Ionic channel gating under electromagnetic exposure: a stochastic model

289

Bioelectrochemistry and Bioenergetics, 29 (1993) 289-304 Elsevier Sequoia S.A., Lausanne

JEC BB 01581

Ionic channel gating under electromagnetic exposure : a stochastic model Guglielmo D'Inzeo, Stefano Pisa and Luciano Tarricone * Department of Electronic Engineering, University "La Sapienza" of Rome, Via Eudossiana 18, 00184, Rome (Italy) (Received 24 January 1992 ; in revised form 15 August 1992)

Abstract Researchers interested in the biological effects of electromagnetic (EM) fields are focusing their attention on the behavior of transmembrane ionic channels and on their kinetic properties . Theoretical studies of the biochemical dynamic properties of the channels have suggested the development of a modelistic approach considering the membrane channel as a non-deterministic state machine. Its behavior is fully described by a set of states, a matrix of transition rates, and a vector for the probability of the machine to be in each single state at a certain instant . In this work a stochastic model is developed, generating random processes where the probability for each state is an aleatory variable . The model can be applied to both voltage- and ligand-dependent channels, both unexposed and exposed to EM fields . The response of the model, for voltage-dependent channels such as K+, Nat and Ca" in a voltage-clamp situation, is analyzed for sinusoidal EM fields in the ELF range. The results obtained appear more satisfactory than those presented in earlier papers using similar approaches, as this model shows the sensitivity of the channel response to both the frequency and amplitude of the EM stimulation.

1 . INTRODUCTION

Most experimental observations have revealed the existence of effects of electromagnetic (EM) fields on biological systems . Many examples can be found in the literature [1], both of macroscopic systems (like a whole complex organism) and of microscopic systems (neural cells, cell membranes, membrane channels) . A different behavior of the biological systems has been noted depending on the amplitude and the frequency of the applied EM fields [2]. For instance, the existence of both amplitude and frequency windows has been demonstrated: at particular values of

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frequency and amplitude, biological organisms are more sensitive to EM stimulation. However, the mechanism and the site of the interaction is not clear [2]. Some recent theoretical and experimental works have considered the hypothesis that the EM field acts on membrane proteinic channels : some researchers have supposed direct action of the electric field on proteins [3-6], and others have supposed action on gating particles [7,8] . Action on the conducting properties of membrane channels has also been observed [9-11] . The creation and development of new techniques to characterize membrane behavior, such as the "patch-clamp" technique [12,13], has aroused interest in the importance of the kinetic mechanisms of channel gating on the functionality of the cell . Recent works focus attention on further possibilities of analysis of the kinetics of a protein channel by studying the current fluctuations of a single channel [14]. The growing interest in single-channel records has led researchers to consider theoretical studies of membrane channels ; some models of channel behavior have been presented [15-17] . These models simulate the physiological behavior of the membrane channels, but they have not been used to simulate the channel behavior under exposure to EM fields . This work proposes a stochastic model to study theoretically the behavior of a single channel, both exposed and unexposed to EM stimulation . Based on a state machine model of channel microscopic behavior, a method has been developed, which enables the properties of membrane conductance to be calculated from a single-channel simulation . An analysis, as general as possible, of the effects of the EM field on the channel is attempted . 2. MODEL

2.1. The state-machine model of the channel

Models for macroscopic and microscopic currents based on a single-channel process have already been proposed [15-17] . These algorithms apply to both voltage-clamped and unclamped channels, and simulate the behavior of voltagedependent sodium (Na') and potassium (K+) channels . A deeper knowledge of the physiological characteristics of a protein channel can be obtained by analyzing the single-channel currents (Fig . 1(a)). Observations have shown that the state of a protein channel in a membrane is "all-or-none" : a channel appears open (conducting) or closed (non conducting), looking at it from outside [12]. The possibility of different open states with different conductances [12] and of inactivated states (non conducting but with physiological properties different from the closed state) has also been observed [18]. The channel is considered as a non-deterministic state-machine (Fig . 2), whose behavior is defined by a set of N states, a transition rate matrix Q, whose elements q;j are the transition rates regulating the kinetics of the process, and taken so that N

E q ,1 =0Vi i=1

(1)

291

%WO

p,,,,+wy sty"^N"&

4 M,..".h,,y,. # ,M

MA~~~ rM"Mr}

LY

4 pA 50 ms

~r4L~udy~Vlvgb `MM~M~A$

-v r'?VwM1~W

*04

4

A

(a)

P

I

ni n

(b)

f/ms

Fig. 1 . (a) A typical example of a single-channel record from a patch-clamp measurement ; (b) a single-channel record simulated via software : the probability that the channel is open flips from zero to one and the time a channel remains in one of the two states is called "dwell time" .

and, finally, a vector p(O) whose N elements are the occupancies for each state at the starting time t=0 (p;(t) is the "state-occupancy at t", i.e. the probability for the channel to be in the state i at the instant t) [19-22] . 2.2. The random technique In general, for every single channel, the occurrence of transitions among these states is a stochastic process ; the probability of finding the channel in a given state can be considered an aleatory variable [15,19-22]. Supposing the channel as a zero-order Markov chain, stationary and ergodic, at each instant of time t, it is possible to calculate the value of occupancies for each state, generating a random process whose aleatory variable is the dwell-time in a single state . In order to do this, it is important to find the dwell-time probability distributions in a voltageclamp situation (Fig . 3), which is the probability that a channel, in a given state at

Fig. 2 . A generalized N-state machine scheme for a membrane channel . Each node represents a state, each arrow a transition from one to another and labels on the arrows are the transition rates .



292 V

V2 V1

ti t2 t Fig. 3. (c) The typical voltage-clamp signal used in the simulations . At the time t 1 a voltage step is applied. Both V1 , V2 , t 1 and t 2 can be chosen if preferred .

an instant, can dwell in this state for a certain interval of time . These distributions of dwell-times have the form : d1 = exp(-q l,t)

(2)

when there is only one path leaving the i state and arriving at the j state ; otherwise, the form is

-

d;=Ew k exp( g1kt)

k

(3)

where wk is a weighting factor [19-21] . We can now derive the random variable, the dwell-time in a certain state T D , by generating a random number r. For instance, from eqn . (2), substituting r for d 1 we obtain 1 =--lnr 0
Some earlier works [15] propose implementation of this theoretical approach in a stochastic model for ionic channels in a voltage-clamp situation . Here, we are interested in a dynamic extension of the proposed model, to investigate the behavior of ionic channels when exposed to EM fields . Equations (2) and (3) of the model, in a dynamic situation just like that considered with EM sinusoidal stimulation, should be rearranged, because the stationary solution of these equations is no longer valid [15,16] . For instance, in a two-state model, where a is the opening rate and /3 is the closing rate, for the open state occupancy we have dpo _ dt

- PP0

(5)

In a dynamic model, this gives P0(t) =p 0(0) exp {

ftp(t) dt ,0 l

(6)

293

Now we can consider a series of transitions, each one divided by a time interval whose length is decided using eqn . (4) (this equation may be considered as approximately equivalent to eqn . (6) if /3(t) changes smoothly) . Every time a transition occurs, we generate a random number and calculate the corresponding dwell-time. The zero-order Markov chain hypothesis is now clear : at a certain time, we calculate TD instantaneously, considering only the system's situation at that moment, i.e . membrane voltage and temperature . The starting state is also determined using a random technique : at a certain voltage, it is possible to calculate the steady state occupancies p,(oo) for the conducting and non-conducting subsets [19-22] . Now, we suppose that at the starting time, the channel is in the steady state ; so, we generate a random number r. If, for example, the total probability for the conducting set of states is more than r, then we let the channel be in a conducting state, otherwise we consider the channel to be non-conducting at the starting time t = 0 . When the voltage signal is applied to the membrane channel, in the algorithm the voltage change (Fig . 3) is considered as such a quick change in the system's environment that, at the first instant after the step, the transition rates are calculated for the new applied voltage . In this way, we consider membrane channels as if they interacted with the environment only on making a transition : it is at that moment that they "nose around" their electrical surroundings and choose their TD future time . Dividing all the states into two subsets of conducting and non-conducting states, we can always associate with each state a boolean value of current through the channel. For instance, if we have a three-state model, with an open, a closed and an inactivated state, we gather together the inactivated and the closed states, since they are both non-conductive . Figure 1(b) shows a simulation trace of the channel behavior generated by this algorithm . Good agreement between experimental and simulated data can be observed for the dwell times of open and closed states . Repeating the simulation N times, we build up a random process of N events. Summing up the traces of each event and dividing by N, the final vector is the time-evolution of the open state probability . That is, we have the current response of the channel to a voltage step. As we supposed ergodicity, we can consider ensemble averaging as well as temporal averaging. The information we obtain from the ensemble averaging technique, already shown, is comparable with a long-term observation of a single event ; the equivalent length of this observation depends strictly on the number N of events used to create the process . The larger their number, the sharper the information we obtain . An example of a simulation of a voltage-clamped channel is shown in Fig . 4(a). Because of the instantaneous sensitivity to change of the transmembrane voltage, independently of the way the changes occur and of their causes, this model can be used to study the effect of different chemical, EM, or temperature perturbations on a single channel . Another interesting aspect is that, as the state-machine approach is general, the algorithm can be applied to each type of channel, both ligand- and voltage-dependent .



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2.3. Unperturbed behavior

The model has been applied to three kinds of voltage-dependent channels : K +, Na' and calcium (Ca 2+ ). For K + and Na' kinetics, H-H equations were chosen [23]. The modeling scheme for the K+ channel is a two-state one:

n f3 )

Q=( a

O E=C

n

n

where C (0) represents the closed (open) state . For a n and On the following holds [23]: an

0 .01(V+ 10) exp[(V+ 10)/10] - 1

Tf

On =0 .125 exp(V/80) Tf

(7)

(8)

where V is the membrane voltage, and the temperature dependence is expressed

by Tf =

3(T-6)/io

( 9)

with T in degrees centigrade [23]. For Na+, the H-H equations also hold [23] . A three-state model is supposed : am Ph C~ OE ~I Sm

ah

The m kinetics rules fluctuations between open (0) and closed (C) states, the h kinetics is for the inactivating (I) process . For the transition rates,

am

0 .1(V + 25) exp[(V+25)/10] -1

(10)

Pm=4 exp(V/18)Tf

(11)

a h = 0 .07 exp(V/20) Tf

(12)

Ph

1 exp[(V+ 30)/10] + 1

Tf

( 13)

Equation (9) has been used to calculate Tf . For the Ca 2+ channel, we propose a H-H like set of equations . Akaike [24] suggested a kinetic scheme, composed of three states : open, closed and inactivated . Values for the transition rates and their dependences on the voltage were calcu-



295

lated in this work by fitting data presented in ref . 24. By means of a numerical least-squares interpolation, we obtained the following equations : 10.0 am

1 .0+6671 .6 exp[

(14) 1 .6 log(V+ 50)]

R m =0 .06(V+50) exp[-0 .03(V+50)]

(15)

a h = - 0 .075 exp[0 .21 log(V+ 50)] + 0 .21

(16)

0 .8 13h

1 .0 + 374163 .73 exp[ -2 .8362 log( V+ 50)]

(17)

The dependence on temperature is the same as observed for the previous channels . 2.4. The electromagnetic stimulation

In our model, each kind of EM stimulation can be simulated : there are no limits to the form of the stimulating field (pulsed, modulated, and so on) . Here, we suppose the field to be sinusoidal and consider its action as the effect of a sinusoidal signal superimposed on the voltage-clamping step (Fig . 3). The analysis is accomplished in the ELF range for the EM stimulation . The reason why we consider the EM field in this way requires some explanation . An EM stimulus on an excitable membrane causes structural changes in the bilayer and it generally changes the conductive properties of the membrane [3-6,25-27] . Direct action of the EM stimulus on the channel proteins can only take place by action on charged particles or dipoles, through the following mechanisms . (1) Proteins have charged poles ; these charged zones can be free or saturated, depending on the composition of the solution containing the protein . A peptide in a protein has a permanent electric dipole moment of some debyes . The singlepeptide dipoles, all together, give the whole protein a dipolar moment [25,26] . (2) EM fields can cause movements and new distributions of the charges on the membrane surface or just around it . This can modify the electrical properties of proteins inside the membrane . (3) The channel kinetics depends on the membrane voltage . An EM field modifies this voltage and, as a consequence, the values of transition rates too . Conformational variations take place, i .e. there are changes in the occupancies of the energetic states [5,6]. A quantitative analysis of the effects on the membrane voltage of an EM field has been made [3-6,27]. The voltage AVM produced at the membrane ends can be calculated using the following formula : 1 .5ROE cos 0 AVM= (18) 1/2 1 1+(wr) 21

296

where Ro is the radius of a cell (eqn. (18) holds for a membrane of a cell considered as a sphere), E is the amplitude and w is the pulsation of the external sinusoidal field, 0 is the angle between the external field and the normal to the membrane surface at the point where we calculate AV M , and T is a time constant for the membrane, considered as an RC circuit. From the preceding discussion, it seems reasonable to introduce a direct effect of sinusoidal EM stimulation, at a molecular level, on the channel protein, and in particular on the transition rates of the channel . As we said at the beginning of this section, we simulate the effect of EM stimulation by superimposing a sinusoidal signal on the clamping voltage . From eqn . (18) we can see that at low frequencies AVM is proportional to E. That is why, in the following, we use the voltage units to represent the effect of the EM field on the behavior of the membrane channel . In eqn. (18), the presence of a time constant r suggests that we should also study the relaxation times of the channel ; this problem will lead us to choose a frequency range like the ELF range . An overview of studies on the dielectric properties of proteins is available in the literature [26], giving relaxation times of the order of nanoseconds . The slowest conformational changes in proteins have been found to be about 1 As [4-6] . Now, we suppose a stimulus with a very long period compared with the protein relaxation time . So, we assume that the protein is able to register and react to the stimulus nearly instantaneously . The transition rates take, moment by moment, the new values corresponding to the membrane voltage at that instant. We can consider the stimulation as if it were composed of a succession of short steps, very low in amplitude and of short duration, simulating a sinusoid; every time a transition occurs, the values of the transition rates are calculated using the value of membrane voltage at that moment ; these values are then used to obtain the corresponding dwell-time . As a consequence, in the equations for the dependence on voltage of the transition rates, the membrane voltage is now the superimposition of a sinusoidal signal on the clamping d .c. level. A quick glance at the equations involved reveals the non-linear behavior of the device "protein channel" . The channel works just like an electronic device, with its own d .c. supply and a small signal input . Now, it should be clear that an analysis in the ELF range means that the conditions we imposed on the stimulating wave and relaxation times are valid . In this range of frequency, linearization of the exponent in eqn . (6) with the method proposed is actually feasible, and a formula like eqn . (2) holds again . 3 . RESULTS AND DISCUSSION

In the following, all values of transition rates are calculated for a temperature of 6°C. Macroscopic and microscopic analyses are possible using several options included in our algorithm . A macroscopic analysis is feasible by generating a number of events and studying the resulting random process . Microscopic evaluations are accomplished by carrying out a statistical study on a single event . As a first step in



297

0 .6 0.55 0 .5 0 .45 d 0.4 0 .35 0.3 z 0 .25 w Q 0 .2 0 .15 0 .1 0 .05 0 0

10

15

20

25

30

t/ MS

(b)

Fig. 4. The variation for the open state occupancy in a simulation for the potassium channel : (a) the channel is voltage-clamped at 40 mV for 5 ms, then a step at -80 mV is applied ; (b) a 100 Hz, 20 mV signal is applied to the channel, the clamp is at 40 mV for the first 5 ms, then a step at -40 mV is applied.

the analysis of the model and its functionality, we checked its behavior in the situation of a simple d .c. voltage-clamp : we studied the values for open-state occupancy and compared them with experimental data on the voltage-current characteristics of the studied channels available in the literature [23,24]. Figure 4(a), (b) shows an example of the resulting behavior over time of a voltage-clamped K+ channel, first unexposed (a) and then with ELF stimulation (b). A decrease in the maximum value and in the decay time of the open probability can be observed for the exposed data (Fig . 4(b)). An example of the response to a voltage step of the two other channels starting from a closed state in an unexposed condition is presented in Fig . 5(a), (b). An important point to be noted is the different response to a voltage step in the potassium channel vs. the sodium and calcium channels . The K + channel has no inactivated state, whereas the Na' and the Ca 21 channels have an inactivated state . That is why, for Na'



298 0 .6 0.55 0 .5 0.45

J m

0.4 0 .35

O IL IL

0.3

z IL

w 0

0.25 0.2 0 .15 0 .1 0.05 0 0

40

45

50

(a) 0 .240.22 0 .2 0 .18 0 .16 0 .14 0 .12 0 .1 0 .08 0 .06 0 .04 0 .02 0 0 (b)

10

15

20 t/ms

25

30

35

40

Fig. 5. The channels' responses : (a) calcium to a voltage-step from 100 to 30 mV ; (b) sodium to a voltage-step from -90 to -40 mV.

and Ca", after a peak in the open probability, we can see damping behavior, which is not observed in the K+ response . As stated previously, the test of the model consists of a measurement of the open-state occupancy p o in a d .c.-clamping situation . The quantity is calculated by time averaging the p o shape, which presents, as shown in Fig . 4(a), (b), a fluctuation around an average value in the steady state . The same averaging technique has been used for ELF EM stimulation . The only precaution necessary is that time intervals containing an adequate number of periods of the stimulating signal be used. The results of the simulations can be summarized in diagrams representing values of p o for the channel at different values of d .c. clamping voltage. The graphs (Fig. 6(a), (b), (c)) were verified by comparison with the shape of an experimental voltage-current diagram (what is really important is the existence of a proportional link between the current flowing through the channel and po). For the K+ channel (Fig . 6(a)) experimental data are available in ref. 18, for the Na' (Fig. 6(b)) channel in ref. 28, and for the Ca 21 channel (Fig . 6(c)) in



299 I .0

0 .8

r ka p a a 0 a

0.6

z w a O

0.

0.

-120

-100

-80

-60

-40

-20

0

20

40

60

CLAMPING VOLTAGE/mV

(0)

0.35

0.30

„ 0.25

ka

a

0,20

aO C

Z

0.15

w a 0 0.10

0.05

0 -60 (c)

-40

-20

0 20 40 60 CLAMPING VOLTAGE/mV

80

100

120

Fig. 6. The open state occupancy versus membrane voltage-clamp values for different channels : (a) potassium ; (b) sodium ; (c) calcium .

3 00

(b)

m

M

J

Fig. 7 . Variations of the channels' open state probability for several values of the clamping voltage and field applitude : (a) potassium for a field at 100 Hz ; (b) sodium for a field at 100 Hz ; (c) calcium for a field at 100 Hz ; (d) potassium for a field at 50 Hz .

ref. 29 . There is good agreement between our theoretical results obtained by computer simulations and the experimental results . For the K + and Na' channels the agreement is satisfactory but rather predictable, since H-H theories have by now been tested widely . For the Ca21 channel, the results are particularly interesting because the values of a m , S m, a h and Ph were computed using a numerical interpolation ; despite this the theoretical results show a very good match with the experimental data . As for microscopic results, the well known exponential distributions for the open and closed state dwell-times [22] were obtained, and a satisfactory test to calculate the time constants in a voltage-clamp at -80 mV (in the H-H sign convention [231) on an unexposed K + channel has been carried out . The effects of the EM stimulation are summarized using three-dimensional graphics (Fig . 7(a), (b), (c), (d)), showing the amplitude of the EM field and the d .c.-clamping voltage in the x-y coordinates, and the value of poexp - PO on the z axis, where poexp is

301

the open state occupancy when EM radiation is considered . In other words, the graphics show the changes in average conductance due to an EM field . Each diagram is for a particular value of frequency . Here, only four diagrams at 50 and 100 Hz are shown, but other data were collected at 1, 50 and 100 Hz . At particular voltages (different for the three kinds of channel) fields of equal amplitudes can induce variations of completely different magnitude . Percentage values vary up to 60%-70%, although the large majority of changes fall between 2% and 10% . Moreover, it is particularly interesting to observe that at some amplitudes of the field the effects on the conducting properties are more evident : these amplitude windows can be observed in Fig . 7(a) (the hole at about 50 mV), in Fig. 7(c) (peak at about 50 mV), and in Fig . 7(d) (three different peaks). Attention should be paid to the fact that very low amplitude fields are able to affect the open state occupancy, sometimes more significantly than stronger fields can . Another result is the different response of each channel at different frequencies . A methodology for theoretical analysis of the open and closed state occupancies, not too different from that proposed here, has already been developed [7,8]. Moving from thermodynamics, these papers give an analytical solution to the problem of finding the voltage-dependence for the activation and inactivation transition rates . For stimulating fields in the r .f . range, the kinetics of "gating-particles" is studied [8], while at lower frequencies an H-H-like theory is developed with Fourier analysis [7] . In both cases, these papers consider time-averaging of the transition rates for an interval corresponding to a period of the stimulating signal . Afterwards, a change in the occupancies is revealed compared with the absence of

0 0.624 0.626 0.628 0.63 0.632 0.634 0.636 0.638 0.64 0.642 0.644 OPEN PROBABILITY Fig. 8. Distribution of the open state probability (p 0 ) at the steady state in a potassium channel clamped at -60 mV .

302

stimulation . This change is due to the EM stimulation . This kind of technique does not take into account the frequency of the field (it only differentiates between microwave (MW) and low frequencies as for the kinetic model to be considered) . We know, however, that great sensitivity of the channel response to EM frequency has been demonstrated experimentally [1,2]. The technique proposed here reveals the sensitivity to frequency, because it deals with "ensemble-averaging" of random processes, of which the single events register quicker or slower oscillations in the membrane voltage . It is therefore necessary to deal with channel kinetic phenomena from a non-deterministic point of view . The model has been able to simulate the behavior of protein channels, showing frequency and an amplitude sensitivity in the channel response to EM stimulation . Studies on the fluctuations of p o (noise) have also been performed . The relation between current noise and the kinetics of a single channel are well known [14]. Figure 8 shows the distribution we obtained for p o at the steady state . We notice a peak and a Poisson-like distribution . That is what we expected theoretically : since we used a large number of events (1000) to generate the process, the binomial distribution, which would be the actual distribution for the process, can also be considered a Poisson distribution [19]. The results presented deal with ELF ; in a future development, this analysis will be extended to study what happens over large frequency ranges . The microwave range is of particular interest, and preliminary studies on Ach-receptor channels exposed to microwaves suggest that the model presented can also be used to investigate EM effects at these frequencies [9]. 4 . CONCLUSIONS

From the previous discussions, we want now to focus attention on some conclusions . First, this work presents a study of the behavior of a protein voltagedependent membrane channel exposed to EM fields . The technique used in the model is based on a sort of "flashing" behavior of the channel : it can register the electromagnetic environment at a certain instant, and at that moment decides its future for a period of time . The electric situation at the membrane sides influences the length of the interval of time . In the time between two successive transitions, the channel is practically insensitive to external electrical stimulation . We tried to hypothesize different mechanisms of channel response, but each attempt proved unsuccessful . Another conclusion is the need to deal with a stochastic approach to these phenomena, instead of a deterministic one . We obtained evidence of both the different response at different frequencies, and the existence of different sensitivity of one channel at several clamping voltages . The deterministic methods have been unable to reproduce the frequency sensitivity of a membrane channel . A parameter to be taken very carefully into consideration is the temperature . The transition rates are all temperature dependent, and an important development of the model might be the study of the channel's response at different tempera-

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Lures, or of the time-variation of the temperature during a single process . Afterwards, important extensions of the model could be developed to study the effects of microwaves on channels, in particular on ligand-dependent channels . Finally, an important future development of studies of protein channels using the model presented is the possibility of a theoretical analysis of channel noise . Studies on the spectra of current fluctuations, in which interest is growing, can be easily accomplished with a simple manipulation of the results of computer simulations. Since the model is general, further developments are expected, for example simulation of ligand-dependent channels or analysis of noise power spectra for several channels exposed to EM fields. REFERENCES 1 C. Polk and E . Postow (Eds .), CRC Handbook of Biological Effects of Electromagnetic Fields, CRC Press, Boca Raton, FL, 1986 . 2 E. Postow and M.L . Swicord, Modulated fields and "window" effects, In C . Polk and E. Postow (Eds.), CRC Handbook of Biological Effects of Electromagnetic Fields, CRC Press, Boca Raton, FL, 1986, pp . 425-461 . 3 T.Y. Tsong and D.R. Astumian, Electroconformational coupling and the effects of static and dynamic EF on membrane transport, Annu . Rev . Physiol., 80 (1988) 1-56 . 4 H.V. Westerhoff, F. Kamp, T .Y. Tsong and D .R. Astumian, Interaction between enzyme catalysis and non-stationary EF . In M. Blank and E . Findl (Eds.), Mechanistic Approach to Interactions of Electric and Electromagnetic Fields with Living Systems, Plenum, New York, 1987, pp . 203-216 . 5 T .Y. Tsong and D .R. Astumian, Electroconformational coupling and membrane protein function, Prog. Biophys . Mol . Biol., 50 (1987) 1-45 . 6 T.Y. Tsong, D.R. Astumian and F . Chauvin, Interaction of membrane proteins with static and dynamic EF via electroconformational coupling, In M . Blank and E . Findl (Eds.), Mechanistic Approach to Interactions of Electric and Electromagnetic Fields with Living Systems, Plenum, New York, 1987, pp . 187-202 . 7 A.C . Cain, Biological effects of oscillating electric fields : role of voltage-sensitive ion channels, Bioelectromagnetics, 2 (1981) 23-32 . 8 A.C . Cain, A theoretical basis for MW and RF field effects on excitable cellular membranes, IEEE Trans. Microwave Theory Tech., 28 (1980) 142-149 . 9 G. D'Inzeo, P. Bernardi, F . Eusebi, F. Grassi, C. Tamburello and B . Zani, Microwave effects on acetylcholine-induced channels in cultured chick myotubes, Bioelectromagnetics, 9 (1988) 363-372 . 10 T.M. Philippova, V.I. Noselov, M .F. Bistrova and S.I . Aleksev, Microwave effects on camphor binding to rat olfactory epithelium, Bioelectromagnetics, 9 (1988) 347-354 . 11 P . Bernardi, G . D'Inzeo, F . Eusebi and C. Tamburello, The patch-clamp technique in the study of EM fields effects on biological structures, Alta Frequenza, LVIII (1989) 341-347 . 12 E. Neher and B . Sakmann, Single-channel currents recorded from membrane of denervated frog muscle fibers, Nature, 260 (1976) 799-802 . 13 R.W. Aldrich and G . Yellen, Analysis of nonstationary channel kinetics . In E . Neher and B. Sakmann (Eds.), Single-Channel Recording, Plenum, New York, 1983, pp . 292-299 . 14 H. Mino and K. Yana, A parametric modeling of membrane current fluctuations with its application to the estimation of the kinetic properties of single ionic channels, IEEE Trans . Biomed . Eng ., 36 (1989) 1028-1037 . 15 L.J . De Felice and J .R. Clay, Membrane current and membrane potential from single-channel kinetics . In E . Neher and B . Sakmann (Eds .), Single-Channel Recording, Plenum, New York, 1983, pp. 323-342 .

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