276 Bioehimica el Biophysica Acta, 464 (1977) 276--286
(c) Elsevier/North-Holland Biomedical Press
BBA 77574
TRANSPORT ACROSS EPITHELIA A KINETIC EVALUATION GENE BARNETT a* and VOJTECH LI(~KO b a Deparmenl of Pharmaceutical Chemistry, School o f Pharmacy and b Cardiovaseular Research Instilule, School o f Medicine, Universily o f California, San Francisco, Calif. 94143 (U.S.A.)
(Received July 19th, 1976)
Summary C o m p a r t m e n t a l analysis of t h r e e m o d e l s for solute t r a n s p o r t across epithelial tissue is p r e s e n t e d . T h e simplest m o d e l describes o n l y o n e tissue c o m p a r t m e n t , t h e s e c o n d i n c o r p o r a t e s t h e n o t i o n o f a p o r e as a parallel p a t h w a y and t h e t h i r d m o d e l i n t r o d u c e s a serial c o m p a r t m e n t c o r r e s p o n d i n g t o non-epithelial p o r t i o n s o f t h e tissue. E x p e r i m e n t a l d a t a were o b t a i n e d , using a m o d i f i e d Ussing a n d Z e r a h n t e c h n i q u e ( ( 1 9 5 1 ) A e t a Physiol. S t a n d , 23, 1 1 0 - - 1 2 7 ) , for salicylate t r a n s p o r t across rat j e j u n u m in vitro and a n a l y z e d in t e r m s o f t h e s e three m o d e l s . T h e c o n c l u s i o n s b a s e d solely on t h e m a t h e m a t i c a l analysis o f this r a t h e r simple e x p e r i m e n t are: the tissue is n o t a h o m o g e n e o u s p e n e t r a t i o n barrier as o f t e n c o n s i d e r e d . T r a n s p o r t is n o t limited b y u n s t i r r e d layers either at t h e tissue surfaces or w i t h i n t h e tissue itself. Salicylate is n o t passively transp o r t e d . Parallel t r a n s p o r t p a t h w a y s d o exist.
Introduction Various m o d i f i c a t i o n s of the m e t h o d of Ussing and Z e r a h n [1] (Schultz and Z a l u s k y [ 2 ] ) have b e e n e x t e n s i v e l y used in t h e past d e c a d e for t h e s t u d y o f t h e t r a n s p o r t o f a v a r i e t y of s u b s t a n c e s across tissues such as frog skin, t o a d bladder, m a m m a l i a n intestine and m o u s e skin. Since this is an in vitro t e c h n i q u e , t h e fluxes in either d i r e c t i o n s can be o b t a i n e d . In general, o n e observes (Fig. 1) a f t e r a t r a n s i e n t p e r i o d , a linear rise o f t h e c o n c e n t r a t i o n o f s u b s t a n c e t r a n s p o r t e d t h r o u g h t h e tissue [3--7 ]. F r o m these o b s e r v a t i o n s the s t e a d y - s t a t e flux o f t h e s u b s t a n c e is d e t e r m i n e d b y the slope a o f t h e linear p o r t i o n o f the curve. T h e d i r e c t i o n a l flux r a t i o so d e t e r m i n e d is a useful quantit:y t o c h a r a c t e r i z e t h e t r a n s p o r t s y s t e m . A n o t h e r * Present address: National Institute of Drug Abuse, Rockville, Md., U.S.A.
277
•
SLOPE(7" - ~ 7 Z > D,~, Ou Z T
TIME
Fig. 1. T y p i c a l c u r v e of s o l u t e t r a n s p o r t e d i n t o the r e c e i v i n g ( s i n k ) c o m p a r t m e n t l i n e a r p o r t i o n o ( f l u x ) a n d i n t e r c e p t of t h e e x t r a p o l a t e d s t r a i g h t line r (lag t i m e ) .
w i t h tile slope o f t h e
observable quantity is the lag time r. While a curve such as shown in Fig. 1 is usually described by the classical equation for diffusion applied to transport across a h o mo g ene ous barrier (cf. ref. 2) [8], we will show that such an equation has only limited validity. A closer look at those curves which allow one to observe b o t h the transient and the steady-state behavior suggests that there might be more parameters that one can determine, thus permitting a bet t er understanding of the details involved in the absorption process. A com p art m ent al approach seems to be a natural technique applicable to this end. In the following we review a nuinber o f simple models with which we analyze the experimental data report ed here. C o m p a r t m e n t a l models of absorption
(A) Model of tissue compartment A schematic diagram of the model is shown in Fig. 2. In the diagram M represents the a m o u n t of the transporting solute in the mucosal bathing solution, S is the a m o u n t in the sero~al solution and T is the a m o u n t of solute in the tissue. Th e Greek characters represent the corresponding rate coefficients, which are assumed constant, in units of time -~. The system is defined in terms o f linear differential equations as follows
7
Fig. 2. M o d e l A: T i s s u e ( T ) c o m p a r t m e n t . See t e x t f o r d e t a i l e d d e s c r i p t i o n . Fig. 3. M o d e l B: T i s s u e ( T ) w i t h p o r e . See t e x t f o r d e t a i l e d d e s c r i p t i o n .
7
278 M = - - a M + fiT f = a M - - (¢~ + ~,)T + 5S
(1)
= 3 ` T - 5S. E x p e r i m e n t a l l y the t r a n s p o r t can be d e t e r m i n e d in either d i r e c t i o n , t h u s , we have t w o sets of solutions t o t h e e q u a t i o n s and t w o sets of c h a r a c t e r i s t i c slopes a n d lag t i m e s f o r the t w o e x p e r i m e n t a l curves. F o r t r a n s p o r t f r o m M to S, we define M as t h e source c o m p a r t m e n t and S as t h e sink c o m p a r t m e n t b y setting M = 0, 5S < < 7T. T h e a m o u n t o f solute in the sink c o m p a r t m e n t S is t h e n given b y the t r a d i t i o n a l e q u a t i o n S = o s ( t - - rs) + Osrs e x p ( - - t / ~ s )
(2)
T h e f l o w into S is c h a r a c t e r i z e d b y t h e slope os and t h e lag t i m e rs, viz. =
~3"
and
1
rs -
(3)
N o t e t h a t t h e e x p o n e n t i a l c o e f f i c i e n t is precisely the r e c i p r o c a l o f T~. F o r t r a n s p o r t in t h e reverse d i r e c t i o n (S = 0, a M < < / 3 T ) the increase as a f u n c t i o n of t i m e for t h e a m o u n t o f solute M is l~I = Om(t
--
T m
)+
OmT
m
exp(--t/r m)
(4)
T h e c o r r e s p o n d i n g slope Om and lag t i m e rn~ are Om-
and
Tm
(5)
For this model then we find the ratio of slopes r~j, the directional flux ratio, and the ratio of lag times r r are r . = ~a7
and
rr =
1
(6)
( B ) M o d e l o f tissue w i t h p o r e T h e s c h e m a t i c d i a g r a m o f this m o r e c o m p l i c a t e d m o d e l is given in Fig. 3 w h e r e , in a d d i t i o n t o t h e s y m b o l s o f Fig. 1, t h e p o r e is c h a r a c t e r i z e d b y t h e rate c o e f f i c i e n t s ~? and $. T h e kinetic e q u a t i o n s are n o w
= --(~ + r~)M + fiT + ~S = a M - - (/3 + 7 ) T + 5Z = ~M + 3`T-
(7)
(5 + ~)S
F o r t r a n s p o r t f r o m M to S t h e a s s u m e d source and sink c o n d i t i o n s are given b y M = 0, 5S < < 3`T, ~S < < r~M. T h e t r a d i o n a l e q u a t i o n no longer h o l d s in its original f o r m as t h e solute increase in S is n o w given b y t h e following e q u a t i o n S = (~s(t - - -rs) + Os~-S e x p [ - - ( f i + 3`)t]
(8)
w h e r e t h e e x p o n e n t i a l c o e f f i c i e n t (~ + 3') is n o t equal to t h e r e c i p r o c a l o f t h e lag time. In this m o d e l t h e slope and lag t i m e are given b y t h e following relations
279
G-/3+Ta7 +~
and
T~
_
1
fi+7
_
~
~(/3+7)+a7
(9)
In these m o r e c o m p l i c a t e d expressions for Os and Ts we can i d e n t i f y the tissue and pore c o m p o n e n t s since the first t e r m in each case is the same e x p r e s s i o n t h a t was o b t a i n e d in m o d e l A (tissue o n l y ) , Eqn. 3, while t h e second t e r m is the c o n t r i b u t i o n o f the pore. We see t h a t t h e i n t r o d u c t i o n o f a p o r e results in an additive c o n t r i b u t i o n t o the slope and a decrease in the lag t i m e as c o m p a r e d to t h e previous m o d e l . F o r t r a n s p o r t in the o p p o s i t e d i r e c t i o n {S = 0, aM < < fiT, T~M < < ~S) the expression for M is analogous to Eqn. 8 and the resulting slope and lag t i m e are
am
-
7+~
+~
and
7 m
-
7 +;3
~(7 +/3) +8/3
(10)
F o r this m o d e l the slope ratio and lag t i m e ratio are considerably m o r e complicated, viz.
°
8/3+ ~(v +/3)
and
~v 8/3 + ~(v +/3) r~ = - ~ . ~7 + ~(/3 + 7)
(11)
(C) Model o f epithelial and extraepithelial compartments with pore T h e schematic diagram for the system is p r e s e n t e d in Fig. 4. This m o d e l is an e x t e n s i o n o f m o d e l B (tissue with p o r e ) b y including an additional c o m p a r t m e n t X with e x c h a n g e rate coefficients p and v. However, the i n t e r p r e t a t i o n is s o m e w h a t altered as E d e n o t e s o n l y the epithelial cellular c o m p a r t m e n t and X d e n o t e s the extracellular c o m p a r t m e n t o f the c o m p l e t e tissue. With this m o r e extensive m o d e l the two-sidedness o f the tissue is n o w m o r e a c c u r a t e l y represented. T h e general system of linear kinetic e q u a t i o n s f o r this m o d e l is t h e n /~/= --(a + ~/)M + fiE + ~X
/~=aM--(/3+7)E+SX 2 = r~M + 7 E -- (8 + } + p ) X
(12)
~9 = p x For the specific case o f t r a n s p o r t M to S the e.quations simplify slightly after i n t r o d u c t i o n o f the source and sink c o n d i t i o n s (M = O, vS < < pX), b u t the solution o f these e q u a t i o n s is still m u c h m o r e c o m p l i c a t e d t h a n was the ease o f the previous models. N o w t h e r e are t w o e x p o n e n t i a l t e r m s in the f u n c t i o n for S, describing t h e a c c u m u l a t i o n o f solute, i.e. S = O s ( t - - T~) + O~[T~lexp(~t) + T~2exp(k2t) T h e slope o~ o f the straight line p o r t i o n o f t h e S curve is given b y
(13)
280
M ;E;x
s
Fig, 4. Model C: Epithelial (E) and extraepithelial (X) c o m p a r t m e n t s with pore. See text for detailed description.
_
+ v)(6
+ ~ + #) -
%j
and the lag time, which can be expressed as a sum, is
1
[(~+y +~ +~ +~)(~ +v)]
The c o m p o n e n t s of r s are defines as
x2
[_!:
]
and
where Xl and X2 are bot h real negative solutions to the characteristic equation corresponding to the system of differential equations, i.e. Eqn. 12 after source and sink conditions are introduced, viz. ~,2 J- (~ q- y q- 5 "{" ~ "b //)~k q- (~ q- 7 ) ( 5
+ ~ "{-~/)--'~ = 0
(17)
In this model, the expressions for the slope and lag time are, of course, even more complicated. Nevertheless, a similar separation of various c o m p o n e n t s is obvious from Eqns. 9, 14 and 15. The slope os has the tissue plus pore components which are now multiplied by a "shielding f a c t o r " due to the extracellular c o m p a r t m e n t X, which has the value less than unity (cf. the second factor in Eqn. 14). The lag time has a different "shielding f a c t o r " (again due to the X c o m p a r t m e n t ) which modifies only the tissue c o m p o n e n t of r~ (cf. the factor in brackets in Eqn. 15). For transport in the reverse direction the solution to Eqn. 12 is again analogous to Eqn. 13 but the as and r~ are replaced by
+ ~)---~}~j l]) ] O'n= [V~+~ + ~] [(~ + ~ + u(7 ~)(V
(18)
and
1 [-(~+7+5+~+p)(~,~f)j
~
(19)
281
In the formulas c o r r e s p o n d i n g to rml and rm2 the last t e r m in brackets in Eqn. 16 is replaced b y tl~e last t e r m o f Eqn. 19. Simplification of model C. Various c o n d i t i o n s for t r a n s p o r t across epithelia m a y exist that i n t r o d u c e some simplification o f m o d e l C and t h u s allow a b e t t e r intuitive view o f the t r a n s p o r t system as described b y this r a t h e r complicated model. T h e solutions X1 and X2 o f the characteristic e q u a t i o n , Eqn. 17, which a p p e a r as the e x p o n e n t i a l coefficients in Eqn. 13 are simplified if the c o n d i t i o n holds t h a t 5 < < (~ + p), because t h e n the t e r m 75 is negliglible. This w o u l d follow for a solute, passively t r a n s p o r t e d , having a p a r t i t i o n c o e f f i c i e n t m u c h d i f f e r e n t t h a n u n i t y w h e r e the lipid exit r o u t e f r o m the extracellular c o m p a r t m e n t X is m u c h slower t h a n the t w o a q u e o u s routes. T h e n ~, -~ --(/3 + 7)
and
~2 -~ --(~ + P)
(20)
where S(t) is still a t w o - e x p o n e n t i a l f u n c t i o n o f course, and has a lag time, e.g. rs, given b y ~ 1 r ~ +. 1 . -~(~+~)+~+~+~ . . . . .
.
(21)
This c o n d i t i o n applies regardless o f the relative m a g n i t u d e s o f X, and X2, i.e. w h e t h e r t h e y m'e equal or d i f f e r e n t . (a) Thick X compartment. T h e rate coefficients a, fl, 7, 5, ~, ~, p and v can be expressed as A P / V w h e r e A is the tissue area across which t r a n s p o r t occurs, P is the tissue p e r m e a b i l i t y which can include b o t h active and passive c o m p o nents, and V is the v o l u m e o f t h e driving c o m p a r t m e n t . In t h e case of t h e extracellular c o m p a r t m e n t X, which is m u c h larger t h a n the epithelial c o m p a r t m e n t E , the respective volumes are Vx > > V~. It t h e n follows t h a t since ~ and ~, contain V~ in the d e n o m i n a t o r and ~ and/~ are inversely p r o p o r t i o n a l to V~,
m a y hold and t h e n f r o m Eqn. 16 v~,~ 0
and
1 r~2 ~ + p -
(22)
thus the f u n c t i o n describing S will behave as a o n e - e x p o n e n t i a l f u n c t i o n , Eqn. 2. As a c o n s e q u e n c e o f this c o n d i t i o n , X being a very large p o r t i o n o f the tissue, even if the tissue does have t r a n s p o r t via a pore p a t h w a y as well as across the epithelial p a t h w a y the process is describable b y m o d e l A (tissue with no pore) and the e x p o n e n t i a l c o e f f i c i e n t characterizes o n l y t h e extracellular comp a r t m e n t X and contains no i n f o r m a t i o n a b o u t the epithelial p o r t i o n o f the tissue. (b) Thin X compartment. This is the o p p o s i t e o f the situation discussed in (a) as we n o w have V~ < < V e and ~+~/ 1/X forces us to e x c l u d e m o d e l B since, a c c o r d i n g to Eqns. 9 a n d 10 r m u s t be less t h a n 1/X = 1/(/3 + "),). T h e r e f o r e , while n e i t h e r m o d e l A n o r m o d e l B is s a t i s f a c t o r y , we w o u l d like to investigate t h e Bossibility t h a t m o d e l C is in f a c t a valid d e s c r i p t i o n o f this s y s t e m . In an e f f o r t to resolve the d i f f i c u l t y o f n o t being able to a n a l y z e the experim e n t a l d a t a b y m o d e l C we have carried o u t a series o f digital s i m u l a t i o n s o f e x p e r i m e n t s utilizing Eqn. 13, g e n e r a t i n g curves S(t) similar to Fig. 1 b y intro-
284
ducing a normal scatter into the c o m p u t e d data. For these simulations we have c h o s e n values of ~, ~, 7, 5, ~, ~, 12 and u, non-negative parameters, such that o and r w o u l d be characteristic of the observed values by linear or non-linear analyses (i.e. o ~- 0.1 n m o l • cm -2 • min -~ and T ~ 15 rain). Obviously there is no u n i q u e s o l u t i o n to this problem but through extensive searching we can classify t w o types of solutions, namely such that Xl > > X2 or the X~ ~ X2. These two classes of solutions can be exemplified by the t w o sets o f parameters included in Table III * When the simulated data with 0% standard deviation of the normal scatter were analyzed by m o d e l C, o n l y in the case of X~ ~ k2 could we recover the "experimental" parameters used for data generation. Using scattered data with 5% standard deviation no fitting was obtained. This was also the case with our efforts reported above to attempt an analysis o f the experimental data in Table II by m o d e l C. Clearly, the curves obtained by the t y p e o f experiment reported here do n o t contain sufficient i n f o r m a t i o n to allow one to estimate the experimental parameters for m o d e l C. Nevertheless, the analysis by m o d e l B o f the data generated by simulation of model C ( s h o w n in Table IV) was carried o u t more successfully. For each case reported in Table IV, the values are the averages of five simulations with data scattered at the 5% standard deviation level. In the case of k~ > > X2, we f o u n d that rs < 1/Xs which was statistically significant at the level of P < 0 . 0 0 5 . This is in contradiction to the results of the analysis of experimental data as reported in Table II. However, the analysis of the case X~ ~ k2(kl ~> X2) we
III
TABLE TWO
EXAMPLES
PARAMETERS
Model
parameters
OF OF
VALUES
MODELC
OF
MODEL
LEADING
PARAMETERS
TO o s=0.1
AND
mnol'cm
a r e i n u n i t s o f m i n -1 . F o r u n i t s o f " S i m u l a t e d
Parameters
kl
Model c( 7 5
/1
> ; > X2
i,
"SIMULATED
"min -I
andr s=
experimental"
EXPERIMENTAI," 15 min
parameters
s e e T a b l e II.
X 1 ":k 2
0.15
• 10 -3
0.16
0.3
• 10-1
0.12
0.3 0.3
• 10 -! • 1 0 -1
0.13 0.175.
0.15
• 10 -4
0.36
0.3
• 10 -2
O.55
10 -l
0.65 0.28
3.0 0.15
-2
• 10 -3
10 -1 - 10-4 • 10-2 10 -I - 10-3 •
"Simulated experimental" 0.0895
(l s
14.25 0.0491
7s 1 7s2 Ts ks 1 ks2
14.30 ---0.0597 --3.0333
* By restricting tion
0.0982 16.96 --1.035 15.93 ---0.075 --0.263
the parameters further to simulate o f t h e t y p e h I >> X2 . As will b e s e e n b e l o w ,
a passive transport lnodel we arrived at the solut h i s is n o t a v a l i d d e s c r i p t i o n o f t h e s y s t e m .
285 TABLE
IV
"SIMULATED EXPEII, IMENTAL" PARAMETERS ESTIMATES OF CURVES W1TIIMODEL PARAMETERS OF TABLE III TO MODEL B
BY
FITTING
SIMULATED
F o r u n i t s see T a b l e II. )'1 > > c7s rs 1/A s
0.0898 14.6 17.0
~2 + 0.0079 + 3.4 ~ 4.6
h i - X2 0.1061 19.1
+- 0 . 0 0 6 1 -+ 2 . 3
la.0
+ 2.8
found that rs > l/ks at the level of significance P < 0.005. This observation is fully compatible with the results of our non-linear analysis of the experimental data (Table II) using the functional form of model B as a device to test the data for compatibility with models A and B. As m ent i oned above, the observation that rs ¢ l/k s allows us to exclude model A as a valid description of this system. F u r t h e r m o r e , as noted above, the fact that 7s > l/ks allows us to exclude model B as well. Therefore, on the basis o f these observations, we conclude that model C is the minimal compatible model that may be appropriate for a description o f this system of intestinal anion transport. This t y p e of analysis has also been applied to the transport of m e t h o t r e x a t e across mouse skin where model B was f ound appropriate to describe the system and to rationalize the pH dependence of solute flux. [10]. Conclusions Due to the mathematical nature of the foregoing analysis we enunciate our conclusions and related implications in detail here. We wish to stress that these findings are based solely on the mathematical analysis of the experi m ent and therefore stand as i nde pe nde nt evidence to those biochemical and physicochemical studies which lead to similar conclusions. (1) The tissue is not a hom oge ne ous barrier to solute transport. This follows from our finding that model A is not a valid description for the sytem of salicylate transport across rat jejunum. As a consequence, analysis via the classical diffusion equation [8], which forces the system to obey model A, does not extract complete information when applied to this t y p e of experiment. (2) Transport is not rate limited by unstirred layers either at the tissue surfaces or within the tissue. This t y p e of process would be described by model A. While we have suggested that model C may be appropriate, the considerations leading to Eqn. 22 make it clear that the X c o m p a r t m e n t is not " t h i c k " as that would again lead back to an A t y p e model. (3) Multiple transport pathways exist for this system. The finding that model C is the simplest possible description of the tissue requires that there are at least parallel transport pathways. (4) Salicylic acid is not passively transported. The simulation studies could n o t r ep r o d u ce the experimental observations when the parameters were restricted to obey a passive model. Clearly this type of analysis stands self-justified in view of these results. To
286
make these observations directly would require a considerable a m o u n t of experimentation b e y o n d ~he rather simple procedure applied here. The value of the co mp ar tm e nt a l analysis is further enhanced as it suggests a more complete definition of the experinlent (and tissue) through additional measurements that would be desirable. In particular, it is advantageous to do flux measurements in both directions across the tissue. Measurements of the am ount of solute in the tissue, especially as a function of time, would allow further definition of the parameters. If techniques to determine cellular and extracellular amounts as a function of time were developed it could then allow one to obtain all the parameters introduced by model C. Finally, this t ype of analysis has a broad applicability to many experimental tissue and solute systems. Acknowledgements This work was supported by N.I.H. Grant GM 16496 to Professor L.Z. Benet in whose laboratory the experimental work was performed. References 1 2 3 4 5 6 7 8 9 10
Ussing, ft. and Z e r a h n , K. ( 1 9 5 1 ) A e t a Physiol. S e a n d . 23, 110- 127 S e h u l t z , S. a n d Z a l u s k y , R. ( 1 9 6 4 ) 3. G e n . Physiol. 47, 5 6 7 - - 5 8 4 Q u a y , J. a n d A r m s t r o n g , W. ( 1 9 6 9 ) A m . J. Physiol. 2 1 7 , 6 9 4 7 0 2 Field, M., F r o m m , D. a n d MeColl, L ( 1 9 7 1 ) A m . J. Physiol. 220, 1 3 8 8 - - 1 3 9 6 J a c k s o n , M., Shiau, Y., Bane, S. a n d F o x , M. ( 1 9 7 4 ) J. G e n . Physiol. 63, 187 213 B a r n e t t , G., ttui~ S. and B e n e t , L. ( 1 9 7 6 ) S u b m i t t e d to A m . J. Physiol Wallaee, S., R u n i k i s , J. a n d S t e w a r t , W. ( 1 9 7 6 ) d. P h a r m . P h a r m a e o l . , in the press J o s t , W. ( 1 9 5 2 ) D i f f u s i o n in Solids, L i q u i d s a n d Gases, p. 37, A c a d e m i c Press, N e w Y o r k D i x o n , W. ( 1 9 7 4 ) BMD B i o m e d i c a l C o m p u t e r P r o g r a m s , U n i v e r s i t y California Press, Berkeley Walaee, S., B a r n e t t , G. and L i e k o , B. ( 1 9 7 6 ) S u b m i t t e d to J. P h a r m a e o k i n . B i o p h a r m .
(c) Elsevier/North-Holland Biomedical Press
BBA 77574
TRANSPORT ACROSS EPITHELIA A KINETIC EVALUATION GENE BARNETT a* and VOJTECH LI(~KO b a Deparmenl of Pharmaceutical Chemistry, School o f Pharmacy and b Cardiovaseular Research Instilule, School o f Medicine, Universily o f California, San Francisco, Calif. 94143 (U.S.A.)
(Received July 19th, 1976)
Summary C o m p a r t m e n t a l analysis of t h r e e m o d e l s for solute t r a n s p o r t across epithelial tissue is p r e s e n t e d . T h e simplest m o d e l describes o n l y o n e tissue c o m p a r t m e n t , t h e s e c o n d i n c o r p o r a t e s t h e n o t i o n o f a p o r e as a parallel p a t h w a y and t h e t h i r d m o d e l i n t r o d u c e s a serial c o m p a r t m e n t c o r r e s p o n d i n g t o non-epithelial p o r t i o n s o f t h e tissue. E x p e r i m e n t a l d a t a were o b t a i n e d , using a m o d i f i e d Ussing a n d Z e r a h n t e c h n i q u e ( ( 1 9 5 1 ) A e t a Physiol. S t a n d , 23, 1 1 0 - - 1 2 7 ) , for salicylate t r a n s p o r t across rat j e j u n u m in vitro and a n a l y z e d in t e r m s o f t h e s e three m o d e l s . T h e c o n c l u s i o n s b a s e d solely on t h e m a t h e m a t i c a l analysis o f this r a t h e r simple e x p e r i m e n t are: the tissue is n o t a h o m o g e n e o u s p e n e t r a t i o n barrier as o f t e n c o n s i d e r e d . T r a n s p o r t is n o t limited b y u n s t i r r e d layers either at t h e tissue surfaces or w i t h i n t h e tissue itself. Salicylate is n o t passively transp o r t e d . Parallel t r a n s p o r t p a t h w a y s d o exist.
Introduction Various m o d i f i c a t i o n s of the m e t h o d of Ussing and Z e r a h n [1] (Schultz and Z a l u s k y [ 2 ] ) have b e e n e x t e n s i v e l y used in t h e past d e c a d e for t h e s t u d y o f t h e t r a n s p o r t o f a v a r i e t y of s u b s t a n c e s across tissues such as frog skin, t o a d bladder, m a m m a l i a n intestine and m o u s e skin. Since this is an in vitro t e c h n i q u e , t h e fluxes in either d i r e c t i o n s can be o b t a i n e d . In general, o n e observes (Fig. 1) a f t e r a t r a n s i e n t p e r i o d , a linear rise o f t h e c o n c e n t r a t i o n o f s u b s t a n c e t r a n s p o r t e d t h r o u g h t h e tissue [3--7 ]. F r o m these o b s e r v a t i o n s the s t e a d y - s t a t e flux o f t h e s u b s t a n c e is d e t e r m i n e d b y the slope a o f t h e linear p o r t i o n o f the curve. T h e d i r e c t i o n a l flux r a t i o so d e t e r m i n e d is a useful quantit:y t o c h a r a c t e r i z e t h e t r a n s p o r t s y s t e m . A n o t h e r * Present address: National Institute of Drug Abuse, Rockville, Md., U.S.A.
277
•
SLOPE(7" - ~ 7 Z > D,~, Ou Z T
TIME
Fig. 1. T y p i c a l c u r v e of s o l u t e t r a n s p o r t e d i n t o the r e c e i v i n g ( s i n k ) c o m p a r t m e n t l i n e a r p o r t i o n o ( f l u x ) a n d i n t e r c e p t of t h e e x t r a p o l a t e d s t r a i g h t line r (lag t i m e ) .
w i t h tile slope o f t h e
observable quantity is the lag time r. While a curve such as shown in Fig. 1 is usually described by the classical equation for diffusion applied to transport across a h o mo g ene ous barrier (cf. ref. 2) [8], we will show that such an equation has only limited validity. A closer look at those curves which allow one to observe b o t h the transient and the steady-state behavior suggests that there might be more parameters that one can determine, thus permitting a bet t er understanding of the details involved in the absorption process. A com p art m ent al approach seems to be a natural technique applicable to this end. In the following we review a nuinber o f simple models with which we analyze the experimental data report ed here. C o m p a r t m e n t a l models of absorption
(A) Model of tissue compartment A schematic diagram of the model is shown in Fig. 2. In the diagram M represents the a m o u n t of the transporting solute in the mucosal bathing solution, S is the a m o u n t in the sero~al solution and T is the a m o u n t of solute in the tissue. Th e Greek characters represent the corresponding rate coefficients, which are assumed constant, in units of time -~. The system is defined in terms o f linear differential equations as follows
7
Fig. 2. M o d e l A: T i s s u e ( T ) c o m p a r t m e n t . See t e x t f o r d e t a i l e d d e s c r i p t i o n . Fig. 3. M o d e l B: T i s s u e ( T ) w i t h p o r e . See t e x t f o r d e t a i l e d d e s c r i p t i o n .
7
278 M = - - a M + fiT f = a M - - (¢~ + ~,)T + 5S
(1)
= 3 ` T - 5S. E x p e r i m e n t a l l y the t r a n s p o r t can be d e t e r m i n e d in either d i r e c t i o n , t h u s , we have t w o sets of solutions t o t h e e q u a t i o n s and t w o sets of c h a r a c t e r i s t i c slopes a n d lag t i m e s f o r the t w o e x p e r i m e n t a l curves. F o r t r a n s p o r t f r o m M to S, we define M as t h e source c o m p a r t m e n t and S as t h e sink c o m p a r t m e n t b y setting M = 0, 5S < < 7T. T h e a m o u n t o f solute in the sink c o m p a r t m e n t S is t h e n given b y the t r a d i t i o n a l e q u a t i o n S = o s ( t - - rs) + Osrs e x p ( - - t / ~ s )
(2)
T h e f l o w into S is c h a r a c t e r i z e d b y t h e slope os and t h e lag t i m e rs, viz. =
~3"
and
1
rs -
(3)
N o t e t h a t t h e e x p o n e n t i a l c o e f f i c i e n t is precisely the r e c i p r o c a l o f T~. F o r t r a n s p o r t in t h e reverse d i r e c t i o n (S = 0, a M < < / 3 T ) the increase as a f u n c t i o n of t i m e for t h e a m o u n t o f solute M is l~I = Om(t
--
T m
)+
OmT
m
exp(--t/r m)
(4)
T h e c o r r e s p o n d i n g slope Om and lag t i m e rn~ are Om-
and
Tm
(5)
For this model then we find the ratio of slopes r~j, the directional flux ratio, and the ratio of lag times r r are r . = ~a7
and
rr =
1
(6)
( B ) M o d e l o f tissue w i t h p o r e T h e s c h e m a t i c d i a g r a m o f this m o r e c o m p l i c a t e d m o d e l is given in Fig. 3 w h e r e , in a d d i t i o n t o t h e s y m b o l s o f Fig. 1, t h e p o r e is c h a r a c t e r i z e d b y t h e rate c o e f f i c i e n t s ~? and $. T h e kinetic e q u a t i o n s are n o w
= --(~ + r~)M + fiT + ~S = a M - - (/3 + 7 ) T + 5Z = ~M + 3`T-
(7)
(5 + ~)S
F o r t r a n s p o r t f r o m M to S t h e a s s u m e d source and sink c o n d i t i o n s are given b y M = 0, 5S < < 3`T, ~S < < r~M. T h e t r a d i o n a l e q u a t i o n no longer h o l d s in its original f o r m as t h e solute increase in S is n o w given b y t h e following e q u a t i o n S = (~s(t - - -rs) + Os~-S e x p [ - - ( f i + 3`)t]
(8)
w h e r e t h e e x p o n e n t i a l c o e f f i c i e n t (~ + 3') is n o t equal to t h e r e c i p r o c a l o f t h e lag time. In this m o d e l t h e slope and lag t i m e are given b y t h e following relations
279
G-/3+Ta7 +~
and
T~
_
1
fi+7
_
~
~(/3+7)+a7
(9)
In these m o r e c o m p l i c a t e d expressions for Os and Ts we can i d e n t i f y the tissue and pore c o m p o n e n t s since the first t e r m in each case is the same e x p r e s s i o n t h a t was o b t a i n e d in m o d e l A (tissue o n l y ) , Eqn. 3, while t h e second t e r m is the c o n t r i b u t i o n o f the pore. We see t h a t t h e i n t r o d u c t i o n o f a p o r e results in an additive c o n t r i b u t i o n t o the slope and a decrease in the lag t i m e as c o m p a r e d to t h e previous m o d e l . F o r t r a n s p o r t in the o p p o s i t e d i r e c t i o n {S = 0, aM < < fiT, T~M < < ~S) the expression for M is analogous to Eqn. 8 and the resulting slope and lag t i m e are
am
-
7+~
+~
and
7 m
-
7 +;3
~(7 +/3) +8/3
(10)
F o r this m o d e l the slope ratio and lag t i m e ratio are considerably m o r e complicated, viz.
°
8/3+ ~(v +/3)
and
~v 8/3 + ~(v +/3) r~ = - ~ . ~7 + ~(/3 + 7)
(11)
(C) Model o f epithelial and extraepithelial compartments with pore T h e schematic diagram for the system is p r e s e n t e d in Fig. 4. This m o d e l is an e x t e n s i o n o f m o d e l B (tissue with p o r e ) b y including an additional c o m p a r t m e n t X with e x c h a n g e rate coefficients p and v. However, the i n t e r p r e t a t i o n is s o m e w h a t altered as E d e n o t e s o n l y the epithelial cellular c o m p a r t m e n t and X d e n o t e s the extracellular c o m p a r t m e n t o f the c o m p l e t e tissue. With this m o r e extensive m o d e l the two-sidedness o f the tissue is n o w m o r e a c c u r a t e l y represented. T h e general system of linear kinetic e q u a t i o n s f o r this m o d e l is t h e n /~/= --(a + ~/)M + fiE + ~X
/~=aM--(/3+7)E+SX 2 = r~M + 7 E -- (8 + } + p ) X
(12)
~9 = p x For the specific case o f t r a n s p o r t M to S the e.quations simplify slightly after i n t r o d u c t i o n o f the source and sink c o n d i t i o n s (M = O, vS < < pX), b u t the solution o f these e q u a t i o n s is still m u c h m o r e c o m p l i c a t e d t h a n was the ease o f the previous models. N o w t h e r e are t w o e x p o n e n t i a l t e r m s in the f u n c t i o n for S, describing t h e a c c u m u l a t i o n o f solute, i.e. S = O s ( t - - T~) + O~[T~lexp(~t) + T~2exp(k2t) T h e slope o~ o f the straight line p o r t i o n o f t h e S curve is given b y
(13)
280
M ;E;x
s
Fig, 4. Model C: Epithelial (E) and extraepithelial (X) c o m p a r t m e n t s with pore. See text for detailed description.
_
+ v)(6
+ ~ + #) -
%j
and the lag time, which can be expressed as a sum, is
1
[(~+y +~ +~ +~)(~ +v)]
The c o m p o n e n t s of r s are defines as
x2
[_!:
]
and
where Xl and X2 are bot h real negative solutions to the characteristic equation corresponding to the system of differential equations, i.e. Eqn. 12 after source and sink conditions are introduced, viz. ~,2 J- (~ q- y q- 5 "{" ~ "b //)~k q- (~ q- 7 ) ( 5
+ ~ "{-~/)--'~ = 0
(17)
In this model, the expressions for the slope and lag time are, of course, even more complicated. Nevertheless, a similar separation of various c o m p o n e n t s is obvious from Eqns. 9, 14 and 15. The slope os has the tissue plus pore components which are now multiplied by a "shielding f a c t o r " due to the extracellular c o m p a r t m e n t X, which has the value less than unity (cf. the second factor in Eqn. 14). The lag time has a different "shielding f a c t o r " (again due to the X c o m p a r t m e n t ) which modifies only the tissue c o m p o n e n t of r~ (cf. the factor in brackets in Eqn. 15). For transport in the reverse direction the solution to Eqn. 12 is again analogous to Eqn. 13 but the as and r~ are replaced by
+ ~)---~}~j l]) ] O'n= [V~+~ + ~] [(~ + ~ + u(7 ~)(V
(18)
and
1 [-(~+7+5+~+p)(~,~f)j
~
(19)
281
In the formulas c o r r e s p o n d i n g to rml and rm2 the last t e r m in brackets in Eqn. 16 is replaced b y tl~e last t e r m o f Eqn. 19. Simplification of model C. Various c o n d i t i o n s for t r a n s p o r t across epithelia m a y exist that i n t r o d u c e some simplification o f m o d e l C and t h u s allow a b e t t e r intuitive view o f the t r a n s p o r t system as described b y this r a t h e r complicated model. T h e solutions X1 and X2 o f the characteristic e q u a t i o n , Eqn. 17, which a p p e a r as the e x p o n e n t i a l coefficients in Eqn. 13 are simplified if the c o n d i t i o n holds t h a t 5 < < (~ + p), because t h e n the t e r m 75 is negliglible. This w o u l d follow for a solute, passively t r a n s p o r t e d , having a p a r t i t i o n c o e f f i c i e n t m u c h d i f f e r e n t t h a n u n i t y w h e r e the lipid exit r o u t e f r o m the extracellular c o m p a r t m e n t X is m u c h slower t h a n the t w o a q u e o u s routes. T h e n ~, -~ --(/3 + 7)
and
~2 -~ --(~ + P)
(20)
where S(t) is still a t w o - e x p o n e n t i a l f u n c t i o n o f course, and has a lag time, e.g. rs, given b y ~ 1 r ~ +. 1 . -~(~+~)+~+~+~ . . . . .
.
(21)
This c o n d i t i o n applies regardless o f the relative m a g n i t u d e s o f X, and X2, i.e. w h e t h e r t h e y m'e equal or d i f f e r e n t . (a) Thick X compartment. T h e rate coefficients a, fl, 7, 5, ~, ~, p and v can be expressed as A P / V w h e r e A is the tissue area across which t r a n s p o r t occurs, P is the tissue p e r m e a b i l i t y which can include b o t h active and passive c o m p o nents, and V is the v o l u m e o f t h e driving c o m p a r t m e n t . In t h e case of t h e extracellular c o m p a r t m e n t X, which is m u c h larger t h a n the epithelial c o m p a r t m e n t E , the respective volumes are Vx > > V~. It t h e n follows t h a t since ~ and ~, contain V~ in the d e n o m i n a t o r and ~ and/~ are inversely p r o p o r t i o n a l to V~,
m a y hold and t h e n f r o m Eqn. 16 v~,~ 0
and
1 r~2 ~ + p -
(22)
thus the f u n c t i o n describing S will behave as a o n e - e x p o n e n t i a l f u n c t i o n , Eqn. 2. As a c o n s e q u e n c e o f this c o n d i t i o n , X being a very large p o r t i o n o f the tissue, even if the tissue does have t r a n s p o r t via a pore p a t h w a y as well as across the epithelial p a t h w a y the process is describable b y m o d e l A (tissue with no pore) and the e x p o n e n t i a l c o e f f i c i e n t characterizes o n l y t h e extracellular comp a r t m e n t X and contains no i n f o r m a t i o n a b o u t the epithelial p o r t i o n o f the tissue. (b) Thin X compartment. This is the o p p o s i t e o f the situation discussed in (a) as we n o w have V~ < < V e and ~+~/ 1/X forces us to e x c l u d e m o d e l B since, a c c o r d i n g to Eqns. 9 a n d 10 r m u s t be less t h a n 1/X = 1/(/3 + "),). T h e r e f o r e , while n e i t h e r m o d e l A n o r m o d e l B is s a t i s f a c t o r y , we w o u l d like to investigate t h e Bossibility t h a t m o d e l C is in f a c t a valid d e s c r i p t i o n o f this s y s t e m . In an e f f o r t to resolve the d i f f i c u l t y o f n o t being able to a n a l y z e the experim e n t a l d a t a b y m o d e l C we have carried o u t a series o f digital s i m u l a t i o n s o f e x p e r i m e n t s utilizing Eqn. 13, g e n e r a t i n g curves S(t) similar to Fig. 1 b y intro-
284
ducing a normal scatter into the c o m p u t e d data. For these simulations we have c h o s e n values of ~, ~, 7, 5, ~, ~, 12 and u, non-negative parameters, such that o and r w o u l d be characteristic of the observed values by linear or non-linear analyses (i.e. o ~- 0.1 n m o l • cm -2 • min -~ and T ~ 15 rain). Obviously there is no u n i q u e s o l u t i o n to this problem but through extensive searching we can classify t w o types of solutions, namely such that Xl > > X2 or the X~ ~ X2. These two classes of solutions can be exemplified by the t w o sets o f parameters included in Table III * When the simulated data with 0% standard deviation of the normal scatter were analyzed by m o d e l C, o n l y in the case of X~ ~ k2 could we recover the "experimental" parameters used for data generation. Using scattered data with 5% standard deviation no fitting was obtained. This was also the case with our efforts reported above to attempt an analysis o f the experimental data in Table II by m o d e l C. Clearly, the curves obtained by the t y p e o f experiment reported here do n o t contain sufficient i n f o r m a t i o n to allow one to estimate the experimental parameters for m o d e l C. Nevertheless, the analysis by m o d e l B o f the data generated by simulation of model C ( s h o w n in Table IV) was carried o u t more successfully. For each case reported in Table IV, the values are the averages of five simulations with data scattered at the 5% standard deviation level. In the case of k~ > > X2, we f o u n d that rs < 1/Xs which was statistically significant at the level of P < 0 . 0 0 5 . This is in contradiction to the results of the analysis of experimental data as reported in Table II. However, the analysis of the case X~ ~ k2(kl ~> X2) we
III
TABLE TWO
EXAMPLES
PARAMETERS
Model
parameters
OF OF
VALUES
MODELC
OF
MODEL
LEADING
PARAMETERS
TO o s=0.1
AND
mnol'cm
a r e i n u n i t s o f m i n -1 . F o r u n i t s o f " S i m u l a t e d
Parameters
kl
Model c( 7 5
/1
> ; > X2
i,
"SIMULATED
"min -I
andr s=
experimental"
EXPERIMENTAI," 15 min
parameters
s e e T a b l e II.
X 1 ":k 2
0.15
• 10 -3
0.16
0.3
• 10-1
0.12
0.3 0.3
• 10 -! • 1 0 -1
0.13 0.175.
0.15
• 10 -4
0.36
0.3
• 10 -2
O.55
10 -l
0.65 0.28
3.0 0.15
-2
• 10 -3
10 -1 - 10-4 • 10-2 10 -I - 10-3 •
"Simulated experimental" 0.0895
(l s
14.25 0.0491
7s 1 7s2 Ts ks 1 ks2
14.30 ---0.0597 --3.0333
* By restricting tion
0.0982 16.96 --1.035 15.93 ---0.075 --0.263
the parameters further to simulate o f t h e t y p e h I >> X2 . As will b e s e e n b e l o w ,
a passive transport lnodel we arrived at the solut h i s is n o t a v a l i d d e s c r i p t i o n o f t h e s y s t e m .
285 TABLE
IV
"SIMULATED EXPEII, IMENTAL" PARAMETERS ESTIMATES OF CURVES W1TIIMODEL PARAMETERS OF TABLE III TO MODEL B
BY
FITTING
SIMULATED
F o r u n i t s see T a b l e II. )'1 > > c7s rs 1/A s
0.0898 14.6 17.0
~2 + 0.0079 + 3.4 ~ 4.6
h i - X2 0.1061 19.1
+- 0 . 0 0 6 1 -+ 2 . 3
la.0
+ 2.8
found that rs > l/ks at the level of significance P < 0.005. This observation is fully compatible with the results of our non-linear analysis of the experimental data (Table II) using the functional form of model B as a device to test the data for compatibility with models A and B. As m ent i oned above, the observation that rs ¢ l/k s allows us to exclude model A as a valid description of this system. F u r t h e r m o r e , as noted above, the fact that 7s > l/ks allows us to exclude model B as well. Therefore, on the basis o f these observations, we conclude that model C is the minimal compatible model that may be appropriate for a description o f this system of intestinal anion transport. This t y p e of analysis has also been applied to the transport of m e t h o t r e x a t e across mouse skin where model B was f ound appropriate to describe the system and to rationalize the pH dependence of solute flux. [10]. Conclusions Due to the mathematical nature of the foregoing analysis we enunciate our conclusions and related implications in detail here. We wish to stress that these findings are based solely on the mathematical analysis of the experi m ent and therefore stand as i nde pe nde nt evidence to those biochemical and physicochemical studies which lead to similar conclusions. (1) The tissue is not a hom oge ne ous barrier to solute transport. This follows from our finding that model A is not a valid description for the sytem of salicylate transport across rat jejunum. As a consequence, analysis via the classical diffusion equation [8], which forces the system to obey model A, does not extract complete information when applied to this t y p e of experiment. (2) Transport is not rate limited by unstirred layers either at the tissue surfaces or within the tissue. This t y p e of process would be described by model A. While we have suggested that model C may be appropriate, the considerations leading to Eqn. 22 make it clear that the X c o m p a r t m e n t is not " t h i c k " as that would again lead back to an A t y p e model. (3) Multiple transport pathways exist for this system. The finding that model C is the simplest possible description of the tissue requires that there are at least parallel transport pathways. (4) Salicylic acid is not passively transported. The simulation studies could n o t r ep r o d u ce the experimental observations when the parameters were restricted to obey a passive model. Clearly this type of analysis stands self-justified in view of these results. To
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make these observations directly would require a considerable a m o u n t of experimentation b e y o n d ~he rather simple procedure applied here. The value of the co mp ar tm e nt a l analysis is further enhanced as it suggests a more complete definition of the experinlent (and tissue) through additional measurements that would be desirable. In particular, it is advantageous to do flux measurements in both directions across the tissue. Measurements of the am ount of solute in the tissue, especially as a function of time, would allow further definition of the parameters. If techniques to determine cellular and extracellular amounts as a function of time were developed it could then allow one to obtain all the parameters introduced by model C. Finally, this t ype of analysis has a broad applicability to many experimental tissue and solute systems. Acknowledgements This work was supported by N.I.H. Grant GM 16496 to Professor L.Z. Benet in whose laboratory the experimental work was performed. References 1 2 3 4 5 6 7 8 9 10
Ussing, ft. and Z e r a h n , K. ( 1 9 5 1 ) A e t a Physiol. S e a n d . 23, 110- 127 S e h u l t z , S. a n d Z a l u s k y , R. ( 1 9 6 4 ) 3. G e n . Physiol. 47, 5 6 7 - - 5 8 4 Q u a y , J. a n d A r m s t r o n g , W. ( 1 9 6 9 ) A m . J. Physiol. 2 1 7 , 6 9 4 7 0 2 Field, M., F r o m m , D. a n d MeColl, L ( 1 9 7 1 ) A m . J. Physiol. 220, 1 3 8 8 - - 1 3 9 6 J a c k s o n , M., Shiau, Y., Bane, S. a n d F o x , M. ( 1 9 7 4 ) J. G e n . Physiol. 63, 187 213 B a r n e t t , G., ttui~ S. and B e n e t , L. ( 1 9 7 6 ) S u b m i t t e d to A m . J. Physiol Wallaee, S., R u n i k i s , J. a n d S t e w a r t , W. ( 1 9 7 6 ) d. P h a r m . P h a r m a e o l . , in the press J o s t , W. ( 1 9 5 2 ) D i f f u s i o n in Solids, L i q u i d s a n d Gases, p. 37, A c a d e m i c Press, N e w Y o r k D i x o n , W. ( 1 9 7 4 ) BMD B i o m e d i c a l C o m p u t e r P r o g r a m s , U n i v e r s i t y California Press, Berkeley Walaee, S., B a r n e t t , G. and L i e k o , B. ( 1 9 7 6 ) S u b m i t t e d to J. P h a r m a e o k i n . B i o p h a r m .
