Eur. Phys. J. E 2, 161–168 (2000)

THE EUROPEAN PHYSICAL JOURNAL E EDP Sciences c Societ`
a Italiana di Fisica Springer-Verlag 2000

Noise measurements in bilayer lipid membranes during electroporation A. Ridi1 , E. Scalas2 , and A. Gliozzi1,a 1 2

Istituto Nazionale per la Fisica della Materia and Dipartimento di Fisica, Universit` a di Genova, Via Dodecaneso 33, 16146, Genova, Italy Dipartimento di Scienze e Tecnologie Avanzate, Universti` a del Piemonte Orientale, Corso Borsalino 54, 15100, Alessandria, Italy Received 1 June 1999 and Received in final form 1 October 1999 Abstract. A study of voltage fluctuations in bilayer lipid membranes during electroporation and under current-clamp conditions is presented. Qualitative considerations based on the electroporation theory are used in order to explain the phenomenon on long time scale. Indeed, the current-clamp condition induces a feedback mechanism on the pore formation and therefore on the macroscopic conductance. Voltage fluctuations can thus be recorded. These fluctuations are nonstationary long-living and have a flicker power spectrum over nearly four decades of frequency between about 10−2 and 102 Hz. The study of the fluctuations in the time domain has been performed by introducing an electrical model of the system formed by the membrane and the circuit under current-clamp configuration. The analysis of the time series gives a characteristic time of 100 ms for the circuitry response to the fragments of electroporation signals with characteristic times faster than 100 ms. During electroporation, the response to an external periodic stimulus in the frequency range 10−1 –10 Hz shows that the system behaves linearly, even if voltage fluctuations are present. PACS. 87.16.Dg Membranes, bilayers, and vesicles – 87.68.+z Biomaterials and biological interfaces – 87.15.Ya Fluctuations

1 Introduction The phenomenon of membrane electropermeabilization has been known for many years [1,2]. It consists of an increase in membrane conductance and permeability to ions and macromolecules induced by the application of an electric field. Electropermeabilization is becoming more and more important in view of its applications to biotechnology such as electrofusion of cells and electroinsertion of foreign molecules [3–8]. Experiments on cells have shown that strong electric pulses stimulate molecular transport and increase membrane conductance up to 1 S/cm2 . It has been shown that the phenomenon is related to the lipid bilayer structure of the cell membrane. Thus, a quantitative study of electropermeabilization was made possible by experiments on model systems in well-controlled conditions, mainly bilayer lipid membranes (BLMs) [1,8–16]. Qualitatively, the phenomenon of electropermeabilization can be described as follows. When a BLM of capacitance Cm and resistance Rm is subjected to a voltage step change ∆U the current response I(t) is initially analogous to that of a passive parallel RC circuit and, eventually, the a

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current reaches the reference value I0 = ∆U/Rm but, at a certain time, current fluctuations appear which are followed by membrane rupture (irreversible breakdown). If the potential step is applied for short times, the membrane conductance temporarily increases and when the potential is set to zero again, the membrane recovers its initial leak conductance. This phenomenon is known as reversible breakdown. In the case of irreversible breakdown, the membrane lifetime after the potential step is not deterministic, but randomly varies for different membranes [1]. The whole set of experimental observations available up to now corroborate the so-called electroporation theory. According to this theory the membrane is in a metastable state with a population of nucleating voltage-dependent pores. The nucleation of pores is analogous to the homogeneous nucleation of a two-dimensional phase and when the radius of the pores overcomes a critical value, dependent on the applied potential difference U , the membrane breaks. The stochastic pore evolution is described by a Smoluchowsky-type equation ([1] and references therein). In this framework, it is possible to predict the membrane average lifetime which is a decreasing function of the applied voltage. For instance, the average lifetime of a dipalmitoyl phosphatidylcholine (DPPC) membrane passes

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Fig. 1. Sketch of the experimental set-up. a) The membrane is in the feedback branch of a high impedance operational amplifier. b) The model configuration used in this paper is shown with the membrane replaced by a time-dependent RC circuit.

from 10 s to 1 ms as U varies from 0.2 V to 0.4 V [12]. In the case of a diphytanoyl phosphatidylcholine (DPhPC) membrane the lifetime decrease is even steeper [13]. Although electro-optical relaxation methods using optical membrane probes and polarised light have provided new insights into the molecular mechanism of electric pore formation in lipid membranes, molecular details of pore formation are not yet available [5,6]. Therefore, new experimental approaches can provide complementary information on the electropermeabilization process. So far there has not been any detailed study of the electrical fluctuations induced by electroporation on long time scales (up to some hours). In order to prevent membrane rupture for long times, it is necessary to perform measurements in current-clamp. In such a configuration, the membrane is on the feedback branch of a high impedance operational amplifier (see Fig. 1). In this situation, if the membrane is stimulated by a ramp, it exhibits a transition to a higher conductance state followed by the onset of transmembrane voltage fluctuations. The fluctuations can live for long times because of the feedback mechanism. Indeed, when the membrane is under current-clamp conditions, the rise of conductance due to the opening of the pores causes a sudden decrease of the transmembrane potential which yields an increase of the average lifetime. Here, we describe the properties of long-living transmembrane voltage fluctuations during electroporation under current-clamp conditions.

Our previous work [14–16] has been devoted to the first conductance transition as a function of membrane composition, pH and ionic concentration. In particular, we have found that these parameters strongly influence the average value of the transition potential and of the conductance jump, as well as the distributions of these two quantities. As for membrane composition, both the polar head groups and the hydrophobic chain regions are relevant. Another noticeable result is that the fluctuations disappear and the membrane recovers its initial conductance value when the stimulating current is set again to zero. It is important to stress that the onset of fluctuations is a stochastic and not a deterministic phenomenon, in agreement with the electroporation theory summarised above. In particular, for the DPhPC membranes investigated here, the values of the first transition potential cluster around 400 mV [16]. In this paper, we show that, if the input current ramp is followed by a constant current stimulus, the membrane in current-clamp fluctuates up to 4 h. It turns out that transmembrane voltage fluctuations U (t) show a flicker power spectrum. This indicates that the long-time electroporation dynamics in pure lipid membranes seems to be characterised by a continuous set of time constants [17]. Past investigations found current flicker noise in BLMs containing molecules acting as channels; however, in that case the power spectrum density is four to five orders of magnitude smaller than in the present case, for a frequency range between 10−2 to 10 Hz. A Lorentzain spectrum characterises the opening and closing dynamics of the channel molecules [18]. We remark that we have not introduced any ionophore in the membrane and carefully prevented any contamination.

2 Materials and methods In Figure 1, a sketch of the experimental apparatus is plotted. In the experiments, performed at 25 ◦ C, a computergenerated signal stimulated a bilayer lipid membrane (BLM). The electric signals were measured by means of two Ag/AgCl electrodes. The current-voltage (I(t) vs. U (t)) characteristics and the voltage fluctuations were recorded under current-clamp conditions with the membrane in the feedback branch of a high-impedance (1015 Ω) operational amplifier (Burr Brown 3528 CM), which acts as a current-voltage converter (Fig. 1). Voltage-fluctuation recordings were computer assisted. To avoid aliasing, an Ithaco 4302 dual 24 dB/octave filter was used; its channels were connected in series to obtain an attenuation of 48 dB/octave. The sampling frequencywas 100 Hz and the filter cut-off frequency was set at 20 Hz. BLMs were formed at 25 ◦ C by hydrophobic apposition of two diphytanoyl phosphatidylcholine (DPhPC) monolayers on a circular hole, 160 µm in diameter, according to the Montal and M¨ uller technique. Monolayers were spread on the water surface from lipids dissolved in n-hexane (BDH, Milan, Italy) at a concentration of 10 mg/ml. DPhPC was obtained from Avanti Polar Lipids (Alabaster, Al, USA). Analytical grade KCl

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(Carlo Erba, Milan) solutions were prepared, at 0.1 M concentration, by using a water purification system (Millipore Milli-Q system including a terminal 0.22 µm filter). The salt was previously baked at 500 ◦ C for 1 h in order to prevent organic contamination. Before every experiment, great care was taken to avoid impurities and therefore after careful washing in sulfochromic solution, the cell was boiled in Milli-Q water for at least 1 h. The teflon film (12 µm thick) containing the hole was sealed to the two half-cells by means of two O-rings. In this way, silicon grease, which could contaminate the membrane, was no longer necessary. For the same reason, pretreatment of the hole with squalene was not used.

3 Experimental results It is important to remark that, even if current is the controlled quantity in our experimental protocol, the phenomenon of electroporation is driven by the transmembrane voltage [1]. Current-voltage characteristics under current-clamp conditions were obtained by stimulating the membrane with a current ramp, usually with a slope of 0.17 pA/s. Figure 2 shows that the membrane undergoes a transition to a higher conductance state. According to the electroporation theory, this rise in conductance is due to the opening of pores. The conductance increase causes a decrease of the transmembrane potential which, in turn, stabilises the membrane resulting in an increase of the membrane average lifetime. At the first transition the membrane resistance decreases by a factor of about 50 from 1011 Ω to 5 · 109 Ω. When a voltage U is applied to a membrane, the specific capacitance varies according to the relationship
 (1) C(U ) = C0 1 + αU 2 , −2

−2

−2

where α is a coefficient in the range of 10 V –10 V [14] depending on the amount of solvent present in the membrane. In our experimental conditions we obtain that α is in the order of 1 V−2 and that C0 = 0.6 µF/cm2 , suggesting that small amounts of solvent are still present in the membrane. During the conductance transition, the relative capacitance variation ∆C/C is about 6%, while during the fluctuations a small change in the specific capacitance can be detected. These changes can be related to electrostriction phenomena as predicted by equation (1). Indeed, when several I/U cycles are performed, the specific capacitance becomes 0.75 µF/cm2 and no further capacitance variation can be detected within 1% accuracy [14]. These results indicate that during voltage fluctuations the capacitance changes are negligible with respect to conductance changes. This fact is in agreement with previous experimental findings on electroporation experiments [1]. In order to analyse the time evolution of the fluctuations, the membrane was stimulated by a current ramp followed by a constant current (see Fig. 3, inset). The voltage response in Figure 3 shows that voltage fluctuations appear after a sudden decrease in the conductance.

Fig. 2. Uo /I characteristics of a DPhPC membrane, stimulated by a current ramp with a slope of 0.17 pA/s lasting 100 s. There are two different behaviours: at the beginning the membrane acts as a passive RC circuit, whereas at I = 3.2 pA and Uo = 460 mV there is a conductance transition leading to a decrease of transmembrane potential and to the onset of voltage fluctuations. The ionic solution used is 0.1 M KCl.

The voltage fluctuations of more than 60 membranes have been recorded for 10 minutes and for different input current levels: 9, 17, 34, and 70 pA. Five membranes have been observed for at least 1 hour at 17 pA. In one experiment, the fluctuations have been followed for up to 4 hours under a 17 pA stimulus.

3.1 Frequency domain Let us now characterize the time series within the framework of the classical theory of stochastic processes developed by Kolmogorov [19]. The first step is the computation of the so-called two-point correlation function, CU U (t, τ ), defined as CU U (t, τ ) = E[U (t)U (t + τ )],

(2)

where U is the transmembrane potential, τ is the time lag, and E is the ensemble average. The tow-point correlation function gives information on the memory of the time series. Under suitable hypotheses [19], equation (2) can be approximated by time averages and, after sampling, a good

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Fig. 4. Non-time stationary voltage-voltage autocorrelation function CU U (t, τ ) vs. the initial time t and the delay time τ for different subsets of the time series represented in Figure 3. It appears that the autocorrelation functions depend on the initial time, thus showing the time-domain nonstationary behaviour of the system.

As a consequence, the power spectrum density of the voltage fluctuations, S(f, t), defined as the Fourier transform F of the two-point correlation function,   S(f, t) = F CU U (t, τ ) =  +∞ 1 √ CU U (t, τ )e−j2πf τ dτ , (4) 2π −∞ Fig. 3. Time-dependent voltage fluctuations of a membrane stimulated by the input current represented in the inset. The experiment is performed under current-clamp; the initial ramp is used to induce electroporation which is shown to occur by the sharp potential decrease. The analysis of voltage fluctuations is performed when the current is kept constant and the membrane is in the electroporated state.

estimator of CU U (t, τ ) is the following: CU U (j, k) =

N 1  U (i + j)U (i + j + k), N i=1

(3)

where t = js, τ = ks, s being the sampling time and i, j, k being integers, with N < M/2 and M is the total number of sampled points. For stationary and ergodic time-series, the two-point correlation is a monotone decreasing function of the time lag τ and does not depend on t. Moreover, it has an absolute maximum for zero time lag. In Figure 4, we show that this is not the case for the time series plotted in Figure 3. Indeed, the two-point correlation function estimate does depend on the time t and it may have an absolute maximum for non-zero time lag. The former property means that the time series is not stationary, whereas the latter property indicates that this non-stationry behaviour is due to the presence of trends, which, in our case, cannot be easily removed.

is non-stationary as well and depends on the time t. However, for every t, the power spectrum density has a flicker behaviour: S(f, t) =

B(t) , fγ

(5)

with γ ≈ 1; B(t) is a time-dependent scaling factor. The power spectrum density of the voltage fluctuations at a constant current of 9 pA is plotted in Figure 5, in this case γ = 0.99 ± 0.01. The membrane was subjected to the same protocol described above: after stimulation with a ramp to induce fluctuations, the current was then kept constant. As the response of the membrane to a constant current is far from linear, an intriguing question is how the membrane reacts to a periodic stimulus. For this purpose, we have performed a set of experiments with a different protocol: sine-wave signals of different amplitude and frequency have been sent to the input branch of the operational amplifier circuit and superimposed to a constant voltage of 340 mV. Again, an initial ramp was needed to induce membrane electroporation and to cause fluctuations [20]. The membrane response has been characterised by the output power, Pout determined as a function of the input power, Pin . Essentially, the power of the input stimulus is given by the square amplitude of the sine wave. The power of the output signal is determined by Fouriertransforming the time series and measuring the height of the periodic peak over the 1/f noise component. As shown

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Fig. 5. The power spectrum density (PSD) of the voltage fluctuations of a membrane held at a constant current of 9 pA during electroporation. The initial ramp to induce electroporation had a slope of 0.09 pA/s and lasted 100 s. The slope of the PSD on a log-log scale is 0.99 ± 0.01.

in Figure 6, top, there is a linear dependence of the output power, Pout , as a function of the input power, Pin . Moreover, this linear behaviour is well reproducible, because successive measurements on the same membrane give similar results. If the input frequency is changed, the slope of the Pout vs. Pin line changes, but we still have a linear relation in the frequency range 0.1–10 Hz. The linear dependence of Pout vs. Pin , and the reproducibility of this dependence indicate that the system is linear and timeinvariant with respect to an external periodic stimulus, even during electroporation. The output voltage is a superposition of fluctuations and a periodic response. With these experimental findings in mind, we can explain the slope change as a function of frequency using the transfer function H(f ) of the circuit plotted in Figure 1. A linear time-invariant system is completely characterised by its impulse response h(t) [19]. Indeed, for a generic input signal x(t) the output signal y(t) is given by the following convolution: y(t) = h(t)∗ x(t).

(6)

The transfer function, H(f ), is defined as the Fourier transform of the impulse response, h(t). By Fouriertransforming equation (6), we get Y (f ) = H(f )X(f )

(7)

as the input power is proportional to |X(f )|2 and the output power to |Y (f )|2 , we have that Pout = |H(f )|2 Pin .

(8)

The solid line in Figure 6, bottom, is a fit of the experimental points (obtained by the slope of Pout vs. Pin at

Fig. 6. Top: Output power, Pout vs. input power, Pin , for three independent runs measured on a single membrane. The membrane is stimulated by a 1 Hz sinusoidal wave which is superimposed to a constant current used to induce electroporation. Bottom: square modulus transfer function vs. frequency for the same membrane. The circles denote the experimental points; the solid line is the non-linear fit of the experimental data with the square modulus of equation (9).

different frequencies) with the transfer function of the circuit in Figure 1: H(f ) =

Rm 1 + jωRi Ci · , Ri 1 + jωRm Cm

(9)

where Ri = 1010 Ω and Ci = 300 pF are the input resistance and capacitance, and Rm and Cm are the membrane resistance and capacitance. In other words, even if the membrane resistance is fluctuating, in the investigated frequency range (0.1–10 Hz) an external periodic signal sees an effective membrane resistance with a parallel capacitance. For the experimental data in Figure 6 (bottom), we get the following values from a non-linear fit of the square modulus of equation (9): Cm = 200 pF

and

Rm = 6.8 · 109 Ω.

The capacitance is in the same order of magnitude as that directly measured before electroporation, whereas the

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resistance order of magnitude is that measured during electroporation.

3.2 Time domain The analysis of the fluctuations in the time domain can be performed introducing an electrical model of the system formed by the membrane and the circuit under currentclamp configuration. The differential equation to be solved is the following (cf. Fig. 1): Rm Ci

dUi Rm (t) dUo + Ui , = Uo + τm (t) · dt Ri dt

(10)

where Ci is the input capacitance, Ui and Uo are the input and output voltages, τm (t) is the decay time of the membrane and the other symbols are as previously defined. In i the experimental conditions in which dU dt = 0, the first term in the left-hand side vanishes and the current across the membrane is constant. In order to reproduce a resistance jump, the following conditions on the parameters can be imposed: Rm (t) = R1 , Rm (t) = R2 ,

t ≤ 0, t > 0,

(11)

with R2 < R1 , if the variation of Rm is very fast with respect to R2 Cm . The solution of equation (10) is then   Uo (t) = Ii ∆R exp(−t/R2 Cm ) + R2 , (12) where ∆R = R1 − R2 is the resistance jump. The experimental voltage variation corresponding to a decrease in resistance (opening of the pores) can be fitted by a single exponential, whereas the inverse process (closing of the pores) is rarely characterised by a single exponential curve, and, therefore, it is more difficult to analyse in terms of the previous simple model. Moreover, the closing time during electroporation is much longer and the conditions specified in equation (11) cannot be applied. Therefore, we shall restrict the analysis to the opening process only. As our physical model is based on the electroporation phenomenon, in which the pore opening process has time constants of 1–2 µs, much shorter than the time constant of the system (0.1–1 s), the hypothesis of instantaneous variation in resistance is a good approximation. If the time scale of Figure 3 is expanded, it is possible to see that fluctuations are due to relatively smooth decays and increases of transmembrane voltage. Let us now analyse a single voltage decay (Fig. 7). The curve can be fitted by an exponential function such as Uo (t) = Uo (0) + A exp(−t/R2 Cm ).

(13)

Comparing equation (13) with equation (12) we get Uo (0) − Uo (∞) = Ii ∆R = A.

Fig. 7. Analysis of a single voltage decay at constant current, magnified from Figure 3. The experimental decay is represented by black dots; the continuous curve is an exponential fit. In the inset, the whole fluctuation is plotted. Uo coincides with the transmembrane potential U , as the membrane is connected to the virtual mass of the operational amplifier.

(14)

We have analysed records of 90 s in order to get distribution of the parameter A and of the time constant τm = R2 Cm . Control experiments, performed by measuring the transfer function of the circuit stimulated with white noise and several RC values on the feedback branch of the amplifier have shown that parasitic capacitance do not affect the time constant of the fluctuations. The analysis has been performed for 170 voltage exponential decays on the time series (Fig. 3), and the results (Fig. 8), indicate that the distribution is unimodal and gives the following average values: τm = (135 ± 6) ms, A = (45 ± 3) mV. From these averages, we can compute the average values of R1 and R2 by means of equation (14) and by the definition of τm . For a membrane capacitance Cm = 200 pF and an input current Ii = 17 pA, we get R2 = (0.67 ± 0.02) · 109 Ω and R1 = (3.3 ± 0.4) · 109 Ω. It is worth observing that this independent analysis yields a typical resistance of a membrane in the electroporated state, which is in the same order of magnitude obtained with equation (9) (the transfer function of the system). Moreover, these results indicate that the feedback mechanism prevents not only membrane rupture, but also complete membrane resealing.

4 Discussion and conclusions Most of the work on noise in lipid membranes has been focused on conductance noise determined by channel transitions between different conductance states. However, there are other noise components with different origin, whose study is not as soundly established as channel

A. Ridi et al.: Noise in BLMs during electroporation

Fig. 8. Analysis of the fluctuations, using equations (13) and (14), of 90 s records of the time series represented in Figure 3. Top: histogram of the decay time, τm : the time bins are 25 ms. Bottom: amplitude histogram: the amplitude bins are 10 mV. N is the number of events.

conductance noise [18]. For this reason, great attention was paid to avoid misinterpretation of experimental data. Several experiments were performed in order to control that fluctuations are really related to electroporation. In a test, the membrane has been replaced by an RC passive circuit (R = 1010 Ω and C = 100 pF); in another experiment no membrane was formed on the hole (corresponding to a resistance of 106 Ω between the electrodes). In a third set of experiments either no current or a level of

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1 pA were used, two situations in which we could not observe the conductance transition. In all these cases, voltage fluctuations were below the resolution of our measuring apparatus (500 µV). It is important to remark that the other common sources of membrane voltage noise (Johnson noise, shot noise and 1/f noise due to current through open channels) are at least four orders of magnitude lower in amplitude than the flicker fluctuations analysed in the present paper ([18] and references therein). In artificial and biological membranes containing channels, noise analysis allows the evaluation of microscopic quantities such as the conductances of the various channel conformational states. On the contrary, in this case, information on the microscopic pore behaviour does not straightforwardly follow from the noise spectrum. The main result of this study is that long-living voltage fluctuations in electroporated BLMs have a flicker power spectrum over at least four decades of frequency. Fluctuations with such a power spectrum have been found in a great number of physical systems such as film resistors and solid state devices. As these systems are different, it is likely that the detailed origin of flicker noise is different as well [17,21]. A possible explanation for the 1/f behaviour could be based on mechanical vibrations. However, as we said before, large voltage fluctuations do not spontaneously appear, at least within our observation time scale (around 1 h). As a further remark, for membranes formed on holes with diameters ranging from 50 µm to 200 µm, we were not able to detect any significant difference in the power spectrum density of the voltage fluctuations. This set of experiments was started with the objective of finding characteristic times for the pore dynamics. Actually, we were aware that we could not observe the fast (less than 1 ms) time constants typical of most opening processes, due to the circuit cut-off. However, Lorentzian spectra were expected for the slow closure processes. We were surprised to find a non-stationary response and 1/f spectra. We could observe a non-stationary time series, as, for the first time, it has been possible to follow voltage membrane fluctuations for some hours. An intriguing question is why the direct analysis of the time-series gives a characteristic time constant, τm of 100 ms, while the power spectrum gives 1/f noise (i.e. absence of a characteristic time). This could point to the presence of some slow processes masking the time constants associated to the membrane plus circuitry time response [22]. On the other side, the physical meaning of such a non-stationary behaviour has still to be understood; slow non-stationary fluctuations could be due to pore interactions or mechanical forces generated by the electric field and their relationship to membrane deformation and changes in rheological parameters [2,23]. The analysis of the membrane response to a periodic stimulus in the frequency range of 0.1 Hz to 10 Hz shows that the system behaves linearly, even if fluctuations are present. In other words, in this frequency range, the system transfer function does not depend on the

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instantaneous value of the membrane resistance. On the other hand, in current-clamp, slower stimuli do influence the electroporated state as waves of 2 · 10−3 Hz frequency are able to induce electroporation [16]. Therefore, we do not expect that the observed linear response extends to very low frequency regions. Finally, the membrane response to an external electrical stimulus is the same always, even after hours from the beginning of the experiment. This suggests that the non-stationary timedependent behaviour of the voltage fluctuations (Fig. 4) is not associated to a change in the other electrical membrane parameters. This body of results shows that a current-clamp analysis of the electroporation process is feasible and, in the future, may give relevant information provided that: the membrane behaviour is decoupled from the circuit and the time series studies are performed by methods more sophisticated than the simple spectral analysis. It will be interesting to study the effect of membrane composition on the fluctuation. For instance, the addition of molecules which favour pore formation, e.g., lysophosphatidylcholine could change the frequency range where 1/f fluctuations are present. Moreover, this kind of experimental work could be extended to the skin whose electrical properties, dominated by stratum corneum, are similar to those of multilamellar lipid membranes [24]. Another important goal is the extension of microscopic theories [1,23, 25] of electroporation in order to include long-time effects observed in current-clamp conditions. This work has been partially supported by the Italian Ministry for Scientific and Technological Research (MURST) and by the University of Genova.

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