A Perturbation Model for Electromagnetic-Field Interaction with

10 A Perturbation Model for Electromagnetic-Field Interaction with Excitable Cellular Membranes CHARLES A . CAIN

Downloaded by EMORY UNIV on August 25, 2015 | http://pubs.acs.org Publication Date: August 4, 1981 | doi: 10.1021/bk-1981-0157.ch010

Department of Electrical Engineering, Bioacoustics Research Laboratory, University of Illinois at Urbana-Champaign, Urbana, IL 61801

Voltage sensitive ion channels for Na + , K~+, and Ca++ in the lipid bilayer membrane of excitable cells are protein macromolecules with several intraconvertible conformations, one of which is open and the others closed or nonconducting. Charged or dipolar components of the channel protein (or protein complex) move in response to perturbation of the membrane electric field and act as voltage sensors which trigger "gating" or conformational substates resulting in the channel being either open or closed. Graded changes in membrane conductance is the result of the average behavior of a large number of channels which open or close stochastically in response to changes in the electric field within the lipid bilayer and background thermal agitation. Some ion channels are exquisitely sensitive to small perturbations in the membrane electrostatic field. Moreover, the probability of a channel being in a given state is a highly nonlinear function of the applied electric field. Because of these nonlinearities, it will be shown that an alternating (AC) electric field might bias the probability of the channel being in a given state. The possibility that the voltage sensing charges (or dipoles) incur a steady or DC displacement in an alternating electric field is the basis for the theory discussed herein. The model predicts that an AC field of sufficient magnitude will affect the average conductance of the membrane to particular ion species. The resulting biological effect would depend on the magnitude of the conductance changes and the physiological function of the affected channel. A model, based on a perturbation analysis of the highly successful empi rical formulation of Hodgkin and Huxley (1), has been developed which makes predictions of the effect of oscillating fields on a particular nerve membrane system (2). In the present paper, a theoretical model will be presented along with some of the predicted effects of AC electric fields on the HodgkinHuxley (HH) model of squid axon membranes.

0097-6156/81/0157-0147$05.00/0 © 1981 American Chemical Society

In Biological Effects of Nonionizing Radiation; Illinger, K.; ACS Symposium Series; American Chemical Society: Washington, DC, 1981.

148

BIOLOGICAL EFFECTS OF NONIONIZING RADIATION

Theory A Channel Model. Consider a two s t a t e v o l t a g e sensor f o r an i o n i c channel i l l u s t r a t e d d i a g r a m a t i c a l l y as a(V) A

-

Β

(closed)

b(V)

(open)

where rate constants a(V) and b(V) are f u n c t i o n s of membrane po t e n t i a l . Assume Q(V) i s the e q u i v a l e n t charge c a r r i e d across the membrane when a v o l t a g e sensor moves ( r o t a t e s ? ) from A to B. Let η be the occupancy of s t a t e Β [ f r a c t i o n of v o l t a g e s e n s i t i v e d i poles or charged groups i n the open s t a t e ] and l e t r\ be the oc cupancy of s t a t e A. I f pressure and temperature remain unchanged, an e q u i l i b r i u m constant defined as Κ = Hc/-o w i l l be a f u n c t i o n o n l y of membrane p o t e n t i a l . From f i r s t p r i n c i p l e s of thermody namics [e.g., see Schwarz ( 3 ) ] :


0

c

d ln(K) dV

Q(V) kT

m V

Λ

Assume f o r s i m p l i c i t y that Q(V) = constant = Ze where e i s the elementary charge and Ζ i s a p o s i t i v e number. I n t e g r a t i n g Eq. (1) then g i v e s

\ Κ = —

= exp[Ze(V-V)/kT] Ο

where V i s the membrane p o t e n t i a l at which Κ = 1 [or r\ - r-] . T h i s i s immedicately r e c o g n i z a b l e as the Boltzmann r e l a t i o n . More over, s i n c e η + n = 1, then c

0

c

η (V) =

1

-

(2)

1 + exp[-Ze(V-V)/kT] and η (γ) = 1 +

1 exp[+Ze(V-V)/kT]

(3).

Thus, the f r a c t i o n of v o l t a g e sensors i n a g i v e n s t a t e can be c a l c u l a t e d i f the membrane p o t e n t i a l i s known. Now, assume the a c t u a l opening and c l o s i n g of the 2 - s t a t e channel i s a f i r s t order process such that a(V) Λ

channel open

-

$(V)

Β

channel open


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149

where the r a t e constants a ( V ) and 6 ( V ) depend on the s t a t e of the v o l t a g e sensor as f o l l o w s : BOO = - η

(5)

ο

and


a(V)

=k n 2

(6),

Q

The constants and k- r e f l e c t an assumed d i r e c t p r o p o r t i o n a l i t y of the r a t e constants f o r opening and c l o s i n g the channel on the occupancies of the "open" and " c l o s e d " s t a t e s r e s p e c t i v e l y of the v o l t a g e sensor. Let ρ be the f r a c t i o n of the channels (not v o l t a g e sensors) i n the open s t a t e . For a f i r s t order process, ρ i s the s o l u t i o n of the d i f f e r e n t i a l equation

jfc = 3 0 0

[1-p] - a ( V ) p

(7).

The r a t e constants α and 3 determine both the r a t e of change and f i n a l s t e a d y - s t a t e value of p. The steady s t a t e v a l u e of ρ a f t e r a step input i s n p

g(V) + 300

=

(8)

a(V)

oo

where ρ reaches p- e x p o n e n t i a l l y w i t h time constant τ = ρ

(9) a ( V ) + β(ν)

The preceding model provides the b a s i s f o r a d i s c u s s i o n of how AC e l e c t r i c f i e l d s might i n t e r a c t w i t h a v o l t a g e s e n s i t i v e ion channel i n such a way as to b i a s the conductance of the chan n e l i n a p a r t i c u l a r d i r e c t i o n . In what f o l l o w s , the b a s i c assump t i o n made i s t h a t the v o l t a g e sensing process i s c o n s i d e r a b l y f a s t e r then the p h y s i c a l opening and c l o s i n g of the channel. With t h i s assumption, an AC f i e l d which might a f f e c t the v o l t a g e sens ing process would cause a much slower change i n channel conduc tance. S e n s i t i v i t y t o an AC F i e l d . V i s replaced by

Assume the membrane p o t e n t i a l

V ( t ) = V + V cosu) t ο m ο

(10)

where V i s the DC membrane p o t e n t i a l and V i s the peak amplitude of the a l t e r n a t i n g component of a p p l i e d memBrane p o t e n t i a l [see Barnes and Hu (4) or MacGregor ( 5 ) ] . q


150


I f the channel opens or c l o s e s w i t h a time constant Tp and the v o l t a g e sensor has a r e l a x a t i o n time T , then assume Tp » T and l/Tp « ω t

(12).

ο Because of the n o n l i n e a r form of Eq. (11), an AC f i e l d w i l l cause the average or DC v a l u e of r| (a)t) t o s h i f t i n a p a r t i c u l a r d i r e c t i o n depending on the unperturbed v a l u e of Based on the above argument, i t i s c l e a r that a change i n the average v a l u e of η (or η ) w i l l b i a s the r a t e of opening or c l o s i n g of a p a r t i c u l a r channel because of Eqs. (5) and ( 6 ) . The change i n s t a t e of the channel w i l l then progress a t a r a t e slower than the v o l t a g e sensing processes which t r i g g e r e d the change. 0

0

0

M o d i f i e d Hodgkin-Huxley Model. I n the HH model, the membrane current I , w r i t t e n as a f u n c t i o n of V i s expressed by the system of coupled equations given i n Table I . I n these equations V i s the displacement of membrane p o t e n t i a l from the r e s t i n g v a l u e ( d e p o l a r i z a t i o n n e g a t i v e ) . Constants CM, g-, g f l , "Sl> K » Na> and V i are e x p l a i n e d i n d e t a i l i n (1). I n Table I , g5 n and g j i potassium and sodium conductances, respec t i v e l y . The dimensionless dynamical q u a n t i t i e s m, n, and h a r e s o l u t i o n s of the given f i r s t order d i f f e r e n t i a l equations and vary between zero and u n i t y a f t e r a change i n membrane p o t e n t i a l . The α and β r a t e constants a r e assumed t o depend o n l y on the i n s t a n taneous v a l u e of membrane p o t e n t i a l . Note that the equations i n Table I have a s i m i l a r mathemat i c a l s t r u c t u r e as the much s i m p l e r 2-state channel model developed e a r l i e r i n t h i s paper. One major d i f f e r e n c e should be emphasized; namely, that the r a t e constants of the model depend on the phys i c a l movement of charge so a r e not instantaneous f u n c t i o n s of membrane p o t e n t i a l as assumed i n the HH model. However, i f the v o l t a g e sensing process i s s u f f i c i e n t l y f a s t e r than the dynamics of channel opening and c l o s i n g , then the assumption of an i n stantaneous dependence of a s and β s on V i s reasonable (see Discussion). V

a

m

K

n

a r e

t n e

a

1

Some Numerical R e s u l t s .

1

F i g u r e 1 i s a p l o t of the


V

10.

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151



TABLE I . The Hodgkin-Huxley equations which d e s c r i b e the r e l a t i o n s h i p between t o t a l membrane c u r r e n t I and transmembrane p o t e n t i a l V i n the squid g i a n t axon [see ( 1 ) ] .

1

C

+

" M of

s /

(

V

"V

+

%

Α

(

ν

&

+

"V

*1

( V

-

V

where dn/dt = a ( l - n) n

- m) - β m m

dm/dt = dh/dt =

V

1

- h)

-\

h

and α

η

= 0.0KV + 10)/(exp

6

n

= 0.125 exp(V/80)

a

m

= 0.1(V + 25)/(exp

$

«

m

h

V

V

+

+

1 Q Q

2 5 Q

- 1)

- 1)

= 4 exp(V/18)

= 0.07 exp(V/20)



V =IOmV, avg. value m

V -V

= IOmV


0

7Γ/4

7Γ/2

37Γ/4

7Γ

57Γ/4

37Γ/2

J_

77Γ/2

27Γ

ajt (rad) Figure 1. Instantaneous values of η (ω{) when V — V = 10 mV and V isOmV or 10 mV. ~ , the average value of ψ,(ωί) when V . = 10 mV, is shown. 0

0

0

m

WI


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p r o b a b i l i t y η ( ω Ο of a v o l t a g e sensor being i n s t a t e Β (open) over a f u l l p e r i o d of the a p p l i e d a l t e r n a t i n g membrane p o t e n t i a l of angular frequency ω and peak amplitude V of 10 mV. Equation (11) i s evaluated a t room temperature [290K where kT/e = 25 mV] w i t h Ζ = 5 and a t a DC membrane p o t e n t i a l 10 mV more p o s i t i v e than V [ V - V = 10]. I f V = 0, the p r o b a b i l i t y η that the v o l t a g e sensor i s i n the open s t a t e i s 0.881 under these c o n d i t i o n s . The v a r i a t i o n i n t h i s p r o b a b i l i t y f o r nonzero v a l u e s of V i s not symmetric about the unperturbed v a l u e of η , e.g., the average v a l u e of Ώ (ωί), "%> i s 0.808 when V = 10 as shown w i t h the dashed l i n e i n F i g . 1. Thus an a p p l i e d AC p o t e n t i a l r e s u l t s i n a decrease i n the average p r o b a b i l i t y of the v o l t a g e sensor being open. F i g u r e 2 shows the changes i n η as a f u n c t i o n of V f o r f i v e d i f f e r e n t v a l u e s of V - V w i t h Ζ = 5 and Ζ = 20. A graph of r\ u s i n g the same parameters would look i d e n t i c a l t o F i g . 2 except t h a t curves f o r n e g a t i v e values of V - V would appear a t the top of the drawing. Note t h a t , as one would expect, higher v a l u e s of Ζ r e s u l t i n both steeper changes i n η w i t h i n c r e a s i n g V and a lower t h r e s h o l d f o r changes t o occur. Estimated v a l u e s of Ζ f o r a c t u a l membrane channels range from 3 t o 20 (see D i s c u s sion) . A p l o t of "η and n as a f u n c t i o n of V - V w i t h V as a parameter i s given i n F i g . 3. Thus, an AC f i e l d b i a s e s the aver age occupancy of a p a r t i c u l a r s t a t e of the v o l t a g e sensor i n a d i r e c t i o n depending on the unpreturbed v a l u e of η or η . Since the r a t e constants 3 0 0 and a(V) a s s o c i a t e d w i t h opening and c l o s ing of the channel were assumed to be p r o p o r t i o n a l t o η and T l , the r e s p e c t i v e occupancies of the "open" and " c l o s e d s t a t e of t h e v o l t a g e sensor, then an AC f i e l d r e s u l t s i n a steady s h i f t i n these r a t e constants. T h i s would r e s u l t i n a change i n the prob a b i l i t y of the channel being open f o r a given DC membrane poten t i a l . T h i s e f f e c t i s the b a s i s f o r the HH p e r t u r b a t i o n model which f o l l o w s . Using the same argument i m p l i c i t i n Eqs. (11) and (12), the average values of the α and 3 r a t e constants of the HH equations (Table I ) w i l l be s h i f t e d i n an AC f i e l d i n a way q u a l i t a t i v e l y s i m i l a r to the DC s h i f t s of F i g . 3 [see ( 2 ) ] . Since the α and 3 r a t e constants determine both the r a t e of change i n m, n, and h and the f i n a l steady s t a t e v a l u e of these q u a n t i t i e s , an AC f i e l d - i n d u c e d b i a s of the r a t e constants w i l l a f f e c t g- and gas w e l l as the membrane p o t e n t i a l V [see ( 2 ) ] . Such an e f f e c t i s shown i n F i g . 4 where a 10 ms pulsed AC component of membrane e l e c t r i c f i e l d i s a p p l i e d t o the model. The change i n g , g * and V i n response to the p u l s e can be obtained by n u m e r i c a l l y s o l v i n g the HH equations (Table I ) w i t h i n i t i a l c o n d i t i o n s being the r e s t i n g s t a t e values f o r V, m, n, and h. The AC f i e l d i s assumed to be a p p l i e d u n i f o r m l y along the axon so t h a t dV/dx = 0. A f t e r each i t e r a t i o n through the modified HH equations, new estimates f o r the average v a l u e s of the a ( o ) t ) s and αίωΟ'β a r e obtained n u m e r i c a l l y . ο

m

0

m

0

m

0


0

m

0

m

Q

c

Q

0

m

0

c

Q

m

0

0

0

c

11

a

K

f


N a



154


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155

EM-Field Interaction with Cellular Membranes •

ι

ι

ι

I

ι

I

I

I

I

I

ι

ι

ι

I Τ — I — I — I — I — I — I — I — Γ

> Ε -2 -3 - 1 4

-ι ι ι I ι ι ι ι I ι ι ι ι—I—ι—ι—I—I—L


1—ι—I

I

J

I

I

L-

-τ—ι—ι—ι—Γ

10 t—PULSE ON(V =25.0mV) m

15 PULSE OFF TIME (m sec)

Figure 4. Solutions of the modified ΗΗ equations (see text) with a pulsed alter nating component of membrane potential. Computed response is for a 10-ms pulse with V = 25 mV. m

-25,

-20h

10

20

30 V (mV)

40

50

m

Figure 5. Threshold depolarization for generating an action potential as a function of V . Threshold voltages were determined from the modified HH model. Trace A : V applied over interval — oo < t < oo; .Trace B: V applied over interval 0 < t < 10 ms. Stimulus applied at t = 0. TO

m

m


156

BIOLOGICAL EFFECTS OF NONIONIZING

RADIATION

Since an a p p l i e d AC f i e l d produces a steady-state decrease i n g j j and an increase i n g (see F i g . 4 ) , i t has a h y p e r p o l a r i z ing or i n h i b i t o r y e f f e c t on the axon. This i s i l l u s t r a t e d i n F i g . 5 where the t h r e s h o l d d e p o l a r i z a t i o n r e q u i r e d t o generate an a c t i o n p o t e n t i a l i s p l o t t e d vs V , the applied AC f i e l d . Note t h a t even a pulsed f i e l d of 10 ms a p p l i e d simultaneously w i t h the ex c i t a t o r y d e p o l a r i z a t i o n w i l l r a i s e the t h r e s h o l d p o t e n t i a l s i g n i f i c a n t l y . I t should be noted that the e f f e c t of an AC f i e l d de pends on the d e t a i l s of the p a r t i c u l a r model and i s not necessar i l y i n h i b i t o r y . I t i s i n t e r e s t i n g , however, that an i n h i b i t o r y e f f e c t i s opposite to that which would be produced by heating so perhaps could be more r e a d i l y r e c o g n i z a b l e e x p e r i m e n t a l l y . a

K

m


Discussion As pointed out by Hodgkin and Huxley, the extreme steepness of the r e l a t i o n between i o n i c conductance and membrane p o t e n t i a l i n the squid g i a n t axon i m p l i e s the presence i n the membrane of p o l a r molecules having l a r g e charges or d i p o l e moments (1). I n order to e x p l a i n t h e i r experimental r e s u l t s , the equivalent num ber of e l e c t r o n i c charges Ζ c a r r i e d across the membrane when a v o l t a g e sensor goes from the closed to the open s t a t e must be about 6 f o r a c t i v a t i o n of the N a channel CI,6). Moreover, a t an a p p r o p r i a t e s t a t i c membrane p o t e n t i a l , i t was determined ex p e r i m e n t a l l y that g oc exp[v/4.17] where ν i s the displacement from the membrane p o t e n t i a l . Thus, a 4 mV d e p o l a r i z a t i o n w i l l cause an e - f o l d i n c r e a s e i n sodium conductance. I f the v o l t a g e sensor i s assumed to be a d i p o l e , then the above p r o p o r t i o n a l i t y becomes g exp[yv/kTd] where μ i s the d i p o l e moment of t h e v o l t a g e sensor and d i s the d i s t a n c e over which ν i s dropped. I f ν = 4 mV causes an e - f o l d change i n g and d - 100A (membrane t h i c k n e s s ) , then a t room temperature kT - 4 0 ( 1 0 ~ ) J so that μ ~ 10-26 c.m. or about 3000 Debye. This i s a r a t h e r e x t r a o r d i n a r y d i p o l e moment s i n c e μ f o r an average p r o t e i n i s a t l e a s t an order of magnitude l e s s , e.g., μ - 300 D f o r hemoglobin. When one considers that the v o l t a g e sensor most probably occupies a s m a l l f r a c t i o n of the volume of the p r o t e i n complex comprising the channel, these estimates of the d i p o l e moment of the v o l t a g e sensor a r e even more i n t e r e s t i n g . For the simple 2-state channel model discussed e a r l i e r , values of Ζ from 3 to 20 r e s u l t i n estimated d i p o l e moments of the v o l t a g e sensor from 1400 t o 9600 D r e s p e c t i v e l y . These values of Ζ span the range of values estimated i n the l i t e r a t u r e f o r d i f f e r e n t v o l t a g e s e n s i t i v e i o n i c channels (6). One consequence of these l a r g e values of μ i s that i t i s p o s s i b l e f o r the e l e c t r i c a l energy of i n t e r a c t i o n between the d i p o l e and the f i e l d t o be greater than the thermal energy kT a t r e l a t i v e l y s m a l l p e r t u r bations i n the membrane f i e l d ( l e s s than 1 mV f o r some channels). Induced AC f i e l d s i n membranes from RF or microwave i r r a d i a t i o n have been estimated to be t h i s l a r g e i n some models (4,5). +

N a

α

N a

N a

22


10.

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157

A key assumption i n the t h e o r e t i c a l d i s c u s s i o n given e a r l i e r i s that a v o l t a g e sensing process e x i s t s w i t h a r e l a x a t i o n time (or d i s t r i b u t i o n of times) f a s t e r than the f a s t e s t time constant f o r i o n i c conductance changes. The f a s t e s t conductance changes i n the squid axon, w i t h time constant T on the order of 100 y s , i n v o l v e opening of the N a channel or sodium a c t i v a t i o n (1). A v o l t a g e sensor w i t h a minimum r e l a x a t i o n time of 100 ys would not respond, i n the sense d i s c u s s e d h e r e i n , to AC f i e l d s of frequency greater than (2πτ )~ or about 2 kHz. P i c k a r d and Rosenbaum, based on s e v e r a l p h y s i c a l arguments, place the upper frequency f o r p o s s i b l e i n t e r a c t i o n w i t h the v o l t age sensors s o l i d l y i n the microwave r e g i o n Ç7). On the other hand, i n squid (8,9,10) and a number of other p r e p a r a t i o n s (1115), one can detect s m a l l asymmetrical displacement c u r r e n t s i n response to step changes i n membrane p o t e n t i a l which, based on a number of l i n e s of evidence (6) appear to be a s s o c i a t e d i n some way w i t h the g a t i n g process i t s e l f . Moreover, i n the squid axon Rojas and Keynes (16) and Keynes and Rojas (17) were a b l e to show that the r e l a x a t i o n time constant of t h i s " g a t i n g c u r r e n t " was very n e a r l y T over a broad range of membrane p o t e n t i a l s . How ever, i t i s not c l e a r whether the measured charge movements are due to r e o r i e n t a t i o n of the v o l t a g e sensors d i r e c t l y or t o d i s placement c u r r e n t s a t t r i b u t a b l e t o l a r g e r p r o t e i n c o n f o r m a t i o n a l changes ("gating") t r i g g e r e d by p e r t u r b a t i o n of the v o l t a g e sen sors by the a p p l i e d f i e l d . I f the l a t t e r case i s t r u e , t h i s sug gests that v o l t a g e sensing and i o n g a t i n g are two separate pro cesses each of which may have d i f f e r e n t c h a r a c t e r i s t i c r e l a x a t i o n times or d i s t r i b u t i o n of times. The assumption of two separate processes i s inherent i n the 2-state model d i s c u s s e d h e r e i n . Due to c e r t a i n behaviors anomalous to the HH model obtained from some p r e p a r a t i o n s i n v o l t a g e clamp experiments (18,_19), a number of models have been proposed whereby N a a c t i v a t i o n and i n a c t i v a t i o n are coupled i n t o a m u l t i s t a t e sequence r a t h e r than proceeding independently as suggested by HH m-h dynamics. F o r example, consider the f o l l o w i n g k i n e t i c scheme suggested by Gold man and Hahin (20) m

+

1


πι

m

+

-

-

R

C I 7 where R, C, and I are the r e s t i n g , conducting, and i n a c t i v a t e d s t a t e s r e s p e c t i v e l y and and A2 are a c t i v a t e d but non-conducting s t a t e s . In a s i m u l a t i o n based on t h i s scheme, i t was necessary for one r e l a x a t i o n time to be at l e a s t an order of magnitude f a s t e r than T i n order to f i t the time course of sodium conduc tance d u r i n g steps i n p o t e n t i a l (20). T h i s k i n e t i c scheme has other i n t e r e s t i n g consequences. Due to the s e q u e n t i a l nature of t h i s model, o n l y one v o l t a g e sensor i s r e q u i r e d as opposed to separate sensors i m p l i e d f o r the h and m processes i n the HH A

±

A

2

yr

m



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O F NONIONIZING

RADIATION

model. Furthermore, t r a n s i t i o n through e i t h e r or both of the A s t a t e s i n the above k i n e t i c scheme may d i r e c t l y i n v o l v e v o l t a g e sensing, a separate process from a c t u a l channel g a t i n g . In another k i n e t i c model suggested by Jakobsson, the conducting s t a t e i s preceded by an " e x c i t e d s t a t e " which i s entered w i t h a r a t e constant p r o p o r t i o n a l to the time d e r i v a t i v e of membrane p o t e n t i a l (19). Thus, membrane p o t e n t i a l p e r t u r b a t i o n s which a r e r a p i d can r e s u l t i n very s m a l l r e l a x a t i o n times. This model very s u c c e s s f u l l y p r e d i c t s a wide range of v o l t a g e clamp data which are w e l l d e s c r i b e d by the HH equations as w e l l as behavior anomalous to the HH model which have been seen e x p e r i m e n t a l l y . Furthermore, a very r a p i d i n i t i a l process may be a necessary c o n d i t i o n f o r any k i n e t i c model t o f i t some experimental v o l t a g e clamp data which i s anomalous t o the HH model (E. Jakobsson, p e r s o n a l communication). Thus, although relaxât ion times f a s t e r than TJJJ r e l a t i n g to channel g a t i n g have not been detected e x p e r i m e n t a l l y i n asymmetrical displacement c u r r e n t experiments, there a r e theor e t i c a l reasons f o r expecting that such r a p i d processes do occur. Research on the b i o e f f e c t s of nerve and muscle exposure to electromagnetic f i e l d s has r e s u l t e d i n a l i t e r a t u r e f i l l e d w i t h c o n t r a d i c t i n g phenomenology. Thus, a number of r e s e a r c h e r s have published evidence which supports (21,22,23,24) or does not support (25-,-6,27-,-8) the hypothesis that AC f i e l d s can a f f e c t exc i t a b l e c e l l s by nonthermal mechanisms. With few e x c e p t i o n s , l i t t l e a t t e n t i o n has been g i v e n to p o s s i b l e u n d e r l y i n g mechanisms of i n t e r a c t i o n . I t i s hoped that t h i s paper w i l l help to p r o v i d e a r a t i o n a l b a s i s f o r design of experiments based on s p e c i f i c hypotheses which can be t e s t e d i n the l a b o r a t o r y . Literature Cited 1. 2. 3. 4. 5. 6. 7.

Hodgkin, A. L.; Huxley, A. F. A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 1952, 117, 500-544. Cain, C. A. A theoretical basis for microwave and RF field effects on excitable cellular membranes, IEEE Trans. Microwave Theory Tech. 1980, MTT-28, 142-147. Schwarz, G. On dielectric relaxation due to chemical rate processes. J. Phys. Chem. 1967, 71, 4021-4030. Barnes, F. S.; Hu, C. H. Model for some nonthermal effects of radio and microwave fields on biological membranes. IEEE Trans. Microwave Theory Tech. 1977, MTT-25, 742-746. MacGregor, R. J. A possible mechanism for the influence of electromagnetic radiation on neuroelectric potentials. IEEE Trans. Microwave Theory Tech. 1979, MTT-27, 914-921. Aimers, W. Gating currents and charge movements in excitable membrane. Rev. Physiol. Biochem. Pharmacol. 1978, 82, 97-190. Pickard, W. F.; Rosenbaum, F. J. Biological effects of microwaves at the membrane level: two possible athermal electrophysiological mechanisms and a proposed experimental test.


10.

8. 9. 10.


11.

12. 13. 14. 15. 16.

17.

18. 19. 20. 21. 22. 23.

CAIN

EM-Field


159

Math. Biosci. 1978, 39, 235-253. Armstrong, C. M.; Bezanilla, F. Charge movement associated with the opening and closing of the activation gates of the Na channels. J. Gen. Physiol. 1974, 63, 533-552. Keynes, R. D.; Rojas, E. Kinetics and steady state properties of the charged system controlling sodium conductance in the squid giant axon. J. Physiol. 1974, 239, 393-434. Meves, H. The effect of holding potential on the asymmetry currents in squid giant axons. J. Physiol. 1974, 243, 847867. Neumcke, B.; Nonner, W.; Stämpfli, R. Asymmetrical displacement current and its relation with the activation of sodium current in the membrane of frog myelinated nerve. Pflügers Arch. 1976, 363, 193-203. Schneider, M. F.; Chandler, W. K. Voltage-dependent charge movement in skeletal muscle: a possible step in excitationcontraction coupling. Nature 1973, 242, 244-246. Aimers, W.; Best, P. M. Effects of tetracaine on displacement currents and contraction in frog skeletal muscle. J. Physiol. 1976, 262, 583-611. Rudy, B. Sodium gating currents in Myxicola giant axons. Proc. Roy. Soc. Lond. (Biol.) 1976, 193, 469-475. Adams, D. J.; Gage, P. W. Gating currents associated with sodium and calcium currents in an Aplysïa neuron. Science 1975, 192, 783-784. Rojas, E; Keynes, R. D. On the relation between displacement currents and activation of the sodium conductance in the squid giant axon. Phil. Trans. R. Soc. 1975, 270, 459482. Keynes, R. D.; Rojas, E. The temporal and steady-state relationships between activation of the sodium conductance and movement of the gating particles in the squid giant axon. J. Physiol. 1976, 255, 157-190. Goldman, L. Kinectics of channel gating in excitable membrane. Q. Rev. Biophys. 1976, 9, 491-526. Jakobsson, E. A fully coupled transient excited state model for the sodium channel. J. Math. Biology 1978, 5, 121-142. Goldman, L.; Hahin, R. Initial conditions and the kinetics of the sodium conductance in Myxi,cola giant axons. J. Gen. Physiol. 1978, 72, 879-898. Frey, A. H.; Seifert, E. Pulse modulated UHF energy illumination of the heart associated with change in heart rate. Life Sci. 1968, 7, 505-512. Bawin, S. M.; Kaczmarek, L. K.; Adey, W. R. Effects of modulated VHF fields on the central nervous system. Ann. NY Acad. Sci. 1975, 247, 74-81. Tinney, C. E . ; Lords, J. L.; Durney, C. H. Rate effects in isolated turtle hearts induced by microwave irradiation. IEEE Trans. Microwave Theory Tech. 1976, MTT-24, 18-24.


160

24. 25. 26. 27.


28.

BIOLOGICAL

EFFECTS OF NONIONIZING RADIATION

Seaman, R. L; Wachtel, H. Slow and rapid responses to CW and pulsed microwave radiation by individual Aplysia pacemakers. J. Microwave Power 1978, 13, 77-86. Clapman, R. M.; Cain, C. A. Absence of heart-rate effects in isolated frog heart irradiated with pulse modulated microwave energy. J. Microwave Power 1975, 10, 411-419. Liu, L. M.; Rosenbaum, F. J.; Pickard, W. F. The insensitivity of frog heart to pulse modulated microwave energy. J. Microwave Power 1976, 11, 225-232. Chou, C. K.; Guy, A. W. Effect of 2450 MHz microwave fields on peripheral nerves. Dig. Tech. Papers (IEEE Int. Microwave Symp., Boulder, CO), 1973, 318-320. Courtney, K. R.; Lin, J. C.; Guy, A. W.; Chou, C. K. Microwave effect on rabbit superior cervical ganglion. IEEE Trans. Micrwoave Theory Tech. 1975, MTT-23, 809-813.

RECEIVED October 31, 1980.


A Perturbation Model for Electromagnetic-Field Interaction with

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