The equivalence of fluctuation analysis and chemical relaxation measurements: a kinetic study of ion pore formation in thin lipid membranes

ELOPHYSKAL CHEhWXRY 2 (L974) L97-207.0

NORTH-HOLLAND PUBLISHING COMPANY

EQUIVALENCE OF FLUCTUATiON ANALYSIS AND CHEMICAL RELAXATION MEASUREMENTS: A KINETIC STUDY OF ION FORE FORMATION IN TEiJlV LiPiD MEMBRANES

THE

I-U?. ZiNGSHElM and E. NEHER Max-Ranck-Inrtitzxt

fw biophysikatkche

Chemie

(Karl-Fn~edrch-Bonhoeffer-Institur).

D 34 G~ttingen-Nikolausberg,

Germany

Received 11 June 1974

(1) Autocorrelation

measurements

were made of the current fluctuations

by gramicidin A in black lipid membranes. (2) Relaxation experiments using a vohqe

due to ion-conducting

channels produced

jump were made on the same sysrem.

(3) Recipracd time constants were determined by both methods over a IO8 fold range in membrane conductivity. starting from the single channel level. (4) The mean squared amplitudes of the fluctuations were determined from the autocorrelation

(5) The data

were tentatively rationalized on the basis of a hypothetical

dimerization

functions.

reaction. assuming that

gramicidin A dimen form conducting channels. The same fonvard rate constants are obtained, via (3). by both methods (1) and (2). The backward rate constant agrees exceUently witi direct mevurements of the mean life time of a conducting channel. (6) The unit channel conductance and - assumintz a dimerization - the equilibrium constant tin be obtained - aeain from the fluctuation amplitude distribution.

formation. On the basis of this assumption measurements of ion current noise have been introduced into

1. rntroduction of reaction rate constants in systems by chemical relaxation methods

The determination

homogeneous

has become a standard practice during the past decade [ 11. The fluctuation dissipation theorem [2] indicates an alternative approach, which in principle should lead to equivalent resufts. The theorem relates the average time course of the decay of spontaneous microscopic

fluctuations

to the time course observed during equia small but macroscopic perturbation. The autocorrelation function ot random concentration fluctuations arising from a simple process - i.e., a single reaction step - is an exponential, which has the same time constant as the relaxation response of the libration

after

system. Only very few experimental attempts towards an application of this theorem for kinetic measurements in homogeneous systems have been reported c3,41. in the particular case of membranes one may regard the ionic currents as directly proportional to the number of ionic channels being created by a chemical trans-

membrane

physiology

[5]

as an alternative

to the

traditional relaxation studies by the voltage clamp method [6].Noise measurements have led io an estimate of the unit conductance step associated with the opening of an “Acetylcholine gate” at the neuromuscularjunction [7,8]. It was also claimed that the would allow one to decide between various models fo the action currents in nerve membranes [9- 111. Discrete, stepwise conductance changes, which represent the opening and closing of individual ionic channels, may be obtained in black lipid membranes. Such channels are produced by a number of polypep. tides, the antibiotic gramicidin A presently being the best characterized ion chawel former [ 121. The random superposition of many single events leads to higher membrane conductance values. The current noise may be analyzed and the results be related to the properties of the unit channel. We have chosen this model system for our study because

it appears

ideally

suited

for testing

the validit

198

H.P. Zingsheim and E. Neher. Fluctuation study of ion pores in membranes

of fluctuation analysis. incidentially, it may also serve as a unique illustration of the fluctuation dissipation theorem as applied to chemical reactions. Bamberg and J&ger [ 131 have recently studied the kinetics of ion pore formation by gramicidin A with a relaxation technique (voltage clamp)_ The parameter, which was assumed to perturb the equilibrium between the channels and the nonconducting species is ihe voltage dependent membrane thickness [ 12, 141. The authors analyzed their results on the basis of a hypothetical reaction scheme assuming a dimerization process as suggested by Urry et al. [ 151. A further assumption is that only grarnicidin A dimers form conducting channels and that every dimer is an ion pore. The results of the present study show that the same rate constants are obtained by correlation analysis and by the relaxation technique. The analysis of the fluctuation amplitudes yields the value for the unit conductance step and, provided the assumption on the reaction mechanism is correct, the equilibrium constant of the reaction. The data of Bamberg and fiuger as well as those of our study do not contradict the dimerization scheme. NevertheIess, for the purpose of this investigation the question of the correct model was largely irrelevant. Our intention is not to present a full kinetic analysis of the ion channel formation by gramicidin A. The object is primarily the demonstration of the equivalence of both kinetic methcds in a case which is sufficiently clear and where measurements are sufficiently sensitive to allow comparison of the statistical analysis with direct measurements of individual molecular events.

2. Experimental

solutions were equilibrated prior to use. 2.2. htrumentation

The lipids used were dioleoyl-La-lecithin (Supetco), glyceryl monooleate (Sigma Chem. Co) and cholesterol (Sigma Chem. Co). Valine grarnicidin A was a sampIe kindly provided by Dr. E. Grell from this institute_ n-Decane (reference substance for gas chromatography) was obtained from Merck. Ethanol was pa_ grade from Merck. KC1 and NaCl were Suprapur grades from Merck. All chemicals were used without further purification. The water used was triple destilled (twice from alkaline and acid KMNOq, respectively) and the salt

and filtered

and experimental procedure

MembranFe were formed by conventional techniques [ 161 over 0.3- 1.5 mm diameter openings of polypropylene vessels placed in a 30 ml outer container made of glass. The concentrations of the Iipid solutions were approx.: Dioleoyl-La-lecithin, 20 mM; dioleoylLa-lecithin: cholesterol, 20 : 20 mM; glyceryl-monooleate, 10 mM. Gramicidin A was added to the aqueous phase of the outer container in approx. 5 @? aliquots from 5 X 1O-g-5 X 10S6 M stock solutions in ethanol. Experiments were performed at room temperature (2 l-23°C). Two sintered Ag-AgCl pellets were used as electrodes. The membrane current was measured with a standard feedback circuit employing a model 380 K operational amplifier from Function Modules Inc. This amplifier was selected on the basis of its subpicoampere specifications with respect to current noise and current dl-ift, combined with low input capacitance and low input voltage noise. Membrane current noise measurements were performed at 100 mV applied potential. Relaxation experiments were performed using voltage steps from either 50 or 150 mV to the 100 mV level, supplied by a HewlettPackard Model 33 10 A function generator. The bandwidth was limited to 100 Hz during single channel experiments (1 GQ feedback resistor) and to 20 kHz during high conductance experiments (100 kSZ feedback resistor). The current signals were directly recorded on a strip chart recorder and an oscilloscope and simultaneously stored by an FM-magnetic tape recorder (Hewlett-Packard

2.1. Materials

with ndecane

Model

3960)

after passage

high pass filter with a time constant of 13 s and a further step of amplification. For correfation analysis the current recordings were subsequently replayed at higher speeds into a “Fabritek” Model 1074 Instrument Computer (Nicolet Instruments Corporation) equipped with a hIode SD-75 A auto/cross correlation digitizer. As this instrument utilizes only one sample of the signal waveform per sweep, the same noise trace was replayed repeatedly (with varying starting points) until the autocorrelation function showed no significant improvement by doubling the number of sweeps through a

H.P. Zingsheim and E.

Neher.Fh~tuation study of ion pores

in membranes

_

199

(5 12- 1024 sweeps)_ The relaxation experiments, however, were made “on line” using the Fabritek instrument in the averaging mode, averaging from 2 to 16

of Bamberg and L5uger [ 131. The described manner of plotting the data follows from a hypothetical reaction scheme, involving 2 dimerization reaction [ 131:

reIaxation responses_ All curves shown in this report are direct rcpraductions from the original results.

A-t-A

3. Analysis of experimental 3. I. The

data

steps The current signal at the single channei level represents a superposition of independent square pulses of varying duration. The autocorrelation function of such a signal is an exponential with a time constant equal to the mean duration of the pulses. This results from the following considerations: (a) The autocorrelation function of superimposed independent signals is equal to the sum of the autocorrelation functions of the individual signals. (b) The autocorrelation function of a random sequence of unit square pulses of ftied duration T is a symmetric triangular function centered at zero delay time and extending to f-T. (c) For a signal consisting of a random sequence of unit pulses possessing a distribution of durations, represented by ~(7’) =p(O) exp (-T/r,,) (121 one will obtain a distribution of triangular autocorrelation functions which, when superimposed according to (a), give rise to an exponential With a time constant rO_ At higher levels of conductance, individual pulses cannot be resolved. The above consideration is invalid, because the individual events may no longer be regardThe autocorrelation

function is,

however, still an exponential, provided the fluctuation5 arise from a single process. The time constant depends on the actual nature of the interzction [IO). 3.2. The dependence of the time constantson the

membraneconductance The reciprocal values of the time constants l/r (s-1) obtained by either fluctuation analysis or relaxation measurements were plotted versus the square root of the specific membrane conductivity A(S2-1 cm-z)_ This was done for ease of comparison with the results as

II)

kd

where A represents monomer and A, dimer. k, and kd are the forward and backward raie constants, respectively_

autocorrelationfunction of unit conductance

ed as independent.

kr = A,,

N=N,+2NZ,

(2)

where N (Mol cm-z) is the total molar concentration of gramicidin A moIecules in the membrane. N, and N2 aye the respective concentrations of monomers and dimers. The solution of the linearized rate equation is an exponential with a time consta* [I ] I/r=kd

+4k,PJ,.

(3)

By virtue of the law of mass action and the fact that the superposition of NZ single channels leads to an average conductivity X=N~N,A

(4)

one obtains l/r=k,,

+4~k~k,/N+/%,

where NL is Avogadro’s

channel

the unit

conductance.

3.3. Fluctuation Let lation tively. lation

(5) number and A (C2-t)

ampliiudes

G(O) and G(m) be the values of the autocorrefunction at zero and infinite delay time respecIt follows from the definition of the autocorrefunction that 9(O) is equal to the mean squared

amplitude and a(-) is equal to the square of the mean amplitudell71. D ue to (4), 9(O) - G(w) is proportional to N$ - p2 = c&, the variance of the distri-

bution of the dimer coricentration around the average concentration EZ _ in our measurements @(a) was usually zero, because the dc component of the current signal was removed by a high pass filter. Q(O) - +‘(a) was measured in units of (membrane current)2 and 4 denotes this quantity. hloreover, it is evident that the relationship *

* For footnote see next pap.

200

H-P. Zitzgsheim ami E. Neher, Fluctudion

study of ion pores in membranes

should be valid for independent channels [IS]. Generally this will be true only for very low conductivity values close to the single channel level. At higher conductivities, gh/x should decrease by a factor depending on the actual nature of the interdependence between the channels_ Following again the hypothesis of a simple dimerization reaction, one obtains:

This follows from o& = kT/(d2G/dN$,

where G is the Helmholtz free energy of the reaction, k the Boltzmann constant and T tie absolute temperature [ 19]_ K = k,/k, represents the equilibrium constant. G is expanded into a Taylor series around the equilibrium value in the usual way, including also second order terms. Together with the equilibrium condition dC/dN* = 0, with eqs. (2) and (4) this procedure yields eq. (6a). A plot of x/o: versus fi should yield a straight line due to

Fig. 1. Recordings of unit conduc-ace steps produced by gramicidin A in black lipid membranes formed from dioleoyl-L-crledthin in ndecane. Upper trace: at 1 Sf NaCl; lower trace: at I M KCI. The arrows poin:z towards infrequently occurring conductance steps of nonuniform magnitude_

casionally occurring small conductance

4. Results 4. I. Unit channel conductance Fig. 1 shows examples of unit channel recordings obtained with dioleoyl-L-cc-lecithin membranes. Below we summarize the unit channel conductance values A (at 100 rnV1 for the membrane systems investigated in this study (table 1). Our values were obtained from histograms of current amplitudes, measured by an A/D converter connected to a PDP-1 l/20 digital

channels (see fig. 1) are included, the average unit channel conductance dif‘&s from the most frequent va!ue by less than loo/o. Any distinction between integral and differential conductance definitions is irrelevant for this study, as the unit channel conductance of gramicidin A is practical!y ohmic in 1 M salt solutions at potentials up to 200 mV [12]. Table 1

Unit channel conductance LO” A (n-t)

Bbmbrane iipid in n-decane

computer. The most frequent value is given. If the oc* The normalization used is as follows: of is obtained from a: by dividingthis value by the membrane area and the square of the applied potential. The validity of thii procedure becomes evident, if the operation !r.Icoked upon as the superposition of the contributions of many independent membranes, so that the area adds up to 1 cm2-CTkevarianceof the sum of independent processes is equai to the sum of the variaxes.) T#UIS,uz has the dimension SZ* cm-.

diokoyl-I

-a-lecithin

a)

1 hf NaCl

1.95

1.65

2.3

dioleoyl-L-orlecithin f cholesterol a) g~yceryl monoohte

LhCKCl

b)

a) This study.

4.1

6) Data by Hladky and Haydon [ 12]- The v&e

agrea within = 5%_

of this study

4.2. The dissociation mre constant of ion channel fonnarion by gramicidin A As long as the assumption is correct that there exists only a single, exclusively conducting species (the dimer) the interpretation of the mean channel life time is immediately evident. It equals the backward rate constant k, of the reaction leading to the conducting channel.

Fig. 2a shows a life time histogram obtained by manual measurement of 800 recorded single channels in dioIeoyl-L-u-lecithin membranes at I M N&l. The appropriately scaled histogram is superimposed OR a direct reproduction of the autocorrelation function obtained from the same experiment_ From the histogram kd = 1.52 + 0.05 s-t _The autocorretation function yields kd = I.56 f 0.08 s-1 _ The analogous information for 1 M KC1 is presented in fig. 2b, with k, = 1.32 t 0.05 s-1 from the histogram (300 channeIs) and kd = I.28 + 0.08 s-t from the autocorrelation function. The corresponding values at I M KC1 (by autocorrelation only) for glyceryl monooleate are k, = 2.45 t- 0.07 s-t and for dioieoyl-L-cr-lecithin pIus cholesterol kd = 2.2 + 0. I s-1. 4.3. 77te dependerrceof the :ime ComtantS on the membrane canducriviry 4.3. I. Dioleoyl-L-at-lecithin membranes At higher conductivity levels individual single channels can no longer be resolved. However, it is still possible to obtain reliable autocorreiiition functions. Their time constants T are, within experimental error, equal to the time constants of the relaxation responses at the same conductivity. Furthermore, it is evident that any conductivity dependence of I/r, if extrapolated to zero conductivity (the single channel level) should yield kd for either method. The results presented in fig. 3 have been plotted in the manner described earlier. For a summarization of the numerical resdts see table 2.

FI 2. Histograms of we chnnel lifetimes. Ike autocorrehtioon functions obtained from the same experiments are superimposed; (a) at 1 M N&t. (b) at 1 M KCI.

4.3.2. Other lipids Correlation analysis of the conductivity dependence of the time constants wru also applied to black lipid membranes formed at I M KC1 from glyceryl monooIeate and I: I mixtures of dioleoyl-L-a-Iecitbin and cholesterol. In contrast to dioleoyl-L-a-fecithin membranes no statistically si&icant variation of I/r with 6 was observed for gIycery1 monooleate membranes (fig. 4a) over the same conductance range. A small variation may be deduced From the considerably scattered data in the case OF the mixed lipid membranes (fig. 4b).

ND+ K+ Na+ a) b)

K+ Ic+ K+ It+

conduc tlvity dcpcndencc, rclnxn1ion conductivity dcpcndcncc, rclnxation, hy Dnn1bcrg ml mgcr [ 13 1 single channel autocorrclation casductivity dcpendcncc, autocortclation single channel autocorrelation conductivity dcpcndcnee, autococrclotion 2.2

2.20

2.4

2.4~5

1.6 I.4

1.3 1.B

I3 I.4

oz5

CO.1

22 88

6.2 2.9

0.428

0,964 0,392

0.930 0,982

0.957 0.896

Nn+ IL+

conduc~~~tydc~cnd~n~, uutocorrelation 7.0 2.5

1.56 1.28

Ntl+ K+

1.32

sin@ ahonncl nutocorrclation

l~istafirnrn

1.52

‘xv

Nil+ K+

lo”‘%

single chnnncl lift time

&I

(s-k) (K'ai2s-1)

cation

(1M)

mothod

2nd order

AnaIysis of the constwts

10-‘3 K kd

10=6k

3rd order

0.2

CO.1

14 63

4.7 1.6

4.7 1.9

not applicable

not iipplicablc

14 120

288 0,8

2,7 a.7

not invcstis~tcd

no1 invcsti~~tcd

2.1 2.1

1.6 1.9

1.7 I.5

(M-'cd) (s?) (hi+ c:~s”‘)

Abb~ev~~ttons: D,t: DIoleoyl.L-a_IecitBIni GMO: Glyceryt inonooi~nte; Ck C~~o~esfcroi. r y: Correlation coeffident of the linear rcgmsion. GF 125 mV, data corrcclcd for A 5 L65 X 10”” ST’. b) 205 mV, datacorrected for A * iI65 x 10”” a.“‘.

D,L*Ch

GM0

D,L.

lipid

rncnlbr~nc

Experimental conditions

T*blc 2 Summuo JT the numerical results

0.960 0.989

0.930 0.931

Oa9SI O.gEO

'w

2.6

4.4

1.6 231

'1O"A W’)

2.8

co.1

16 1.7

UPK (Mm’ cm?)

Anntysis of fluctunlion ~rn~li~lide~ for llypottict. d~~criz~t~on

Ef. P.

Zingsheimand E.

Neher.

Fiucruarionrrudy of

ion pores

in

membranes

103

Fig. 3. Dependence of ‘he reciprocal time constxtts on membrane conductivity. I/r is plotted versus & (see text). (*) Correlation zmnalysis,(0) relaxation experiment. (a) At 1 hI NaCl, (b) at I hl KCL The arrows indicate the vJues obtained from single channel

measurements.

4.4. Slowflucmatiorzs Slow current fluctuations

introduced

additionaI inclined base lines of the autocorrelation functions were observed frequently_ The slow current fluctuations result from at least two undesired effects: (a) Lateral convection in the membrane, combined with an inhomogeneous gramicidin A distribution over the membrane and the adjacent bulk liquid interfaces_ (b) A combination of (a) plus capacitive transients upon the Fusion of solvent lenses with the torus 1161 produces large instantaneous current excursions proportional to the size of the lenses. The slow fluctuations showed amplitudes up to 10 times the normal current fluctuation amplitudes. Even SO, this means that at high levels of conductivity the membrane current was constant to within 1 part in at least 104 over time periods of the order of 10 s. For an example, see fig. 5. The slow fluctuations not only interfered with our measurements, they also completely masked any expected contributions by latere! diffusion of gramicidin A in the plane of very small membranes (10 pm diameter). High pass filters were used to suppress the undesired components. As a rule data were rejected, whenever the filter time constant had to be sm+ler than 10 t. Nevertheless, we frequentIy had to put up with inclined baselines. The slow fluctuations were nearly absent in dioleoyI-L-cclecithin f cholesterol membranes and most pronounced

time constants > 10 T. Correspondingly,

A

1.

0.1

0.2

0.3 K

m-‘cm-‘I

Fig. 4. AS tig 3, L M KCI. Correhtion analysis only.
HP- Zingsheirn and E. Neher. FIucruabm study of tin pores in membranes

204

B

A %’ IlO"'

A')

I

Delay

lima

1st

3.2

J 0

OeIoy

time

a 3.2

(5)

Fig. 5. Examples of autocorrelation functions obtained from glyceryl monooleate membranes in 1 hl KCI. In both ca%es a hi& pass fidter with a time constant of 13 set was used. The inserts show strip chart recordings of the actual noise traces (the amplified ac componi?nt of the membrane conductivity) from which the autocorrelation functions were obtained; (a) in the absence of slow fluctuations, (b) in the presence of sIow fluctuations. This is a typical example for data which were rejected.

in glyceryl monooleate membranes. Lateral convection could be minimized by restricting the amount of lipid solution around the hole in the polypropylene vessel to a minimum. 4.5. Amplitude analysis Fig_ 6 shows the experimental results for dioleoylL-cu-Iecithin membranes in 1 M KC1 and for glyceryl monooleate membranes in 1 IMKCI. The data are plotted according to eq. (6b). The solid iines represent the expected result when K is calculated from the kinetic parameters (figs. 3b, 4a). For the remaining two types of membrane, namely dioleoyi-L-a-lecithin in 1 M NaCI and dioleoyl-L-&-I-lecithin + choIestero1 in 1 M KC1 agreement with eq. (6) is poor. Although the plots are linear and for low conductivity values 4/X equals A within
lyze the experimental results, there remain three further points: (a) The channels are independent at the unit conductance level. This was already investigated by Hladky and Haydon (121, who showed that the frequency of occurrence of 0, 1,2, .._ etc., simuitaneous-

1.L

‘.

t.2

”

1.0

‘.

01

5. Discussion Although we have already pointed out a number of assumptions which must be made in order to ana-

0.2

Q3

qr

X

IQ-%l-'l

Fig. 6. Dependence of X/ai on fi (see text). Upper part: Dioteoyl-L-a-lecithin membranes in 1 M KQ. Lower part: Glyceryl monooleate membranes in 1 M KCI. The arrows indicate the reciprocal values of the unit channel conductances-

H-P_ Zingsheim and 15’. Neher. Ftucruarion rtltdy of ion pores in membranes

ly open channels is described by a Poisson distribution.

(b) The autocorrelation function arises from a single process, provided slow fluctuations, which represent convective motions in the membrane, are absent. Pure diffusion in our system is sucficiently slow and may be neglected: for a membrane diameter of O-l- 1 mm and an assumed upper limit of IO-8 cm’ s-l for the lateral diffusion coefficient of gramicidin the time constants due to diffusion would be of the order of hours. Thus, there is only an extremely small probability that an existing channel is terminated by diffusion out of the membrane into the torus. (c) The rate of association is not diffusion limited. This was shown by Bamberg and tiuger [ 131. As figs. 3,4, and table 2 show, the results (I/r, k,, kd) obtained by a relaxation method and by statistical analysis of the membrane current noise are in good agreement. Moreover, the particularly beneficial feature of the studied model system lies in the possibility of independently obtaining two important parameters, namely the backward rate constant kd and the unit channel conductance A by direct measurements. Thus, for low conductance values we may also compare l/kd with the mean channel lifetime and gx/x with A (see also table 2). Again, there is excellent agreement. At low levels of conductivity (< LOA n-1 cm-*) both ka and A may be measured within * 10%. At this level of conductance the channels may be regarded as independent. Less than 2% of the gramicidin molecules in the membrane form conducting channels. On the other hand, at a membrane conductance of 10-t Q--t cm-2 approx. 40% of the gramicidin molecules form channels. (Values for dioleoyEL-a-lecithin membranes at 1 M KCI.) The comparison of our results (k,) for dioleoyl-I_cr-lecithin membranes in 1 M NaCl with the results of Bamberg and LSuger [ 131 shows agreement only within a factor of 2-3, even if the latter results are recalculated for the correct value of A (see table 2). (The authors assumed A = 2.4 X LO-” 52-t. This is the value for glyceryl monooleate membranes in 1 M NaCl [12]. The correction reduces their k, by approx. 30%.) There are at least two possible explanations based on experimental observations: (a) Frequently, and almost exclusively for dioteoylL&-lecithin membranes in I M NaC1, erratic current bursts were observed_ They were strongly voltage de-

20.5

pendent and much larger and faster than the normal current fluctuations due to gramicidin A or alamethicin [20]. Under these conditions the time constants were shortened by a factor of 2-S in both correlation and relaxation measurements. The nature of this contamination is not clear. An inorganic precipitate is suggested by the partial success due to filtering of the salt solutions as opposed to roasting of the NaCl. (b) If the gramicidin A, which spreads at the oil water and membrane water interfaces [2 1,221, is not homogeneously distributed over the plane of the membrane, the specific membrane conductivity x will always be underestimated. This leads to an overestimation of k,. Correspondingly, we observed abnormally short time constants whenever the conductivity was still rising steeply after gramicidin A had been added to a membrane which previously had given normal results. The effect is potentially more serious with larger membranes. Large membranes are needed in relaxation studies for reasons pointed out by Bamberg and i3uger [ 131. The poor agreement of a:/x with the prediction by eq. (6) at high conductivity values for dioleoyl-la-lecithin membranes in 1 hl NaCl may also be caused by these effects. On the other hand the large scatter of the time constants from dioleoyl-l-cy-lecithincholesterol membranes most likely reflects small differences in membrane thickness, due to slight variations in membrane composition. It is known that Eid depends strongly on the membrane thickness [ 121. Generally, a most important factor influencing the reproducibility of the measurements was the age of the membranes. It is very difficult to obtain truly equilibrated black lecithin membranes of a well defined, minimum thickness [ 161. No problems in this respect are encountered with the glyceryl-monooleate membranes, which reach their equilibrium thickness within a few seconds after drainage [ 16, 141. Although the ion pore properties of the gramicidin A induced membrane conductance have been thoroughly characterized 112,231, the mechanism of pore formation has not yet been established unequivocally. The dimerization reaction used in rationalizing the kinetic data must be regarded as hypothetical as long asother alternatives have not been excluded. Even the order of the reaction is difficult to obtain from the presently available data. A linear regression analysis of our data

HP. Zingsheim and E_ Neher. Flucruaticm sfuiy

206

points and of those published by Barnberg and Euger shows that the data fit a 3rd order trimolecular reaction just as well (see table 2). The only reliable kinetic clue that the 2nd order is more likeiy comes from

comparing the k, values (obtained by extrapolation of concentration

dependence

of the time constants)

with the independently obtained values (from sin@ channel measurements)_ Agreement is poor for 3rd order. This comparison was not possible for Bamberg and LSuger, though, because the mean single channel life time was not known to them. The above reservations also apply to the interpretation of the fluctuation amplitudes at higher conductivity levels. The same assumption concerning the reaction scheme had to be made again for the derivation of eq. (6). Moreover, the equilibrium constant K in eq. (6) presently cannot be obtained from independent measurements of the concentrations of the ncnconducting species in the membrane. The presence of a second, slow time constant in the autocorrelation function (due to the described slow fluctuations) affected the accuracy

of the amplitude

-of the time constants.

ly absent in the membranes le-,ithin + cholesterol

analysis more than that

As slow fluctuations

were near-

formed

from dioleoytl-a-

the amplitude

analysis on this

system is more rehable than the kinetic analysis. The opposite is true for pure dioleoyl-L-a-lecithin membranes. We conclude that measurements of the ion current noise in membranes can be at least as reliable as relaxation studies. Due to their essentially two-dimensicnaf geometry, membranes (and interfaces) are inherently well adapted to the main requirement for fluctuation analysis, namely that the number of observed events be sufficiently smaI.l. Apart from the practical advantage of obtaining kinetic and equilibrium information whenever the membrane current is measured, a further important advantage of fluctuation measurements lies in the fact that the system need not be perturbed. The choice of suitable perturbing parameters is rather restricted if fast membrane transport processes are to be investigated_ In the case of black lipid membranes the reported method eliminates disturbing capacitive transients due to torus relaxations and makes kinetic measurements feasible, which would otherwise be impossible [24].

of

ion pores in membranes

in our study the concentration fluctuations frequently were of the same order of magnitude as the average concentrations. Nevertheless, we observed many artifacts of which the most important examples have been discussed. The experience gained by this work at first sight seems pertinent only to a rather special type of membrane conductance which trans-

duces concentration fluctuations into electrical noise. Many of the difficulties are, howevet, caused by the properties of the black lipid membrane [16]. The probIems will therefore reappear in the expected attempts to apply correlation analysis to processes (diffusion, binding) measured by optical transducers, i.e., fluorescent probes.

Acknowledgement

We are particularly grateful to Dr. K. Cnidig and our attention to the conand for many helpful discussions. We are also indebted to Dr. K.H_ Tews for the computer program used in obtaining our ampJ.itude histograms. Dr. J. Hendrix

for drawing

cept of correlation

analysis

References

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