MECHANISM FOR CHANNEL GATING IN EXCITABLE BILAYERS Harold Lecar, Gerald Ehrenstein, and Ramon Latorre* Laboratory of Biophyrics, 1R National Institute of Neurological and Communicative Disorders and Stroke National Institutes of Health Bethesda, Maryland 20014 Excitation and Gating
Electrical excitation in the nervous system is produced by the transient increase of the permeability of a particular ion through the membrane. Action potential generation, for example, is often described in terms of a sodium conduction pathway having a negative-resistance current-voltage characteristic.', ' The negative resistance and its attendant instability are the result of a steeply voltage-sensitive ionic conductance which approaches to new values with a voltage-dependent relaxation time.' As Mueller and Rudin'.' first showed, certain additives, such as EIM (excitability inducing material, a protein extracted from aerobacter bacteria), alamethicin and monazomycin, endow synthetic lipid bilayer membranes with the type of voltage-dependent conductance necessary for excitability. FIGURE 1 shows the voltage-dependent conductance and relaxation time for
FIGURE1. Normalized voltage-dependent conductance and conductance relaxation time for an EIM-doped lipid bilayer membrane (From Ehrenstein er a1.0 By permission of the Jourrial of General Physiology.)
MEMBRANE POTENTllL lmvl
* Present address: Department of Physiology and Pharmacology, Duke University, Durham, North Carolina 27710. 304
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The curves are reminiscent of the Hodgkin-Huxley parameters which describe axonal excitation.’, ’ Observations on Single Channels
Much of what we know about the excitable bilayers comes from the fortunate circumstance that it is often possible to measure conductance changes at extremely low concentrations of additive. Bean and his coworkers’ observed that for low levels of EIM added to the solution bathing a bilayer the conduc-
-
( 1 -
V = 30mV
FIGURE2. Conductance jumps for a single EIM
b
r
-
-. V+ 45mV
channel in an oxidized cholesterol membrane. (From By perrnisEhrenstein et sion of the Jortrnal of General Physiology.) 0.1“0 d
d
[
-8‘
V*70mV
c _
20 n c .
tance changes appeared as discrete steps. These steps represent the change in current through individual ion-conducting units. All the units seemed to mho o r about be identicals and to conduct a high flux of ions (4 x lo’ ions per sec with 100 mV across the membrane). The conductance steps thus represent discrete channels through the membrane; channels, not domains, because the conductances are very uniform’; channels, not carriers, because of the high flux per unit.3 By diluting the EIM solution still further, we eventually were able to observe single channels which remained in isolation for long times, often an hour o r more.’ With such isolated conducting channels, one can vary the membrane potential, observe a pulse train, and learn whether the amplitudes of the jumps or their frequency vary in a significant way over the negativeresistance region. One can answer the question of whether the negative resistance is a transport property of each individual channel or whether it is a statistical result of the random channel jumps. EIM jumps recorded with various membrane potentials on a single channel are shown in FIGURE2. These jumps in current are caused by transitions of the channel from a low-conductance state to a high-conductance state. We call these states “closed” and “open.” The conductance in the open state is a property of the EIM channel independent of the lipid used. The closed-state conductance can vary depending upon the lipid (up to one-fourth the open conductance for EIM in oxidized cholesterol). Thus, when a channel is closed by application of membrane potential, the channel structure is altered, but the channel does not break up completely.
Annals New York Academy of Sciences
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Transport through a Single Channel
The jump amplitudes plotted as a function of voltage are shown in FIGURE 3 for various alkali cations. These curves are current-voltage curves for the ion transport through a single channel. The curves are approximately linear, but it is more important to note that they are monotonically increasing through the region of voltage corresponding to negative resistance. Thus, there is nothing remarkable about the transport process, and the origin of excitability resides in the conductance transitions of the EIM channels.
..= = 0. A'
0
I-
z
w
% ' u 3
VOLTAGE,
I
I
60
80
J 100
mv
FIGURE 3. Current-voltage curves for a single EIM channel. Measurement is current-jump amplitude as a function of steady potential. All measurements in 100 mM chloride solutions of ions indicated. (From Latorre et al. By permission of the Journal of General Physiology.)
The single-channel current-voltage curves can also be used to measure 3 give the relative permeionic selectivity. The slopes of the curves in FIGURE abilities of the channel to the various alkali cations. The permeabilities determined from these slope measurements (and confirmed by bi-ionic potential measurements on many-channel membranes) are in proportion to the free-solution mobilities of the ions.' Thus the EIM channels do not select among cations. Because of this and because EIM channels have such a large conductance per site, one may reasonably conclude that the channels are aqueous, possibly negative-charge-lined aqueous pores, since they are cation-selective. This conclusion has been confirmed by Latorre, Alvarez, and Verdugo,1° who found the temperature-dependence of open EIM channel conductance to be the same as that of aqueous electrolyte. Open alamethicin channels also appear to behave as aqueous pores." Hence, although these gated channels possess the major property of excitation-gated ion fluxes-they do not have the high selectivity of the sodium o r potassium channels found in natural excitable membranes.
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Voltage-Dependent Transition Probabilities
From a long train of conductance jumps, one can measure the fraction of time that a channel stays open at a given p ~ t e n t i a l .If~ all the channels are identical, we might expect the time average of the probability of a single channel being open to be equal to the steady-state average of the fraction of channels open for a many-channel membrane at the same voltage. FIGURE 4 shows this comparison. The points are time-averages for single- or few-chan-
VOLTAGE
mV
FIGURE4. Fraction of time spent in open-conducting state as a function of membrane potential. Data taken on few-channel membranes with number of channels indicated next to experimental point. Different symbols indicate different cations. Dashed curve shows fraction of open channels as predicted from Equation 5. nel membranes. The curve is the normalized voltage-dependent conductance of a many-channel membrane. This experiment shows that the permeability is voltage-dependent because the channel has two conducting states and the relative probability of being in either state is a function of voltage. Statistical Mechanics of the Channels
We can show that both the sigmoid voltage-dependent conductance and the bell-shaped relaxation time have their origin in the simplest possible picture of the channel transitions. Imagine a channel structure that has two states. Let us say that the energy difference between the two states is given by the sum of some intrinsic configuration energy and the work needed for the electric field to push charges around or twist dipoles in going from one state to the other. That is AW = AWconf AWeiec and AWelec= aV
+
where V is membrane potential and a is a constant. Since at the potential VO,for which half the channels are in each state, the AWelccjust cancels AWoonf,we can write the energy as AW(V) = a(v - V,) (3)
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If we denote the total number of channels by N, the number open by no and the number closed by no, we can say that for thermal equilibrium at any potential (4) no/nc = exp (-AW/kT) where T is absolute temperature and k is Boltzmann's constant. Since no no = N, Equation 4 leads to the following expression for the fraction of channels open at
+
Substitution for AW from Equation 3 immediately gives the sigmoid conductance function of FIGURW 1 and 4. One further assumption is needed to get the bell-shaped relaxation time. If we denote the rate for a closed channel to open as a and that for an open channel to close as p, we know from the principle of detailed balance that a/P = n$nc = exp
[
vO)l We now assume that a and 0 have opposite voltage-dependence, that is a = X exp (-a(V
- Vo)/2kT)
p
= X exp (+a(V
- Vo)/2kT)
(7)
where X is an arbitrary rate. In this case the relaxation time, 7, can be written as 7(V) = (a
+
@)-I
=
(>$X) sech [a(V
- Vo)/2kT]
(8)
This shows that the bell-shaped time constants, analogous to the well-known Hodgkin-Huxley time constants for axonal excitation,3 are inherent in this type of voltage-dependent transition. Equation 7 stated the only assumption, a specific form for the rate constants. This assumption can be tested experimentally. One of the nice features of single-channel jump measurements is that one can measure forward and backward transition rates independentlyP The measurements are done by statistical analysis of the distribution of jump durations. The distribution of jump widths (open channel times) is exponential, indicating that the channels obey Poisson statistics; i.e., the chance of an open channel closing or a closed channel opening is uniform in time. For a Poisson distribution it can be shown that the mean open (closed) time is equal to the reciprocal of the rate constant for closing (opening). FIGURE 5 shows the results of rate constant measurements on three different membranes, each having one or a few channels. The log plot shows that the rates are exponential functions of potential in rough agreement with Equation 7. These experiments demonstrate that the energy of each conductance state is linear in the applied electric field. Complications; Other Channels
EIM channels in oxidized cholesterol bilayers exhibit simple gating behavior. The channel jumps between two states. The kinetics indicate a simple first-order Poisson process, with transition rates varying as an exponential function of potential. Latorre, Alvarez, and Verdugo" have shown that the
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FIGURE 5 . Opening and closing transition rates as a function of voltage for three different membranes: ( A ) one, three, and five channels; (B) two, four, and six channels; (C)one channel. Varying number of channels was necessitated by birth or death of channels during the course of a long experiment. (From Latorre et al. By permission of the Journal of General Physiology.)
two conductance states of EIM have very different temperature coefficients for ionic transport. This suggests that the open and closed (not completely closed) states have physically different modes of transport. Bean'' has shown that EIM in different lipids gives different closed-state conductances and multiple step-height jumps. Thus, although EIM behavior in other lipids resembles that in oxidized cholesterol, there are some differences. This provides interesting possibilities for the study of the interaction between the protein channel structure and the underlying lipid matrix. FIGURE 5 shows one observation having to do with a kind of local viscosity; the different channels, all in oxidized cholesterol, have the same voltage-dependence, but do not have the same absolute rate scale. The energy difference between open and closed state is thus uniform throughout the membrane, and probably a property of the channel, but the activation energy for the transition is not uniform, possibly because of inhomogeneities in the interaction between channel and lipid. There are other channels for which single-channel jumps have been observed. Alamethicin, described by Eisenberg, Hall, and Mead," produces a channel capable of transitions between half a dozen sublevels of conductance. These sublevels represent the stable states of the channel structure, probably various stable states of agglomeration of alamethicin subunits. Channel jumps have also been studied for bilayers doped with keyhole limpet hernocyanin. These jumps lead to a picture of discrete conductance states, but with a nonlinear current-voltage relation prevailing in each state, possibly due to a set of finer, more rapid structural changes." Despite the complexity caused by the large number of conductance states, the alamethicin and hemocyanin preparations are valuable because they contain gating channels made of molecules for which there is structural information. Inactivation and Molecular Pictures of Gating
Baumann and M ~ e l l e r 'have ~ described the kinetics of membrane channels made of subunits that are tipped into the membrane by the electric field,
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and are then free to aggregate into polar-lined pore structures. One of the attractive features of this model is that it provides a natural description of inactivation-the process by which a channel, such as the sodium channel of a nerve axon, opens and then spontaneously closes again during excitation. In the subunit (or oligomer) model, inactivation is a result of the relaxation of the population of channels to nonconducting states of aggregation. EIM-doped bilayers have an electric-field behavior opposite to that of alamethicin. As can be seen from FIGURE1, the channels are open when the membrane potential is zero and closed for high potential. The Figure shows only positive potentials but there is a similar, although not identical, voltage-dependent conductance region observed for negative potentials. Recent experiments’: show that this two-sided gating is a property of individual channels and not a superposition of the responses of oppositely oriented channels. If an EIM channel is made of subunits, like an alamethicin channel, then we might envision a situation in which a large electric field induces a closed state consisting of monomers assembled in some channel-precursor structure (possibly at the membrane-solution interface). Removal of the field allows the entire assembly to turn into the membrane to form a stable pore. One can think of the closed state as a group of loosely connected barrel staves floating at the surface of the membrane and the open state as the reconstituted barrel. One consequence of this “barrel-stave” picture, as illustrated in FIGURE 6 , is that it gives a natural explanation of why EIM has two distinct closed states for high fields in either direction across the membrane. In the picture, high fields in either direction orient the membrane dipoles to push the EIM subunits out towards a membrane surface. This two-sided gating leads to
FIGURE 6 . Highly schematic picture of hypothetical EIM channel structure made of subunits. Arrows indicate direction of dipole moment. State B is open state in absence of field, with subunits forming a pore. State A is closed state corresponding to one direction of transmembrane field (upward in figure), and state C is closed state corresponding to opposite direction of transmembrane field.
Lecar et al.: Channel Gating in Excitable Bilayers
311
conductance inactivation when the voltage is switched from one polarity to the other. The channels are open transiently because they must pass through the open state in going between the two closed c~nfigurations.~~ This mechanism for inactivation can be regarded as a simple three-state model for inactivating channels such as the axon sodium channel. The need for a three-state model in the description of axon kinetics has recently been discussed by Goldman and Schauf.Ia The barrel-stave model can also explain the fmding of Latorre et al.” that the temperature-dependence of the conductance of the open channel is the same as the temperature-dependence of the mobility of,the bathing electrolyte, but the conductance of the closed channel decreases with increasing temperature. In the model, the array of EIM subunits at the membrane interface might affect the membrane structure, resulting in a partially-open channel. As temperature is increased, the lipids become more mobile, and can more effectively disrupt the ionic pathway, thus reducing the conductance. The channel-precursor states of the model are necessary to explain the observation that a membrane may contain one, and only one, channel over a period of many minutes. If an oligomer always broke up into disconnected monomers, the number of channels would fluctuate considerably. Channels in Natural Membranes
There is considerable evidence for discrete conducting channels in natural excitable membranes. Experiments with blocking agents, such as tetrodotoxin, provide an estimate of the minimal conductance blocked by a single drug molecule.” The recently discovered gating currents” represent the actual charge movement or dipole reorientation associated with the gating process. There is in addition a body of indirect evidence for the existence of conducting channels3 Direct observation of single-channel fluctuations is difficult for natural membranes. It was possible to observe the channel jumps in bilayers because the conductance of an undoped lipid bilayer is so low that current fluctuations associated with leakage pathways cause relatively small background noise. For natural membranes, the most promising approach at present is the measurement of the electrical noise produced when many channels undergo transitions. For a conductance relaxation process, such as EIM gating, the electrical noise produced by the random opening and closing of the channels has a characteristic frequency spectrum that can readily be identified. Electrical noise spectra have been studied for the acetylcholine-induced gating of the postsynaptic for the electrical gating of the axon membrane:’ and for photoreceptors.** The study of channel noise promises to yield much of the same kind of detailed information that we now have about the gating process in doped bilayers. References 1. HODGKIN, A. L. & A. F. HUXLEY. 1952. J. Physiol. 117: 500. 2. COLE,K. S. 1968. Membranes, lons and Impulses. University of California
Press. Berkeley, California.
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Annals New York Academy of Sciences
G. & H. LECAR.1972. Ann. Rev. Biophys. Bioeng. 1: 347. 3. EHRENSTEIN, 4. MUELLER, P. & D. 0. RUDIN.1963. J. Theor. Biol. 4: 268. 5. MUELLER, P. & D. 0. RUDIN.1969. Curr. Top. Bioenerg. 3: 157. 6. EHRENSTEIN, G., R. BLUMENTHAL, R. LATORRE & H. LECAR.1974. J. Gen. Physiol. 63: 707. 7. BEAN,R., W. C. SHEPHERD, H. CHAN& J. T. EICHNER.1969. J. Gen. Physiol. 53: 741. 8. LATORRE, R., G. EHRENSTEIN & H. LECAR.1972. J. Gen. Physiol. 60: 72. G., H. LECAR& R. NOSSAL.1970. J. Gen. Physiol. 55: 119. 9. EHRENSTEIN, R., 0. ALVAREZ & P. VERDUGO. 1974. Biochirn. Biophys. Acta 367: 10. LATORRE, 361. 1 1 . EISENBERG, M., J. E. HALL& C. A. MEAD.1973. J. Membr. Biol. 14: 143. 12. BEAN,R. C. 1972. J. Membr. Biol. 7: 15. 13. BAUMANN, G. & P. MUELLER. 1974. J. Supramol. Biol. In press. 14. ALVAREZ, O., E. DIAZ& R. LATORRE. 1975. Biochim. Biophys. Acta. In press. 15. EHRENSTEIN, G. & H. LECAR.1975. Biophys. J. 15: 167a. L. & C. L. SCHAUF. 1973. 3. Gen. Physiol. 61: 361. 16. GOLDMAN, 17. KEYNES,R. D., J. M. RITCHIE& E. ROJAS. 1971. J. Physiol. 213: 235. C. M. & F. BEZANILLA. 1974. J. Gen. Physiol. 63: 533. 18. ARMSTRONG, 19. KATZ,B. & R. MILEDI.1971. Nature New Biol. 232: 124. 20. ANDERSON, C. R. & C. F. STEVENS. 1973. J. Physiol. 235: 655. 21. SACHS, F. & H. LECAR.1973. Nature New Biol. 246: 214. A. A. & L. J. DE FELICE. 1974. Progr. Biophys. Mol. Biol. 28: 191. 22. VERVEEN, 23. FISHMAN, H. 1973. Proc. Natl. Acad. Sci. U S A . 7 0 876. 24. HAGINS, W. A. 1965. Cold Spring Harbor Symp. Quant. Biol. 30: 403.
Discussion DR. MUELLER:I would like to emphasize that the conductance of these membranes is controlled by the fluctuations between the zero conductance level and the first step because that is where the very large change of the conductance occurs. The later step is only a three- to fivefold change in the conductance, whereas during this first part, if you study many channel kinetics, the conductance varies over three, four, or five orders of magnitude. In the sphingomyelin membranes one finds high-order kinetics, whereas in the oxidized cholesterol system, where the conductance varies only between three- or fourfold one has first-order kinetics. After the channel has assembled in the field, it can move back and forth, with the possibility that the lipid heads may tend to close up the opening, resulting in a partially closed channel; and yet there are other high-order mechanisms superimposed on all that. DR. LECAR: There are a number of interesting experiments involving changes in the lipids. EIM can be made to look like alamethicin if you take it out of oxidized cholesterol; in fact alamethicin can be made to look more like EIM in some lipids. Further, there is the possibility of having a time delay in activating the channels in a membrane without the use of a first-order variable raised to a power as in the Hodgkb-Huxley equations. In that type of kinetic description you have to go to a very high power to explain long time delays, and that seems unreasonable especially in the models where the power
Lecar et al.: Channel Gating in Excitable Bilayers
313
is associated with the number of particles involved. Furthermore, the power is different in different species. The Frankenhaeuser-Huxley equations that describe the frog node have the sodium conductance proportional to mz instead of to m', and potassium conductance proportional to n' instead of n'. Perhaps the gatingcurrent or noise-spectrum experiments will provide new information on the physical origin of the time delays. DR. PRESSMAN: I would like to suggest another possiblity with respect to the barrel stave concept developed by Mueller and others from alamethicin properties. If these staves are attached to an ionophore in the shape of a doughnut which also serves as an anchor in one side of the membrane, then when the staves move across in the field they would arrive oriented to form the channel, eliminating the time-consuming aggregation process. The doughnut would be able to recognize the ions and confer on this channel a degree of ion-selectivity that most of the channels described, up to now at this conference do not. Finally, within that doughnut there is a large amount of material that could serve as the mechanism for the chemistry involved in the inactivation of those channels which do inactivate. DR. FINKELSTEIN: In the case of alamethicin you can change the delay times, the lag in the S shape behavior, etc., by the amount of material you add to the membrane because the absolute rates are going to be determined by the number of monomers that you have in the system and hence, the more you have the faster things go. In the EIM case, where you have a prefabricated channel, the kinetics for the macroscopic system are reflected in the kinetics of the single channel and you don't have that degree of play.
Electrical excitation in the nervous system is produced by the transient increase of the permeability of a particular ion through the membrane. Action potential generation, for example, is often described in terms of a sodium conduction pathway having a negative-resistance current-voltage characteristic.', ' The negative resistance and its attendant instability are the result of a steeply voltage-sensitive ionic conductance which approaches to new values with a voltage-dependent relaxation time.' As Mueller and Rudin'.' first showed, certain additives, such as EIM (excitability inducing material, a protein extracted from aerobacter bacteria), alamethicin and monazomycin, endow synthetic lipid bilayer membranes with the type of voltage-dependent conductance necessary for excitability. FIGURE 1 shows the voltage-dependent conductance and relaxation time for
FIGURE1. Normalized voltage-dependent conductance and conductance relaxation time for an EIM-doped lipid bilayer membrane (From Ehrenstein er a1.0 By permission of the Jourrial of General Physiology.)
MEMBRANE POTENTllL lmvl
* Present address: Department of Physiology and Pharmacology, Duke University, Durham, North Carolina 27710. 304
Lecar et al.:
Channel Gating in Excitable BiIayers
305
The curves are reminiscent of the Hodgkin-Huxley parameters which describe axonal excitation.’, ’ Observations on Single Channels
Much of what we know about the excitable bilayers comes from the fortunate circumstance that it is often possible to measure conductance changes at extremely low concentrations of additive. Bean and his coworkers’ observed that for low levels of EIM added to the solution bathing a bilayer the conduc-
-
( 1 -
V = 30mV
FIGURE2. Conductance jumps for a single EIM
b
r
-
-. V+ 45mV
channel in an oxidized cholesterol membrane. (From By perrnisEhrenstein et sion of the Jortrnal of General Physiology.) 0.1“0 d
d
[
-8‘
V*70mV
c _
20 n c .
tance changes appeared as discrete steps. These steps represent the change in current through individual ion-conducting units. All the units seemed to mho o r about be identicals and to conduct a high flux of ions (4 x lo’ ions per sec with 100 mV across the membrane). The conductance steps thus represent discrete channels through the membrane; channels, not domains, because the conductances are very uniform’; channels, not carriers, because of the high flux per unit.3 By diluting the EIM solution still further, we eventually were able to observe single channels which remained in isolation for long times, often an hour o r more.’ With such isolated conducting channels, one can vary the membrane potential, observe a pulse train, and learn whether the amplitudes of the jumps or their frequency vary in a significant way over the negativeresistance region. One can answer the question of whether the negative resistance is a transport property of each individual channel or whether it is a statistical result of the random channel jumps. EIM jumps recorded with various membrane potentials on a single channel are shown in FIGURE2. These jumps in current are caused by transitions of the channel from a low-conductance state to a high-conductance state. We call these states “closed” and “open.” The conductance in the open state is a property of the EIM channel independent of the lipid used. The closed-state conductance can vary depending upon the lipid (up to one-fourth the open conductance for EIM in oxidized cholesterol). Thus, when a channel is closed by application of membrane potential, the channel structure is altered, but the channel does not break up completely.
Annals New York Academy of Sciences
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Transport through a Single Channel
The jump amplitudes plotted as a function of voltage are shown in FIGURE 3 for various alkali cations. These curves are current-voltage curves for the ion transport through a single channel. The curves are approximately linear, but it is more important to note that they are monotonically increasing through the region of voltage corresponding to negative resistance. Thus, there is nothing remarkable about the transport process, and the origin of excitability resides in the conductance transitions of the EIM channels.
..= = 0. A'
0
I-
z
w
% ' u 3
VOLTAGE,
I
I
60
80
J 100
mv
FIGURE 3. Current-voltage curves for a single EIM channel. Measurement is current-jump amplitude as a function of steady potential. All measurements in 100 mM chloride solutions of ions indicated. (From Latorre et al. By permission of the Journal of General Physiology.)
The single-channel current-voltage curves can also be used to measure 3 give the relative permeionic selectivity. The slopes of the curves in FIGURE abilities of the channel to the various alkali cations. The permeabilities determined from these slope measurements (and confirmed by bi-ionic potential measurements on many-channel membranes) are in proportion to the free-solution mobilities of the ions.' Thus the EIM channels do not select among cations. Because of this and because EIM channels have such a large conductance per site, one may reasonably conclude that the channels are aqueous, possibly negative-charge-lined aqueous pores, since they are cation-selective. This conclusion has been confirmed by Latorre, Alvarez, and Verdugo,1° who found the temperature-dependence of open EIM channel conductance to be the same as that of aqueous electrolyte. Open alamethicin channels also appear to behave as aqueous pores." Hence, although these gated channels possess the major property of excitation-gated ion fluxes-they do not have the high selectivity of the sodium o r potassium channels found in natural excitable membranes.
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Voltage-Dependent Transition Probabilities
From a long train of conductance jumps, one can measure the fraction of time that a channel stays open at a given p ~ t e n t i a l .If~ all the channels are identical, we might expect the time average of the probability of a single channel being open to be equal to the steady-state average of the fraction of channels open for a many-channel membrane at the same voltage. FIGURE 4 shows this comparison. The points are time-averages for single- or few-chan-
VOLTAGE
mV
FIGURE4. Fraction of time spent in open-conducting state as a function of membrane potential. Data taken on few-channel membranes with number of channels indicated next to experimental point. Different symbols indicate different cations. Dashed curve shows fraction of open channels as predicted from Equation 5. nel membranes. The curve is the normalized voltage-dependent conductance of a many-channel membrane. This experiment shows that the permeability is voltage-dependent because the channel has two conducting states and the relative probability of being in either state is a function of voltage. Statistical Mechanics of the Channels
We can show that both the sigmoid voltage-dependent conductance and the bell-shaped relaxation time have their origin in the simplest possible picture of the channel transitions. Imagine a channel structure that has two states. Let us say that the energy difference between the two states is given by the sum of some intrinsic configuration energy and the work needed for the electric field to push charges around or twist dipoles in going from one state to the other. That is AW = AWconf AWeiec and AWelec= aV
+
where V is membrane potential and a is a constant. Since at the potential VO,for which half the channels are in each state, the AWelccjust cancels AWoonf,we can write the energy as AW(V) = a(v - V,) (3)
308
Annals New York Academy of Sciences
If we denote the total number of channels by N, the number open by no and the number closed by no, we can say that for thermal equilibrium at any potential (4) no/nc = exp (-AW/kT) where T is absolute temperature and k is Boltzmann's constant. Since no no = N, Equation 4 leads to the following expression for the fraction of channels open at
+
Substitution for AW from Equation 3 immediately gives the sigmoid conductance function of FIGURW 1 and 4. One further assumption is needed to get the bell-shaped relaxation time. If we denote the rate for a closed channel to open as a and that for an open channel to close as p, we know from the principle of detailed balance that a/P = n$nc = exp
[
vO)l We now assume that a and 0 have opposite voltage-dependence, that is a = X exp (-a(V
- Vo)/2kT)
p
= X exp (+a(V
- Vo)/2kT)
(7)
where X is an arbitrary rate. In this case the relaxation time, 7, can be written as 7(V) = (a
+
@)-I
=
(>$X) sech [a(V
- Vo)/2kT]
(8)
This shows that the bell-shaped time constants, analogous to the well-known Hodgkin-Huxley time constants for axonal excitation,3 are inherent in this type of voltage-dependent transition. Equation 7 stated the only assumption, a specific form for the rate constants. This assumption can be tested experimentally. One of the nice features of single-channel jump measurements is that one can measure forward and backward transition rates independentlyP The measurements are done by statistical analysis of the distribution of jump durations. The distribution of jump widths (open channel times) is exponential, indicating that the channels obey Poisson statistics; i.e., the chance of an open channel closing or a closed channel opening is uniform in time. For a Poisson distribution it can be shown that the mean open (closed) time is equal to the reciprocal of the rate constant for closing (opening). FIGURE 5 shows the results of rate constant measurements on three different membranes, each having one or a few channels. The log plot shows that the rates are exponential functions of potential in rough agreement with Equation 7. These experiments demonstrate that the energy of each conductance state is linear in the applied electric field. Complications; Other Channels
EIM channels in oxidized cholesterol bilayers exhibit simple gating behavior. The channel jumps between two states. The kinetics indicate a simple first-order Poisson process, with transition rates varying as an exponential function of potential. Latorre, Alvarez, and Verdugo" have shown that the
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FIGURE 5 . Opening and closing transition rates as a function of voltage for three different membranes: ( A ) one, three, and five channels; (B) two, four, and six channels; (C)one channel. Varying number of channels was necessitated by birth or death of channels during the course of a long experiment. (From Latorre et al. By permission of the Journal of General Physiology.)
two conductance states of EIM have very different temperature coefficients for ionic transport. This suggests that the open and closed (not completely closed) states have physically different modes of transport. Bean'' has shown that EIM in different lipids gives different closed-state conductances and multiple step-height jumps. Thus, although EIM behavior in other lipids resembles that in oxidized cholesterol, there are some differences. This provides interesting possibilities for the study of the interaction between the protein channel structure and the underlying lipid matrix. FIGURE 5 shows one observation having to do with a kind of local viscosity; the different channels, all in oxidized cholesterol, have the same voltage-dependence, but do not have the same absolute rate scale. The energy difference between open and closed state is thus uniform throughout the membrane, and probably a property of the channel, but the activation energy for the transition is not uniform, possibly because of inhomogeneities in the interaction between channel and lipid. There are other channels for which single-channel jumps have been observed. Alamethicin, described by Eisenberg, Hall, and Mead," produces a channel capable of transitions between half a dozen sublevels of conductance. These sublevels represent the stable states of the channel structure, probably various stable states of agglomeration of alamethicin subunits. Channel jumps have also been studied for bilayers doped with keyhole limpet hernocyanin. These jumps lead to a picture of discrete conductance states, but with a nonlinear current-voltage relation prevailing in each state, possibly due to a set of finer, more rapid structural changes." Despite the complexity caused by the large number of conductance states, the alamethicin and hemocyanin preparations are valuable because they contain gating channels made of molecules for which there is structural information. Inactivation and Molecular Pictures of Gating
Baumann and M ~ e l l e r 'have ~ described the kinetics of membrane channels made of subunits that are tipped into the membrane by the electric field,
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and are then free to aggregate into polar-lined pore structures. One of the attractive features of this model is that it provides a natural description of inactivation-the process by which a channel, such as the sodium channel of a nerve axon, opens and then spontaneously closes again during excitation. In the subunit (or oligomer) model, inactivation is a result of the relaxation of the population of channels to nonconducting states of aggregation. EIM-doped bilayers have an electric-field behavior opposite to that of alamethicin. As can be seen from FIGURE1, the channels are open when the membrane potential is zero and closed for high potential. The Figure shows only positive potentials but there is a similar, although not identical, voltage-dependent conductance region observed for negative potentials. Recent experiments’: show that this two-sided gating is a property of individual channels and not a superposition of the responses of oppositely oriented channels. If an EIM channel is made of subunits, like an alamethicin channel, then we might envision a situation in which a large electric field induces a closed state consisting of monomers assembled in some channel-precursor structure (possibly at the membrane-solution interface). Removal of the field allows the entire assembly to turn into the membrane to form a stable pore. One can think of the closed state as a group of loosely connected barrel staves floating at the surface of the membrane and the open state as the reconstituted barrel. One consequence of this “barrel-stave” picture, as illustrated in FIGURE 6 , is that it gives a natural explanation of why EIM has two distinct closed states for high fields in either direction across the membrane. In the picture, high fields in either direction orient the membrane dipoles to push the EIM subunits out towards a membrane surface. This two-sided gating leads to
FIGURE 6 . Highly schematic picture of hypothetical EIM channel structure made of subunits. Arrows indicate direction of dipole moment. State B is open state in absence of field, with subunits forming a pore. State A is closed state corresponding to one direction of transmembrane field (upward in figure), and state C is closed state corresponding to opposite direction of transmembrane field.
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conductance inactivation when the voltage is switched from one polarity to the other. The channels are open transiently because they must pass through the open state in going between the two closed c~nfigurations.~~ This mechanism for inactivation can be regarded as a simple three-state model for inactivating channels such as the axon sodium channel. The need for a three-state model in the description of axon kinetics has recently been discussed by Goldman and Schauf.Ia The barrel-stave model can also explain the fmding of Latorre et al.” that the temperature-dependence of the conductance of the open channel is the same as the temperature-dependence of the mobility of,the bathing electrolyte, but the conductance of the closed channel decreases with increasing temperature. In the model, the array of EIM subunits at the membrane interface might affect the membrane structure, resulting in a partially-open channel. As temperature is increased, the lipids become more mobile, and can more effectively disrupt the ionic pathway, thus reducing the conductance. The channel-precursor states of the model are necessary to explain the observation that a membrane may contain one, and only one, channel over a period of many minutes. If an oligomer always broke up into disconnected monomers, the number of channels would fluctuate considerably. Channels in Natural Membranes
There is considerable evidence for discrete conducting channels in natural excitable membranes. Experiments with blocking agents, such as tetrodotoxin, provide an estimate of the minimal conductance blocked by a single drug molecule.” The recently discovered gating currents” represent the actual charge movement or dipole reorientation associated with the gating process. There is in addition a body of indirect evidence for the existence of conducting channels3 Direct observation of single-channel fluctuations is difficult for natural membranes. It was possible to observe the channel jumps in bilayers because the conductance of an undoped lipid bilayer is so low that current fluctuations associated with leakage pathways cause relatively small background noise. For natural membranes, the most promising approach at present is the measurement of the electrical noise produced when many channels undergo transitions. For a conductance relaxation process, such as EIM gating, the electrical noise produced by the random opening and closing of the channels has a characteristic frequency spectrum that can readily be identified. Electrical noise spectra have been studied for the acetylcholine-induced gating of the postsynaptic for the electrical gating of the axon membrane:’ and for photoreceptors.** The study of channel noise promises to yield much of the same kind of detailed information that we now have about the gating process in doped bilayers. References 1. HODGKIN, A. L. & A. F. HUXLEY. 1952. J. Physiol. 117: 500. 2. COLE,K. S. 1968. Membranes, lons and Impulses. University of California
Press. Berkeley, California.
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G. & H. LECAR.1972. Ann. Rev. Biophys. Bioeng. 1: 347. 3. EHRENSTEIN, 4. MUELLER, P. & D. 0. RUDIN.1963. J. Theor. Biol. 4: 268. 5. MUELLER, P. & D. 0. RUDIN.1969. Curr. Top. Bioenerg. 3: 157. 6. EHRENSTEIN, G., R. BLUMENTHAL, R. LATORRE & H. LECAR.1974. J. Gen. Physiol. 63: 707. 7. BEAN,R., W. C. SHEPHERD, H. CHAN& J. T. EICHNER.1969. J. Gen. Physiol. 53: 741. 8. LATORRE, R., G. EHRENSTEIN & H. LECAR.1972. J. Gen. Physiol. 60: 72. G., H. LECAR& R. NOSSAL.1970. J. Gen. Physiol. 55: 119. 9. EHRENSTEIN, R., 0. ALVAREZ & P. VERDUGO. 1974. Biochirn. Biophys. Acta 367: 10. LATORRE, 361. 1 1 . EISENBERG, M., J. E. HALL& C. A. MEAD.1973. J. Membr. Biol. 14: 143. 12. BEAN,R. C. 1972. J. Membr. Biol. 7: 15. 13. BAUMANN, G. & P. MUELLER. 1974. J. Supramol. Biol. In press. 14. ALVAREZ, O., E. DIAZ& R. LATORRE. 1975. Biochim. Biophys. Acta. In press. 15. EHRENSTEIN, G. & H. LECAR.1975. Biophys. J. 15: 167a. L. & C. L. SCHAUF. 1973. 3. Gen. Physiol. 61: 361. 16. GOLDMAN, 17. KEYNES,R. D., J. M. RITCHIE& E. ROJAS. 1971. J. Physiol. 213: 235. C. M. & F. BEZANILLA. 1974. J. Gen. Physiol. 63: 533. 18. ARMSTRONG, 19. KATZ,B. & R. MILEDI.1971. Nature New Biol. 232: 124. 20. ANDERSON, C. R. & C. F. STEVENS. 1973. J. Physiol. 235: 655. 21. SACHS, F. & H. LECAR.1973. Nature New Biol. 246: 214. A. A. & L. J. DE FELICE. 1974. Progr. Biophys. Mol. Biol. 28: 191. 22. VERVEEN, 23. FISHMAN, H. 1973. Proc. Natl. Acad. Sci. U S A . 7 0 876. 24. HAGINS, W. A. 1965. Cold Spring Harbor Symp. Quant. Biol. 30: 403.
Discussion DR. MUELLER:I would like to emphasize that the conductance of these membranes is controlled by the fluctuations between the zero conductance level and the first step because that is where the very large change of the conductance occurs. The later step is only a three- to fivefold change in the conductance, whereas during this first part, if you study many channel kinetics, the conductance varies over three, four, or five orders of magnitude. In the sphingomyelin membranes one finds high-order kinetics, whereas in the oxidized cholesterol system, where the conductance varies only between three- or fourfold one has first-order kinetics. After the channel has assembled in the field, it can move back and forth, with the possibility that the lipid heads may tend to close up the opening, resulting in a partially closed channel; and yet there are other high-order mechanisms superimposed on all that. DR. LECAR: There are a number of interesting experiments involving changes in the lipids. EIM can be made to look like alamethicin if you take it out of oxidized cholesterol; in fact alamethicin can be made to look more like EIM in some lipids. Further, there is the possibility of having a time delay in activating the channels in a membrane without the use of a first-order variable raised to a power as in the Hodgkb-Huxley equations. In that type of kinetic description you have to go to a very high power to explain long time delays, and that seems unreasonable especially in the models where the power
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is associated with the number of particles involved. Furthermore, the power is different in different species. The Frankenhaeuser-Huxley equations that describe the frog node have the sodium conductance proportional to mz instead of to m', and potassium conductance proportional to n' instead of n'. Perhaps the gatingcurrent or noise-spectrum experiments will provide new information on the physical origin of the time delays. DR. PRESSMAN: I would like to suggest another possiblity with respect to the barrel stave concept developed by Mueller and others from alamethicin properties. If these staves are attached to an ionophore in the shape of a doughnut which also serves as an anchor in one side of the membrane, then when the staves move across in the field they would arrive oriented to form the channel, eliminating the time-consuming aggregation process. The doughnut would be able to recognize the ions and confer on this channel a degree of ion-selectivity that most of the channels described, up to now at this conference do not. Finally, within that doughnut there is a large amount of material that could serve as the mechanism for the chemistry involved in the inactivation of those channels which do inactivate. DR. FINKELSTEIN: In the case of alamethicin you can change the delay times, the lag in the S shape behavior, etc., by the amount of material you add to the membrane because the absolute rates are going to be determined by the number of monomers that you have in the system and hence, the more you have the faster things go. In the EIM case, where you have a prefabricated channel, the kinetics for the macroscopic system are reflected in the kinetics of the single channel and you don't have that degree of play.
