e d i u m ( B o u r d e a u a n d Lau, 1990) a n d n u m e r i c a l simulation o f the same p r o t o c o l are shown in Fig. 4. In the simulation, NaC1 in the serosal m e d i u m is r e p l a c e d by i m p e r m e a n t species (e.g., m a n n i t o l ) at time 5-15 min, with a linear r a m p o f 1 m i n o n each side. Final
Cytoplasm
Serosal
mV
+
-56.1
-56.1
0.0
10-:~cm~/cm2 ep. 1.38
v
mM
c~ Na K Cl Ca IMP OSM
140.0 5.0 149.0 2.0 0.0 296.0 C~a
Ca
Table IV contains the steady state solution for the closed-circuit solution. T h e m e m b r a n e potential o f the principal cell is - 5 6 . 1 mV, an i n t e r m e d i a t e value between ~bps a n d qbpm in the o p e n circuit case. This c h a n g e in the m e m b r a n e potential gives an increased Ca influx f r o m apical m e m b r a n e , e n d i n g u p with C Pa = 120 nM, h i g h e r t h a n 93.5 nM in the open-circuit case. T h e increase in CP a decreases the Na influx across the apical m e m b r a n e . T h e r e is an increase in the e l e c t r o c h e m i c a l g r a d i e n t for the entry o f Na as qbPm is increased. But this is n o t sufficient to offset the effect o f C Pa o n the Na channel. As a result, the n e t Na t r a n s p o r t across the apical m e m b r a n e is decreased. This leads to a decrease in Na c o n c e n t r a t i o n in cytoplasm. Concurrently, K conc e n t r a t i o n is increased. Cytoplasmic C1 c o n c e n t r a t i o n also increases, because the basolateral m e m b r a n e potential b e c o m e s less negative. Cell v o l u m e increases slightly as C1 enters the cell across the basolateral m e m brane.
216
T A B L E IV Equilibrium Solution for Closed-circuited Epithelium between Equal Ringer Solutions
Jk
120.0 • 10 3 mp
11.9 141.7 32.9 109.5 296.0 CaBP ~M
2.12 ps
140.0 5.0 149.0 2.0 0.0 296.0 C~i~
38.70
(~B r
523.5
ms pmol/s,cm2 ep.
Na K c~ Ca
766.0 -289.4 0.063 1.62 Na/K pump
Jk
766.0 -289.4 0.063 1.62 Cotransporter
0 0 0 0 Ca pump Na/Ca exchange
pmol/s.cm2 ep.
Na K C1 Ca ER flux
790.5 -527.0
-20.9 -20.9 -41.8
-0.61
1.83
v" J ~ k
vr.J~~
-(t.20
V"JPp~n,p
pmol/s"cm2 ep.
0.83
1.07
1.90
C~acl is 50, 30, 14, a n d 7 m M for the c o r r e s p o n d i n g curves. Steady state C eva increases as C ~aCl is d r o p p e d . CPa shows an o v e r s h o o t at the initial p e r i o d o f onset o n decreasing CNa b o t h experimentally a n d n u m e r i cally. This o v e r s h o o t is caused by the kinetic characteristic o f the Ca c h a n n e l s o n the ER. D e p e n d i n g o n h o w fast a n d the extent o f C NaC1 s changes, the o v e r s h o o t can either r e t u r n to the steady state m o n o t o n i c a l l y (C~acl = 50 m M ) , or show d a m p i n g oscillation a n d gradually r e t u r n to the high level o f state steady level (C~aCl = 14 m M ) . It is interesting to see how these details o f the e x p e r i m e n t are c a p t u r e d by the simulation. Cytoplasmic Ca increases little as Na c o n c e n t r a t i o n in the b a t h i n g m e d i u m drops 50 m M % f r o m its backg r o u n d level o f 140 mM. This m e a n s the principal cell can k e e p C ~a at almost a c o n s t a n t level d u r i n g physio-
A Model of Calcium Dynamics in CCT
2.5
7 2.0
O f~
O O
1.5
(mM)
1.0 t
0.5
'~,, 14 ',, -". . . . . . . . . . . . .
0.0
~: 30 .. . . . . ;.- 5 0 - - -. V - - .,
. 0
2
4
6
8
10
900. ,..,
800.
~,
BATH
NaCI,
150mM
12
14
16
18
20
(rain)
Time
I
BATH NaCl,
I
11~ ' / ~
500 400" 300
0
200. '
100-
o
a
0.10
~
0.08
-
150mM
600.
9+
Simulation with only C~a decreased but not C1 shows a similar profile of CPa, in a g r e e m e n t with experimental observations (Bourdeau and Lau, 1990). If NaC1 is decreased to 5 mM, C P rises to a toxic level of ~ 8 VLM. Further decrease o f serosal NaC1 leads to uncontrolled Ca increase; which is caused by the massive Ca entry f r o m the apical m e m b r a n e with a severely hyperpolarized potential a n d the almost complete annihilation of N a / C a exchange. U n d e r similar circumstance, e x p e r i m e n t a l results s h o w CP a stabilized in the OLM range. This difference between the m o d e l a n d the e x p e r i m e n t a l data can be explained by factors not included in the model. O n e possible m e c h a n i s m is t h r o u g h the downregulation of apical Ca entry caused by saturation or o t h e r feedback regulations. T h e o t h e r
y
Lumen NaCI
(raM) .......
7_0_ .........
~t 0 . 0 6
0
---"~
TIME
, 30
.. . . . 35 40
45
O fl
(rain)
FIGURE 4. Effect on C Pa of isosmotically decreasing NaCI in the bath. (a) Numerical simulation. Serosal NaCI is replaced by impermeant species (e.g., mannitol) at time 5-15 min, with a linear ramp of 1 min on each side. Final serosal NaCI is 50, 30, 14, and 7 mM. (b) Experimental result. Serosal NaC1 is replaced by mannitol completely in three different experiments in the cortical connecting tubule. Data from Bourdeau and Lau (1990) with permission.
o 0.02
logical changes in C NaCI- T h e increase in C p b e c o m e s m u c h m o r e substantial, as C~ao falls below 20% of the physiological range. T h e initial Ca accumulation is caused by the reversal of N a / C a exchange, transporting Ca into the cell (Fig. 2 c). This reversal in direction, however, does not necessarily hold for the behavior o f the system at the steady state. As Ca accumulates in the cytoplasm, the apical Na entry is inhibited due to the effect of CP a on the apical Na channel. As a consequence, cytoplasmic Na level drops a n d the apical m e m b r a n e hyperpolarizes. T h e hyperpolarization of apical m e m b r a n e increases the apical entry o f Ca, contributing to the secondary Ca accumulation. This secondary Ca accumulation, together with a decreased cytoplasmic Na level, decreases the Ca entry f r o m the basolateral m e m b r a n e via the N a / C a exchanger, or even reverse its direction of Ca transport. At steady state, the sustained elevation of Ca is m a i n t a i n e d mainly by increased Ca entry t h r o u g h the apical m e m b r a n e . TANG AND STEPHENSON
,
10 5
0.00 0
2
4
6
8
10
Time
b
6"
217
35
~ 0.04
~ -
.... , ..... .,.,/ ..... i , 0 5 10 15 20 25
16
18
20
2.5
2.0
~
1.5
Et
14
3.0
g 0 0
12
(re.in)
Lumen NaCI IOmM
1.o
.- ....
0.5 ~ o
0.0 -0.5
-1.0 0
i
!
I
I
i
I
I
i
|
2
4
6
8
10
12
14
16
18
20
Time (mi-) FtcuP.~ 5. Effect on C~a of isosmotically decreasing NaC1 in the lumen. In a, mucosal NaC1 is replaced by i m p e r m e a n t species (e.g., mannitol) at time 5-15 min, with a ramp of 1 min on each side. Final mucosal NaC1 concentrations are 70, 35, 10, and 5 raM. (b) Components of Ca fluxes across the basolateral m e m b r a n e with mucosal NaC1 at 10 mM; flux by N a / C a exchange (solid line), by Ca-ATPase (dashed line).
is t h r o u g h the insertion of new Ca-ATPase into the basolateral m e m b r a n e . Decreasing NaCl in the luminal medium. Alternatively to s decreasing C NAG1, one can lower C NaG1 m a n d observe its effect on C Pa" Experimental results obtained by Bourdeau and Lau show that such a m a n e u v e r actually decreases C Pa, i.e., its effect on C P is the opposite of decreasing C ~aC] (Bourdeau and Lau, 1990). O u r numerical simulations behave similarly. In Fig. 5 a the numerical results are shown. NaCI in the l u m e n is isosmotically replaced by i m p e r m e a n t species as in Fig. 4 to different levels. Steady state C Pa drops as C Nmcl is d r o p p e d . W h e n C N~cl " is d r o p p e d by 50%, C cPa drops f r o m 93.5 nM to 63.7 nM. In the e x t r e m e case, when C Nmac]is 5 mM, C CPa drops to a very low level of 8.1 nM. In experimental ctata, the d r o p of C Pa is less substantial, even when C Nmac]is at 0. T h e r e are two possible reasons for this discrepancy: (a) the e x p e r i m e n t a l dye (fura-2) is inaccurate for CP a below 100 nM; (b) low C cP might increase apical Ca conductance. O t h e r possibilities of intracellular Ca regulation may be also involved. As Na in the l u m e n drops, so does the Na flux into the cytoplasm f r o m the apical surface. T h e N a / K p u m p in the basolateral m e m b r a n e will still operate in the same m o d e as governed by Eq. 4, pushing the steady state C YPadownward. T h e decrease in C YPaalters the activity of N a / C a exchange, making it m o r e effective in transporting Ca out of the cell (Fig. 2 a). As m o r e C cP~ is transported out of the cell than that enters, C ~ decreases. In Fig. 5 b we show a typical case of the Ca m fluxes across the basolateral m e m b r a n e where C NaC~ is at 10 mM. T h e Ca outward flux carried by N a / C a exchange increases f r o m close to zero to a stable value of 2.0 p m o l / s 9 cm 2 ep. This increase causes the d r o p in C cPa, and the Ca flux carried by the Ca-ATPase decreases f r o m 1.2 p m o l / s 9 cm 2 ep. to 0.06 p m o l / s 9 cm 2 ep. correspondingly. T h e decrease in Ca-ATPase activity can not completely c o m p e n s a t e the net increase in N a / C a exchange activity. T h e net increase in Ca outward current is a b o u t 0.7 p m o l / s 9 cm 2 ep. Several o t h e r events occur to up-regulate C cPa u n d e r these circumstances, in addition to the downregulation of Ca-ATPase. O n e is the increase in apical Ca influx. This increase is induced by the increase in the Ca concentration gradient across the apical m e m b r a n e . T h e other is the feedback regulation of Ca on the apical Na channel. As C~a drops, the Na channel conductance increases, according to Eq. 21. This increase in Na entry will offset the net decrease in C Pa induced by decreasing C N'~ac~.This loop of interaction actually prevents the d r o p in c P~ and h e n c e a d r o p in C Pa t h r o u g h the action of N a / C a exchange. T h e changes in m e m b r a n e potentials also contribute to the upregulation of C P , which will be detailed later. All of these 218
a
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Time (rain) 6. Ca oscillations of the principal cell in Na free medium. Extracellular NaC1 is isosmotically replaced by impermeant species (e.g., TEA-CI) from 140 to 0 mM in 4.5-14.5 min, with a linear ramp of 0.5 rain from each end. Stable Ca oscillation is induced, as reported from experiments in Koster et al. (1993). (a) Ca in cytoplasm; (b) Ca inside ER (solid line) and buffered Ca inside ER (dashed line, scaled by a factor of 1/10). FIGURE
regulations are feedback controls, induced by the primary action, and acting to diminish its final effect, but the net effect on the system will still be a decrease in CcP~. Ca Oscillations in the Closed-Circuit Epithelium Koster et al. (1993) r e p o r t e d that in the cultured principal cell of CCT, C P can show an oscillatory response, o induced by lowering C N~ below 5 mM. T h e oscillation frequency is ~ 0 . 6 4 / m i n and the p e a k is a r o u n d 600 nM. Koster's e x p e r i m e n t was d o n e with cultured cells in a h o m o g e n e o u s m e d i u m , where the distinction between mucosal and serosal c o m p a r t m e n t s no longer exists. T h e mathematical description of the cultured cell
A Model of Calcium Dynamics in CCT
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FIGURE 7. Other intracellular responses of the principal cell in Na-free medium. Extracellular Na profile is the same as in Fig. 6. (a) Membrane potential; (b) electrolyte concentrations; (c) Ca fluxes carried by Ca-ATPase and Na/Ca exchange; (d) fraction of the IP3R channels in the activated (/~c) state.
is actually the same as for the closed-circuit epithelium, since ~)pm = qbps, and the o t h e r c o m p o n e n t s of the system in Eq. 20 are u n c h a n g e d . T h e m e m b r a n e potential of the principal cell (qbp) is d e t e r m i n e d by the comb i n e d apical a n d basolateral m e m b r a n e fluxes as in Eq. 22. As a principal cell is d e t a c h e d f r o m the epithelium a n d put in a culture of the same ionic composition, apical Ca entry will increase substantially as the apical m e m b r a n e b e c o m e s m o r e hyperpolarized. To maintain a p r o p e r basal level of cytoplasmic Ca, the principal cell can either increase the a m o u n t of basolateral Ca-ATPase, or down-regulate the apical Ca-conductive channels. In the simulation, we increase the maximal ps basolateral Ca rate P ..... Ca f r o m 1.9 p m o l / s 9 cm '~ ep. to 2.55 p m o l / s 9 cm z ep. In the simulation, the decrease in C N~a is isoosmotically replaced by i m p e r m e a n t species coupled to C1 219
TANG
AND
STEPHENSON
ion. As CNa~is completely replaced, CPa oscillation begins immediately (Fig. 6), and stabilizes after the first several spikes. Peak C P is ~1.25 txM and frequency is a b o u t 0.5/min. T h e oscillation in CPa is also reflected in phase by the profile of CaBP, Ccar , and C a B r (Fig. 6 b). T h e r e is excessive release f r o m the ER when the oscillation begins, and excessive uptake of Ca as C Noa is restored (Fig. 6 b). In the cytoplasmic Ca profile, the first spike is substantially higher than the o t h e r spikes. As C N~a is restored, C Pa shows a slight dip before it returns to its steady state, which is caused by reversal of N a / C a o exchange as CNa is withdrawn (Fig. 7 c). Both of these features have b e e n observed in e x p e r i m e n t a l results (Fig. 1 in Koster et al., 1993). T h e primary source for the periodic C P~ increase is the periodic release f r o m the internal Ca store, ER, not by periodic entry of Ca f r o m extracellular m e d i u m . As
shown by Fig. 6 b, Ca released f r o m the ER in a period is sufficient to explain the maximal c P~ spike. T h e transient spiking is an internal characteristic of the cytoplasm-ER network, primarily caused by the biphasic property of the IP3R. W h e n C Pa is low, C CPa induces IP3R to open, causing further Ca release f r o m the ER (positive feedback); when C P is high, it inhibits IP~R f r o m o p e n i n g (negative feedback). From a kinetic point of view, the positive feedback loop has a faster time scale than that of the negative feedback loop. In a certain range of Ca in the cytoplasm-ER network, the positive feedback loop (Ca induces Ca release) and the
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The effect of decreasing extracellular Na to different
concentrations. Extracellular NaCl is replaced by TEA-CI or choline-C1 from 140 to 20 mM linearly in 2-3 min; held at 20 mM in 3-8 rain; and decreased further to 10 mM in 8-13 min, 5 mM in 13-18 min, and 2.5 mM in 18-25 rain. (a) Free Ca (solid line) and buffered Ca (dashed line, scaled by a factor of 1/10) in the cytoplasm; (b) Ca fluxes carried by the Ca-ATPase (dashed line) and the Na/Ca exchange (solid line). 220
negative feedback loop (Ca inhibits Ca release) work h a n d in h a n d to generate the oscillatory pattern. The contribution from a decreased C~a is to increase the net a m o u n t of Ca in the ER-cytoplasmic network. As the total a m o u n t of Ca in the cytoplasm-ER network increases to a certain value, oscillation begins. Fig. 7 shows the cellular events in Na free m e d i u m . W h e n C~a is d r o p p e d , m e m b r a n e potential is hyperpolarized by m o r e than 20 mV. This hyperpolarization is caused by the withdrawal of the sodium gradient across the m e m b r a n e . Extracellular Ca enters the cell due to this hyperpolarization. This increase in Ca influx is the main source for the increased Ca in the cytoplasm-ER network that drives the Ca oscillation. A second source of initial Ca accumulation is through the N a / C a exchanger. As C ~a is depleted, CPa decreases to ahnost zero. These changes in the Na concentration across the cell m e m b r a n e at first e n h a n c e Ca uptake by the cell t h r o u g h N a / C a exchange (CPa > C ~ ) . At steady state, however, N a / C a exchange almost stops functioning because of the disappearance of the Na gradient. C o m p a r e d with its earlier role of actively transporting Ca inwards, the inhibition of N a / C a exchange actually decreases the net Ca accumulation inside the cell. Therefore, the contribution of the N a / C a exchanger to the steady state intracellular Ca is, counter-intuitively, negative. Although C P regulates Na permeability in the apical portion of the m e m b r a n e , we see no detectable potential fluctuations in Na free m e d i u m (Fig. 7 a). Since Na concentrations are zero in both intracellular and extracellular media, changes in Na conductance will not affect m e m b r a n e potential. For o t h e r cellular responses, C p increases f r o m 142 to 153.8 raM; C1 leaves the cell slowly, causing cell volume to decrease slightly (result not shown). As argued before, the kinetic properties of the IP3R O m a n d a t e C Pa oscillation over a certain range of C Na' In our model, Ca starts to oscillate when C Noa is below 10 raM, and continues until C ~ is zero. Fig. 8 shows C Pa 9 o dynamics when C ~a is sequentially decreased. As C N~ first drops from 140 to 20 mM, cP~ increases slightly from 120 nM to a steady state of 181 nM. T h e small spike within 2-4 m i n in front of the oscillations is typical and has b e e n seen in Fig. 4. As C~, reaches 10 mM, C CPa shows oscillations with decreasing amplitude, which will eventually settle down to a stable oscillation with peak C Pa level at 508 nM and at a frequency of 0.58/min. Further decrease in C N~a increases the oscillation amplitudes, as shown in Fig. 8 with C ~ = 5 mM in 13-18 min and with C~a -- 2.5 mM in 18-24 min. O For C N~ = 5 mM, the oscillation frequency is 0 . 5 3 / m i n a n d the amplitude 1.18 txM; for C N~ ~ = 2.5 mM, the frequency is 0 . 5 1 / m i n and the amplitude 1.25 IxM. T h e oscillation frequency changes only slightly in the oscil-
A Model of Calcium Dynamics in C C T
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lation r a n g e as C Na changes, in a g r e e m e n t with experim e n t a l observations (Koster et al., 1993). T h e r e are, however, significant increases in the a m p l i t u d e as C ~, o decreases. W h e n CNa is fixed at below 10 raM, CPa oscillations are stable over time a n d r e t u r n to the basal level b e t w e e n cycles. With the restoration o f Na in the media, Ca oscillation disappears immediately. cP~ buffers b i n d to Ca quickly a n d oscillate in phase with C P (Fig. 8 a). T h e buffered Ca in the cytoplasm is ~ 2 0 times the free Ca over all o f the oscillatory phase. T h e cytoplasmic Ca buffers are n o t saturated as Ca oscillates; otherwise the linear correlation between free Ca a n d b u f f e r e d Ca will be violated. For Ca oscillation to
occur, the positive a n d negative f e e d b a c k loops via IP3R have to be triggered. Especially, free cytoplasmic Ca has to reach a certain critical value for the positive f e e d b a c k l o o p to function. T h e buffered cytoplasmic Ca level has to reach its c o r r e s p o n d i n g level, in the equilibrium o f b i n d i n g a n d u n b i n d i n g with the free Ca. Since the buffered Ca represents the majority o f Ca in cytoplasm, for Ca oscillation to occur, the a m o u n t o f n e t Ca entry n e e d e d is d e t e r m i n e d mainly by the Ca buffering capacity in the cytoplasm a n d that inside the ER. As C Noa is r e d u c e d , the Ca flux carried by the N a / C a e x c h a n g e varies. Since the f u n c t i o n o f the N a / C a exc h a n g e r is d e t e r m i n e d by multiple factors, it is h a r d to
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FIGURE 9. Response of the principal cell to decreasing extracellular Ca with feedback regulation. The feedback regulation of Ca on the apical Na permeability is responsible for maintaining intracellular Ca homeostasis. Ca in both the lumen and the bath are reduced sequentially from 2 mM (0-2 rain) to 1 mM (3-11 rain) and then to 0.1 mM (12-20 rain), with 1 min ramp in each case. Membrane potentials and intracellular Na and K concentrations adjust slightly in response (a and b). Cell can retain a relatively high level of C~a even at very low extracellular Ca (0.1 raM) (c), due to the reversal of the Na/Ca exchange. (d) shows the components of J~:aflux across the basolateral membrane. 221
TANG
AND
STEPHENSON
predict which way the Ca flux will vary. T h e complex response pattern of the N a / C a exchanger is shown in Fig. 8 b. Initially, the N a / C a exchanger is transporting Ca inward to the cytoplasm. When C Noa is d r o p p e d to 10 mM, this flux becomes positive, i.e., transporting Ca out of cytoplasm. As CPa oscillates, N a / C a exchange can show oscillatory changes in direction, as shown here in the 8-13-min region; or it can stay at the negative domain (for C Noa < 2.5). We can conclude here that N a / C a exchange is not essential for the Ca oscillation; its role is secondary compared with the change in m e m b r a n e potential. In addition, it is not necessary for N a / C a exchange to alternate its direction during Ca oscillations. The basolateral Ca-ATPase activity is increased significantly as C ~ is dropped, mainly to p u m p out the extra Ca entry caused by the hyperpolarized membrane. In the oscillatory mode, the basolateral Ca-ATPase
operates at a close to maximal rate. Especially, it shows saturation at the peak of C P spike. This saturation of the Ca-ATPase is one characteristic of Ca oscillations observed in the simulation. Because of the relative high density o f the basolateral Ca-ATPase and a low basal level of cytoplasmic IP~, the basolateral Ca-ATPase can equilibrate Ca release from ER through the IP3R, thus maintaining an equilibrium cytoplasmic Ca at relatively higher levels, c o m p a r e d to other cell types where the Ca-ATPase is not present or at low levels. Therefore, for Ca oscillation to occur u n d e r this experimental protocol, the basolateral Ca-ATPase has to be partially saturated. It is, however, not a necessary condition for intracellular Ca oscillations u n d e r other experimental conditions, nor does it hold for other epithelial cell types. Many manipulations, such as blocking the baso]ateral Ca-ATPase, increasing IP~ cytoplasmic Ca level, or changes on the ER Ca p u m p or m e m b r a n e leakage,
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