- Email: [email protected]
[7] Nonstationary noise analysis and application to patch clamp recordings
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NONSTATIONARY NOISE ANALYSIS
131
from the bath solution. They represent the liquid junction potentials that are present at the pipette tip before patch formation. In addition, corrections have to be applied when the bath solution is being changed during a measurement (i.e., after seal formation). In that case the following rules apply. (1) The new solution should never get into contact with the bare silver/silver chloride wire of the reference electrode. This requirement is best met by using a salt bridge. (2) The "best" salt bridge is a 3 M KCI bridge with an abrupt KC1-bath fluid boundary at its tip (see above). This bridge does not require any additional potential corrections, but it may lead to KC1 poisoning of the bath or become contaminated by solutions used previously. (3) Local solution changes (microperfusion by puffer pipette, O tool or sewer pipe arrangements) as well as recessed KC1 bridges require additional corrections, which (together with the simple liquid junction potential correction) are approximately given by Eqs. (6)-(8). It should be stressed that all equations given here represent approximate corrections, since liquid junction potentials are thermodynamically ill-defined. This is particularly relevant for Eqs. (6) and (7) where the sum of two liquid junction potentials appears.
[7] N o n s t a t i o n a r y N o i s e A n a l y s i s a n d A p p l i c a t i o n t o Patch Clamp Recordings
By STEFANH. HEINEMANNand FRANCOCONTI Introduction Noise analysis was the first type of measurement that yielded reliable quantitative estimates of single-channel parameters ~-3 and thereby provided evidence for the very existence of channel proteins embedded in the lipid matrix of excitable membranes. It is still a powerful tool for investigation of ion transport mediated by channels. The more direct way of measuring single-channel properties is the recording of unitary events using the patch clamp technique.4,5 Singlechannel recordings provide the richest information on the kinetics of the ' B. Katz and R. Miledi, Nature(London) 226, 692 (1970). 2 F. Conti, L. J. DeFelice, and E. Wanke, J. Physiol. (London) 248, 45 (1975). 3 E. Neher and C. F. Stevens, Annu. Rev. Biophys. Bioeng. 6, 345 0977). 4 E. Neher and B. Sakmann, Nature (London) 260, 799 (1976). 50. P. Hamill, A. Marty, E. Neher, B. Sakmann, and F. J. Sigworth, PfluegersArch. 391, 85 (1981).
METHODS IN ENZYMOLOGY,VOL 207
Copyright© 1992by AcademicPress,Inc. Alldghtsof reproductionin any formreserved.
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ELECTROPHYSIOLOGICAL TECHNIQUES
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conformational transitions which underlie the operation of individual channel proteins, but they are extremely difficult to perform when these kinetics are fast for the channel under investigation, as in the case of the voltage-activated sodium channel. Furthermore, they can only be analyzed easily for membrane patches containing few channels, possibly only one channel, and analysis requires a long time. For this reason most studies of single sodium channel events have been limited to fixed temperature conditions and to the narrow voltage range in which the frequency of channel openings is not too high. 6-s Also, only single-channel events larger than the background noise can be evaluated directly, whereas macroscopic fluctuations can reveal much smaller unitary signals, as in the case of gating noise experiments where charges translocated in an elementary conformational transition of voltage-gated ion channels are measured. 9,1° Analysis of nonstationary fluctuations of macroscopic currents are a classic way of estimating single-channel conductance. 2,1~-1a Whole-cell recordings from small cultured cells provide an ideal preparation for this type of study, owing to the very high ratio between channel noise and background noise which is obtained with this technique. ~4 The greatest advantages of this method as compared to single-channel recordings are (1) it yields fairly accurate estimates of the single-channel conductance for most preparations; (2) enough useful recordings can be obtained from the same preparation for several different experimental conditions; and (3) the analysis does not involve any subjective selection of events, it requires much shorter time, and it is easily automated by computer programs. In patch clamp recordings, transient artifacts due to instabilities of the seal as well as rundown and drift phenomena, especially while measuring under extreme environmental conditions, are not uncommon. Therefore it is advisable to introduce further improvements to the method of nonstationary noise analysis as used earlier. Among these are (1) a better strategy for analysis of recordings to obtain variance estimates, (2) a more rigorous fitting procedure of variance versus current plots to account for weights of the data, and (3) automatic, objective tests for discarding "bad" records. As an application of the analysis method we shall report some studies of nonstationary fluctuations of sodium currents in bovine adrenal chromaf6 F. J. Sigworth and E. Neher, Nature (London) 287, 447 (1980). 7 E. M. Fenwick, A. Marty, and E. Neher, J. Physiol. (London) 331, 577 (1982). 8 R. Horn and C. A. Vandenberg, J. Gen. Physiol. 84, 505 (1984). 9 F. Conti and W. Sttihmer, Eur. Biophys. J. 17, 53 (1989). lOS. H. Heinemann, F. Conti, and W. Stiihmer, this volume [22]. 11 F. Conti and E. Wanke, (2. Rev. Biophys. 8, 451 (1975). ~2F. J. Sigworth, Nature (London) 270, 265 (1977). i~ F. J. Sigworth, J. Physiol. (London) 307, 97 (1980). 14A. Many and E. Neher, in "Single-Channel Recording" (B. Sakmann and E. Neher, eds.), p. 107. Plenum, New York, 1983.
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NONSTATIONARY NOISE ANALYSIS
133
fin cells for estimating the temperature and pressure dependence of the conductance of voltage-activated sodium channels. Nonstationary Noise Analysis The purpose of noise analysis is to relate macroscopic observables, such as the total ionic current, to microscopic parameters like the single-channel current i, thc number of functional channels in the membrane N, and the probability that thc channels are open under a given condition, P o ~ . For a homogeneous population of statistically independent channels, the mean, I(t), and the curent variance, al(t) 2, arc given by 15,16 I(t) = Nipov~(t)
0-~(t)2 = N i 2 p o p e . ( t ) [ 1 -- Pove,(t)]
(1) (2)
Provided that I and 0-2 are estimated for various open probabilities, i and N can be determined by data fitting procedures according to 0"1(I) 2 = I i - I 2 / N
(3)
as first introduced by Sigworth. 12'13'17It is customary to convert i estimates to a single-channel conductance y, obtained by knowledge of the specific reversal potential, E ~ , and the potential set by the voltage clamp, Eeom, ), = i/(Eoo m -- E~v)
(4)
Note that the latter equation implies a linear relationship between i and the voltage only if ), is assumed to be constant. Effect of Series Resistance
In practice the single-channel current in Eq. (3) may vary because the variable mean current I ( t ) gives rise to a variable voltage drop across the series resistance R s which is determined by the electrical access from the pipette electrode to the cell membrane. For small values of I R s we have i = i* - 3 , * I R ,
(5)
where the asterisk denotes quantities ideally measurable for a membrane potential equal to Ecom. Thus, it follows 0"~ = i * I -
y * I 2 R s -- I 2 / N
=i*I
l + F m ~ R , i2 N
(6)
15 G. Ehrenstein, H. Lecar, and R. Nossal, J. Gen. Physiol. 55, 119 0970). 16T. Begenisieh and C. F. Stevens, Biophys. J. 15, 843 (1975). 17 F. J. Sigworth, in "Membranes, Channels and Noise" (R. S. Eisenberg, M. Frank, and C. F. Stevens, eds.), p. 24. Plenum, New York, 1984.
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ELECTROPHYSIOLOGICAL TECHNIQUES
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where F , ~ = ?*N is the maximal macroscopic conductance. This result implies that a finite value of R, does not affect the estimates of i* derived from nonstationary noise analysis, provided a uniform voltage clamp exists all over the cell membrane. For small cells without big membrane invaglnations this is a fair approximation, because R, is mainly determined by the pipette tip opening. 14 Equation (6) can be written formally as Eq. (3): a~ = i * I - I2]N *
(7)
showing that R s causes a wrong estimate of the number of channels by the factor (1 + N y R , ) -1. In most cases this correction factor is expected to be close to unity, except for whole-cell measurements from large cells where N exceeds 1000. For example, ?* = 20 pS, Rs = 5 Mf~, and N = 1000 would yield an error of 10% in the estimate of N. A correct fitting procedure according to Eq. (7) requires proper account to be taken of the errors involved in the determination of a 2 a n d / , errors which vary very strongly as a function of L Another important weighting factor in fitting a 2 versus I data is due to the strong correlation expected to exist between a 2 measurements separated in time by intervals of the order of the time constants which characterize the time course of the voltage clamp responses. The theoretical considerations underlying the fitting procedure that we have used to cope with these problems are detailed in a later section. Bandwidth Limitations
Because the bandwidth of the recordings is limited, the measured variance will be smaller than the true variance. To estimate the effect of bandwidth consider the case of a channel whose gating is described by a single relaxation process with the time constant z. The variance of the channel noise is then the total integral of a Lorentzian spectrum a 2 = fo** 1 + So (f/fN) 2 d f = ~--Sof~r (8) 2 wherefis the frequency andfu = I/(2rcr). Further assuming a filter with an infinitely sharp cutoff at f~ the measured variance is a~m)= Sof~ t a n - l(fJfN) (9)
yielding a relative deviation from a 2 of a2 - cram)= 1 - 2 tan_l(fc/f~.) a2
7[
(10)
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NONSTATIONARYNOISEANALYSIS
135
Thus, with ft¢= 1 kHz (z ~ 160/tsec) and a recording bandwidth of 10 kHz, the measured variance is only 6.3% smaller than the theoretical variance. The deviation is 9.1% for a one-pole filter. Stimulation Protocol Following the succession of operations actually performed in the offline analysis of the data we shall describe the protocol of analysis in two steps: (1) determination of mean currents and variances and (2) fitting of these data according to Eq. (7). Before, however, we describe the stimulation protocol tailored to noise measurements. The protocol of voltage stimulation for noise measurements was designed to optimize the subsequent off-line analysis of the data. It consisted of repeated alternating sequences of test stimulations m from a stationary holding potential, En, o f - 90 mV to test potentials between - 10 and + 30 m V m a n d control stimulations having the same temporal pattern but voltage amplitudes of only ___30 mV relative to a control holding potential o f - 100 inV. Each test sequence consisted of 20 identical stimulations, and each control sequence consisted of 4 stimulations. Successive stimulations were repeated after a pause period of about 1 sec. For a typical noise measurement at any test potential these alternating sequences were applied successively at least 7 times for collecting a total of at least 140 test records, yielding a theoretical 10% accuracy of the variance measurements and, therefore, of the single-channel conductance estimates (see below). Thus, the measurement lasted about 3 min. In between successive noise measurements at different membrane potentials and/or under different temperature or pressure conditions, standard protocols of stimulation were applied to collect information about the macroscopic current-voltage relationship and the voltage dependence of channel inactivation. These protocols allowed us to characterize the size of the overall drifts which had occurred in the preparation during the noise measurements, owing to rundown or, in the case of whole-cell recordings, to diffusion of intracellular substances initiated by the pipette perfusion. 7 Analysis of Noise Records Very generally, fluctuation analysis yields estimates of the parameters which characterize elementary events with an accuracy increasing linearly with the square root of the number of independent samples and therefore with the square root of the time during which fluctuations are measured. When applied to biological preparations this basic principle finds itself in conflict with the intrinsic property of these systems of undergoing steady, irreversible modifications. Therefore, it is mandatory to organize the analysis of the data so that there is a minimum contamination of the estimated
136
ELECTROPHYSIOLOGICAL TECHNIQUES
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properties of reversible fluctuations from systematic irreversible changes of the system.
Difference Records Ideally one could calculate the ensemble variance as the mean-square deviation of each record from the mean. However, particularly in patch clamp recordings, slow drifts in the electrode potentials, changes in cell capacitance, washout phenomena, and irreversible shifts in the voltage dependence of ion channels are known to occur as function of time.14 Such linear drifts are largely eliminated if the variance is calculated from the ensemble average of the squared differences of successive records, ~(t)18:
a~ = ( ~ ( t ) 2)/2
(1 I)
For the estimate of the variance according to Eq. (11) all couples of successive records [i.e., (1,2), (2,3), . . . ] can be used rather than only the nonoverlapping ones, because it can be shown that by this procedure the accuracy of the variance estimates is improved by a factor of 2/(3 m) (see Appendix I). An example of analysis of an experiment with large leak currents and a changing, uncompensated capacitance is shown in Fig. 1.
Discarding "Bad" Records All test and control records were analyzed for background noise by calculating the variance of the baseline current a~ sampled during the first few milliseconds of the records at the holding potential. A first upper limit for the maximum acceptable variance in any record is initially set by the analyzer in order to discard records that for some reason (transient deterioration of the pipette seal, extraneous electrical artifacts, etc.) have an obvious abnormal background noise. For the remaining records the mean, (a~), and the root mean square (rms) deviation of a~ from the mean, Aa~, was calculated and used for a second selection, based this time on the objective principle that any record with a~ > (a~) + 4Aa~ is most likely affected by some artifactual extra noise. In the majority of our measurements both of these tests were passed by all records, but a few percent of noise sequences contained one to four records to be discarded according to these tests. In all these cases a~ was at least 50% higher than its maximum allowed value, clearly legitimating the selection performed by the algorithm.
Analysis of Leak Records First a further check for the absence of occasional artifacts was performed by discarding any record which differed at any point from the next is F. Conti, B. Neumke, W. Nonner, and R. St/lmpfli, J. PhysioL (London) 308, 217 (1980).
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NONSTATIONARYNOISEANALYSIS
137
to
B
E __! C F
3 ms
FIG. 1. Steps in nonstationary noise analysis of records in response to test pulses to - 10 mV at 14°. (A) Averaged leak record (N= 72) yielding a leak reversal potential of 18 mY. The maximal accepted baseline noise was 5.96 pA (rms); the estimated baseline variance was 20.5 pAL Four leak records did not pass the criterion and were discarded. (B) Consecutive individual current traces during test pulses without leak correction. (C) Difference records with a boundary of 7 times the expected standard deviation [see Eq. (17)] determining the criterion for discarding records with excessivelylarge noise. (D) Two covariance functions centered at the time to the peak current, tw and at 3t~, respectively,showing a correlation time of 1.1 msec. (E) Leak-corrected mean current (N= 219); one record had to be discarded. (F) Ensemble variance record. one by m o r e than expected on the basis o f the previous m e a s u r e m e n t of (try) (apart f r o m a constant difference arising f r o m a baseline shift). The analysis of the remaining records was accomplished in two steps. F o r each record, the difference between the m e a n baseline current, I~n, and the m e a n current, I c, during the second half o f the m a i n control segment at voltage E c is c o m p u t e d in order to obtain an estimate o f the leak reversal potential, EL:
1 ~ ICiEn-InEc EL=-~L i=' -ffi Z i ~
(12)
Then, the ensemble m e a n o f all leak records ~ ( t ) was calculated 1
~vL
( ~ ( t ) ) =~LL ~ ~L(t)
(13)
i--I
as illustrated in Fig. 1A. The purpose o f calculating EL is to correct possible artifacts in the analysis o f test records occurring when successive records
138
ELECTROPHYSIOLOGICAL TECHNIQUES
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have differences in the mean baseline current which are statistically unexpected, indicating changes in leak or seal conductance, 7L, between the records. It would be incorrect to cope with these changes by simply assuming that the difference record has a constant bias, because the variation of leak current is expected to be given by AIL = ATL(E-- EL)
(14)
so that it will be different for the holding potential and the test potential.
Analysis of Test Records The first step in analyzing test records is simply to obtain an ensemble average from all of them, which is then corrected for linear leakage and capacitive components by subtracting the ensemble average of the leak records appropriately scaled. The mean record obtained in this way was then used as a first estimate of the time course of the mean currents to perform a further test of the quality of each individual record and discard those which contained deviations from adjacent records much larger than statistically expected. This test was based on the expectation that at any time t, Prob(Itf~(t)l > 512(a~+ (a2))] '/2) < 10 -4 (15) so that any couple of records for which It~(t)l > 7(try+ (a~)) '/2 can be safely discarded as being affected by some artifact. At this state, before any actual measurement of al, the latter quantity could only be estimated theoretically from its upper limit:
a~ = iI(l -Po~.~) < il
(16)
For estimating i we also had to rely on already analyzed data yielding a reasonable estimate for the single-channel conductance. Therefore, the test for accepting any couple of records was [t~(t)l < 7[ ? ( E ~ - E ~ ) I + ( tr~)] u2
(17)
In Fig. 1C consecutive difference records are shown together with the envelope determined by the criterion of Eq. (17). In 70% of the measurements this test was passed by all test records. In the worst experiment 5 records out of 120 were discarded. For the final analysis only the selected records were used to evaluate the mean current (Fig. 1E). Then, for each pair of successive selected records a "pure fluctuation record" was constructed by taking the difference of the original records. If the difference in the time averaged baseline current of the two records exceeded its maxim u m expected random fluctuation [+4((a~)ToB) ~/2, where B is the bandwidth of the recordings and TOthe duration of the recorded baseline
[7]
NONSTATIONARY NOISE ANALYSIS
139
trace], the difference record was corrected on the basis of the assumption that the baseline change was due to a change in leak conductance. The fluctuation records were then used to compute the ensemble variance according to Eq. (11) (see Fig. IF) and autocovariance functions centered at different times, ti, d~(t~,t) = ( ~ ( t , ) ~ ( t ) ) (18) Figure 1D shows two covariances centered at the time to the peak current, to, and at 3tp, respectively. The covariances can be used for consistency checks that the measured noise has correlation features expected for the fluctuations under investigation. This is of particular importance, for example, for gating noise experiments, where one wants to verify that no ionic currents contaminate the gating c u r r e n t s . 9 The covariances are also used as a criterion for grouping the variance versus current data in bins as discussed in the next section. Analysis of Variance versus C u r r e n t Plots For the correct attribution of weights to the estimates of I and a 2 two important considerations must be made. (1) Each estimate of I and a~ is affected by an error proportional to a~ and to a2, respectively, which depends on I. Therefore, estimates which were obtained near the peak of the mean current, Ip, must be weighted less than those obtained for small values of I if the maximal Po~-, is smaller than 0.5. (2) Fluctuations in the current measured at any time are strongly correlated with those measured within time intervals of the order of the intrinsic correlation times of channel fluctuations, which also characterize the macroscopic current kinetics. For a constant sampling interval, a plot of all (/, cry) estimates would show a much higher density of points for I values collected during the slow phase of the response, attributing excessive independent weight to data that are not independent. To account at least partially for correlations the mean current and variance data are grouped into a smaller number of bins corresponding to the division of the duration of the main pulse segment into successive intervals of variable length such that (1) I does not vary by more than Ip/20 within each interval and (2) the interval length does not exceed the estimated correlation time, to. Mean variances within any bin were plotted against the corresponding mean current. The standard error of any such individual data point was then estimated taking into account correlations of data only within each bin (see Appendix II). This assured that the data points were now approximately statistically independent such that least-squares fitting methods could be used. The data, weighted according to the inverse of the sum of the squares of the standard errors of the
140
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ELECTROPHYSIOLOGICAL TECHNIQUES
500
-
4oo 300 0 e--
•=
200
10o I
I
I
I
250
500 Current [pA]
750
1000
FIG. 2. Variance versuscurrent for the experiment shown in Fig. 1. The result of the fit was ~,= 10.2 pS, Pn= ----0.32, N= 4630 channels. The curve represents a nonconstrained fit. The diamond showsthe estimated background variance during the baselinebeforethe pulse. variance and the mean, were finally fitted by the relationship of Eq. (7) employing an effective variance least-squares method ~9which accounts for both errors in I and a 2 and allowing a constant background variance, expected to be close to ( a 2). A plot of variance versus current corresponding to the raw data shown in Fig. 1 is given in Fig. 2, yielding an estimate of the single-channel current of 1.18 pA. The data fit according to Eq. (7) yield an estimate of the current flowing through an open channel at the test voltage. This value was converted to an estimate of the apparent single-channel conductance by dividing it by the driving force applied to the channels, using estimates of the reversal potential, E~,, obtained from macroscopic current-voltage measurements. If patches had a too small number of channels, current voltage plots were not reliable, and E,~, was assumed to have the standard value of + 5 5 mV, the mean of measurements from large patches and whole cells. Reliable estimates of the number of channels could only be obtained for voltage steps which elicited large currents, that is, such that the open probability near the peak was larger than 0.2. This is seen in Fig. 3, which shows the analysis of current fluctuations at low potential and high temperature. An estimated maximal open probability of 0.1 does not allow an unconstrained fit according to Eq. (7). For this purpose we introduced two possible constraints; first, we fixed optionally the zerocurrent point to the measured baseline noise variance. The other constraint was a fixed number of channels, which can be estimated roughly. This does -
19j. Orear, Am. J. Phys. 50, 912 (1982).
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NONSTATIONARY NOISE ANALYSIS
141
1500
~o. 1000 O
500
I
I
500 1000 Current [pA]
I
1500
FIG. 3. Variance versus current from a measurement at 27 ° with test pulses to - 30 mV. The correlation time was 0.56 reset. The result of the fit was y = 13.8 pS, Pm~ =0.10. Because the open probability of the channels was so low, no unconstrained fit could be performed. The background noise level was fixed to the value estimated from the baseline (diamond), and the number of channels was set to 9000.
not appreciably affect the estimate of i*, but it obviously excludes the possibility of estimating N independently. It should be stressed that, no matter how laborious the whole procedure might appear, the actual computer time required by our PDP 11/73 computer to perform the whole analysis, starting from 200 records of raw sampled data of 256 points each and ending with an estimate for 7 and for N* was 1 min for step 1 and 10 sec for step 2. With computers used in the laboratory nowadays the process could be faster by another order of magnitude such that the time for analysis poses no real limit. Application to P a t c h C l a m p Data Recordings of single-channel events undoubtedly contain more information than noise measurements about the gating kinetics and the conductance of specific ion channels. 2° However, in spite of the perfected recording technique currently available, there is still a broad variety of cases where application of single-channel analysis is not feasible. Among the possible reasons are a too small signal-to-noise ratio making it impossible to identify single channel openings and the amount of time needed to analyze such recordings, particularly if ion channel screening is desired. To illustrate the application of nonstationary noise analysis, we investigated the effect of temperature and hydrostatic pressure on the single20 F. J. Sigworth and J. Zhou, this volume [52].
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ELECTROPHYSIOLOGICAL TECHNIQUES
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channel conductance of voltage-activated sodium channels in bovine adrenal chromaftin cells. The experimental conditions were such that singlechannel analysis would have failed because of prohibitively large background noise (high-pressure recordings) or because of too short durations of the single-channel open state (high temperature). Materials and M e t h o d s
Chromatfin cells from the medulla of bovine adrenal glands were prepared and cultured as described by Fenwick et al. 7 All measurements were performed in either whole-cell configuration or on excised outside-out patches, in both instances using the methods described by Hamill et a l : Standard patch pipette holders were used in all experiments, except for those on the effect of pressure, for which we used a specially designed holder and cable connection to the headstage of the recording amplifier (EPC-7, List Electronics, Darmstadt, Germany). The methods for pressurization of cells and membrane patches under patch clamp control are described by Heinemann et al. 2~ Pipettes with large tip openings and resistances in the range of 1 to 3 Mf~ were selected in order to reduce the access resistance, R,, for whole-cell measurements. However, we did not make any further attempt to reduce the effective value of Rs by electronic compensation, since (as discussed above) the most important artifact generated by R, in whole-cell recordings of voltage clamp currents turns out to have no consequences for the estimates of single-channel conductances from nonstationary noise analysis. Current records were sampled and stored on-line on a Winchester disk with the aid of a PDP 11/73 computer, which also generated the voltage stimulation protocol. Before being sampled (at a rate of 25 kHz) the current records were passed through an 8-pole Bessel filter with a - 3 dB cutoff frequency at 10 kHz. Solutions. In most of the experiments described here the extracellular solution had the following composition (in mM): 140 NaC1, 2.8 KC1, 2 MgC12, 1 CaCI2, 10 HEPES-NaOH, pH 7.2. In some control experiments with whole cells Co 2÷ was used to replace Ca 2+ in order to ensure that our records of sodium currents were not contaminated by calcium currents. We found no obvious difference in the results obtained with the Ca2+-free solutions, most likely because the contribution of the sodium channels to the total currents is in any case overwhelming. Furthermore, it is known that calcium channels subside after 10 min of cell dialysis: Co 2+ was used 21S. H. Heinemann,W. Sttahmer,and F. Conti,Proc. Natl. Acad. Sci. U.S.A. 84, 3229(1987).
[ 7]
NONSTATIONARYNOISEANALYSIS
143
routinely in measurements from excised outside-out patches, where a significant contribution from few calcium channels might have occasionally and unpredictably biased our data. The pipette solution which was invariably used had the following composition (in mM): 70 CsC1, 70 CsF, 1 MgC12, 0.5 CaC12, 10 HEPES-NaOH, 11 EGTA-NaOH, pH 7.2.
Effect of Temperature and Pressure on Sodium Channels The dependence of the ionic permeabilities of excitable membranes on intensive thermodynamic parameters like temperature and pressure, besides having important physiological consequences on the adaptation of biological systems to different environments, yields information about the molecular mechanisms underlying the operation of ion channels. Changes with temperature and pressure in the time course of the relaxation of ionic currents following a step in membrane potential in a classic voltage clamp experiment can be directly ascribed to variations of the rate constants of the conformational transitions of ion-channel proteins, and, within any assumed kinetic scheme, can be interpreted unambiguously in terms of activation enthalpies and activation volumes of these transitions. ~ However, variations in the absolute values of the ionic currents yield only information about the product of two quantities which can both be strongly dependent on the external parameters. One is the total number of activatable ion channels; the other is the conductance or permeability of a single open channel. Most literature on temperature effects has assumed that N is constant, apart from irreversible rundowns of the biological preparation. However, slow inactivation phenomena, by which the number of normally functioning ion channels strongly depends on the holding membrane potential, are known.23a4 It has also been clearly demonstrated that transitions of functional ion channels to conformational states from which they cannot be easily resumed for normal activity are brought about by decreasing temperature25 or increasing pressure. 26 The only possibility to quantify these phenomena reliably is to measure the single-channel conductance directly. The effect of temperature on the conductance of single sodium channels was estimated by averaging over 14 experiments at temperatures between 9* and 28 ° . Measurements at various temperatures on the same 22 F. Conti, Neurol. Neurobiol. 20, 25 (1986). 23 j. M. Fox, Biochim. Biophys. Acta 426, 245 (1976). 24 B. Rudy, J. Physiol. (London) 283, 1 (1978). 25 D. R. Matteson and C. M. Armstrong, J. Gen. Physiol. 79, 739 (1982). 26 F. Conti, R. Fioravanti, J. R. Segal, and W. St0hmer, J. Membr. Biol. 69, 23 (1982).
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ELECTROPHYSIOLOGICAL TECHNIQUES
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TABLE I PRESSURE EFFECT ON SODIUM CHANNELS a
Potential
Pressure
Temperature
~, Nch
(pS)
pope.
(msec)
89 70 88
3850 2170 1792
12.9 11.9 11.7
0.16 0.13 0.13
0.64 1.12 0.64
21.0 21.6 21.3
128 89 126
7370 3000 2600
11.5 12.6 11.5
0.29 0.26 0.33
0.48 0.64 0.24
0.1 41.7 0.1
21.6 22.0 21.7
114 114 114
5550 2350 2060
11.4 13.9 14.3
0.21 0.18 0.23
0.64 1.04 0.56
0.1 40.0 0.1
16.1 16.3 16.0
67 72 72
1670 910 920
10.5 11.6 11.5
0.27 0.22 0.30
0.64 1.28 0.64
(m V )
(MPa)
(* C)
N,~
-
10 -10 - 10
0.1 45.5 0.1
23.2 23.8 23.5
-
10 10 - 10
0.1 43.3 0.1
-10 - 10 - 10 10 10 10
a The effect of hydrostatic pressure on the tingle-channel conductance ~,, the maximal open probability po~.,, the correlation time z, and the estimated number of functional channels under investigation Na~. Nr~ denotes the number of raw current traces which passed all tests for unexpectedly large noise and that were used for the final computation of the mean currents and ensemble variances. cell very rarely succeeded because o f the long t i m e required to obtain a new equilibrium. T h e determined t e m p e r a t u r e dependence o f 7Na o f 1.3 (__+0.1)/10 ° at - 10 m V test potential compares well with values reported for other ion channels measured by single-channel analysis. 2~ The determination o f the effect o f hydrostatic pressure on the sodium channel is experimentally m o r e involved, because various experimental manipulations are needed to pressurize the patch c l a m p holder with the pipette a n d the attached cell. 2~ However, because changes in hydrostatic pressure could be obtained m u c h faster than changes in temperature, experiments could be p e r f o r m e d on single cells, which is o f particular i m p o r t a n c e to show the reversibility o f the effects. In Table I four such experiments with pressure applications o f approximately 40 M P a (1 M P a --- 10 a t m ) are listed. It is seen that within the experimental error the single-channel conductance is not affected b y this pressure (1 1.9 ___ 1.2 pS at a m b i e n t pressure and 12.5 _ 1.0 pS at 4 0 . 0 - 4 5 . 5 MPa) as reported for the alamethicin channel 27 and the acetylcholine receptor channel. 2~ The correlation times increase on pressure application in a reversible m a n n e r as already shown by the m e a s u r e m e n t o f m e a n current kinet27L. J. Brunet and J. E. Hall, Biophys. J. 44, 39 (1983).
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NONSTATIONARYNOISEANALYSIS
145
its. 26'25'29 The increase of the time constant by the factor of 1.8 ± 0.1
between 0.1 and 43 MPa corresponds to an apparent activation volume of 60 A 3. The consistent irreversible decrease in the number of functional channels amounting to approximately 50% of the initial number on pressurization to 40 MPa is remarkable. This phenomenon was anticipated by the earlier measurements of peak currents. However, it could not be demonstrated conclusively, because no reliable information about the singlechannel conductance was available. Conclusions The method of nonstationary noise analysis was automated employing statistical considerations regarding the "quality" of individual records. The procedure presented allows a rapid analysis of noise records obtained under nonideal experimental conditions based on objective selection criteria. Although single-channel analysis has surpassed noise analysis in many instances, there are still regimes where it cannot be successfully applied for various reasons. In this sense nonstationary noise analysis will retain its value for electrophysiological research in particular, since ever fainter electrical signals are being investigated in biological membranes. To demonstrate the methods we measured the temperature and oressure dependence of the sodium channel conductance. In both respects the sodium channel showed features similar to other ion channels (e.g., the acetylcholine receptor channel), namely, having a weak temperature dependence of 7N, of 1.3/10 ° and no appreciable pressure dependence in the range explored. Both findings are in accord with the physical picture of a rather free ion diffusion through the channel pore which, unlike the channel gating mechanism, does not involve protein rearrangements associated with measurable activation volumes. Appendix I: Errors of Variance Estimates In this appendix we calculate the error involved in the estimate of the variance using difference records. Let us first write the measured signal of record i as sum of its expectation, assumed to be independent of i, and a fluctuating term: ~i(t) = Xo(t) + ~x,(t)
(A.I. 1)
2a j. V. Henderson and D. L. Gilbert, Nature (London) 258, 351 (1975). 29 S. H. Heincmann, F. Conti, W. Stf~hmer, and E. Neher, J. Gen. Physiol. 90, 765 (1987).
146
[7]
ELECTROPHYSIOLOGICAL TECHNIQUES
Arising from the superposition of a large number of independent elementary fluctuations, 5x~(t) can be safely assumed to be a Gaussian stochastic variable with zero mean. Using angular brackets to denote expectation values and omitting the explicit writing of the time dependence, unless necessary, we have (~i) = xo; (t~xi) = 0; Var(~) = (t~x2) = oa; (t~x~) = 3a 4. It follows for the variance of the squared fluctuations Var(cix2) = (ti~) - (cix2) 2 = 2 a '
(A.I.2)
Now we consider three possible strategies for estimating a 2.
Case I: Independent Records Using upper bars to denote ensemble averages over N records we have 1
~¢
= ~ . ~ ~, = xo + ~x
(A.I.3)
If all ~x~ and ~xj are statistically independent the followin_~__equations hold: (tfx, cixj) -- 0, (~xt~x.)z_)_=oa/N, (ti~ ) = ~, (Ox2) = oa/N, (tfx2tf~) = 0 4, (tfx~) = 3tr 4, (cix 4) -- 3tr4/N. Now we introduce V¢, the experimental estimate of a:: 1
N
V¢_ N _ 1 ~ (¢ _~)2 i--I
(V¢)-
N°a -- N(°a/N) -- oa N- 1
Var(V¢)- (V~) - (V e) 2 ((:~ , ~ ) ~ ) + N2(ax 4) - 2N(,~x ~ :~ , ~ ) = ( N - - 1)2
(A.I.5)
- - tr 4
(A.I.6)
Thus, as expected, the standard error squared of the "variance estimate from N records," 2a 4 Var(V¢) - N - 1 (A.I.7) is 1/(N-- 1) times the variance of the "squared fluctuation of only one record."
Case II: Nonoverlapping Difference Records Now we define the difference between two consecutive records: r/i =
~i -
~+l = ~xi - 6xi+l
(A.I.8)
[ 7]
NONSTATIONARY NOISE ANALYSIS
147
It can be shown that (qi) = 0, ( ~ ) = 2OZ, (~/~+1) = 6a4, (~/~) = 12a4, and Var(~) = (q4) _ 4t74 = 8a4. We estimate oZ experimentally by taking the average 1 N/2-1 V~ -----~ ~ ~,+~ (g.I.9) Because no overlapping traces are used for averaging, q2i+~ is independent o f % + v It follows that (V,) = oZ and ~r/2-l 4a4 Var(V,)=~-5 ~ Var(t/2,+,)- N
(A.I. 10)
i--0
Therefore, in this case the error in the experimental estimate of oZ is the same that would occur in Case I if we processed N]2 + 1 independent records without drifts.
Case IlL" Overlapping Difference Records In the case of overlapping difference records, which is what we did in our analysis, we take all r/i values for estimating OZ; that is, we use the information of all pairs of consecutive records. Let us call this estimate V~: 1
N--i
V; -- 2(N-- l) ~
~
(A.I. 1 l)
i-- 1
Again we obtain (V~) = oz. The variance, however, is
Var(V~) ---- 4 ( N - 1)2
-- a 4
(A.I.12)
After some algebra one obtains 3N - 4 0"4 Var(V~)---- -(N-1)2 3a 4 N
(A.I.13)
for large values of N. Thus, in this case the error in the estimate of oa is equivalent to that obtainable from the processing of approximately 2/3 N independent records without drift. Compared with using nonoverlapping difference records only, the precision for the estimate of oZ is increased by a factor of 2/(31/2). Appendix II: Variance of T i m e Averages
Let I(ti), I ( t 2 ) , • • • , I(t~) be n measurements of the stochastic variable I(t) at different times. Letting c~jbe the cross-correlation between I(ti) and
148
ELECTROPHYSIOLOGICAL TECHNIQUES
[7]
I(tj), co = (<~I(OOI(tj)) where tff indicates the
deviation from the expectation value. If we take the time average of I(t) in the interval (q, t~), we get a new random variable I* = 1 ~,"=1 I(t~). The variance of I* is VarO')
=
JI(tj) i--I
1
(A.II.I)
co ld
Assume that for the set of times under consideration (q, t2, • • . , tn) which constitute a bin the covariance can be written as:
c°=~exp(
-Iti-tJI }tc
(A.II.2)
where tc is the correlation time and ¢u (=tr2) is assumed to be fairly independent of i within the bin of the data. Then, tr2 ~j~ Var0t) = ~-i ._, exp ia
{
-
't~-ti' }
(A.II.3)
tc
Equation (A.II.3), showing incidentally that Var(I t) > ~/n, can be applied directly to estimate the error of the time-averaged current within each bin, given the knowledge of the correlation time t~ obtained from autocovariance measurements. To calculate the error of the time average of the estimated variance within a bin we proceed in a similar manner, but we need first to evaluate the autocovariance, ~bo, of the estimated variances:
(t~I2(ti))(~I2(b)) If, as assumed throughout these appendices, t~I(t) is a Gaussian (~o = (c~I2(t/)c~I2(b)) -
(A.II.4) stochastic
variable, it can be shown that
qbo = 2c 2 = 2(~I(t,)JI(b))2
(A.II.5)
Equation (A.II.5) shows that the correlation between squared fluctuations dies out twice as steeply as that of simple fluctuations. Thus, by the same argument leading to Eq. (A.II.3), " exp { - 2 l t ,t~- t j , } Var(~ff') = ~2 a ' .~ Zd
(A.II.6)
NONSTATIONARY NOISE ANALYSIS
131
from the bath solution. They represent the liquid junction potentials that are present at the pipette tip before patch formation. In addition, corrections have to be applied when the bath solution is being changed during a measurement (i.e., after seal formation). In that case the following rules apply. (1) The new solution should never get into contact with the bare silver/silver chloride wire of the reference electrode. This requirement is best met by using a salt bridge. (2) The "best" salt bridge is a 3 M KCI bridge with an abrupt KC1-bath fluid boundary at its tip (see above). This bridge does not require any additional potential corrections, but it may lead to KC1 poisoning of the bath or become contaminated by solutions used previously. (3) Local solution changes (microperfusion by puffer pipette, O tool or sewer pipe arrangements) as well as recessed KC1 bridges require additional corrections, which (together with the simple liquid junction potential correction) are approximately given by Eqs. (6)-(8). It should be stressed that all equations given here represent approximate corrections, since liquid junction potentials are thermodynamically ill-defined. This is particularly relevant for Eqs. (6) and (7) where the sum of two liquid junction potentials appears.
[7] N o n s t a t i o n a r y N o i s e A n a l y s i s a n d A p p l i c a t i o n t o Patch Clamp Recordings
By STEFANH. HEINEMANNand FRANCOCONTI Introduction Noise analysis was the first type of measurement that yielded reliable quantitative estimates of single-channel parameters ~-3 and thereby provided evidence for the very existence of channel proteins embedded in the lipid matrix of excitable membranes. It is still a powerful tool for investigation of ion transport mediated by channels. The more direct way of measuring single-channel properties is the recording of unitary events using the patch clamp technique.4,5 Singlechannel recordings provide the richest information on the kinetics of the ' B. Katz and R. Miledi, Nature(London) 226, 692 (1970). 2 F. Conti, L. J. DeFelice, and E. Wanke, J. Physiol. (London) 248, 45 (1975). 3 E. Neher and C. F. Stevens, Annu. Rev. Biophys. Bioeng. 6, 345 0977). 4 E. Neher and B. Sakmann, Nature (London) 260, 799 (1976). 50. P. Hamill, A. Marty, E. Neher, B. Sakmann, and F. J. Sigworth, PfluegersArch. 391, 85 (1981).
METHODS IN ENZYMOLOGY,VOL 207
Copyright© 1992by AcademicPress,Inc. Alldghtsof reproductionin any formreserved.
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ELECTROPHYSIOLOGICAL TECHNIQUES
[7]
conformational transitions which underlie the operation of individual channel proteins, but they are extremely difficult to perform when these kinetics are fast for the channel under investigation, as in the case of the voltage-activated sodium channel. Furthermore, they can only be analyzed easily for membrane patches containing few channels, possibly only one channel, and analysis requires a long time. For this reason most studies of single sodium channel events have been limited to fixed temperature conditions and to the narrow voltage range in which the frequency of channel openings is not too high. 6-s Also, only single-channel events larger than the background noise can be evaluated directly, whereas macroscopic fluctuations can reveal much smaller unitary signals, as in the case of gating noise experiments where charges translocated in an elementary conformational transition of voltage-gated ion channels are measured. 9,1° Analysis of nonstationary fluctuations of macroscopic currents are a classic way of estimating single-channel conductance. 2,1~-1a Whole-cell recordings from small cultured cells provide an ideal preparation for this type of study, owing to the very high ratio between channel noise and background noise which is obtained with this technique. ~4 The greatest advantages of this method as compared to single-channel recordings are (1) it yields fairly accurate estimates of the single-channel conductance for most preparations; (2) enough useful recordings can be obtained from the same preparation for several different experimental conditions; and (3) the analysis does not involve any subjective selection of events, it requires much shorter time, and it is easily automated by computer programs. In patch clamp recordings, transient artifacts due to instabilities of the seal as well as rundown and drift phenomena, especially while measuring under extreme environmental conditions, are not uncommon. Therefore it is advisable to introduce further improvements to the method of nonstationary noise analysis as used earlier. Among these are (1) a better strategy for analysis of recordings to obtain variance estimates, (2) a more rigorous fitting procedure of variance versus current plots to account for weights of the data, and (3) automatic, objective tests for discarding "bad" records. As an application of the analysis method we shall report some studies of nonstationary fluctuations of sodium currents in bovine adrenal chromaf6 F. J. Sigworth and E. Neher, Nature (London) 287, 447 (1980). 7 E. M. Fenwick, A. Marty, and E. Neher, J. Physiol. (London) 331, 577 (1982). 8 R. Horn and C. A. Vandenberg, J. Gen. Physiol. 84, 505 (1984). 9 F. Conti and W. Sttihmer, Eur. Biophys. J. 17, 53 (1989). lOS. H. Heinemann, F. Conti, and W. Stiihmer, this volume [22]. 11 F. Conti and E. Wanke, (2. Rev. Biophys. 8, 451 (1975). ~2F. J. Sigworth, Nature (London) 270, 265 (1977). i~ F. J. Sigworth, J. Physiol. (London) 307, 97 (1980). 14A. Many and E. Neher, in "Single-Channel Recording" (B. Sakmann and E. Neher, eds.), p. 107. Plenum, New York, 1983.
[7]
NONSTATIONARY NOISE ANALYSIS
133
fin cells for estimating the temperature and pressure dependence of the conductance of voltage-activated sodium channels. Nonstationary Noise Analysis The purpose of noise analysis is to relate macroscopic observables, such as the total ionic current, to microscopic parameters like the single-channel current i, thc number of functional channels in the membrane N, and the probability that thc channels are open under a given condition, P o ~ . For a homogeneous population of statistically independent channels, the mean, I(t), and the curent variance, al(t) 2, arc given by 15,16 I(t) = Nipov~(t)
0-~(t)2 = N i 2 p o p e . ( t ) [ 1 -- Pove,(t)]
(1) (2)
Provided that I and 0-2 are estimated for various open probabilities, i and N can be determined by data fitting procedures according to 0"1(I) 2 = I i - I 2 / N
(3)
as first introduced by Sigworth. 12'13'17It is customary to convert i estimates to a single-channel conductance y, obtained by knowledge of the specific reversal potential, E ~ , and the potential set by the voltage clamp, Eeom, ), = i/(Eoo m -- E~v)
(4)
Note that the latter equation implies a linear relationship between i and the voltage only if ), is assumed to be constant. Effect of Series Resistance
In practice the single-channel current in Eq. (3) may vary because the variable mean current I ( t ) gives rise to a variable voltage drop across the series resistance R s which is determined by the electrical access from the pipette electrode to the cell membrane. For small values of I R s we have i = i* - 3 , * I R ,
(5)
where the asterisk denotes quantities ideally measurable for a membrane potential equal to Ecom. Thus, it follows 0"~ = i * I -
y * I 2 R s -- I 2 / N
=i*I
l + F m ~ R , i2 N
(6)
15 G. Ehrenstein, H. Lecar, and R. Nossal, J. Gen. Physiol. 55, 119 0970). 16T. Begenisieh and C. F. Stevens, Biophys. J. 15, 843 (1975). 17 F. J. Sigworth, in "Membranes, Channels and Noise" (R. S. Eisenberg, M. Frank, and C. F. Stevens, eds.), p. 24. Plenum, New York, 1984.
134
ELECTROPHYSIOLOGICAL TECHNIQUES
[7]
where F , ~ = ?*N is the maximal macroscopic conductance. This result implies that a finite value of R, does not affect the estimates of i* derived from nonstationary noise analysis, provided a uniform voltage clamp exists all over the cell membrane. For small cells without big membrane invaglnations this is a fair approximation, because R, is mainly determined by the pipette tip opening. 14 Equation (6) can be written formally as Eq. (3): a~ = i * I - I2]N *
(7)
showing that R s causes a wrong estimate of the number of channels by the factor (1 + N y R , ) -1. In most cases this correction factor is expected to be close to unity, except for whole-cell measurements from large cells where N exceeds 1000. For example, ?* = 20 pS, Rs = 5 Mf~, and N = 1000 would yield an error of 10% in the estimate of N. A correct fitting procedure according to Eq. (7) requires proper account to be taken of the errors involved in the determination of a 2 a n d / , errors which vary very strongly as a function of L Another important weighting factor in fitting a 2 versus I data is due to the strong correlation expected to exist between a 2 measurements separated in time by intervals of the order of the time constants which characterize the time course of the voltage clamp responses. The theoretical considerations underlying the fitting procedure that we have used to cope with these problems are detailed in a later section. Bandwidth Limitations
Because the bandwidth of the recordings is limited, the measured variance will be smaller than the true variance. To estimate the effect of bandwidth consider the case of a channel whose gating is described by a single relaxation process with the time constant z. The variance of the channel noise is then the total integral of a Lorentzian spectrum a 2 = fo** 1 + So (f/fN) 2 d f = ~--Sof~r (8) 2 wherefis the frequency andfu = I/(2rcr). Further assuming a filter with an infinitely sharp cutoff at f~ the measured variance is a~m)= Sof~ t a n - l(fJfN) (9)
yielding a relative deviation from a 2 of a2 - cram)= 1 - 2 tan_l(fc/f~.) a2
7[
(10)
[7]
NONSTATIONARYNOISEANALYSIS
135
Thus, with ft¢= 1 kHz (z ~ 160/tsec) and a recording bandwidth of 10 kHz, the measured variance is only 6.3% smaller than the theoretical variance. The deviation is 9.1% for a one-pole filter. Stimulation Protocol Following the succession of operations actually performed in the offline analysis of the data we shall describe the protocol of analysis in two steps: (1) determination of mean currents and variances and (2) fitting of these data according to Eq. (7). Before, however, we describe the stimulation protocol tailored to noise measurements. The protocol of voltage stimulation for noise measurements was designed to optimize the subsequent off-line analysis of the data. It consisted of repeated alternating sequences of test stimulations m from a stationary holding potential, En, o f - 90 mV to test potentials between - 10 and + 30 m V m a n d control stimulations having the same temporal pattern but voltage amplitudes of only ___30 mV relative to a control holding potential o f - 100 inV. Each test sequence consisted of 20 identical stimulations, and each control sequence consisted of 4 stimulations. Successive stimulations were repeated after a pause period of about 1 sec. For a typical noise measurement at any test potential these alternating sequences were applied successively at least 7 times for collecting a total of at least 140 test records, yielding a theoretical 10% accuracy of the variance measurements and, therefore, of the single-channel conductance estimates (see below). Thus, the measurement lasted about 3 min. In between successive noise measurements at different membrane potentials and/or under different temperature or pressure conditions, standard protocols of stimulation were applied to collect information about the macroscopic current-voltage relationship and the voltage dependence of channel inactivation. These protocols allowed us to characterize the size of the overall drifts which had occurred in the preparation during the noise measurements, owing to rundown or, in the case of whole-cell recordings, to diffusion of intracellular substances initiated by the pipette perfusion. 7 Analysis of Noise Records Very generally, fluctuation analysis yields estimates of the parameters which characterize elementary events with an accuracy increasing linearly with the square root of the number of independent samples and therefore with the square root of the time during which fluctuations are measured. When applied to biological preparations this basic principle finds itself in conflict with the intrinsic property of these systems of undergoing steady, irreversible modifications. Therefore, it is mandatory to organize the analysis of the data so that there is a minimum contamination of the estimated
136
ELECTROPHYSIOLOGICAL TECHNIQUES
[7]
properties of reversible fluctuations from systematic irreversible changes of the system.
Difference Records Ideally one could calculate the ensemble variance as the mean-square deviation of each record from the mean. However, particularly in patch clamp recordings, slow drifts in the electrode potentials, changes in cell capacitance, washout phenomena, and irreversible shifts in the voltage dependence of ion channels are known to occur as function of time.14 Such linear drifts are largely eliminated if the variance is calculated from the ensemble average of the squared differences of successive records, ~(t)18:
a~ = ( ~ ( t ) 2)/2
(1 I)
For the estimate of the variance according to Eq. (11) all couples of successive records [i.e., (1,2), (2,3), . . . ] can be used rather than only the nonoverlapping ones, because it can be shown that by this procedure the accuracy of the variance estimates is improved by a factor of 2/(3 m) (see Appendix I). An example of analysis of an experiment with large leak currents and a changing, uncompensated capacitance is shown in Fig. 1.
Discarding "Bad" Records All test and control records were analyzed for background noise by calculating the variance of the baseline current a~ sampled during the first few milliseconds of the records at the holding potential. A first upper limit for the maximum acceptable variance in any record is initially set by the analyzer in order to discard records that for some reason (transient deterioration of the pipette seal, extraneous electrical artifacts, etc.) have an obvious abnormal background noise. For the remaining records the mean, (a~), and the root mean square (rms) deviation of a~ from the mean, Aa~, was calculated and used for a second selection, based this time on the objective principle that any record with a~ > (a~) + 4Aa~ is most likely affected by some artifactual extra noise. In the majority of our measurements both of these tests were passed by all records, but a few percent of noise sequences contained one to four records to be discarded according to these tests. In all these cases a~ was at least 50% higher than its maximum allowed value, clearly legitimating the selection performed by the algorithm.
Analysis of Leak Records First a further check for the absence of occasional artifacts was performed by discarding any record which differed at any point from the next is F. Conti, B. Neumke, W. Nonner, and R. St/lmpfli, J. PhysioL (London) 308, 217 (1980).
[7]
NONSTATIONARYNOISEANALYSIS
137
to
B
E __! C F
3 ms
FIG. 1. Steps in nonstationary noise analysis of records in response to test pulses to - 10 mV at 14°. (A) Averaged leak record (N= 72) yielding a leak reversal potential of 18 mY. The maximal accepted baseline noise was 5.96 pA (rms); the estimated baseline variance was 20.5 pAL Four leak records did not pass the criterion and were discarded. (B) Consecutive individual current traces during test pulses without leak correction. (C) Difference records with a boundary of 7 times the expected standard deviation [see Eq. (17)] determining the criterion for discarding records with excessivelylarge noise. (D) Two covariance functions centered at the time to the peak current, tw and at 3t~, respectively,showing a correlation time of 1.1 msec. (E) Leak-corrected mean current (N= 219); one record had to be discarded. (F) Ensemble variance record. one by m o r e than expected on the basis o f the previous m e a s u r e m e n t of (try) (apart f r o m a constant difference arising f r o m a baseline shift). The analysis of the remaining records was accomplished in two steps. F o r each record, the difference between the m e a n baseline current, I~n, and the m e a n current, I c, during the second half o f the m a i n control segment at voltage E c is c o m p u t e d in order to obtain an estimate o f the leak reversal potential, EL:
1 ~ ICiEn-InEc EL=-~L i=' -ffi Z i ~
(12)
Then, the ensemble m e a n o f all leak records ~ ( t ) was calculated 1
~vL
( ~ ( t ) ) =~LL ~ ~L(t)
(13)
i--I
as illustrated in Fig. 1A. The purpose o f calculating EL is to correct possible artifacts in the analysis o f test records occurring when successive records
138
ELECTROPHYSIOLOGICAL TECHNIQUES
[7]
have differences in the mean baseline current which are statistically unexpected, indicating changes in leak or seal conductance, 7L, between the records. It would be incorrect to cope with these changes by simply assuming that the difference record has a constant bias, because the variation of leak current is expected to be given by AIL = ATL(E-- EL)
(14)
so that it will be different for the holding potential and the test potential.
Analysis of Test Records The first step in analyzing test records is simply to obtain an ensemble average from all of them, which is then corrected for linear leakage and capacitive components by subtracting the ensemble average of the leak records appropriately scaled. The mean record obtained in this way was then used as a first estimate of the time course of the mean currents to perform a further test of the quality of each individual record and discard those which contained deviations from adjacent records much larger than statistically expected. This test was based on the expectation that at any time t, Prob(Itf~(t)l > 512(a~+ (a2))] '/2) < 10 -4 (15) so that any couple of records for which It~(t)l > 7(try+ (a~)) '/2 can be safely discarded as being affected by some artifact. At this state, before any actual measurement of al, the latter quantity could only be estimated theoretically from its upper limit:
a~ = iI(l -Po~.~) < il
(16)
For estimating i we also had to rely on already analyzed data yielding a reasonable estimate for the single-channel conductance. Therefore, the test for accepting any couple of records was [t~(t)l < 7[ ? ( E ~ - E ~ ) I + ( tr~)] u2
(17)
In Fig. 1C consecutive difference records are shown together with the envelope determined by the criterion of Eq. (17). In 70% of the measurements this test was passed by all test records. In the worst experiment 5 records out of 120 were discarded. For the final analysis only the selected records were used to evaluate the mean current (Fig. 1E). Then, for each pair of successive selected records a "pure fluctuation record" was constructed by taking the difference of the original records. If the difference in the time averaged baseline current of the two records exceeded its maxim u m expected random fluctuation [+4((a~)ToB) ~/2, where B is the bandwidth of the recordings and TOthe duration of the recorded baseline
[7]
NONSTATIONARY NOISE ANALYSIS
139
trace], the difference record was corrected on the basis of the assumption that the baseline change was due to a change in leak conductance. The fluctuation records were then used to compute the ensemble variance according to Eq. (11) (see Fig. IF) and autocovariance functions centered at different times, ti, d~(t~,t) = ( ~ ( t , ) ~ ( t ) ) (18) Figure 1D shows two covariances centered at the time to the peak current, to, and at 3tp, respectively. The covariances can be used for consistency checks that the measured noise has correlation features expected for the fluctuations under investigation. This is of particular importance, for example, for gating noise experiments, where one wants to verify that no ionic currents contaminate the gating c u r r e n t s . 9 The covariances are also used as a criterion for grouping the variance versus current data in bins as discussed in the next section. Analysis of Variance versus C u r r e n t Plots For the correct attribution of weights to the estimates of I and a 2 two important considerations must be made. (1) Each estimate of I and a~ is affected by an error proportional to a~ and to a2, respectively, which depends on I. Therefore, estimates which were obtained near the peak of the mean current, Ip, must be weighted less than those obtained for small values of I if the maximal Po~-, is smaller than 0.5. (2) Fluctuations in the current measured at any time are strongly correlated with those measured within time intervals of the order of the intrinsic correlation times of channel fluctuations, which also characterize the macroscopic current kinetics. For a constant sampling interval, a plot of all (/, cry) estimates would show a much higher density of points for I values collected during the slow phase of the response, attributing excessive independent weight to data that are not independent. To account at least partially for correlations the mean current and variance data are grouped into a smaller number of bins corresponding to the division of the duration of the main pulse segment into successive intervals of variable length such that (1) I does not vary by more than Ip/20 within each interval and (2) the interval length does not exceed the estimated correlation time, to. Mean variances within any bin were plotted against the corresponding mean current. The standard error of any such individual data point was then estimated taking into account correlations of data only within each bin (see Appendix II). This assured that the data points were now approximately statistically independent such that least-squares fitting methods could be used. The data, weighted according to the inverse of the sum of the squares of the standard errors of the
140
[7]
ELECTROPHYSIOLOGICAL TECHNIQUES
500
-
4oo 300 0 e--
•=
200
10o I
I
I
I
250
500 Current [pA]
750
1000
FIG. 2. Variance versuscurrent for the experiment shown in Fig. 1. The result of the fit was ~,= 10.2 pS, Pn= ----0.32, N= 4630 channels. The curve represents a nonconstrained fit. The diamond showsthe estimated background variance during the baselinebeforethe pulse. variance and the mean, were finally fitted by the relationship of Eq. (7) employing an effective variance least-squares method ~9which accounts for both errors in I and a 2 and allowing a constant background variance, expected to be close to ( a 2). A plot of variance versus current corresponding to the raw data shown in Fig. 1 is given in Fig. 2, yielding an estimate of the single-channel current of 1.18 pA. The data fit according to Eq. (7) yield an estimate of the current flowing through an open channel at the test voltage. This value was converted to an estimate of the apparent single-channel conductance by dividing it by the driving force applied to the channels, using estimates of the reversal potential, E~,, obtained from macroscopic current-voltage measurements. If patches had a too small number of channels, current voltage plots were not reliable, and E,~, was assumed to have the standard value of + 5 5 mV, the mean of measurements from large patches and whole cells. Reliable estimates of the number of channels could only be obtained for voltage steps which elicited large currents, that is, such that the open probability near the peak was larger than 0.2. This is seen in Fig. 3, which shows the analysis of current fluctuations at low potential and high temperature. An estimated maximal open probability of 0.1 does not allow an unconstrained fit according to Eq. (7). For this purpose we introduced two possible constraints; first, we fixed optionally the zerocurrent point to the measured baseline noise variance. The other constraint was a fixed number of channels, which can be estimated roughly. This does -
19j. Orear, Am. J. Phys. 50, 912 (1982).
[7 ]
NONSTATIONARY NOISE ANALYSIS
141
1500
~o. 1000 O
500
I
I
500 1000 Current [pA]
I
1500
FIG. 3. Variance versus current from a measurement at 27 ° with test pulses to - 30 mV. The correlation time was 0.56 reset. The result of the fit was y = 13.8 pS, Pm~ =0.10. Because the open probability of the channels was so low, no unconstrained fit could be performed. The background noise level was fixed to the value estimated from the baseline (diamond), and the number of channels was set to 9000.
not appreciably affect the estimate of i*, but it obviously excludes the possibility of estimating N independently. It should be stressed that, no matter how laborious the whole procedure might appear, the actual computer time required by our PDP 11/73 computer to perform the whole analysis, starting from 200 records of raw sampled data of 256 points each and ending with an estimate for 7 and for N* was 1 min for step 1 and 10 sec for step 2. With computers used in the laboratory nowadays the process could be faster by another order of magnitude such that the time for analysis poses no real limit. Application to P a t c h C l a m p Data Recordings of single-channel events undoubtedly contain more information than noise measurements about the gating kinetics and the conductance of specific ion channels. 2° However, in spite of the perfected recording technique currently available, there is still a broad variety of cases where application of single-channel analysis is not feasible. Among the possible reasons are a too small signal-to-noise ratio making it impossible to identify single channel openings and the amount of time needed to analyze such recordings, particularly if ion channel screening is desired. To illustrate the application of nonstationary noise analysis, we investigated the effect of temperature and hydrostatic pressure on the single20 F. J. Sigworth and J. Zhou, this volume [52].
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ELECTROPHYSIOLOGICAL TECHNIQUES
[7]
channel conductance of voltage-activated sodium channels in bovine adrenal chromaftin cells. The experimental conditions were such that singlechannel analysis would have failed because of prohibitively large background noise (high-pressure recordings) or because of too short durations of the single-channel open state (high temperature). Materials and M e t h o d s
Chromatfin cells from the medulla of bovine adrenal glands were prepared and cultured as described by Fenwick et al. 7 All measurements were performed in either whole-cell configuration or on excised outside-out patches, in both instances using the methods described by Hamill et a l : Standard patch pipette holders were used in all experiments, except for those on the effect of pressure, for which we used a specially designed holder and cable connection to the headstage of the recording amplifier (EPC-7, List Electronics, Darmstadt, Germany). The methods for pressurization of cells and membrane patches under patch clamp control are described by Heinemann et al. 2~ Pipettes with large tip openings and resistances in the range of 1 to 3 Mf~ were selected in order to reduce the access resistance, R,, for whole-cell measurements. However, we did not make any further attempt to reduce the effective value of Rs by electronic compensation, since (as discussed above) the most important artifact generated by R, in whole-cell recordings of voltage clamp currents turns out to have no consequences for the estimates of single-channel conductances from nonstationary noise analysis. Current records were sampled and stored on-line on a Winchester disk with the aid of a PDP 11/73 computer, which also generated the voltage stimulation protocol. Before being sampled (at a rate of 25 kHz) the current records were passed through an 8-pole Bessel filter with a - 3 dB cutoff frequency at 10 kHz. Solutions. In most of the experiments described here the extracellular solution had the following composition (in mM): 140 NaC1, 2.8 KC1, 2 MgC12, 1 CaCI2, 10 HEPES-NaOH, pH 7.2. In some control experiments with whole cells Co 2÷ was used to replace Ca 2+ in order to ensure that our records of sodium currents were not contaminated by calcium currents. We found no obvious difference in the results obtained with the Ca2+-free solutions, most likely because the contribution of the sodium channels to the total currents is in any case overwhelming. Furthermore, it is known that calcium channels subside after 10 min of cell dialysis: Co 2+ was used 21S. H. Heinemann,W. Sttahmer,and F. Conti,Proc. Natl. Acad. Sci. U.S.A. 84, 3229(1987).
[ 7]
NONSTATIONARYNOISEANALYSIS
143
routinely in measurements from excised outside-out patches, where a significant contribution from few calcium channels might have occasionally and unpredictably biased our data. The pipette solution which was invariably used had the following composition (in mM): 70 CsC1, 70 CsF, 1 MgC12, 0.5 CaC12, 10 HEPES-NaOH, 11 EGTA-NaOH, pH 7.2.
Effect of Temperature and Pressure on Sodium Channels The dependence of the ionic permeabilities of excitable membranes on intensive thermodynamic parameters like temperature and pressure, besides having important physiological consequences on the adaptation of biological systems to different environments, yields information about the molecular mechanisms underlying the operation of ion channels. Changes with temperature and pressure in the time course of the relaxation of ionic currents following a step in membrane potential in a classic voltage clamp experiment can be directly ascribed to variations of the rate constants of the conformational transitions of ion-channel proteins, and, within any assumed kinetic scheme, can be interpreted unambiguously in terms of activation enthalpies and activation volumes of these transitions. ~ However, variations in the absolute values of the ionic currents yield only information about the product of two quantities which can both be strongly dependent on the external parameters. One is the total number of activatable ion channels; the other is the conductance or permeability of a single open channel. Most literature on temperature effects has assumed that N is constant, apart from irreversible rundowns of the biological preparation. However, slow inactivation phenomena, by which the number of normally functioning ion channels strongly depends on the holding membrane potential, are known.23a4 It has also been clearly demonstrated that transitions of functional ion channels to conformational states from which they cannot be easily resumed for normal activity are brought about by decreasing temperature25 or increasing pressure. 26 The only possibility to quantify these phenomena reliably is to measure the single-channel conductance directly. The effect of temperature on the conductance of single sodium channels was estimated by averaging over 14 experiments at temperatures between 9* and 28 ° . Measurements at various temperatures on the same 22 F. Conti, Neurol. Neurobiol. 20, 25 (1986). 23 j. M. Fox, Biochim. Biophys. Acta 426, 245 (1976). 24 B. Rudy, J. Physiol. (London) 283, 1 (1978). 25 D. R. Matteson and C. M. Armstrong, J. Gen. Physiol. 79, 739 (1982). 26 F. Conti, R. Fioravanti, J. R. Segal, and W. St0hmer, J. Membr. Biol. 69, 23 (1982).
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ELECTROPHYSIOLOGICAL TECHNIQUES
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TABLE I PRESSURE EFFECT ON SODIUM CHANNELS a
Potential
Pressure
Temperature
~, Nch
(pS)
pope.
(msec)
89 70 88
3850 2170 1792
12.9 11.9 11.7
0.16 0.13 0.13
0.64 1.12 0.64
21.0 21.6 21.3
128 89 126
7370 3000 2600
11.5 12.6 11.5
0.29 0.26 0.33
0.48 0.64 0.24
0.1 41.7 0.1
21.6 22.0 21.7
114 114 114
5550 2350 2060
11.4 13.9 14.3
0.21 0.18 0.23
0.64 1.04 0.56
0.1 40.0 0.1
16.1 16.3 16.0
67 72 72
1670 910 920
10.5 11.6 11.5
0.27 0.22 0.30
0.64 1.28 0.64
(m V )
(MPa)
(* C)
N,~
-
10 -10 - 10
0.1 45.5 0.1
23.2 23.8 23.5
-
10 10 - 10
0.1 43.3 0.1
-10 - 10 - 10 10 10 10
a The effect of hydrostatic pressure on the tingle-channel conductance ~,, the maximal open probability po~.,, the correlation time z, and the estimated number of functional channels under investigation Na~. Nr~ denotes the number of raw current traces which passed all tests for unexpectedly large noise and that were used for the final computation of the mean currents and ensemble variances. cell very rarely succeeded because o f the long t i m e required to obtain a new equilibrium. T h e determined t e m p e r a t u r e dependence o f 7Na o f 1.3 (__+0.1)/10 ° at - 10 m V test potential compares well with values reported for other ion channels measured by single-channel analysis. 2~ The determination o f the effect o f hydrostatic pressure on the sodium channel is experimentally m o r e involved, because various experimental manipulations are needed to pressurize the patch c l a m p holder with the pipette a n d the attached cell. 2~ However, because changes in hydrostatic pressure could be obtained m u c h faster than changes in temperature, experiments could be p e r f o r m e d on single cells, which is o f particular i m p o r t a n c e to show the reversibility o f the effects. In Table I four such experiments with pressure applications o f approximately 40 M P a (1 M P a --- 10 a t m ) are listed. It is seen that within the experimental error the single-channel conductance is not affected b y this pressure (1 1.9 ___ 1.2 pS at a m b i e n t pressure and 12.5 _ 1.0 pS at 4 0 . 0 - 4 5 . 5 MPa) as reported for the alamethicin channel 27 and the acetylcholine receptor channel. 2~ The correlation times increase on pressure application in a reversible m a n n e r as already shown by the m e a s u r e m e n t o f m e a n current kinet27L. J. Brunet and J. E. Hall, Biophys. J. 44, 39 (1983).
[7]
NONSTATIONARYNOISEANALYSIS
145
its. 26'25'29 The increase of the time constant by the factor of 1.8 ± 0.1
between 0.1 and 43 MPa corresponds to an apparent activation volume of 60 A 3. The consistent irreversible decrease in the number of functional channels amounting to approximately 50% of the initial number on pressurization to 40 MPa is remarkable. This phenomenon was anticipated by the earlier measurements of peak currents. However, it could not be demonstrated conclusively, because no reliable information about the singlechannel conductance was available. Conclusions The method of nonstationary noise analysis was automated employing statistical considerations regarding the "quality" of individual records. The procedure presented allows a rapid analysis of noise records obtained under nonideal experimental conditions based on objective selection criteria. Although single-channel analysis has surpassed noise analysis in many instances, there are still regimes where it cannot be successfully applied for various reasons. In this sense nonstationary noise analysis will retain its value for electrophysiological research in particular, since ever fainter electrical signals are being investigated in biological membranes. To demonstrate the methods we measured the temperature and oressure dependence of the sodium channel conductance. In both respects the sodium channel showed features similar to other ion channels (e.g., the acetylcholine receptor channel), namely, having a weak temperature dependence of 7N, of 1.3/10 ° and no appreciable pressure dependence in the range explored. Both findings are in accord with the physical picture of a rather free ion diffusion through the channel pore which, unlike the channel gating mechanism, does not involve protein rearrangements associated with measurable activation volumes. Appendix I: Errors of Variance Estimates In this appendix we calculate the error involved in the estimate of the variance using difference records. Let us first write the measured signal of record i as sum of its expectation, assumed to be independent of i, and a fluctuating term: ~i(t) = Xo(t) + ~x,(t)
(A.I. 1)
2a j. V. Henderson and D. L. Gilbert, Nature (London) 258, 351 (1975). 29 S. H. Heincmann, F. Conti, W. Stf~hmer, and E. Neher, J. Gen. Physiol. 90, 765 (1987).
146
[7]
ELECTROPHYSIOLOGICAL TECHNIQUES
Arising from the superposition of a large number of independent elementary fluctuations, 5x~(t) can be safely assumed to be a Gaussian stochastic variable with zero mean. Using angular brackets to denote expectation values and omitting the explicit writing of the time dependence, unless necessary, we have (~i) = xo; (t~xi) = 0; Var(~) = (t~x2) = oa; (t~x~) = 3a 4. It follows for the variance of the squared fluctuations Var(cix2) = (ti~) - (cix2) 2 = 2 a '
(A.I.2)
Now we consider three possible strategies for estimating a 2.
Case I: Independent Records Using upper bars to denote ensemble averages over N records we have 1
~¢
= ~ . ~ ~, = xo + ~x
(A.I.3)
If all ~x~ and ~xj are statistically independent the followin_~__equations hold: (tfx, cixj) -- 0, (~xt~x.)z_)_=oa/N, (ti~ ) = ~, (Ox2) = oa/N, (tfx2tf~) = 0 4, (tfx~) = 3tr 4, (cix 4) -- 3tr4/N. Now we introduce V¢, the experimental estimate of a:: 1
N
V¢_ N _ 1 ~ (¢ _~)2 i--I
(V¢)-
N°a -- N(°a/N) -- oa N- 1
Var(V¢)- (V~) - (V e) 2 ((:~ , ~ ) ~ ) + N2(ax 4) - 2N(,~x ~ :~ , ~ ) = ( N - - 1)2
(A.I.5)
- - tr 4
(A.I.6)
Thus, as expected, the standard error squared of the "variance estimate from N records," 2a 4 Var(V¢) - N - 1 (A.I.7) is 1/(N-- 1) times the variance of the "squared fluctuation of only one record."
Case II: Nonoverlapping Difference Records Now we define the difference between two consecutive records: r/i =
~i -
~+l = ~xi - 6xi+l
(A.I.8)
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NONSTATIONARY NOISE ANALYSIS
147
It can be shown that (qi) = 0, ( ~ ) = 2OZ, (~/~+1) = 6a4, (~/~) = 12a4, and Var(~) = (q4) _ 4t74 = 8a4. We estimate oZ experimentally by taking the average 1 N/2-1 V~ -----~ ~ ~,+~ (g.I.9) Because no overlapping traces are used for averaging, q2i+~ is independent o f % + v It follows that (V,) = oZ and ~r/2-l 4a4 Var(V,)=~-5 ~ Var(t/2,+,)- N
(A.I. 10)
i--0
Therefore, in this case the error in the experimental estimate of oZ is the same that would occur in Case I if we processed N]2 + 1 independent records without drifts.
Case IlL" Overlapping Difference Records In the case of overlapping difference records, which is what we did in our analysis, we take all r/i values for estimating OZ; that is, we use the information of all pairs of consecutive records. Let us call this estimate V~: 1
N--i
V; -- 2(N-- l) ~
~
(A.I. 1 l)
i-- 1
Again we obtain (V~) = oz. The variance, however, is
Var(V~) ---- 4 ( N - 1)2
-- a 4
(A.I.12)
After some algebra one obtains 3N - 4 0"4 Var(V~)---- -(N-1)2 3a 4 N
(A.I.13)
for large values of N. Thus, in this case the error in the estimate of oa is equivalent to that obtainable from the processing of approximately 2/3 N independent records without drift. Compared with using nonoverlapping difference records only, the precision for the estimate of oZ is increased by a factor of 2/(31/2). Appendix II: Variance of T i m e Averages
Let I(ti), I ( t 2 ) , • • • , I(t~) be n measurements of the stochastic variable I(t) at different times. Letting c~jbe the cross-correlation between I(ti) and
148
ELECTROPHYSIOLOGICAL TECHNIQUES
[7]
I(tj), co = (<~I(OOI(tj)) where tff indicates the
deviation from the expectation value. If we take the time average of I(t) in the interval (q, t~), we get a new random variable I* = 1 ~,"=1 I(t~). The variance of I* is VarO')
=
JI(tj) i--I
1
(A.II.I)
co ld
Assume that for the set of times under consideration (q, t2, • • . , tn) which constitute a bin the covariance can be written as:
c°=~exp(
-Iti-tJI }tc
(A.II.2)
where tc is the correlation time and ¢u (=tr2) is assumed to be fairly independent of i within the bin of the data. Then, tr2 ~j~ Var0t) = ~-i ._, exp ia
{
-
't~-ti' }
(A.II.3)
tc
Equation (A.II.3), showing incidentally that Var(I t) > ~/n, can be applied directly to estimate the error of the time-averaged current within each bin, given the knowledge of the correlation time t~ obtained from autocovariance measurements. To calculate the error of the time average of the estimated variance within a bin we proceed in a similar manner, but we need first to evaluate the autocovariance, ~bo, of the estimated variances:
(t~I2(ti))(~I2(b)) If, as assumed throughout these appendices, t~I(t) is a Gaussian (~o = (c~I2(t/)c~I2(b)) -
(A.II.4) stochastic
variable, it can be shown that
qbo = 2c 2 = 2(~I(t,)JI(b))2
(A.II.5)
Equation (A.II.5) shows that the correlation between squared fluctuations dies out twice as steeply as that of simple fluctuations. Thus, by the same argument leading to Eq. (A.II.3), " exp { - 2 l t ,t~- t j , } Var(~ff') = ~2 a ' .~ Zd
(A.II.6)