The surface compartment model (SCM) with a voltage-sensitive channel

Bioeleetrochemistry and Bioenergetics, 10 (1983) 451-465 A section of J. Electroanal. Chem., and constituting Vol. 155 (1983)

45l

Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands

5 7 4 - T H E S U R F A C E C O M P A R T M E N T M O D E L (SCM) W I T H A VOLTAGE-SENSITIVE CHANNEL *

MARTIN BLANK

Department of Physiology, College of Physicians & Surgeons, Columbia University, 630 W. 168 St., New York, N Y 10032 (U.S.A.) (Manuscript received February 5th 1983)

SUMMARY We have used the surface compartment model (SCM) to study the effects of surface concentrations, charges, potentials and capacitances in the electrical double-layer regions of excitable membranes. The ion fluxes that arise during voltage clamp transients have helped to characterize the properties of ion channels [Blank et al., Bioelectrochem. Bioenerg. 9 (1982) 439] and to develop a model for a voltage-sensitive channel [Blank, ibid., 9 (1982) 615]. When a non-selective channel is included in the SCM, it is possible to obtain currents like those normally observed during voltage clamp (i.e. an early inward and late outward current). The early peak and late steady-state currents are functions of clamp voltage, and it is possible to account for many of the observed changes in the currents under particular experimental conditions in terms of changes in membrane properties. The SCM appears to be a useful model for excitable membranes during voltage clamp.

INTRODUCTION

Natural membranes are complex structures, but to the physiologist studying ion transport, they have usually been considered a single barrier. In 1935, Teorell [1] introduced the idea that the membrane potential includes the effects of two-phase boundary potentials at the surfaces, in addition to a diffusion potential. Since that time, much has been learned about electrostatic potentials and ion adsorption at membrane Isolution interfaces [2]. The numbers of fixed surface charges in squid axon membranes have been estimated [3], and the effects of relatively large surface capacitances have been considered [4]. Other surface properties, such as the increased surface concentrations due to ion binding and transference have also been discussed [5]. All of these properties should influence ion transport, but a complete description of their effects is very difficult. Recently, there have been attempts to include some surface properties in the * Because of computer requirements the symbols used in this paper sometimes differ from those given in the list of symbols customarily used in this periodical. 0302-4598/83/$03.00

© 1983 Elsevier Sequoia S.A.

452

analysis of passive ion transport processes. Pilla and co-workers have described the properties of epithelial [6] and red cell [7] membranes in response to an imposed electrical perturbation. Also, Lorenz and Schulze [8,9] have dealt with similar problems on the theoretical level, and have shown the effects of interfacial control and adsorption on charge-transfer processes. As a first approach to studying the simultaneous influences of several surface properties on ion transport, the surface compartment model (SCM) approximation was developed [10,11]. In the model, the electrical double layers at the surfaces of natural membranes are treated as compartments. Specifically, the SCM includes two surface potentials and two surface capacitances in addition to the dielectric capacitance of the membrane. It also deals with ion concentration changes and cation binding and release processes in the surface layers. Using the SCM, it has been possible to show that an early sodium ion influx can occur in an excitable membrane as a result of a non-specific increase in the permeability to cations [ 12]. Initial studies of the dependence of the early sodium ion flux on the parameters of the system, have suggested ranges of physical properties for the voltage-sensitive ion channel. For example, the results indicate that cation mobility is low in the surface compartment region, cation binding is strong and the cation release rate is fast. This information has enabled us to develop a gating mechanism based on the observed charge-sensitive dissociation equilibria of oligomeric proteins [13]. (The dissociation of hemoglobin tetramers into dimers occurs when the surface charge is approximately the same as on the inner face of the squid giant axon.) The model for the channel is an oligomeric protein that is dissociated (i.e. open) on the outer surface, where the charge is high, and associated (i.e. closed) on the inner surface, where the charge is low [14]. Upon depolarization, the shift of negative charge from the outer to the inner surface causes the channel to open. The fraction of open channels can be calculated as a function of the charge density, assuming the constants that apply to the hemoglobin system. Qualitatively, the properties of the oligomeric pore model of the voltage sensitive ion channel are compatible with observed behavior [14], and therefore the model can serve as a basis for introducing channels into the SCM. A VOLTAGE-SENSITIVE C H A N N E L IN THE SCM

To describe the voltage-dependent permeability of an excitable membrane (e.g. squid axon), we have introduced an oligomeric protein (14) channel into the SCM that is subject to charge-sensitive dissociation (opening). To approximate the dependence of the ion permeability G on the (inner) surface charge density o, we have used the following empirical equation:

c =

P2(o/o0)

(1)

+ (o/o0/2 By an appropriate choice of the constants P1, P2 and o0, it is possible to have G vary

453

-7

k

\

"xx

""

3N \2 -9

t.9

o -IC

t i

i

I

"2

(ms)

|

3

Fig. 1. The logarithm of the membrane permeability to sodium GN, as a function of time t (in milliseconds). The numbers on the curves correspond to log GX, the mobility of the negative charge associated with the gating process. The curves were obtained at two clamp voltages. ( ) For + 20 mV; ( - - - - - - ) for - 2 0 mV.

with o over a particular range of permeabilities at given depolarizations. For both sodium and potassium ions, we have chosen o0 as the value of o at zero time and P1 as the value that yields the proper magnitude of G for each ion at zero time. Here, P2 was set to obtain higher membrane permeabilities for both cations upon depolarization. The permeability of the membrane to cations actually depends upon two factors: GX, the rate of movement of charge associated with gating, a n d / ' 2 , which is related to the elevated permeability when the channels open. In Fig. 1 we see the sodium permeability GN as a function of time for different magnitudes of the negative charge mobility in the membrane. (The membrane permeability to potassium, GK, is slightly higher than the sodium permeability when the channels are open.) Measurements of gating currents [15] suggest that the bulk of the surface charge has shifted within 50/xs, so we chose a rate constant of 10 -3 (s m V ) - l , as in the curve labeled 3. The two sets of curves in Fig. 1, which apply to clamp voltages of + 20 mV and - 20 mV, rise in the same way, but fall back to lower values of GN at different rates which depend upon the ionic processes of the SCM. The magnitude of GX was chosen to obtain the correct rate of the initial gating current, but the later currents, which are more sensitive to GX, may be somewhat too low. (The rapid decay rate of

454

0.2

0~

, log P2 -6

-5

l -4

Fig. 2. The inward peak current during voltage clamp, Ip (in mA/cm2), and the time of the peak, t (in milliseconds), versus the logarithm of P2, the constant that sets the permeability of the membrane when the channels open. The clamp voltage is - 2 0 mV.

the permeability at certain values of clamp voltage suggests a possible mechanism for the desensitizing effect of a prolonged subliminal stimulation.) The magnitude of the permeability change also depends upon the value of/)2, which was generally assumed to be several orders of magnitude greater than the resting level. The magnitudes of the peak inward currents that result from different settings of P2 are shown in Fig. 2. Values of about 10 -5 c m / ( s mV) give peak currents and times that are in line with those observed in squid axon [16]. Both GX and P2 were chosen to produce current characteristics similar to those measured during gating and at the peak of the voltage clamp current. The voltage sensitivity of the conductance increase has been established experimentally, and this feature is included in order to study the properties of the SCM under appropriate conditions. The SCM mechanisms to be considered below are, therefore, independent of these assumptions and should be considered separately. THE VOLTAGE CLAMP CURRENTS

Using the SCM program for voltage clamp [12], together with equation (1) for achieving an increase in ion permeability upon depolarization, we obtain the membrane currents shown in Fig. 3. At the values of the parameters indicated in Table 1 and the same initial values as used earlier [12], there is an inward (positive) current, followed by an outward (negative) current, when the SCM is put through a voltage clamp. The currents are qualitatively as observed in voltage clamp experiments on squid axons [16].

455

L TEA

0

out

E

-1

TTX

t (ms) 0

I

Fig. 3. The membrane current I (in n~LA/cm 2) v e r s u s the time t (in n~l|iseconds),at a clamp voltage of - 2 0 inV. The current obtained under standard conditions shows an early inward and a later outward direction. The curve labeled TEA is obtained when G K = 0, and the curve labeled TTX is obtained when G N = O.

When the permeability to potassium is set equal to zero at all times, the current resembles the curves obtained when introducing tetraethylammonium ions (TEA) in place of potassium. When the permeability to sodium is set equal to zero for all time, the current resembles curves obtained when sodium ions are replaced b y choline (or the poison T T X is used). These results suggest that the voltage clamp current can be resolved into an early inward component due to sodium ion and a late outward component due to potassium. However, upon plotting the individual ion fluxes in Fig. 4, we see that the inward and outward currents have approximately the same kinetics, both reaching a peak within 100 /~s and then decaying to a steady-state level. The peak current is higher for the sodium, so the initial current is dominated

TABLE 1 Parameters in SCM calculations Symbol

Standard magnitude

Units

Comments

GX

1× 1× 1× 1× 1× 10 0.8

1/(s mY) c m / ( s mV) c m / ( s mV) cm3/mole s- l p~F / c m 2 /~F/cm 2

Gives proper gating current Gives proper inward peak current

/)2 M BEQ BR

CAP 1,3 CAP2

10 3 10 - s 10 -6 10 3 10 5

Gives maximal ion binding CAP1,3 >> CAP2 Gives observed value

456 20

10

,]N 0A

C

,I/<

-t0 i

I

ictus)

Fig. 4. The individual cation fluxes, J N for sodium and J K for potassium (in nanomoles cm- 2 s- 1), are plotted versus the time. t (in milliseconds). The simultaneous J N (inward) and J K (outward) lead to the standard curve shown in Fig. 3.

b y the sodium ion. The decay rate constants for the two ion fluxes differ, the sodium flux decaying faster, so that later current is dominated by the potassium ion. The two ion currents interact and when removing either of the ions, as in Fig. 3, we do not resolve the current into its original components. If the SCM results shown in Figs. 3 and 4 reflect the processes that occur during excitation in the squid axon, then the assumption that the voltage clamp currents can be resolved into separate ionic currents for sodium (early) and potassium (late) may not be valid. This possibility raises important questions for the SCM single-channel idea. One can effectively eliminate a sodium flux by substituting a much larger choline ion, and the same is true when T E A substitutes for potassium. However, how do additions of nanomolar concentrations of T T X remove the early (i.e. sodium) flux? H o w does a low concentration of pronase effectively eliminate the late (i.e. potassium) flux? The answers to these questions will be considered in the analysis of the SCM responses to changes in the various membrane properties. It should be noted that in all cases, the experimental observations are electrical currents, and we must focus on what happens to the total membrane current rather than the individual ionic contributions, Let us return to the results of Fig. 3 and consider the behavior of the membrane current. The peak current is well defined and can be used to characterize the early inward current. We can approximate the steady-state current by the value at 3 ms. (Extending the solutions to 8 ms gives about the same magnitudes.) These two currents are generally used to summarize the behavior of the voltage clamp currents

457

-~0

-4'0

-2'0

0

2'0

4'0

~0

Fig. 5. The peak and steady-state currents I (in m A / c m 2 ) , versus the clamp voltage, U (in mV). Two sets of curves are shown: ( ) CAP1,3 = 10 pLF/cm2; ( . . . . ) CAP1,3 = 1 0 0 / ~ F / c m 2. In each case the upper curve is the peak current and the lower one is the steady-state current.

800

600

400

20C

-60

-40

-20

0

20

40

60

Fig. 6. The ratio of the sodium permeability at the peak current to the resting permeability, G N / G N o, as a function of the clamp voltage (in mV).

458 under different degrees of depolarization. Figure 5, which shows the currents obtained for different clamp voltages, is qualitatively similar to the curves published for squid axon [16] . (The sodium permeabilities at the peak inward current corresponding to the different clamp voltages are shown in Fig. 6.) The SCM can apparently give results similar to those observed in the real system. DEPENDENCE OF PEAK CURRENTS ON MEMBRANE PROPERTIES We shall now consider the parameters of the SCM listed in Table 1, in terms of the possible links between the elements of the computer model and the membrane properties they are meant to represent. We shall also consider the results of various experimental procedures and try to relate these to the SCM results. It should be remembered that many experimental procedures (e.g. the addition of a proteolytic enzyme), could affect all of the parameters, and that explanations are necessarily simplified. As stated earlier, the mobility of the charge involved in the gating current has been chosen to provide a (rate and magnitude of) negative charge flow between the two surfaces of the membrane that is comparable to observations. We have also assumed that the channels open instantaneously in accordance with new surface charge densities. If the channel opening did not coincide with the arrival of charge, there would be marked changes in the kinetics and magnitude of the peak current. The delay in channel opening can be simulated by decreasing GX, the mobility of the membrane charge, as shown in Fig. 7, where both the peak current and the time of the peak are plotted. From these results it appears that any process which slows the opening of the channels can cause large decreases in the magnitude of the peak

1.5

~(ms)

1.C

mA/cm2)

-

o.~

=/

I 10g GX

=

/

2

3

Fig. 7. "]'he peak current,/p (in mA/cm 2), and the time of the peak, tp (in milliseconds), are plotted v e r s u s the mobility of the negative charge in the membrane. Note that/p becomes negative at log G X - l. The clamp voltage is - 20 inV.

the logarithm of G X ,

459 t.010.8 t~

0.6 0.4 OY I

-8

-6

-7

-5

Fig. 8. The peak current, Ip (in mA/cm2), versus log M, the logarithm of the ionic mobility in the electrical double layer. Note that at values of log M > - 5 , Ip becomes negative. The clamp voltage is 20 mV. -

c u r r e n t a n d d e l a y its a p p e a r a n c e . It c o u l d e v e n l e a d to t h e c o m p l e t e d i s a p p e a r a n c e o f a p o s i t i v e c u r r e n t . ( A p o s s i b l e m e c h a n i s m for t h e a c t i o n o f T T X c o u l d b e i m p e d i n g t h e o p e n i n g of the c h a n n e l . ) A n o t h e r i m p o r t a n t f a c t o r t h a t c a n l e a d to l a r g e d e c r e a s e s in the p e a k c u r r e n t is t h e i o n i c m o b i l i t y in t h e s u r f a c e c o m p a r t m e n t . F i g u r e 8 s h o w s t h e d e c r e a s e in t h e

/

\

X 1

III~--~~

\

,

-

-60

-4o

do

Fig. 9. The peak and steady-state currents, I (in mA/cm 2), as functions of clamp voltage, U (in mV). Two sets of conditions are shown. The solid curves are for standard values of the parameters, while the dashed curves are for the case where both BEQ and BR are equal to zero. In each case the upper curve is the peak current and the lower one is the steady-state current. Note the displacement of both currents to more positive values when BEQ and BR equal zero.

460

1.5 \

\

1.0

aEq •

0.5

o

\

~\\~_

.....

log B£Q, BR Fig. 10. The peak current, Ip, and the steady-state current, l~s, both in r n A / c m 2, as functions of the logarithm of the binding equilibrium constant, B E Q ( ) and the ion release rate constant B R ( - - - - - ) . In both cases all the other parameters are at the standard magnitudes. The clamp voltage is - 20 mV.

peak current as the aqueous ionic mobility increases beyond the plateau level of GN, the ionic permeability of the membrane, shown in Fig. 1. This suggests that one of the effects of applying proteolytic enzymes to membrane surfaces may be the removal of enough adsorbed protein to increase the ionic mobility in the electrical double layer and eliminate a peak current. In addition to affecting ionic mobility, the proteins at membrane surfaces can bind ions and lead to the ionic processes described by the equations of the SCM. If the bound ions are eliminated, there are significant changes in voltage clamp currents (see Fig. 9). Both the peak and steady-state currents increase in the absence of ion binding. (Apparently, ion binding buffers the effects of the ion concentration changes in the surface compartments.) Note that the normally negative steady-state current remains positive over a large range of clamp voltage. The effects of gradual changes in the role of ion binding in the SCM process are shown in Fig. 10. It appears that at - 2 0 mV clamp voltage, ion binding has a significant effect only in the range of binding constants (BEQ) on the order of 0.1 to 1 M - 1. The ion release rate constant (BR) starts to have an effect between 10 3 and l 0 4 S - ] . (At +20 mV clamp voltage, there is virtually no effect on peak current and a small effect in the opposite direction on steady-state current, at the same ranges of parameters.) These effective ranges of BEQ and BR are not too different from those arrived at in the earlier study [12], where the ionic permeability was set at a high level throughout the calculation.

461 THE MEMBRANE CAPACITANCES Here, U23, the electrical potential across the membrane (i.e. between the two surface compartments), is a critical property in controlling the charge flows associated with excitation. This electrical potential affects the distribution of the negative surface charge across the membrane and, upon depolarization, causes the gating current leading to the opening of a channel. It also contributes to the driving force for sodium ions across the membrane. These two effects are inversely related with regard to excitation. The depolarization that leads to the opening of a channel decreases the driving force for sodium ions to enter the cell. These opposing tendencies lead to an optimum in Ip, the peak inward current, as a function of the membrane potential, shown in Fig. 11. The effect of the surface capacitances on Ip is also shown. The magnitude of U23 after depolarization depends upon the magnitudes of the three capacitances in the system (see Fig. 12.). In effect, the capacitances determine how any change in polarization will be distributed across the membrane. The surface capacitances may be especially important to consider in this respect, since relatively slight changes do not greatly affect the total capacitance, but can cause large changes in U23. Consider the line in Fig. 12 that corresponds to CAP2 = 1, and note the large changes in U23 as CAP1, 3 goes from 10 to 2 / ~ F / c m 2. Here, U23 goes from - 4 0 to - 10 mV while the total capacitance goes from 0.8 to 0 . 5 / ~ F / c m 2. Changes in U23 of this magnitude can cause large changes (i.e. a factor of two) in the peak current according to Fig. 11. The effect of the capacitance on U23 and the peak current is also shown in Fig.

100,,.

<

- -

.~ -.

/

°I/

-

-1

-60

a 23 (mY) -40

-20

0

20

40

Fig. ll. The peak current I e (in mA/cm2), versus U23 (in mV), the electrical potential across the membrane. The solid line is for standard conditions and the dashed line is for surface capacitances (CAP1,3) equal to 100 #F/cm 2. Note that U23 = 32 mV at the resting potential and depolarization leads to lower values.

462

----- CAP1,3---~CAP 2

/

2

!

,. /

/

c

iI g -IC

~-"

-2C

/

~

/ /

/

2

/

/ -3(:

/ p

i

/ I J

/ -4C

/ /1

/

*

j

,,,,,,,,,,10 ,~

f

s,~"

Z

~5C ~-..._-____ . . . . . .

I toter CAP ()aF/cm 2) -

6

0.2

0

0.4

~3 ~

0.6

0.8

~oo

1 1.0

t.2

Fig. 12. The electrical potential across the membrane, U23 (in mV), is plotted as a function of the total capacitance ( i n / ~ F / c m 2 ). The total capacitance is composed of two surface capacitances, CAP1,3 ( - - - - - ) and a dielectric capacitance, CAP2 ( ), both in ~tF/cm 2. Combinations of surface (range 2-100 /~F/cm 2) and dielectric (range 0.8-5 ~ F / c m 2) capacitances are shown.

13. The capacitances in the system were varied systematically in two ways, and lp determined as a function of the total capacitance. From Fig. 13A it is apparent that varying the dielectric capacitance has a much larger effect on lp and gives a different curve from the case of varying the surface capacitances. However, when the two sets of results are plotted as functions of U23 (Fig. 13B), the variation of Ip in both cases is related to the electrical driving force across the membrane. The large effects of hydrocarbons on voltage clamp currents [17] is probably due to changes in the dielectric capacitance. Surface capacitances, which can be altered significantly by the adsorption of surface-active materials [18], can also cause marked changes in currents. It is important to consider the possible influences of various physiologic and pharmacologic agents in these terms. (All of the SCM calculations assume constant surface capacitances, while studies at many interfaces have shown that capacitances of electrical double layers are functions of surface potential and ionic strength [19]. However, the variation in surface capacitance over the ranges of potentials and ionic strengths that apply to

463 I |

0.41-

--.(~--

/- ~

-o ~

t!

,.,L U

I

,

I 8

~", 6

~-~

• CAP2:8 o CAP1 3=10

-.-.,

totat CAP (pF/crn 2) i 1.0

i 1.2

~'~

, 1.4

i a i 16 '~ ~1.8

0.6 - ~ .

0.4 0.2

o

-

B I

- 0.2

-50

-40

-30

-20

Fig. 13. (A) The peak current Ip (in m A / c m 2) versus the total membrane capacitance ( i n / t F / c m 2) under two different conditions: (11) when the dielectric capacitance is constant at 0.8 ffF/cm2; (O) when the surface capacitances are constant at 10 ~ F / c m 2. (B) The peak current Ip (in m A / c m 2) versus the electrical potential across the membrane, U23 (in mV), for the same results as shown in (A).

the membrane is not too large, and the assumption of constant capacitances is not apt to affect the results to any great extent.) DISCUSSION

The separation of the ionic currents during an action potential into an early sodium current and a late potassium current has served as the basis for an understanding of the ionic events during the excitation of nerve. The voltage clamp currents on squid axon were first analyzed in this way, and recent patch clamp studies of ionic channels have been interpreted in terms of two separate ion-selective channels. The evidence in both cases has convinced many of the validity of the interpretation. However, it is difficult to conceive of a molecular mechanism for the ion-selective channels that would allow them to function with ideal ion selectivity. There are also problems accounting for some observations within the context of the two-channel idea. For example, in internally perfused sodium-free axons [20], an early outward current at large depolarizations is carried by potassium, presumably through the sodium channel. Also, lithium moves through the sodium channel more effectively than sodium itself [20], implying that the crystal radius determines the mobility. However, thallium with a larger crystal radius (but the same size hydrated

464 radius) moves almost twice as effectively in the potassium channel [21]. The SCM has eliminated the most troubling aspect of the conventional explanation, the need for ideal ion selectivity. The model uses a single channel and the ionic currents are influenced by well-known physical processes in the electrical double-layer regions of membranes. The results of the SCM approach~ as presented here, show that a non-selective increase in membrane permeability (i.e. a single channel for both sodium and potassium), could provide the pattern of ionic currents seen in the voltage clamp of squid axons. In fact, except for differences in the permeability of the resting membrane to these ions, the two cations are assumed to have identical physical properties (i.e. mobility in the double-layer region, ion binding and release rate constants), so that the apparent specificity in the system appears to arise from the asymmetry of the resting concentration gradients. The numerical values used in the calculations [12] are taken from studies of the squid axon and, as discussed above, the parameters in Table 1 have been set with a view to reproducing the observed responses of the membrane permeability to changes in polarization. The few values that have to be assumed, BR, CAP1, 3 and M, are of reasonable magnitudes. All of the values are approximations, as is the surface compartment in comparison to an electrical double layer. However, this simplification of the real situation has enabled us to describe the unusual effects of surface processes on ion flow across membranes. Probably the most interesting and controversial result of the SCM approach is the suggestion that the original separation of the voltage clamp current into an early sodium and a later potassium current may not be justified. Many experiments point to the existence of discrete ion-selective channels, so it will be necessary to show that the SCM approach can provide alternative explanations in terms of the physical properties (e.g. surface charge density, ion binding, surface capacitance) that have been considered here. Although the data in this area are limited, we have already suggested ways to account for several observations earlier in this paper. One further example at this stage may help to show how the SCM approach can be used. Consider that the application of pronase to the inside of an excitable membrane results in selection for sodium channels and the disappearance of potassium channels. This means that the current during voltage clamp is inward in this case. The proteolytic activity of pronase should remove segments of protein (which contain binding sites) and reduce the influence of bound ions. Referring to Fig. 9, we see that removing the bound ions increases the magnitude of the peak current and changes the normally negative steady-state current to positive. This would cause the voltage clamp current to appear as an inward current only, and give the appearance of sodium selectivity even though both ion flows were contributing. The increase in CAP1, 3 due to the loss of an adsorbed layer would also cause a more positive current (see Fig. 5). It appears that pronase treatment could lead to the observed effect even with a single channel. From the examples in the paper, it appears worthwhile to explore the SCM further. Differences in physical properties between sodium and potassium should be considered. The effects of asymmetrical changes (e.g. a change in the properties of

465

only one surface) should also be explored. As we learn more about the effects of various changes in parameters on the membrane current, we should develop greater insight into possible mechanisms of membrane processes. The introduction of surface compartments as approximations to electrical double-layer regions, while initially complicating the membrane transport problem, has led to simplifications in our ideas of membrane function. The SCM has also drawn our attention to physical properties (e.g. surface capacitances) which have not generally been considered as relevant to membrane behavior, and which may suggest mechanisms for the actions of different chemical agents. ACKNOWLEDGEMENTS

This investigation was supported in part by Contracts N00014-80-C-0027 and N00014-83-K-0043 from the Office of Naval Research. REFERENCES 1 T. Teorell, Progr. Biophys., 3 (1953) 305. 2 S. McLaughlin, Curr. Top. Membr. Transp., 9 (1977) 71. 3 D.L. Gilbert in Biophysics and Physiology of Excitable Membranes, W.J. Adelman (Editor), Van Nostrand Reinhold, New York, 1971, p. 359. 4 Y. Woo and L.Y. Wei, J. Biol. Phys., 1 (1973) 36. 5 M. Blank, J. Colloid Sci., 20 (1965) 933. 6 A.A. PiUa and G. Margules, J. Electrochem. Soc., 124 (1977) 1697. 7 R. Schmukler and A.A. Pilla, J. Electrochem. Soc., 129 (1982) 526. 8 W. Lorenz and K.D. Schulze, Z. Phys. Chem. (Leipzig), 262 (1981) 1032. 9 K.D. Schulze and W. Lorenz, Z. Phys. Chem. (Leipzig), 263 (1982) 27. 10 M. Blank and J.S. Britten, Bioelectrochem. Bioenerg., 5 (1978) 535. 11 M. Blank and W.P. Kavanaugh, Bioelectrochem. Bioenerg., 9 (1982) 427. 12 M. Blank, W.P. Kavanaugh and G. Cerf, Bioelectrochem. Bioenerg., 9 (1982) 439. 13 M. Blank, Colloids Surf., 1 (1980) 139. 14 M. Blank, Bioelectrochem. Bioenerg., 9 (1982) 615. 15 R.D. Keynes and E. Rojas, J. Physiol. (London), 239 (1974) 393. 16 A.L. Hodgldn, A.F. Huxley and B. Katz, J. Physiol (London), 116 (1952) 424. 17 D.A. Haydon, J. Requena and B.W. Urban, J. Physiol. (London), 309 (1980) 229. 18 I.R. Miller, Top. Bioelectrochem. Bioenerg., 4 (1981) 161. 19 J.Th.G. Overbeek in Colloid Science, H.R. Kruyt (Editor), Elsevier, Amsterdam, 1971, p. 115. 20 W.K. Chandler and H. Meves, J. Physiol, (London), 180 (1965) 788. 21 B. Hille, J. Gen. Physiol., 61 (1973) 669.