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METHODS IN ENZYMOLOGY
[9.]
processes by using tracer or fluorescence indicator techniques, as well as any which are not electrogenic. In a third instance an experimenter who works with the patch-clamp method can get signals from a cell in its natural surrounding by a cell-attached patch pipet. The worker can, however, also impose different driving forces and channel modifiers on an excised patch, but is restrained to transport processes which proceed through channels and are electrogenic. To analyze the molecular events of each transport process it is necessary to identify and separate the different transport proteins. This is achieved by affinity or photoaffinity labeling and the general procedure of protein separation and purification. Furthermore, reconstitution of a functional transport unit into liposomes is obligatory to ensure the retention of a protein subunit or lipid constituent during the separation procedure just mentioned. Since for some solutes, for instance Na +, HCO3-, and Ca2+, several transporters participate in transepithelial transport, the quantitative contribution of each of them is important. Thus, once the transporters and their location within the cell are known a balance sheet would be helpful. It certainly helps to prevent confusion, if (as in the proximal renal tubule) more than 15 different transporters are located at each cell side, half of them being Na + cotransporters (Fig. 2). In future studies those methods which identify the transporters in situ might be of great value, for instance, electron microscopic localization with monoclonal antibodies. Studies of transporters at the molecular level might utilize recombinant DNA techniques, introduction or attachment of molecules emitting molecular signals, production of mixed detergenttransporter crystals, and X-ray crystallography.
[2] D e t e r m i n a t i o n
o f P a r a c e l l u l a r S h u n t C o n d u c t a n c e in Epithelia
By G. KOTTRA and E. FROMTER Introduction Ever since it became clear that epithelia do not simply consist of two cell membranes in series but may have significant shunt pathways between the cells 1-4 researchers have been concerned with identifying the structural 1H. H. Ussing and E. E. Windlaager, Acta Physiol. Scand. 61, 484 (1964). 2 T. W. Clarkson, J. Gen. Physiol. 50, 695 (1967). 3 E. L. Boulpaep, in "Eleetrophysiology of Epithelial Cells" (G. G-iebiseh, ed.), p. 91. Sehattauer, Stuttgart, Federal Republic of Germany, 1971. 4 E. Frrmter, J. Membr. Biol. 8, 259 (1972).
METHODS IN ENZYMOLOGY, VOL. 191
Copyright© 1990 by Ac~lmnlc 15~s, Inc. All rish~ of ~ o d u c l l o n in any form tl~rved.
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PARACELLULAR SHUNT CONDUCTANCE IN EPITHELIA
5
equivalent of the shunt and quantifying the shunt permeability in relation to the permeability of the transcellular transport pathway. In epithelia (such as that in Necturus gall bladder) which exhibit a relatively simple microscopic structure and may be easily investigated with microelectrodes, the problem has been largely solved m at least so far as ion permeation is concerned--but in others, such as frog skin, the identification and quantification of the shunt path and the determination of its transport properties remains a considerable challenge. The classical approach to quantify the shunt conductance is to measure and compare transepithelial and cell membrane resistances. Since the latter resistances are often ditficult to determine a number of more or less indirect approaches have been devised. These are based on specific assumptions, the validity of which is usually di~cult to ascertain. In this chapter we attempt to critically review the electrophysiological methods which have been used in the past to quantify epithelial shunt conductances. Emphasis is on flat sheet epithelia. A number of specific problems involved with shunt analysis in tubular epithelia has been discussed in a methodological reviews and will be mentioned only briefly here. The methods to be discussed differ in their ability to represent the electrical properties of given epithelia appropriately. All are based on equivalent circuit representations of epithelia which, for various reasons, are gross oversimplifications. This is true, for example, for the circuits of Fig. l a - c (so-called "lumped" models) in which the epithelium is essentially represented by three elements: two resistors (R, and Rb~ for the apical and basolateral cell membrane, respectively) arranged in series and an additional resistor (R~h for the paracellular shunt) arranged in parallel. These models yield erroneous results whenever the resistance of the lateral space cannot be neglected. To account for lateral space effects, the "distributed" model (Fig. ld) of the lateral space and lateral cell membrane has been developed? In this model the shunt pathway consists of the tight junction resistance (Rj) and of a chain of resistors for the lateral intercellular space (r~, with R~ -- Er~). The distributed model is particularly important for leaky epithelia with low-resistance tight junctions. There are other more general shortcomings, however. Virtually all equivalent circuits treat all resistors as constant, although, in reality, the current voltage relation of cell membranes is usually nonlinear even in the narrow range of physiological membrane voltages ( - 9 0 to + 4 0 mY). This is not yet problematic so long as the analysis is restricted to resistance measurements (as in models l a and l c - d of Fig. l) and so long as the 5 E. Fr6mter, Kidneylnt. 30, 216 0986).
6
[2]
METHODS IN ENZYMOLOGY
I Ra
b
I tEl Rsh
Rbt
I
R="~
l
Eu T
Esh
R,h
Rbt~:
I
Za
R,~ Ca Ru~=
i Z tat
Rj
[ZT] q~7]
rtis
Rsh
Cbt
i Fro. 1. (a) Simple lumped model. (b) Lumped model with electromotive forces representing zero-current potentials Ei for each membrane element. (e) Lumped model including capacitances C. and Ct~ to represent the apical and basal cell membrane impedanev. (d) Distributed model; the Z elements denote parallel RC units for evil membrane impedanev5.
measurements are restricted to the linear range (determination of so-called "small signal" conductance or "slope" conductances at the normal operating point of the system). The analysis becomes problematic, however, if (slope) resistances and zero current potentials E0 are used together in one circuit as in Fig. lb and in methods 2.2, 2.3, and 2.4 below. If the resistance is nonlinear, extrapolation of the slope conductance to the zero current membrane voltage does not yield the proper E, or Eb~that may be equated with ion concentration ratios on the basis of the Nernst equation. 6,7 It is not surprising, therefore, that these methods have led to a number of inconsistent results. Another uncertainty comes up in impedance measurements (see methods 3.1 and 3.2 below). It relates to the fact that in reality biological membranes cannot be represented by ideal capacitors (e.g., Ca and Cu) 6 S. I. Helman and S. M. Thompson, Am. J. Physiol. 243, F519 (1982). 7 S. M. Thompson, Am. J. Physiol. 250, C333 (1986).
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PARACELLULAR SHUNT CONDUCTANCE IN EPITHELIA
7
because they may exhibit dielectric loss. In addition it may be sometimes difficult to separate membrane capacitances from polarization effects of unstirred layers, but the latter problem arises only at low-frequency currents (~- 1 to 10 Hz). 1. Methods Based on Equivalent Circuit l a
1. I Quantitative Comparison of Direct Current (DC) Cell Membrane Resistances and DC Transepithelial Resistance This method requires that the transepithelial resistance, RT, and the cell membrane resistances, R, and R~, be measured and compared. In contrast to RT, the determination of R, and R~ is usually quite difficult. Since most epithelial ceils are electrically coupled with their neighbors) a current that is injected into a single cell does not only pass across R, and R~ into the apical and basal fluid compartments, respectively, but spreads also radially within the epithelial cell layer. A theoretical analysis of this problem, the so-called two-dimensional cable analysis,4,9 predicts that radial voltage (V) attenuation within the ceil layer follows a first order Bessel function K 0 which can be defined by two parameters: a proportionality coefficient A and an apparent length constant 2:
V(x) = AKo(x/A)
( 1)
where x is the radial distance. From these parameters two resistances can be calculated: R,, which describes the resistance of the parallel arrangemerit o f / ~ and Rb~, and R,, which represents the coupling resistance for horizontal current flow in the ceil layer: Rz = [(1/Ra) + (l/Ru)]-' -- (2m422)/io Rx = (2~tA)/io
(2) (3)
where io is the injected current. On first view, to define the voltage attenuation within the cell layer may appear quite demanding and time consuming. However, once it has been verified that the voltage as function of distance x from the point of current injection obeys the proper Bessel function, it is sufficient to determine the current-induced voltage change at only two distances x, since two data points suffice to define the Bessel function and hence to determine A and it. In practice it may be preferable to collect more data points in order to average out individual measuring errors. 8 W. R. Loewenstein, Physiol. Rev. 61, 829 (1981). 9 L. Reuss and A. L. Finn, J. Gen. Physiol. 64, 1 (1974).
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METHODS IN ENZYMOLOGY
[2]
Alternatively, one can also try to circumvent cable analysis, and determine Rz by measuring the input resistance of an individual cell after electrical uncoupling. In practice, however, this approach is of little value, since uncoupling by depolarizing current ~° will certainly alter R, and Rb~, thus invalidating the entire concept and the same might happen with chemical uncouplers such as high CO2 pressure. Once Rz has been determined, R. and R~ can be calculated with the help of a, the direct current (DC) voltage divider ratio (VDR), which in the simple models of Fig. l a - c equals R,/Rm. Finally the shunt resistance, Rta, is obtained by comparing the sum o f R a and Rta with RT: R . ~- [(R a "3I- R b l ) R T ] / ( R a "31-Rbl -- RT)
(4)
The method can also be applied to tubular structures where the voltage attenuation along the tubular lumen and along the ceil column has to be measured separately. Both follow double-exponential functions from which the three resistances R,, Rbl , R e, as well as R T can be derived by data-fitting techniques.5,H-'3 Comments. The method is straightforward but technically rather involved since it requires the insertion of at least two (or better three) microelectrodes simultaneously into nearby cells, which must be recognized under the microscope and the distance between which must be accurately measured. In tubular epithelia an even greater number of measurements is necessary, again including intracellular recordings that must be combined with accurate distance measurements. The advantage of the method is that it does not rely on unproved assumptions such as constancy of zero-current potentials, which is a precondition for most of the other methods discussed below. On the other hand, the method has limited power, because it is based on the "lumped" model equivalent circuit. The problem manifests itself in the use of the voltage divider ratio, ct, which is taken to equal RJRb~, while in fact it may be lower ff Rt~ was overestimated due to a significant contribution of the lateral space resistance, Ru~. As a consequence the estimate of R~h will be in error, as has already been discussed in the literature) 4 Problems with the appropriateness of the measured a values have indeed been noticed in various epithelial preparations, including frog skin) 5 which is commonly held to obey the relation Rj >> R~, with Rt~ being negligible. lo S. A. Lewis, D. C. Eaton, and J. M. Diamond, J. Membr. Biol. 28, 41 (1976). i~ T. Anagnostopoulos, J. Teulon, and A. Edelman, J. Gen. Physiol. 75, 553 (1980). n T. Hoshi, tC Kawahara, R. Yokoyama, and K. Suenaga, Adv. Physiol. Sci. 2, 403 (1981). ~3W. B. Guggino, E. E. Windhager, E. L. Boulpaep, and G. Giebisch, J. Membr. Biol. 67, 143 (1982). t4 E. L. Boulpaep and H. Sackin, Curr. Top. Membr. Transp. 13, 169 (1980). ~s W. Nagel, J. F. C,arcia Diaz, and A. Essig, PJluegers Arch. 399, 336 (1983).
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PARACELLULAR SHUNT CONDUCTANCE IN EPITHELIA
9
1.2. Paired Measurements OfRT and VDR in Response to Alteration of One Membrane Resistance This method consists of measuring R T and a while one surface of the epithelium is suddenly perfused with a solution that alters one or more ion transport mechanisms in the respective cell membrane by partial or total ion substitution, or by application of inhibitors or transport facilitators. The method is easier to apply than the above discussed method 1. l, since it requires only one intraceUular microelectrode (for determination of a). On the other hand, it is less reliable, since it depends on the precondition that the solution change affects only one single resistor of the circuit, leaving the others unchanged. If this precondition is fulfilled, and if R a was changed from R a to R ' , four independent data are available from which R,, R~, and R~, as well as R" can be determined. The respective equations which are presented here in the form published p r e v i o u s l y 16 w e r e first derived by Reuss and Finn 9 (see also Refs. 17 and 18): R, = [a(ot - a')RrR~]/(I + a)(1 + a')(RTV-- R~) Rbl = [(a -- a')RrR~]/(1 + a)(1 + a')(RT -- R) R,h = [(a -- a')RrR~]/[(l + ot)R~ -- (1 + a')Rr]
(5) (6) (7)
where t~ = R,/R~, a ' = R~/Rbl, and R~ is the transepithelial resistance measured after the pertubation. The same principles have been applied to tubular epithelia where RT must be determined by cable analysis. In that case, however, the determination of VDR presents a problem, because usually a ~ R,/R~, but changes from an overestimate of R,/R~ near the point of current injection to an underestimate at large distances from the current source. As analyzed previously 19 the error has a minimum close to x = 2, i.e., at around one length constant from the point of current injection. Comments. While technically speaking the method is comparatively simple and the calculations are straightforward the quantitative validity of the analysis depends on two conditions: (1) the accuracy of t~ as a measure of R,/Rbl (cf. lumped versus distributed model!) and (2) whether indeed only one resistor of the equivalent circuit of Fig. l a was altered and the others remained constant. Hence, when performing such experiments the following precautions must be taken. 1. The solution change must be very fast (depending on cell size, in the order of seconds) and it must be ascertained that only instantaneous 1~E. FrOmter and B. Crebler, Pfluegers Arch. 371, 99 (1977). 17 R. A. Frizzell and S. G. Schultz, J. Gen. Physiol. $9, 318 (1972). is T. Anagnostopoulos, J. Physiol. (London) 233, 375 (1973). 19 D. I. Cook and E. FrSmter, Pfluegers Arch. 4113, 388 (1985).
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METHODS IN ENZYMOLOGY
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changes of RT and ot are recorded. If solution change and recording are too slow, intracellular ion concentrations may change and this may secondarily alter the cell membrane resistances, or even the shunt resistance. 2. For each individual epithelium the proper experimental maneuver must be selected. In tight epithelia, such as frog skin, urinary bladder, or cornea, for example, amiloride can be given to suddenly increase Ri, or amphotericin B or gramicidin D to decrease itJ °,16;°,2~ Although such measurements yield valuable estimates of the lumped circuit parameters, some uncertainty remains with regard to the quantitative reliability of the analysis because of possible side effects on Rbl and/or R~. For example, if the basolateral cell membrane contains voltage-dependent conductances, hyper- or depolarization of this membrane might change Rbl- On the other hand, ionophores might not be restricted to the apical cell membrane but might also be taken up into the basolateral cell membrane. Moreover, as amiloride blocks tight junctions in leaky epithelia,22;3 it has been postulated that it exerts similar effects in tight epithelia, m24,25 In leaky epithelia, such as renal proximal tubule, small intestine, or gallbladder, the situation is similar. Here sugars and/or amino acids have been given to activate rheogenic Na+-substrate cotransport and thereby increase the apical cell membrane conductance as well as Ba2+ or high K + concentrations to increase or, respectively, decrease R a or Rb~.~s-2s In the former case, however, besides Ri, also R~ could have changed because of voltage-dependent conductances and/or R~ could have changed in response to changes in the rate of fluid transport. In the latter case, besides possible effects on Rbl from voltage-dependent conductances, Rj could have decreased by the presence of high K + or increased29 in response to Ba2+. We have tried to quantify the possible errors that can be made by neglecting such side effects in Appendix A. Generally speaking one can conclude that the greater the change of a during an experimental maneuver the more accurate the determination of R,a becomes. Also, the greater 2o N. IC Wills, S. A. Lewis, and D. C. Eaton, J. Membr. Biol. 45, 81 (1979). 21 S. A. Lewis and N. IC Wills, J. Membr. Biol. 67, 45 (1982). 22 R. S. Balaban, L. J. Mandel, and D. J. Benos, J. Membr. Biol. 49, 363 (1979). 23 G. Kottra and E. FrOmter, J. Exp. Biol. 106, 217 (1983). 24 R. G. O'Ncil and S. I. Helman, Am. J. Physiol. 231, 164 (1976). 2s L. G. M. Gordon, J. Membr. Biol. 52, 61 (1980). 26 E. FrOmtcr, PfluegersArch. 393, 179 (1982). 2~ R. Gregcr and E. Schlatter, Pfluegers Arch. 396, 315 (1983). E. Bello Reuss, J. Physiol. (London) 370, 25 0986). 29 G. Kottra and E. Fr6mter, PfluegersArch. 415, 7180990).
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PARACELLULAR SHUNT CONDUCTANCE IN EPITHELIA
11
the value of Rsh in comparison to Ra plus Rbm, the more uncertain its accuracy becomes.
1.3. Plot of Gr versus VDR in Response to Alteration of One Cell Membrane Resistance A variant of technique 1.2 was described by Lewis et al. ~° It is based on the following equation which is equivalent to Eqs. (5-7) derived from the circuit in Fig. I a: GT = G,h + Gbl(1 -I- O~)-1
(8)
with GT = 1/Rr, Gb~=- l/Rbl, and G,h----- 1/R~. This equation can be applied to experiments in which RT and ~ are recorded while the apical cell membrane conductance is progressively changed by apical application of an ionophore, such as nystatin. If the data are plotted as Gr versus (1 + o0-I, Eq. (8) predicts a straight line which intercepts with the conductance axis at G,h and has a slope of Gs. Comments. The method is not superior to the approach defined by Eqs. (5-7) above, since the same problems may arise that have been commented on before. In particular it seems questionable whether R s does indeed remain constant during the progressive action of the ionophore as the theory requires, since the experimental design does not fulfill the criterion of instantaneous action and measurement. In fact, data which have been obtained with this method and were published in conformation with Eq. (8) might as well be interpreted as exhibiting a systematic curvilinear relationship, rather than the postulated straight line. 2° 2. Methods Based on Equivalent Circuit l b
2.1. Measurements of Rr, VDR, and of Transepithelial and Cell Membrane Potentials in Response to Alteration of One Membrane Resistance This method does not differ much from method 1.2. Again a solution change is performed which should affect the ionic conductance of only one cell membrane, and the effect on a is recorded as well as the change in transepithelial potential difference, AVT, and the change in membrane potential across the contralateral cell membrane, A V. or AV~. In addition RT must be known. If, for example, the apical membrane is altered but Eu, E,h, Rb~ and R,h remain constant, we obtain R,h = [(1 + a + ~/(1 + a)]Rr where ot = R./Rbl and fl ffi AVbJAVr.
(9)
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METHODS IN ENZYMOLOGY
[2]
This approach differs from method 1.2, in that fl instead of a' is used for the calculations. However, since ~' is also known, the results of both approaches can be directly compared. Comments. Essentially the same preferences and restrictions apply and the same precautions should be observed which have been discussed with regard to method 1.2 above. With method 2.1, however, it is particularly important to make fast solution changes and to record the potential changes instantaneously, since the additional restriction applies that some of the zero-current potentials be constant. The method has been used in Necturus urinary bladder in which the Na + channel blocker amiloride was applied to the apical surface. ~6 A comparison of methods 1.2 and 2.1 in those experiments gave virtually identical results. Alternatively, combining methods 1.2 and 2.1 one obtains five data points from which a total of five circuit parameters can be determined. Hence, ifa solution change affects R, and R,h simultaneously, both R: and R~h can be determined)° However, the validity of such calculations rests entirely on whether or not Rbl and Eb~ remain constant. This assumption, however, cannot be tested independently unless more sophisticated techniques are applied (see below).
2.2. Measurements of Rr, VDR, and of the Relationship between Cell Membrane Potential and Short Circuit Current (1.~) Disregarding the fundamental problems with the circuit of Fig. I b, various methods have been devised for determination of shunt conductances in tight epithelia such as amphibian skin and urinary bladder which generate significant short circuit currents (I,,). If an epithelium is short circuited, current flow across R,h vanishes, and V, = V~. If I~ is now altered by applying, for example, amiloride apically the simplified treatment of circuit lb yields the following equation: V, - Vbl----Ebl -- RblI~
(10)
which predicts a straight-line relationship between Vbland I.~ provided E~ and Ru remain constant) ~,32 From the slope of this relation Ru can be directly determined. If in addition RT and cz are measured in the same experiments, R, and R.h can be readily calculated. In analogy to that, I,~ can be altered by applying ouabain basolaterally to determine R,. Comments. The method is technically rather simple, but the validity of the data depends critically on whether or not the precondition (constancy ~oS. Henin, D. Cremaschi, T. Schettino, G. Meyer, C. L. L. Donin, and F, CoteUi, £ Membr. Biol. 34, 73 (1977). 31 S. G. Schultz, R. A. Frizzell, and H. N. Nellans, £ Theor. Biol. 65, 215 (1977). 32S. I. Helman and R. S. Fisher, J. Gen. Physiol. 69, 571 (1977).
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PARACELLULAR SHUNT CONDUCTANCEIN EPITHELIA
13
of E~) is fulfilled or, in other words, whether the conductances are constant in the entire range between open circuit and short circuit. By applying increasing concentrations of amiloride Nagel 3a attempted to derive the current-voltage ( I - 1I) relation of the basolateral cell membrane with this approach; however, the results showed marked nonlinearity and in some cases even slopes with negative resistances. In addition, comparison of the shunt resistances calculated with methods 1.2 and 2.2 from the same experiments showed discrepancies of between 0.185- and 1.13-fold. a3 This indicates that Eta was not constant or that the circuit of Fig. 1b did not adequately describe the data. 2.3 Plots of Transepithelial Conductance versus I~ in Response to Alteration of One Cell Membrane Resistance
While the previously discussed methods require intraceUular measurements with one or two microelectrodes it was desirable that methods be found which allow the shunt conductance to be determined exclusively from macroscopic measurements. From the viewpoint of principle this is, of course, impossible since one cannot expect to obtain insight into the structure of an unknown membrane by simply studying its input/output relations. On the other hand, it might have been possible that in some cases enough information was already available to justify a purely macroscopic approach. Based on present-day knowledge of epithelial transport a method has been devised to determine the shunt resistance of tight epithelia by appropriately varying the apical cell membrane resistance and simply measuring Gr = 1/R T and the short circuit current, I,~. Provided Rta, R,h, E,, and Eta remain constant the equivalent circuit of Fig. 1b yields Gx = G~ + (E, + Eb~)-'I~
(1 1)
Accordingly, if the preconditions are fulfilled, plotting the data as Gr versus I~ should show a linear relationship with an intercept on the conductance axis equal to G~h = 1/R,a (see Fig. 2a). Alternatively the same information can be obtained from a plot of Vx versus R x according to the following equation where, in the ideal case, the linear relation between Vx and RT intercepts with the resistance axis at R~: 1 -= [VT/(Ea + Eta)] + (RT/Rsh)
(12)
The introduction of this concept goes back to a paper by Yonath and Civan, 34 who apparently used the word shunt in the sense of comprising all 33W. Nagel,J. Membr. Biol. 42, 99 (1978). 34j. Yonathand M. M. Civan, J. Membr. Biol. 5, 366 (1971).
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METHODS IN ENZYMOLOGY
[2]
1.0 g-.,
g-.
l ,-g
E0.5
2'~
/~ (/~A/cm 2)
5'o
o
2's
5'o
/.~ (uA/cm 2)
Flo. 2. UT vs I,~ plots obtained with 1 m M novobiocin on two frog skins. The triangles denote the G,h values determined with amiloride. (From Rick et al.~9)
ion flux mechanisms in parallel with the Na + pump; however, the concept was later reinterpreted by Lewis and collaborators,~° who also derived Eq. (12) as a method for determination of the paracellular shunt conductance.2° This method has already been widely used in tight epithelia such as frog skin, urinary bladder, colon, and others.2°,2~,35-39 Ra was either increased by increasing concentrations of amiloride or decreased by ionophores such as amphotericin B, gramicidin D, nystatin, novobiocin, silver ions, or by stimulation with vasopressin. Comment. The method is subject to the general criticism of the underlying model circuit (Fig. lb; see Introduction). In addition, its application is restricted to epithelia whose properties are already largely known. It is not capable of identifying paracellular shunts in epithelia which have not yet been further investigated. Furthermore, the quantitative validity of the derived data must be considered with precaution, because usually the measurements do not fulfill the criteria of fast perturbation and instantaneous recordings. Indeed, deviations from a straight-line relationship between GT and I,~ have often been observed (see Fig. 2b) and R~ values derived either by decreasing Ra with ionophores or increasing R, with amiloride usually differ from each other2° (see also Fig. 2). 35 L. C. Isaacson, J. Membr. Biol. 30, 301 (1977). 36 p. U. Feig, G. D. Wetz¢l, and H. S. Frazicr, Am. J. Physiol. 232, F448 (1977). 37 W. S. Marshall and S. D. Klycc, J. Membr. Biol. 73, 275 (1983). 3s N. K. Wills and C. Clauscn, J. Membr. Biol. 95, 21 (1987). 39 R. Rick, A. D6rgc, and E. Sessclmann, PfluegersArch. 411, 243 (1988).
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PARACELLULAR SHUNT CONDUCTANCE IN EPITHELIA
15
In Appendix B we report some model calculations which define the validity limits of the method quantitatively. From this we deduce that the method is restricted to the simple model of tight Na+-absorbing epithelia in which R , is exclusively selective for Na + and to the use of ionophores that are also purely selective for Na +. Alternatively, if the cell membrane is permeable to more than one ion species present, an ionophore must be used whose relative selectivity properties are identical with the relative ionic permeabilities of the respective cell membrane in the control state. Note: Even a straight-fine relationship does not prove that the estimate of G,h from the intercept is correct; also, the method is not able to distinguish a paracellular shunt from a transceUular C1- shunt or K + shunt. A special case of method 2.3 is the determination of R.~ by blocking the apical cell membrane conductance with amiloride. If the apical cell membrane was exclusively permeable to Na +, RT in the presence of amiloride should equal R,h. This approach has been widely used in frog skin and other tight epithelia, particularly in O--free solutions. ~ On the other hand, it is also known that K + channels are present in the apical cell membrane of the skin of at least some frog species4~,42 and the question about a possible effect of amiloride on Rj is still open.
2.4. Steady State Measurements of V.r, R.r, VDR, and of the Intracellular Activity of an Ion Species for Which One of the Cell Membranes Is Permselective This method was published recently. 4a It is based on the analysis of current flow in an equivalent circuit such as that depicted in Fig. lb, with the restriction that all current across one cell membrane (say membrane a) is exclusively carried by one ion species (t). If the intracellular activity of i can be measured the following equation allows calculation of R,h: [ Vr t~ ] R ~ = RT 1 -- [Va + (RT/ziF)] ln(~da~,o) 1 + a
(13)
where zi, F, R, and T are valence, Faraday constant, gas constant, and absolute temperature and a~c and a~o are the intra- and extracellular activities of the permeating ion species. A similar equation has been derived for the short-circuited state in which the I~ appears rather than VT. Comment. The main advantage of the method is the fact that only steady state measurements are required. This eliminates a number of 4oj. Warncke and B. Lindemann, J. Membr. Biol. 86, 255 (1985). 4~ W. Zeiske and W. Van Driessche, J. Membr. Biol. 47, 77 (1979). 42 W. Nagel and W. Hirsehmann, J. Membr. Biol. 52, 107 (1980). 43 N. J. Willumsen and R. C. Boucher, Am. J. Physiol. 256, C1054 (1989).
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METHODS IN ENZYMOLOGY
[2]
problems discussed above. On the other hand, as with the preceding two methods (2.2 and 2.3), this method relies on the assumption that membrane resistances are linear in the range between open and closed circuit, which normally is not true. Other, more technical, problems arise if other ion conductances must be blocked to establish the proper measuring conditions or if ion-selective microelectrode readings must be corrected for interferences.
2.5 Determination of R~ from Break Point in the Transepithelial Current Voltage Relation This method is based on a special form of the equivalent circuit lb. When studying transepithelial current-voltage (I-V) relations in frog skin, Helman and Fischer32 observed that the I - Vcurve consisted of linear segments with distinct break points in between, where the slope suddenly changed to a new constant value. These observations suggested that the apical cell membrane rectified and hence could be modeled by a parallel arrangement of two resistors of different magnitude, each having an ideal diode in series to allow either only inflow or outflow of current. If this model is correct the transition point between two segments in the I - V curve should then be the point at which current flow across the apical cell membrane and hence across the entire cell vanishes. One can conclude then that under those conditions all transepithelial current flows across the shunt path so that R,a can be directly calculated by dividing the break point voltage through the applied current. Comments. Although this method is extremely simple and requires only transepithelial measurements it has only rarely been used in the past, probably because the break points in the I - V relations are not always well-enough defined. This may be related to the fact that in principle the applicability of the method may be restricted to epithelia having an apical cell membrane that is exclusively selective for Na ÷. Any other parallel ion conductances may be expected to reduce the sharpness of the break point. 3. Methods Based on Equivalent Circuits l c and d (Impedance
Techniques) Unfortunately, measurements of the frequency-dependent transepithelial impedance alone do not allow one to recognize and quantify the presence of a shunt path. This derives from the impossibility of identifying parallel elements inside a two-terminal network by simply studying its input-output relations. (In contrast, it is well known that serial RC elements inside a two-pole network can be recognized provided they cause
[2]
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PARACELLULAR SHUNT CONDUCTANCE IN EPITHELIA
relaxations in the frequency-dependent spectra that are distant enough to be separated.)
3. I. Measurements of Transepithelial Impedance and of VDR in the Time Domain If a constant current pulse is passed across an epithelium, the equivalent circuit in Fig. l c predicts that the voltage response will rise with a double-exponential time course toward a new steady value44,45:
AVT(t) = [K~(1 - e-U,,) + K2(1 - e-U,,)]I
(14)
where I is the transepithelial current and the parameters K~, K2, z~, and r2 are functions of all circuit elements. Note that the time constants z~ and T2 differ from the intrinsic time constants of the individual cell membranes (za = RaC, and z~ = R~Cbl) except in the ease G,~ = 0 (for details see Ref. 44). The analysis of this time course yields four independent parameters, which do not suffice to determine the five elements of circuit l c. Consequently a further independent measurement is required, w h i c h - - i n addition--must contribute information from inside the cell layer. Such a measurement would be the DC voltage divider ratio. Hence, with these rather simple measurements the circuit is already fully defined. In reality, however, the approach is less promising because fitting such data with two exponential functions is usually difficult unless the time constants differ by nearly one order of magnitude or more. The technique can be improved if not only the DC voltage divider ratio is recorded, but the full rising flank of the intracellular voltage in response to the transepithelial constant current step: AVe(t) = [Ks (1 - e-u',) + K4(1 - e-U")]I
(15)
This time course of the intracellular potential response, which also obeys a double-exponential function and has the same time constants ~, and z2, allows the data to be fitted more easily. This is particularly true for the voltage response across the cell membrane with the smaller RC product, which exhibits a transient overshoot resulting from superposition of a fast positive and a slow negative exponential (Ks < 0, K4 > 0). Comments. In spite of the simple and straightforward principle, impedance measurements in the time domain cannot be recommended for determination of the shunt resistance. The main problem is associated with fitting two exponentials in the data analysis (see above). From a technical K. Suzuki, G. Kottra, L. Kampmann, and E. Fr6mter, PfluegersArch. 394, 302 (1982). 4s S. A. Lewis and J. L. C. DeMoura, J. Membr. Biol. 82, 123 (1984).
18
METHODS IN ENZYMOLOGY
[2]
point of view the method is quite demanding, since analysis of the intracellular voltage rise time requires shielded intracellular microeleetrodes to be used. In addition, compared to the experimental investment the results are of limited use since the underlying equivalent circuit (lumped model) is generally too simple. Once the experimental set-up is available it would always be preferable to perform measurements in the frequency domain, which yields more explicit information.
3.2. Measurements of the Transepithelial Impedance and of the VDR in the Frequency Domain In these experiments, instead of a square wave pulse, a series of sine wave currents of constant amplitude is passed across the epithelium, either as preprogrammed single-frequency bursts, 4° or as pseudo-random binary sequence (PRBS) noise,~ or as multifrequency superimposed sine wave bursts (MSSB), obtained by inverse Fourier transform from a selected sequence of logarithmically spaced frequencies.47 The relative advantages of the different methods have been discussed.47 In the latter two cases the spectral content of the voltage response is obtained by fast Fourier transform techniques: By dividing the cross-power spectrum through the autopower spectrum of the current signal the frequency-dependent impedance is obtained, which can be represented either as a Bode48 diagram (logarithm of amplitude and phase versus logarithm of frequency) or as a Nyquist plot47 (imaginary versus real component of impedance). The data are then fitted with the appropriate equations which describe the impedance of circuit I c and ld from which quantitative estimates of the circuit parameters can be derived.45,49 As in the case of the time domain analysis mentioned above, transepithelial data of this type alone are not sufficient to determine the five elements of the equivalent circuit of Fig. lc. However, under specific conditions (Rj ~ 0o) and after inclusion of additional information (e.g., the DC voltage divider ratio) it is possible to determine not only the elements of Fig. 1c but to analyze even the distributed equivalent circuit of Fig. ld. This means that the lateral intercellular space resistance can be resolved.38.50,5i Further progress and confirmation of the appropriateness of the analysis and the validity of the results were obtained by combining the transepithelial impedance analysis with simultaneous recordings of the freC. Clausen and J. M. Fernandez, Pfluegers Arch. 390, 290 (1981). 47 G. Kottra and E. Fr0mter, Pfluegers Arch. ,102, 409 (1984). 4s C. Clansen, S. A. Lewis, and J. M. Diamond, Biophys. J. 26, 291 (1979). 49 G. Kottra and E. FrOmter, Pfluegers Arch. ,102, 421 (1984). 5o C. Clansen, P. S. Reinach, and D. C. Marcus, J. Membr. Biol. 91, 213 (1986). 51j. R. Pappcnheimer, J. Membr. Biol. 100, 137 (1987).
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PARACELLULAR SHUNT CONDUCTANCE IN EPITHELIA
19
quency-dependent voltage divider ratio? 7 In experiments on Necturus gallbladder such measurements have clearly indicated that the distributed model (Fig. ld) is more appropriate to represent the epithelium than the lumped model, or in other words that R~, cannot be neglected in comparison with Rj in this tissue.49 A problem with the impedance analysis of leaky epithelia, which remained unsolved thus far, was the proper determination of R,. According to our experience R a values showed a large scatter with skewed distribution and did not respond consistently to perturbations that might have been expected to increase Ra. This problem seems to reflect an unfavorable distribution of resistance values in the Necturus gallbladder equivalent circuit under control conditions.52 Indeed, in gallbladder the major current flows across the shunt path and the small amount of current that passes across the cells produces only minor voltage changes across the cell membranes which may be lost in the scatter. However, further progress was achieved by combining the transepithelial impedance and frequencydependent voltage divider measurements with a three-point two-dimensional cable analysis.29 Together with the voltage divider ratio the latter data define R, and R~ more accurately and thereby restrict the variability of the fit to the impedance data. Comments. The combined transepithelial and intracellular impedance analysis has the following advantages. 1. Compared to all other methods discussed above it not only yields a lumped value for the shunt resistance but allows one to differentiate between the resistance of the tight junction Rj and the resistance of the lateral space R~. It thereby enables one to study variations of Rj, for example in response to neurovegetative stimuli, which was not possible previously due to overlapping changes in Rl~. 2. It allows resistance changes of all circuit parameters to be followed with time resolution of a few seconds, for example during apical or basolateral solution changes. This enables the experimenter to test the assumptions on which some of the above-discussed methods to calculate R~ are based. 3. It also allows changes in cell membrane capacitances to be followed that might occur in response to neurovegetative stimuli (insertion/removal of proteins and lipids into the cell membranes). The method has the following drawbacks: (1) From a technical point of view it is extremely involved; (2) due to the specific distribution of cell membrane and shunt resistances in leaky epithelia, the estimated cell membrane resistances exhibit large scatter so that the measurements have 52 p. Weskamp, Pfluegers Arch. 402, RI0 (1984).
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to be supplemented with simultaneously collected two-dimensional cable measurements; and (3) a particular shortcoming is that the method cannot simply be transferred to tubular epithelia, since DC current injection into the tubular lumen via a microelectrode leads to a double-exponential voltage decay along the tubular axis, which complicates the analysis of time or frequency-dependent signals enormously. Conclusions In this chapter we have reviewed the methods used in the past to determine the paracellular shunt conductance in epithelia. These methods differ in their technical complexity and in their appropriateness to represent the electrical properties of given epithelia correctly. Some even rely on assumptions which are difficult to prove so that the results can only be used with great reservation. Among the methods based on the lumped model circuits of Fig. 1a-c, the first method, direct comparison of cell membrane and transepithelial resistances, is the most reliable one, since it is not based on unproved assumptions. On the other hand, this method involves two-dimensional cable analysis with intracellular voltage measurements and microscopic distance measurements which must be performed with low enough current density and extreme care to yield reliable results. All methods based on the equivalent circuit of Fig. lb are technically less demanding but suffer from the uncertainty of whether or not the preconditions (perturbation of only one circuit element while all others remain constant) are indeed fulfilled. In addition, some of these methods are subject to the restriction of being applicable only to tight epithelia which perform rheogenic active transport. On the other hand, they offer the advantage of allowing the shunt conductance to be determined from simple transepithelial measurements without using even a single microelectrode. The applicability of those "macroscopic" methods, however, is restricted to epithelia which fulfill certain criteria (such as permselectivity of the apical cell membrane for Na+), which must be ascertained before USe.
The impedance techniques, finally, have the advantage that they are able to define a distributed equivalent circuit model of epithelia (Fig. ld) which allows the shunt resistance to be separated into its contributions from tight junctions and lateral intercellular spaces, and thereby also provides more reliable values for the apical and basal cell membrane resistances; but they have their own limitations and are technically rather involved. In summary, we conclude that an ideal, simple, and reliable method applicable to all epithelia does not exist. In order to obtain meaningful data
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PARACELLULAR SHUNT CONDUCTANCE IN EPITHELIA
21
the more complicated techniques have to be applied and the results from two or three different approaches must be compared before the data can be accepted as valid. Appendix A Method 1.2 for the calculation of cell membrane and shunt resistances depends on the assumption that only one resistance (generally R,) changes after the experimental perturbation of the system. If this condition is not fulfilled, and hence R~h and/or Rb~ change simultaneously, the calculation leads to erroneous values of all circuit parameters. The possible errors in the determination of R~h were in part analyzed in the past. 9 In this Appendix we analyze the errors made when either Rsh or Rb~ do not remain constant. The general case that both R~ and R~h change simultaneously with Ra leads to unpredictable results and hence will not be considered further here. Case I: R~h Changes, Rb~ Remains Constant
The following notations will be used. The true control values are denoted as R a, R~, R,h, and ot -- RJRb~; the experimental values are denoted as R'~, Rbl, R~h and a ' -- R'/Rbl; and the (erroneously) calculated control values are denoted as R*, R~, R*,h with Ct = Ra/Rb~. The ratio p of the change in G~h and GT will be denoted as p = AGsh/AGT
(A 1)
R* can be calculated from Eq. (5) as Ra*--- {[a(ot' - or)]/[(1 + t~)(1 + or')]) (1/AGr)
(A2)
To calculate the true value of R,, however, the change of G,h must be subtracted from the change in GT: R~ = ([ot(t~' - or)]/(1 + ot)(l + a')]}[1/(AGx - AG~)]
(A3)
Dividing Eq. (A2) by Eq. (A3) leads to R~/R, = (AGT - AG~,)/AGT = 1 - p
(A4)
Sincet~ = R~/Rbl = Ra/Rbi * * it follows that R*olRbl
=
1 - - 19
(A5)
Equations (A4) and (A5) show that a change of G,h into the same direction as the change of Gr (i.e., 0 < p < 1) will lead to an underestimation of the membrane resistances, while a change of G,h into the opposite direction (i.e., p < 0) will overestimate Ra and R s .
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The error in the determinations of R,h can be calculated from
I/RT = (1/R~) + [1/(Ra + R~)] -- (1/R~) + [ l / ( R * + R~)]
CA6)
Inserting R [ = R~(I - p) and R~ = Ru(1 - p) into Eq. (A6) and rearranging leads to
R*~_ 1 R,h [R~h](Ra+ Rta)](l - [I/(I -p)]) + I
(A7)
A n analysis of Eq. (A7) shows that the error in the determination of R.h does not only depend on the factorp, but also on the ratioR,h/(R. + RbO. W h e n thisratiois small, as in the case of leaky epithelia,the error becomes small.
Case II: R~ Changes, R~h Remains Constant Using the same notation as above in the control statewe have R,, Rt~, R~h, and a = Ra/Rb~; in the experimental state we have R~', R ~ = pR~, R~h, and o: ----R~/R~; and the (erroneously) calculated control values are R,*, Rb~,*R.h,* and a ----R~IR~ where p denotes the fractionalchange ofR,. The transepithelialresistancesin the control and experimental stateare
I/Rx = (I/R~ + [I/(R, + R~)] = (I/R,h + (i/[Rb~(I + a)]} (AS) and
I/R~. = (I/R.h) + [I/(R~'+ R Q ] -- (I/R,h) + (I/[PRb~ (I + a')]} (A9) From Eqs. (A8) and (A9) Rb~ can be expressed as Rb~ ----([p(l + a') -- (I + a)]/~(l + a)(l + a')])/AGT
(AI0)
O n the other hand, R ~ can be calculated using the assumption that p -- I: R ~ = ((a' - a)l[(l + a)(l + a')])IAGx
(AI I)
Dividing Eq. (A I I) by Eq. (A I0) and rearranging the resultsyields
R~/Rt~ = [p(a' - a)]l([p(l + a')] - (I + a)}
(A 12)
Since c~ = RJRbl----R ~*/ R u* the same expression is also true for the apical membrane: R ~ / R . =- [p(a" --
a)]/{[p(1 + a')] -- (1 + o0)
(AI3)
The error R ~ R ~ c a n b e calculated from Eq. (A7), case I; however, the expression (1 - p ) must be replaced by the right-hand side ofEq. (AI2):
R~/Roh ---1 [R.h/(Ra + Rb~)] (1 -- [p(1 + a') -- (1 +
a)]/~a'
--
a)]) + 1
(A14)
[9.]
23
PARACELLULAR SHUNT CONDUCTANCE IN EPITHELIA
A:1.3 2.0
R~/R~
METHODS IN ENZYMOLOGY
[9.]
processes by using tracer or fluorescence indicator techniques, as well as any which are not electrogenic. In a third instance an experimenter who works with the patch-clamp method can get signals from a cell in its natural surrounding by a cell-attached patch pipet. The worker can, however, also impose different driving forces and channel modifiers on an excised patch, but is restrained to transport processes which proceed through channels and are electrogenic. To analyze the molecular events of each transport process it is necessary to identify and separate the different transport proteins. This is achieved by affinity or photoaffinity labeling and the general procedure of protein separation and purification. Furthermore, reconstitution of a functional transport unit into liposomes is obligatory to ensure the retention of a protein subunit or lipid constituent during the separation procedure just mentioned. Since for some solutes, for instance Na +, HCO3-, and Ca2+, several transporters participate in transepithelial transport, the quantitative contribution of each of them is important. Thus, once the transporters and their location within the cell are known a balance sheet would be helpful. It certainly helps to prevent confusion, if (as in the proximal renal tubule) more than 15 different transporters are located at each cell side, half of them being Na + cotransporters (Fig. 2). In future studies those methods which identify the transporters in situ might be of great value, for instance, electron microscopic localization with monoclonal antibodies. Studies of transporters at the molecular level might utilize recombinant DNA techniques, introduction or attachment of molecules emitting molecular signals, production of mixed detergenttransporter crystals, and X-ray crystallography.
[2] D e t e r m i n a t i o n
o f P a r a c e l l u l a r S h u n t C o n d u c t a n c e in Epithelia
By G. KOTTRA and E. FROMTER Introduction Ever since it became clear that epithelia do not simply consist of two cell membranes in series but may have significant shunt pathways between the cells 1-4 researchers have been concerned with identifying the structural 1H. H. Ussing and E. E. Windlaager, Acta Physiol. Scand. 61, 484 (1964). 2 T. W. Clarkson, J. Gen. Physiol. 50, 695 (1967). 3 E. L. Boulpaep, in "Eleetrophysiology of Epithelial Cells" (G. G-iebiseh, ed.), p. 91. Sehattauer, Stuttgart, Federal Republic of Germany, 1971. 4 E. Frrmter, J. Membr. Biol. 8, 259 (1972).
METHODS IN ENZYMOLOGY, VOL. 191
Copyright© 1990 by Ac~lmnlc 15~s, Inc. All rish~ of ~ o d u c l l o n in any form tl~rved.
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PARACELLULAR SHUNT CONDUCTANCE IN EPITHELIA
5
equivalent of the shunt and quantifying the shunt permeability in relation to the permeability of the transcellular transport pathway. In epithelia (such as that in Necturus gall bladder) which exhibit a relatively simple microscopic structure and may be easily investigated with microelectrodes, the problem has been largely solved m at least so far as ion permeation is concerned--but in others, such as frog skin, the identification and quantification of the shunt path and the determination of its transport properties remains a considerable challenge. The classical approach to quantify the shunt conductance is to measure and compare transepithelial and cell membrane resistances. Since the latter resistances are often ditficult to determine a number of more or less indirect approaches have been devised. These are based on specific assumptions, the validity of which is usually di~cult to ascertain. In this chapter we attempt to critically review the electrophysiological methods which have been used in the past to quantify epithelial shunt conductances. Emphasis is on flat sheet epithelia. A number of specific problems involved with shunt analysis in tubular epithelia has been discussed in a methodological reviews and will be mentioned only briefly here. The methods to be discussed differ in their ability to represent the electrical properties of given epithelia appropriately. All are based on equivalent circuit representations of epithelia which, for various reasons, are gross oversimplifications. This is true, for example, for the circuits of Fig. l a - c (so-called "lumped" models) in which the epithelium is essentially represented by three elements: two resistors (R, and Rb~ for the apical and basolateral cell membrane, respectively) arranged in series and an additional resistor (R~h for the paracellular shunt) arranged in parallel. These models yield erroneous results whenever the resistance of the lateral space cannot be neglected. To account for lateral space effects, the "distributed" model (Fig. ld) of the lateral space and lateral cell membrane has been developed? In this model the shunt pathway consists of the tight junction resistance (Rj) and of a chain of resistors for the lateral intercellular space (r~, with R~ -- Er~). The distributed model is particularly important for leaky epithelia with low-resistance tight junctions. There are other more general shortcomings, however. Virtually all equivalent circuits treat all resistors as constant, although, in reality, the current voltage relation of cell membranes is usually nonlinear even in the narrow range of physiological membrane voltages ( - 9 0 to + 4 0 mY). This is not yet problematic so long as the analysis is restricted to resistance measurements (as in models l a and l c - d of Fig. l) and so long as the 5 E. Fr6mter, Kidneylnt. 30, 216 0986).
6
[2]
METHODS IN ENZYMOLOGY
I Ra
b
I tEl Rsh
Rbt
I
R="~
l
Eu T
Esh
R,h
Rbt~:
I
Za
R,~ Ca Ru~=
i Z tat
Rj
[ZT] q~7]
rtis
Rsh
Cbt
i Fro. 1. (a) Simple lumped model. (b) Lumped model with electromotive forces representing zero-current potentials Ei for each membrane element. (e) Lumped model including capacitances C. and Ct~ to represent the apical and basal cell membrane impedanev. (d) Distributed model; the Z elements denote parallel RC units for evil membrane impedanev5.
measurements are restricted to the linear range (determination of so-called "small signal" conductance or "slope" conductances at the normal operating point of the system). The analysis becomes problematic, however, if (slope) resistances and zero current potentials E0 are used together in one circuit as in Fig. lb and in methods 2.2, 2.3, and 2.4 below. If the resistance is nonlinear, extrapolation of the slope conductance to the zero current membrane voltage does not yield the proper E, or Eb~that may be equated with ion concentration ratios on the basis of the Nernst equation. 6,7 It is not surprising, therefore, that these methods have led to a number of inconsistent results. Another uncertainty comes up in impedance measurements (see methods 3.1 and 3.2 below). It relates to the fact that in reality biological membranes cannot be represented by ideal capacitors (e.g., Ca and Cu) 6 S. I. Helman and S. M. Thompson, Am. J. Physiol. 243, F519 (1982). 7 S. M. Thompson, Am. J. Physiol. 250, C333 (1986).
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PARACELLULAR SHUNT CONDUCTANCE IN EPITHELIA
7
because they may exhibit dielectric loss. In addition it may be sometimes difficult to separate membrane capacitances from polarization effects of unstirred layers, but the latter problem arises only at low-frequency currents (~- 1 to 10 Hz). 1. Methods Based on Equivalent Circuit l a
1. I Quantitative Comparison of Direct Current (DC) Cell Membrane Resistances and DC Transepithelial Resistance This method requires that the transepithelial resistance, RT, and the cell membrane resistances, R, and R~, be measured and compared. In contrast to RT, the determination of R, and R~ is usually quite difficult. Since most epithelial ceils are electrically coupled with their neighbors) a current that is injected into a single cell does not only pass across R, and R~ into the apical and basal fluid compartments, respectively, but spreads also radially within the epithelial cell layer. A theoretical analysis of this problem, the so-called two-dimensional cable analysis,4,9 predicts that radial voltage (V) attenuation within the ceil layer follows a first order Bessel function K 0 which can be defined by two parameters: a proportionality coefficient A and an apparent length constant 2:
V(x) = AKo(x/A)
( 1)
where x is the radial distance. From these parameters two resistances can be calculated: R,, which describes the resistance of the parallel arrangemerit o f / ~ and Rb~, and R,, which represents the coupling resistance for horizontal current flow in the ceil layer: Rz = [(1/Ra) + (l/Ru)]-' -- (2m422)/io Rx = (2~tA)/io
(2) (3)
where io is the injected current. On first view, to define the voltage attenuation within the cell layer may appear quite demanding and time consuming. However, once it has been verified that the voltage as function of distance x from the point of current injection obeys the proper Bessel function, it is sufficient to determine the current-induced voltage change at only two distances x, since two data points suffice to define the Bessel function and hence to determine A and it. In practice it may be preferable to collect more data points in order to average out individual measuring errors. 8 W. R. Loewenstein, Physiol. Rev. 61, 829 (1981). 9 L. Reuss and A. L. Finn, J. Gen. Physiol. 64, 1 (1974).
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METHODS IN ENZYMOLOGY
[2]
Alternatively, one can also try to circumvent cable analysis, and determine Rz by measuring the input resistance of an individual cell after electrical uncoupling. In practice, however, this approach is of little value, since uncoupling by depolarizing current ~° will certainly alter R, and Rb~, thus invalidating the entire concept and the same might happen with chemical uncouplers such as high CO2 pressure. Once Rz has been determined, R. and R~ can be calculated with the help of a, the direct current (DC) voltage divider ratio (VDR), which in the simple models of Fig. l a - c equals R,/Rm. Finally the shunt resistance, Rta, is obtained by comparing the sum o f R a and Rta with RT: R . ~- [(R a "3I- R b l ) R T ] / ( R a "31-Rbl -- RT)
(4)
The method can also be applied to tubular structures where the voltage attenuation along the tubular lumen and along the ceil column has to be measured separately. Both follow double-exponential functions from which the three resistances R,, Rbl , R e, as well as R T can be derived by data-fitting techniques.5,H-'3 Comments. The method is straightforward but technically rather involved since it requires the insertion of at least two (or better three) microelectrodes simultaneously into nearby cells, which must be recognized under the microscope and the distance between which must be accurately measured. In tubular epithelia an even greater number of measurements is necessary, again including intracellular recordings that must be combined with accurate distance measurements. The advantage of the method is that it does not rely on unproved assumptions such as constancy of zero-current potentials, which is a precondition for most of the other methods discussed below. On the other hand, the method has limited power, because it is based on the "lumped" model equivalent circuit. The problem manifests itself in the use of the voltage divider ratio, ct, which is taken to equal RJRb~, while in fact it may be lower ff Rt~ was overestimated due to a significant contribution of the lateral space resistance, Ru~. As a consequence the estimate of R~h will be in error, as has already been discussed in the literature) 4 Problems with the appropriateness of the measured a values have indeed been noticed in various epithelial preparations, including frog skin) 5 which is commonly held to obey the relation Rj >> R~, with Rt~ being negligible. lo S. A. Lewis, D. C. Eaton, and J. M. Diamond, J. Membr. Biol. 28, 41 (1976). i~ T. Anagnostopoulos, J. Teulon, and A. Edelman, J. Gen. Physiol. 75, 553 (1980). n T. Hoshi, tC Kawahara, R. Yokoyama, and K. Suenaga, Adv. Physiol. Sci. 2, 403 (1981). ~3W. B. Guggino, E. E. Windhager, E. L. Boulpaep, and G. Giebisch, J. Membr. Biol. 67, 143 (1982). t4 E. L. Boulpaep and H. Sackin, Curr. Top. Membr. Transp. 13, 169 (1980). ~s W. Nagel, J. F. C,arcia Diaz, and A. Essig, PJluegers Arch. 399, 336 (1983).
[2]
PARACELLULAR SHUNT CONDUCTANCE IN EPITHELIA
9
1.2. Paired Measurements OfRT and VDR in Response to Alteration of One Membrane Resistance This method consists of measuring R T and a while one surface of the epithelium is suddenly perfused with a solution that alters one or more ion transport mechanisms in the respective cell membrane by partial or total ion substitution, or by application of inhibitors or transport facilitators. The method is easier to apply than the above discussed method 1. l, since it requires only one intraceUular microelectrode (for determination of a). On the other hand, it is less reliable, since it depends on the precondition that the solution change affects only one single resistor of the circuit, leaving the others unchanged. If this precondition is fulfilled, and if R a was changed from R a to R ' , four independent data are available from which R,, R~, and R~, as well as R" can be determined. The respective equations which are presented here in the form published p r e v i o u s l y 16 w e r e first derived by Reuss and Finn 9 (see also Refs. 17 and 18): R, = [a(ot - a')RrR~]/(I + a)(1 + a')(RTV-- R~) Rbl = [(a -- a')RrR~]/(1 + a)(1 + a')(RT -- R) R,h = [(a -- a')RrR~]/[(l + ot)R~ -- (1 + a')Rr]
(5) (6) (7)
where t~ = R,/R~, a ' = R~/Rbl, and R~ is the transepithelial resistance measured after the pertubation. The same principles have been applied to tubular epithelia where RT must be determined by cable analysis. In that case, however, the determination of VDR presents a problem, because usually a ~ R,/R~, but changes from an overestimate of R,/R~ near the point of current injection to an underestimate at large distances from the current source. As analyzed previously 19 the error has a minimum close to x = 2, i.e., at around one length constant from the point of current injection. Comments. While technically speaking the method is comparatively simple and the calculations are straightforward the quantitative validity of the analysis depends on two conditions: (1) the accuracy of t~ as a measure of R,/Rbl (cf. lumped versus distributed model!) and (2) whether indeed only one resistor of the equivalent circuit of Fig. l a was altered and the others remained constant. Hence, when performing such experiments the following precautions must be taken. 1. The solution change must be very fast (depending on cell size, in the order of seconds) and it must be ascertained that only instantaneous 1~E. FrOmter and B. Crebler, Pfluegers Arch. 371, 99 (1977). 17 R. A. Frizzell and S. G. Schultz, J. Gen. Physiol. $9, 318 (1972). is T. Anagnostopoulos, J. Physiol. (London) 233, 375 (1973). 19 D. I. Cook and E. FrSmter, Pfluegers Arch. 4113, 388 (1985).
10
METHODS IN ENZYMOLOGY
[2]
changes of RT and ot are recorded. If solution change and recording are too slow, intracellular ion concentrations may change and this may secondarily alter the cell membrane resistances, or even the shunt resistance. 2. For each individual epithelium the proper experimental maneuver must be selected. In tight epithelia, such as frog skin, urinary bladder, or cornea, for example, amiloride can be given to suddenly increase Ri, or amphotericin B or gramicidin D to decrease itJ °,16;°,2~ Although such measurements yield valuable estimates of the lumped circuit parameters, some uncertainty remains with regard to the quantitative reliability of the analysis because of possible side effects on Rbl and/or R~. For example, if the basolateral cell membrane contains voltage-dependent conductances, hyper- or depolarization of this membrane might change Rbl- On the other hand, ionophores might not be restricted to the apical cell membrane but might also be taken up into the basolateral cell membrane. Moreover, as amiloride blocks tight junctions in leaky epithelia,22;3 it has been postulated that it exerts similar effects in tight epithelia, m24,25 In leaky epithelia, such as renal proximal tubule, small intestine, or gallbladder, the situation is similar. Here sugars and/or amino acids have been given to activate rheogenic Na+-substrate cotransport and thereby increase the apical cell membrane conductance as well as Ba2+ or high K + concentrations to increase or, respectively, decrease R a or Rb~.~s-2s In the former case, however, besides Ri, also R~ could have changed because of voltage-dependent conductances and/or R~ could have changed in response to changes in the rate of fluid transport. In the latter case, besides possible effects on Rbl from voltage-dependent conductances, Rj could have decreased by the presence of high K + or increased29 in response to Ba2+. We have tried to quantify the possible errors that can be made by neglecting such side effects in Appendix A. Generally speaking one can conclude that the greater the change of a during an experimental maneuver the more accurate the determination of R,a becomes. Also, the greater 2o N. IC Wills, S. A. Lewis, and D. C. Eaton, J. Membr. Biol. 45, 81 (1979). 21 S. A. Lewis and N. IC Wills, J. Membr. Biol. 67, 45 (1982). 22 R. S. Balaban, L. J. Mandel, and D. J. Benos, J. Membr. Biol. 49, 363 (1979). 23 G. Kottra and E. FrOmter, J. Exp. Biol. 106, 217 (1983). 24 R. G. O'Ncil and S. I. Helman, Am. J. Physiol. 231, 164 (1976). 2s L. G. M. Gordon, J. Membr. Biol. 52, 61 (1980). 26 E. FrOmtcr, PfluegersArch. 393, 179 (1982). 2~ R. Gregcr and E. Schlatter, Pfluegers Arch. 396, 315 (1983). E. Bello Reuss, J. Physiol. (London) 370, 25 0986). 29 G. Kottra and E. Fr6mter, PfluegersArch. 415, 7180990).
[2]
PARACELLULAR SHUNT CONDUCTANCE IN EPITHELIA
11
the value of Rsh in comparison to Ra plus Rbm, the more uncertain its accuracy becomes.
1.3. Plot of Gr versus VDR in Response to Alteration of One Cell Membrane Resistance A variant of technique 1.2 was described by Lewis et al. ~° It is based on the following equation which is equivalent to Eqs. (5-7) derived from the circuit in Fig. I a: GT = G,h + Gbl(1 -I- O~)-1
(8)
with GT = 1/Rr, Gb~=- l/Rbl, and G,h----- 1/R~. This equation can be applied to experiments in which RT and ~ are recorded while the apical cell membrane conductance is progressively changed by apical application of an ionophore, such as nystatin. If the data are plotted as Gr versus (1 + o0-I, Eq. (8) predicts a straight line which intercepts with the conductance axis at G,h and has a slope of Gs. Comments. The method is not superior to the approach defined by Eqs. (5-7) above, since the same problems may arise that have been commented on before. In particular it seems questionable whether R s does indeed remain constant during the progressive action of the ionophore as the theory requires, since the experimental design does not fulfill the criterion of instantaneous action and measurement. In fact, data which have been obtained with this method and were published in conformation with Eq. (8) might as well be interpreted as exhibiting a systematic curvilinear relationship, rather than the postulated straight line. 2° 2. Methods Based on Equivalent Circuit l b
2.1. Measurements of Rr, VDR, and of Transepithelial and Cell Membrane Potentials in Response to Alteration of One Membrane Resistance This method does not differ much from method 1.2. Again a solution change is performed which should affect the ionic conductance of only one cell membrane, and the effect on a is recorded as well as the change in transepithelial potential difference, AVT, and the change in membrane potential across the contralateral cell membrane, A V. or AV~. In addition RT must be known. If, for example, the apical membrane is altered but Eu, E,h, Rb~ and R,h remain constant, we obtain R,h = [(1 + a + ~/(1 + a)]Rr where ot = R./Rbl and fl ffi AVbJAVr.
(9)
12
METHODS IN ENZYMOLOGY
[2]
This approach differs from method 1.2, in that fl instead of a' is used for the calculations. However, since ~' is also known, the results of both approaches can be directly compared. Comments. Essentially the same preferences and restrictions apply and the same precautions should be observed which have been discussed with regard to method 1.2 above. With method 2.1, however, it is particularly important to make fast solution changes and to record the potential changes instantaneously, since the additional restriction applies that some of the zero-current potentials be constant. The method has been used in Necturus urinary bladder in which the Na + channel blocker amiloride was applied to the apical surface. ~6 A comparison of methods 1.2 and 2.1 in those experiments gave virtually identical results. Alternatively, combining methods 1.2 and 2.1 one obtains five data points from which a total of five circuit parameters can be determined. Hence, ifa solution change affects R, and R,h simultaneously, both R: and R~h can be determined)° However, the validity of such calculations rests entirely on whether or not Rbl and Eb~ remain constant. This assumption, however, cannot be tested independently unless more sophisticated techniques are applied (see below).
2.2. Measurements of Rr, VDR, and of the Relationship between Cell Membrane Potential and Short Circuit Current (1.~) Disregarding the fundamental problems with the circuit of Fig. I b, various methods have been devised for determination of shunt conductances in tight epithelia such as amphibian skin and urinary bladder which generate significant short circuit currents (I,,). If an epithelium is short circuited, current flow across R,h vanishes, and V, = V~. If I~ is now altered by applying, for example, amiloride apically the simplified treatment of circuit lb yields the following equation: V, - Vbl----Ebl -- RblI~
(10)
which predicts a straight-line relationship between Vbland I.~ provided E~ and Ru remain constant) ~,32 From the slope of this relation Ru can be directly determined. If in addition RT and cz are measured in the same experiments, R, and R.h can be readily calculated. In analogy to that, I,~ can be altered by applying ouabain basolaterally to determine R,. Comments. The method is technically rather simple, but the validity of the data depends critically on whether or not the precondition (constancy ~oS. Henin, D. Cremaschi, T. Schettino, G. Meyer, C. L. L. Donin, and F, CoteUi, £ Membr. Biol. 34, 73 (1977). 31 S. G. Schultz, R. A. Frizzell, and H. N. Nellans, £ Theor. Biol. 65, 215 (1977). 32S. I. Helman and R. S. Fisher, J. Gen. Physiol. 69, 571 (1977).
[2]
PARACELLULAR SHUNT CONDUCTANCEIN EPITHELIA
13
of E~) is fulfilled or, in other words, whether the conductances are constant in the entire range between open circuit and short circuit. By applying increasing concentrations of amiloride Nagel 3a attempted to derive the current-voltage ( I - 1I) relation of the basolateral cell membrane with this approach; however, the results showed marked nonlinearity and in some cases even slopes with negative resistances. In addition, comparison of the shunt resistances calculated with methods 1.2 and 2.2 from the same experiments showed discrepancies of between 0.185- and 1.13-fold. a3 This indicates that Eta was not constant or that the circuit of Fig. 1b did not adequately describe the data. 2.3 Plots of Transepithelial Conductance versus I~ in Response to Alteration of One Cell Membrane Resistance
While the previously discussed methods require intraceUular measurements with one or two microelectrodes it was desirable that methods be found which allow the shunt conductance to be determined exclusively from macroscopic measurements. From the viewpoint of principle this is, of course, impossible since one cannot expect to obtain insight into the structure of an unknown membrane by simply studying its input/output relations. On the other hand, it might have been possible that in some cases enough information was already available to justify a purely macroscopic approach. Based on present-day knowledge of epithelial transport a method has been devised to determine the shunt resistance of tight epithelia by appropriately varying the apical cell membrane resistance and simply measuring Gr = 1/R T and the short circuit current, I,~. Provided Rta, R,h, E,, and Eta remain constant the equivalent circuit of Fig. 1b yields Gx = G~ + (E, + Eb~)-'I~
(1 1)
Accordingly, if the preconditions are fulfilled, plotting the data as Gr versus I~ should show a linear relationship with an intercept on the conductance axis equal to G~h = 1/R,a (see Fig. 2a). Alternatively the same information can be obtained from a plot of Vx versus R x according to the following equation where, in the ideal case, the linear relation between Vx and RT intercepts with the resistance axis at R~: 1 -= [VT/(Ea + Eta)] + (RT/Rsh)
(12)
The introduction of this concept goes back to a paper by Yonath and Civan, 34 who apparently used the word shunt in the sense of comprising all 33W. Nagel,J. Membr. Biol. 42, 99 (1978). 34j. Yonathand M. M. Civan, J. Membr. Biol. 5, 366 (1971).
14
METHODS IN ENZYMOLOGY
[2]
1.0 g-.,
g-.
l ,-g
E0.5
2'~
/~ (/~A/cm 2)
5'o
o
2's
5'o
/.~ (uA/cm 2)
Flo. 2. UT vs I,~ plots obtained with 1 m M novobiocin on two frog skins. The triangles denote the G,h values determined with amiloride. (From Rick et al.~9)
ion flux mechanisms in parallel with the Na + pump; however, the concept was later reinterpreted by Lewis and collaborators,~° who also derived Eq. (12) as a method for determination of the paracellular shunt conductance.2° This method has already been widely used in tight epithelia such as frog skin, urinary bladder, colon, and others.2°,2~,35-39 Ra was either increased by increasing concentrations of amiloride or decreased by ionophores such as amphotericin B, gramicidin D, nystatin, novobiocin, silver ions, or by stimulation with vasopressin. Comment. The method is subject to the general criticism of the underlying model circuit (Fig. lb; see Introduction). In addition, its application is restricted to epithelia whose properties are already largely known. It is not capable of identifying paracellular shunts in epithelia which have not yet been further investigated. Furthermore, the quantitative validity of the derived data must be considered with precaution, because usually the measurements do not fulfill the criteria of fast perturbation and instantaneous recordings. Indeed, deviations from a straight-line relationship between GT and I,~ have often been observed (see Fig. 2b) and R~ values derived either by decreasing Ra with ionophores or increasing R, with amiloride usually differ from each other2° (see also Fig. 2). 35 L. C. Isaacson, J. Membr. Biol. 30, 301 (1977). 36 p. U. Feig, G. D. Wetz¢l, and H. S. Frazicr, Am. J. Physiol. 232, F448 (1977). 37 W. S. Marshall and S. D. Klycc, J. Membr. Biol. 73, 275 (1983). 3s N. K. Wills and C. Clauscn, J. Membr. Biol. 95, 21 (1987). 39 R. Rick, A. D6rgc, and E. Sessclmann, PfluegersArch. 411, 243 (1988).
[2]
PARACELLULAR SHUNT CONDUCTANCE IN EPITHELIA
15
In Appendix B we report some model calculations which define the validity limits of the method quantitatively. From this we deduce that the method is restricted to the simple model of tight Na+-absorbing epithelia in which R , is exclusively selective for Na + and to the use of ionophores that are also purely selective for Na +. Alternatively, if the cell membrane is permeable to more than one ion species present, an ionophore must be used whose relative selectivity properties are identical with the relative ionic permeabilities of the respective cell membrane in the control state. Note: Even a straight-fine relationship does not prove that the estimate of G,h from the intercept is correct; also, the method is not able to distinguish a paracellular shunt from a transceUular C1- shunt or K + shunt. A special case of method 2.3 is the determination of R.~ by blocking the apical cell membrane conductance with amiloride. If the apical cell membrane was exclusively permeable to Na +, RT in the presence of amiloride should equal R,h. This approach has been widely used in frog skin and other tight epithelia, particularly in O--free solutions. ~ On the other hand, it is also known that K + channels are present in the apical cell membrane of the skin of at least some frog species4~,42 and the question about a possible effect of amiloride on Rj is still open.
2.4. Steady State Measurements of V.r, R.r, VDR, and of the Intracellular Activity of an Ion Species for Which One of the Cell Membranes Is Permselective This method was published recently. 4a It is based on the analysis of current flow in an equivalent circuit such as that depicted in Fig. lb, with the restriction that all current across one cell membrane (say membrane a) is exclusively carried by one ion species (t). If the intracellular activity of i can be measured the following equation allows calculation of R,h: [ Vr t~ ] R ~ = RT 1 -- [Va + (RT/ziF)] ln(~da~,o) 1 + a
(13)
where zi, F, R, and T are valence, Faraday constant, gas constant, and absolute temperature and a~c and a~o are the intra- and extracellular activities of the permeating ion species. A similar equation has been derived for the short-circuited state in which the I~ appears rather than VT. Comment. The main advantage of the method is the fact that only steady state measurements are required. This eliminates a number of 4oj. Warncke and B. Lindemann, J. Membr. Biol. 86, 255 (1985). 4~ W. Zeiske and W. Van Driessche, J. Membr. Biol. 47, 77 (1979). 42 W. Nagel and W. Hirsehmann, J. Membr. Biol. 52, 107 (1980). 43 N. J. Willumsen and R. C. Boucher, Am. J. Physiol. 256, C1054 (1989).
16
METHODS IN ENZYMOLOGY
[2]
problems discussed above. On the other hand, as with the preceding two methods (2.2 and 2.3), this method relies on the assumption that membrane resistances are linear in the range between open and closed circuit, which normally is not true. Other, more technical, problems arise if other ion conductances must be blocked to establish the proper measuring conditions or if ion-selective microelectrode readings must be corrected for interferences.
2.5 Determination of R~ from Break Point in the Transepithelial Current Voltage Relation This method is based on a special form of the equivalent circuit lb. When studying transepithelial current-voltage (I-V) relations in frog skin, Helman and Fischer32 observed that the I - Vcurve consisted of linear segments with distinct break points in between, where the slope suddenly changed to a new constant value. These observations suggested that the apical cell membrane rectified and hence could be modeled by a parallel arrangement of two resistors of different magnitude, each having an ideal diode in series to allow either only inflow or outflow of current. If this model is correct the transition point between two segments in the I - V curve should then be the point at which current flow across the apical cell membrane and hence across the entire cell vanishes. One can conclude then that under those conditions all transepithelial current flows across the shunt path so that R,a can be directly calculated by dividing the break point voltage through the applied current. Comments. Although this method is extremely simple and requires only transepithelial measurements it has only rarely been used in the past, probably because the break points in the I - V relations are not always well-enough defined. This may be related to the fact that in principle the applicability of the method may be restricted to epithelia having an apical cell membrane that is exclusively selective for Na ÷. Any other parallel ion conductances may be expected to reduce the sharpness of the break point. 3. Methods Based on Equivalent Circuits l c and d (Impedance
Techniques) Unfortunately, measurements of the frequency-dependent transepithelial impedance alone do not allow one to recognize and quantify the presence of a shunt path. This derives from the impossibility of identifying parallel elements inside a two-terminal network by simply studying its input-output relations. (In contrast, it is well known that serial RC elements inside a two-pole network can be recognized provided they cause
[2]
17
PARACELLULAR SHUNT CONDUCTANCE IN EPITHELIA
relaxations in the frequency-dependent spectra that are distant enough to be separated.)
3. I. Measurements of Transepithelial Impedance and of VDR in the Time Domain If a constant current pulse is passed across an epithelium, the equivalent circuit in Fig. l c predicts that the voltage response will rise with a double-exponential time course toward a new steady value44,45:
AVT(t) = [K~(1 - e-U,,) + K2(1 - e-U,,)]I
(14)
where I is the transepithelial current and the parameters K~, K2, z~, and r2 are functions of all circuit elements. Note that the time constants z~ and T2 differ from the intrinsic time constants of the individual cell membranes (za = RaC, and z~ = R~Cbl) except in the ease G,~ = 0 (for details see Ref. 44). The analysis of this time course yields four independent parameters, which do not suffice to determine the five elements of circuit l c. Consequently a further independent measurement is required, w h i c h - - i n addition--must contribute information from inside the cell layer. Such a measurement would be the DC voltage divider ratio. Hence, with these rather simple measurements the circuit is already fully defined. In reality, however, the approach is less promising because fitting such data with two exponential functions is usually difficult unless the time constants differ by nearly one order of magnitude or more. The technique can be improved if not only the DC voltage divider ratio is recorded, but the full rising flank of the intracellular voltage in response to the transepithelial constant current step: AVe(t) = [Ks (1 - e-u',) + K4(1 - e-U")]I
(15)
This time course of the intracellular potential response, which also obeys a double-exponential function and has the same time constants ~, and z2, allows the data to be fitted more easily. This is particularly true for the voltage response across the cell membrane with the smaller RC product, which exhibits a transient overshoot resulting from superposition of a fast positive and a slow negative exponential (Ks < 0, K4 > 0). Comments. In spite of the simple and straightforward principle, impedance measurements in the time domain cannot be recommended for determination of the shunt resistance. The main problem is associated with fitting two exponentials in the data analysis (see above). From a technical K. Suzuki, G. Kottra, L. Kampmann, and E. Fr6mter, PfluegersArch. 394, 302 (1982). 4s S. A. Lewis and J. L. C. DeMoura, J. Membr. Biol. 82, 123 (1984).
18
METHODS IN ENZYMOLOGY
[2]
point of view the method is quite demanding, since analysis of the intracellular voltage rise time requires shielded intracellular microeleetrodes to be used. In addition, compared to the experimental investment the results are of limited use since the underlying equivalent circuit (lumped model) is generally too simple. Once the experimental set-up is available it would always be preferable to perform measurements in the frequency domain, which yields more explicit information.
3.2. Measurements of the Transepithelial Impedance and of the VDR in the Frequency Domain In these experiments, instead of a square wave pulse, a series of sine wave currents of constant amplitude is passed across the epithelium, either as preprogrammed single-frequency bursts, 4° or as pseudo-random binary sequence (PRBS) noise,~ or as multifrequency superimposed sine wave bursts (MSSB), obtained by inverse Fourier transform from a selected sequence of logarithmically spaced frequencies.47 The relative advantages of the different methods have been discussed.47 In the latter two cases the spectral content of the voltage response is obtained by fast Fourier transform techniques: By dividing the cross-power spectrum through the autopower spectrum of the current signal the frequency-dependent impedance is obtained, which can be represented either as a Bode48 diagram (logarithm of amplitude and phase versus logarithm of frequency) or as a Nyquist plot47 (imaginary versus real component of impedance). The data are then fitted with the appropriate equations which describe the impedance of circuit I c and ld from which quantitative estimates of the circuit parameters can be derived.45,49 As in the case of the time domain analysis mentioned above, transepithelial data of this type alone are not sufficient to determine the five elements of the equivalent circuit of Fig. lc. However, under specific conditions (Rj ~ 0o) and after inclusion of additional information (e.g., the DC voltage divider ratio) it is possible to determine not only the elements of Fig. 1c but to analyze even the distributed equivalent circuit of Fig. ld. This means that the lateral intercellular space resistance can be resolved.38.50,5i Further progress and confirmation of the appropriateness of the analysis and the validity of the results were obtained by combining the transepithelial impedance analysis with simultaneous recordings of the freC. Clausen and J. M. Fernandez, Pfluegers Arch. 390, 290 (1981). 47 G. Kottra and E. Fr0mter, Pfluegers Arch. ,102, 409 (1984). 4s C. Clansen, S. A. Lewis, and J. M. Diamond, Biophys. J. 26, 291 (1979). 49 G. Kottra and E. FrOmter, Pfluegers Arch. ,102, 421 (1984). 5o C. Clansen, P. S. Reinach, and D. C. Marcus, J. Membr. Biol. 91, 213 (1986). 51j. R. Pappcnheimer, J. Membr. Biol. 100, 137 (1987).
[2]
PARACELLULAR SHUNT CONDUCTANCE IN EPITHELIA
19
quency-dependent voltage divider ratio? 7 In experiments on Necturus gallbladder such measurements have clearly indicated that the distributed model (Fig. ld) is more appropriate to represent the epithelium than the lumped model, or in other words that R~, cannot be neglected in comparison with Rj in this tissue.49 A problem with the impedance analysis of leaky epithelia, which remained unsolved thus far, was the proper determination of R,. According to our experience R a values showed a large scatter with skewed distribution and did not respond consistently to perturbations that might have been expected to increase Ra. This problem seems to reflect an unfavorable distribution of resistance values in the Necturus gallbladder equivalent circuit under control conditions.52 Indeed, in gallbladder the major current flows across the shunt path and the small amount of current that passes across the cells produces only minor voltage changes across the cell membranes which may be lost in the scatter. However, further progress was achieved by combining the transepithelial impedance and frequencydependent voltage divider measurements with a three-point two-dimensional cable analysis.29 Together with the voltage divider ratio the latter data define R, and R~ more accurately and thereby restrict the variability of the fit to the impedance data. Comments. The combined transepithelial and intracellular impedance analysis has the following advantages. 1. Compared to all other methods discussed above it not only yields a lumped value for the shunt resistance but allows one to differentiate between the resistance of the tight junction Rj and the resistance of the lateral space R~. It thereby enables one to study variations of Rj, for example in response to neurovegetative stimuli, which was not possible previously due to overlapping changes in Rl~. 2. It allows resistance changes of all circuit parameters to be followed with time resolution of a few seconds, for example during apical or basolateral solution changes. This enables the experimenter to test the assumptions on which some of the above-discussed methods to calculate R~ are based. 3. It also allows changes in cell membrane capacitances to be followed that might occur in response to neurovegetative stimuli (insertion/removal of proteins and lipids into the cell membranes). The method has the following drawbacks: (1) From a technical point of view it is extremely involved; (2) due to the specific distribution of cell membrane and shunt resistances in leaky epithelia, the estimated cell membrane resistances exhibit large scatter so that the measurements have 52 p. Weskamp, Pfluegers Arch. 402, RI0 (1984).
20
METHODS IN ENZYMOLOGY
[2]
to be supplemented with simultaneously collected two-dimensional cable measurements; and (3) a particular shortcoming is that the method cannot simply be transferred to tubular epithelia, since DC current injection into the tubular lumen via a microelectrode leads to a double-exponential voltage decay along the tubular axis, which complicates the analysis of time or frequency-dependent signals enormously. Conclusions In this chapter we have reviewed the methods used in the past to determine the paracellular shunt conductance in epithelia. These methods differ in their technical complexity and in their appropriateness to represent the electrical properties of given epithelia correctly. Some even rely on assumptions which are difficult to prove so that the results can only be used with great reservation. Among the methods based on the lumped model circuits of Fig. 1a-c, the first method, direct comparison of cell membrane and transepithelial resistances, is the most reliable one, since it is not based on unproved assumptions. On the other hand, this method involves two-dimensional cable analysis with intracellular voltage measurements and microscopic distance measurements which must be performed with low enough current density and extreme care to yield reliable results. All methods based on the equivalent circuit of Fig. lb are technically less demanding but suffer from the uncertainty of whether or not the preconditions (perturbation of only one circuit element while all others remain constant) are indeed fulfilled. In addition, some of these methods are subject to the restriction of being applicable only to tight epithelia which perform rheogenic active transport. On the other hand, they offer the advantage of allowing the shunt conductance to be determined from simple transepithelial measurements without using even a single microelectrode. The applicability of those "macroscopic" methods, however, is restricted to epithelia which fulfill certain criteria (such as permselectivity of the apical cell membrane for Na+), which must be ascertained before USe.
The impedance techniques, finally, have the advantage that they are able to define a distributed equivalent circuit model of epithelia (Fig. ld) which allows the shunt resistance to be separated into its contributions from tight junctions and lateral intercellular spaces, and thereby also provides more reliable values for the apical and basal cell membrane resistances; but they have their own limitations and are technically rather involved. In summary, we conclude that an ideal, simple, and reliable method applicable to all epithelia does not exist. In order to obtain meaningful data
[2]
PARACELLULAR SHUNT CONDUCTANCE IN EPITHELIA
21
the more complicated techniques have to be applied and the results from two or three different approaches must be compared before the data can be accepted as valid. Appendix A Method 1.2 for the calculation of cell membrane and shunt resistances depends on the assumption that only one resistance (generally R,) changes after the experimental perturbation of the system. If this condition is not fulfilled, and hence R~h and/or Rb~ change simultaneously, the calculation leads to erroneous values of all circuit parameters. The possible errors in the determination of R~h were in part analyzed in the past. 9 In this Appendix we analyze the errors made when either Rsh or Rb~ do not remain constant. The general case that both R~ and R~h change simultaneously with Ra leads to unpredictable results and hence will not be considered further here. Case I: R~h Changes, Rb~ Remains Constant
The following notations will be used. The true control values are denoted as R a, R~, R,h, and ot -- RJRb~; the experimental values are denoted as R'~, Rbl, R~h and a ' -- R'/Rbl; and the (erroneously) calculated control values are denoted as R*, R~, R*,h with Ct = Ra/Rb~. The ratio p of the change in G~h and GT will be denoted as p = AGsh/AGT
(A 1)
R* can be calculated from Eq. (5) as Ra*--- {[a(ot' - or)]/[(1 + t~)(1 + or')]) (1/AGr)
(A2)
To calculate the true value of R,, however, the change of G,h must be subtracted from the change in GT: R~ = ([ot(t~' - or)]/(1 + ot)(l + a')]}[1/(AGx - AG~)]
(A3)
Dividing Eq. (A2) by Eq. (A3) leads to R~/R, = (AGT - AG~,)/AGT = 1 - p
(A4)
Sincet~ = R~/Rbl = Ra/Rbi * * it follows that R*olRbl
=
1 - - 19
(A5)
Equations (A4) and (A5) show that a change of G,h into the same direction as the change of Gr (i.e., 0 < p < 1) will lead to an underestimation of the membrane resistances, while a change of G,h into the opposite direction (i.e., p < 0) will overestimate Ra and R s .
22
METHODS IN E N Z Y M O L O G Y
[2]
The error in the determinations of R,h can be calculated from
I/RT = (1/R~) + [1/(Ra + R~)] -- (1/R~) + [ l / ( R * + R~)]
CA6)
Inserting R [ = R~(I - p) and R~ = Ru(1 - p) into Eq. (A6) and rearranging leads to
R*~_ 1 R,h [R~h](Ra+ Rta)](l - [I/(I -p)]) + I
(A7)
A n analysis of Eq. (A7) shows that the error in the determination of R.h does not only depend on the factorp, but also on the ratioR,h/(R. + RbO. W h e n thisratiois small, as in the case of leaky epithelia,the error becomes small.
Case II: R~ Changes, R~h Remains Constant Using the same notation as above in the control statewe have R,, Rt~, R~h, and a = Ra/Rb~; in the experimental state we have R~', R ~ = pR~, R~h, and o: ----R~/R~; and the (erroneously) calculated control values are R,*, Rb~,*R.h,* and a ----R~IR~ where p denotes the fractionalchange ofR,. The transepithelialresistancesin the control and experimental stateare
I/Rx = (I/R~ + [I/(R, + R~)] = (I/R,h + (i/[Rb~(I + a)]} (AS) and
I/R~. = (I/R.h) + [I/(R~'+ R Q ] -- (I/R,h) + (I/[PRb~ (I + a')]} (A9) From Eqs. (A8) and (A9) Rb~ can be expressed as Rb~ ----([p(l + a') -- (I + a)]/~(l + a)(l + a')])/AGT
(AI0)
O n the other hand, R ~ can be calculated using the assumption that p -- I: R ~ = ((a' - a)l[(l + a)(l + a')])IAGx
(AI I)
Dividing Eq. (A I I) by Eq. (A I0) and rearranging the resultsyields
R~/Rt~ = [p(a' - a)]l([p(l + a')] - (I + a)}
(A 12)
Since c~ = RJRbl----R ~*/ R u* the same expression is also true for the apical membrane: R ~ / R . =- [p(a" --
a)]/{[p(1 + a')] -- (1 + o0)
(AI3)
The error R ~ R ~ c a n b e calculated from Eq. (A7), case I; however, the expression (1 - p ) must be replaced by the right-hand side ofEq. (AI2):
R~/Roh ---1 [R.h/(Ra + Rb~)] (1 -- [p(1 + a') -- (1 +
a)]/~a'
--
a)]) + 1
(A14)
[9.]
23
PARACELLULAR SHUNT CONDUCTANCE IN EPITHELIA
A:1.3 2.0
R~/R~
