Coherent Modes in Biological Systems - American Chemical Society

13 Coherent Modes in Biological Systems Perturbation by External Fields FRIEDEMANN KAISER

Downloaded by CORNELL UNIV on July 22, 2016 | http://pubs.acs.org Publication Date: August 4, 1981 | doi: 10.1021/bk-1981-0157.ch013

Institute of Theoretical Physics, University of Stuttgart, Pfaffenwaldring 57, 7000 Stuttgart - 80, West Germany

In recent years considerable attention has been given to understanding both the stability of biological systems and their interactions with nonionizing electromagnetic waves. Toward this aim a special effort has been made by many authors using either refined experimental techniques or a modelling approach. The problem has gained increasing interest both, through rather exciting experimental results and through theoretical considerations (1). Well documented biological effects exist, arising from irradiations at very low intensities where thermal effects certainly are to be excluded (vid. Refs. 2, 3 for details). The reported nonthermal effects occur in certain frequency regions only. They usually exhibit saturation at rather low intensities. In addition, these effects might be masked by thermal ones. In these cases the energy transfer from the radiation to the system varies slowly with frequency and is largely determined by dielectric losses. A detailed and very successful investigation of the complex structure has shown that there exists no spatial order in these systems (vid. Ref. 3, chapter 2 for details). However, it is wellknown in modern physics that, besides spatial order, other types of order do exist, e.g. order of motion. In many cases this type of order can be described by long range coherence. As early as 1967 Frohlich (4) emphasized that biological systems exhibit a relative stability for some modes of behaviour. In their active state (in vivo) these modes remain very far from thermal equilibrium. One manifestation of the stable behaviour is a coherent excitation, which means that a single mode is strongly excited. Fröhlich has suggested that coherent oscillations (as a special realization of long range coherence) ought to be of great importance in biological systems (4, 5). To give an interpretation of the specific behaviour and stability of biological systems, one should primarily look for basic physical ideas, which may serve as working models to establish physical theories for a description of order and function. The idea of making models is suggested by the enormous complexity, which characterizes biological systems. With the aid of models it is 0097-6156/81/015-0219$05.75/0 © 1981 American Chemical Society

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


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p o s s i b l e to pass over the complexity of the system by r e s t r i c t i n g to the r e l e v a n t mechanisms which might be r e s p o n s i b l e f o r both, the i n t e r n a l f u n c t i o n and the i n t e r a c t i o n w i t h an e x t e r n a l s t i mulus. From our p o i n t of view, the f o l l o w i n g b a s i s seems appropriate for t h e o r e t i c a l considerations: 1. the f u n c t i o n a l complexity of b i o l o g i c a l systems n e c e s s i t a t e s the a p p l i c a t i o n of macroscopic t h e o r i e s , e.g. the concepts of Synergetics (6) and D i s s i p a t i v e S t r u c t u r e s (_7), 2. the existence of some modes of behaviour very f a r from t h e r mal e q u i l i b r i u m r e q u i r e s n o n l i n e a r i n t e r a c t i o n s between the r e l e v a n t subunits, 3. the s p e c i f i c modes, which are thermally decoupled from the remaining part of the system, must be s t a b i l i z e d by a s u i t a b l e energy and/ or matter i n p u t , 4. the i n t e r a c t i o n w i t h e x t e r n a l f i e l d s can lead to a d d i t i o n a l t r a n s p o r t i n s t a b i l i t i e s ( i . e . n o n e q u i l i b r i u m phase t r a n s i t i ons), which can lead to new s t a b l e or unstable s t a t e s of f u n c t i o n w i t h i n the system. I t should be s t r e s s e d that F r o h l i c h ' s concept of coherent o s c i l l a t i o n s agrees w i t h these four r a t h e r general requirements. However, i t should be emphasized that t h e o r e t i c a l physics can only provide us w i t h the r e l e v a n t concepts, which can p o i n t to a number of p o s s i b i l i t i e s , but i t cannot p r e d i c t i n which way the system has evolved. The a c t u a l s t a t e of f u n c t i o n can only be found i n c l o s e c o l l a b o r a t i o n w i t h experiments. General Concept and T h e o r e t i c a l B a s i s Nonlinear Systems, N o n l i n e a r i t i e s p l a y a dominant r o l e i n physics and many other d i s c i p l i n e s . For example, a l l m a t e r i a l laws have a n o n l i n e a r c h a r a c t e r i s t i c . In many cases, the u s u a l l y a p p l i e d l i n e a r i z a t i o n procedures are s u i t a b l e and w e l l e s t a b l i s h e d methods, which might lead to s a t i s f y i n g r e s u l t s . However, n o n l i n e a r systems can e x h i b i t a behaviour, which i s completely absent i n the regime of l i n e a r dynamics. Both, the development and the maintenance of such a behaviour seem to be provided by a general mechanism: nonlinear dissipation. D i s s i p a t i o n as an o r g a n i z i n g f a c t o r can only occur under c e r t a i n c o n d i t i o n s , necessary p r e r e q u i s i t e s are: energy input (open system), n o n l i n e a r dynamics and s t a b i l i z a t i o n f a r away from t h e r mal e q u i l i b r i u m i . e . beyond the regime of l i n e a r i r r e v e r s i b l e t h e r modynamics ( v i d . the requirements of Chapter I ) . O s c i l l a t i n g phenomena and h y s t e r e s i s behaviour ( b i - , m u l t i s t a b i l i t y ) are the most important phenomena i n systems f a r from thermal e q u i l i b r i u m , e x c l u d i n g s p a t i a l s t r u c t u r e s ( p a t t e r n s , t r a v e l l i n g waves). Examples of both types of temporal s t r u c t u r e s can be found i n a l l d i s c i p l i n e s where n o n l i n e a r dynamics p l a y a s i g n i f i c a n t r o l e . In b i o p h y s i c s i t appears that d i s s i p a t i o n prov i d e s a general mechanism and may a l s o account f o r the f u n c t i o n a l



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s t a b i l i t y and s p e c i f i c s e n s i t i v i t y of cooperative systems to environmental i n f l u e n c e s . Nonlinear O s c i l l a t i o n s ( L i m i t C y c l e s ) . We want to r e s t r i c t ourselves to n o n l i n e a r o s c i l l a t i o n s of l i m i t c y c l e type (LC), which means that we are only d e a l i n g w i t h s e l f s u s t a i n e d o s c i l l a t i o n s . This type of n o n l i n e a r o s c i l l a t i o n s can only occur i n nonconservative systems, i t i s a p e r i o d i c process, which i s produced at the expense of a nonperiodic source of energy w i t h i n the system. I t should be s t r e s s e d that o s c i l l a t o r y phenomena are w e l l known i n b i o l o g y . They span a wide range of p e r i o d s , reaching from f r a c t i o n s of seconds (neuronal and EEG a c t i v i t i e s ) and minutes (biochemical o s c i l l a t o r s ) to hours and days ( c i r c a d i a n rhythm...). Besides these cooperative o s c i l l a t i o n s on an i n t r a - , i n t e r - and s u p e r c e l l u l a r l e v e l , the usual o s c i l l a t i o n s on a microscopic b a s i s (e.g. e l e c t r o n i c t r a n s i t i o n s , i n t r a - and i n t e r m o l e c u l a r v i b r a t i o n s , r o t a t i o n a l r e l a x a t i o n , . . . ) must be taken i n t o account. E x t e r n a l l y d r i v e n l i m i t c y c l e s can e x h i b i t a great v a r i e t y of behaviour. Quite g e n e r a l l y , one gets a n o n l i n e a r s u p e r p o s i t i o n of an i n t e r n a l n o n l i n e a r o s c i l l a t i o n w i t h an e x t e r n a l o s c i l l a t o r y pert u r b a t i o n . D e t a i l s of the r e s u l t i n g behaviour ( i n c l u d i n g sub- and superharmonic o s c i l l a t i o n s , entrainment, quenching...) depend on both, the frequency and i n t e n s i t y of the a p p l i e d f i e l d s and on the i n t e r n a l n o n l i n e a r k i n e t i c s of the considered system. To extend the s p e c i a l concept of coherent e l e c t r i c o s c i l l a t i o n s to f u r t h e r phenomena, we assume two r a t h e r general postulates: 1. observable b i o l o g i c a l o s c i l l a t i o n s must be s t a b l e l i m i t cycl e s or coupled sets of n o n l i n e a r modes of t h i s type, 2. s t a b l e l i m i t c y c l e s are a proper d e s c r i p t i o n f o r coherent osc i l l a t i o n s i n a c t i v e systems. To i n c l u d e the d e s c r i p t i o n of e x t e r n a l l y d r i v e n o s c i l l a t i n g systems, a t h i r d p o s t u l a t e i s suggested: 3. e x t e r n a l l y d r i v e n l i m i t c y c l e s are a s u i t a b l e b a s i s f o r a desc r i p t i o n of the e f f e c t s of e x t e r n a l f i e l d s on b i o l o g i c a l and r e l a t e d systems. The l i m i t c y c l e concept r e v e a l s a p o s s i b l e e x p l a n a t i o n of the s p e c i f i c s e n s i t i v i t y of b i o l o g i c a l systems to a weak e x t e r n a l s t i mulus through the f o l l o w i n g mechanism: the i n t e r n a l LC represents a storage of (metabolic) energy, which can be used to b u i l d up a s i g n a l , whereas the e x t e r n a l l y s u p p l i e d f i e l d energy (with approp r i a t e frequency and i n t e n s i t y ) only causes the LC to c o l l a p s e . This t r i g g e r e f f e c t of the e x t e r n a l s i g n a l makes p l a u s i b l e the existence of nonthermal e f f e c t s i n i r r a d i a t e d b i o l o g i c a l systems, though the e x t e r n a l energy supply i s too weak to c r e a t e a response of the system. The LC concept has been a p p l i e d to d e s c r i b e c o l l e c t i v e chemic a l and physicochemical r e a c t i o n s w i t h an o s c i l l a t o r y k i n e t i c s , c i r c a d i a n and c a r d i a c rhythm, b r a i n f u n c t i o n and rhythm e t c . Det a i l s of t h i s concept may be found i n a recent a r t i c l e ( 8 ) , where


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a l a r g e number of both, t h e o r e t i c a l model o s c i l l a t o r s and e x p e r i m e n t a l l y found o s c i l l a t o r y phenomena have been presented. Models


To give an i l l u s t r a t i o n of our concept, we want to present some r a t h e r simple model systems. These models are examples of a t y p i c a l n o n l i n e a r behaviour and they might perhaps serve as an e x p l a n a t i o n of some s p e c i f i c e f f e c t s found i n experiments ( v i d . Refs. J_, 24 3 ) . I t should be emphasized that the models are r a t h e r s p e c u l a t i v e . However, they are based on both, the p h y s i c a l proper t i e s and the r e l e v a n t p h y s i c a l laws, which determine the dynamics of the system under c o n s i d e r a t i o n . Model 1 : G e n e r a l i z e d Van der P o l

Oscillator

The Van der P o l o s c i l l a t o r i s a well-known and best studied example of a l i m i t c y c l e o s c i l l a t i o n . I t has i t s o r i g i n i n non l i n e a r e l e c t r i c c i r c u i t s . We take a g e n e r a l i z e d v e r s i o n of i t , which i n i t s normalized form reads: X - μ(1 - X

2

+ αΧ

4

6

- βΧ ) Χ + Χ = F ( t )

(1)

(" = 4— , Χ = amplitude of o s c i l l a t i o n , e.g. a measure f o r time de pendent p o l a r i z a t i o n i n membranes, bonding s t a t e of ions e t c . ; α, β = 0 i s the u s u a l l y a p p l i e d Van der P o l l i m i t ) . The r.h.s. of the equation r e p r e s e n t s the e x t e r n a l d r i v e ; f o r s i m p l i c i t y we r e s t r i c t to a harmonic one, F ( t ) = F cos Xt. Without the e x t e r n a l f i e l d , two s t a b l e o s c i l l a t i o n s w i t h f r e quencies ω. and and amplitudes Xj and x4 e x i s t ; an u n s t a b l e LC i s s i t u a t e d between the two LCs. The s t a b l e o s c i l l a t i o n s are shown i n F i g u r e 1. By e x t e r n a l means and by parameter v a r i a t i o n s the sys tem can be d r i v e n from one s t a b l e o s c i l l a t i o n to the other one, e x h i b i t i n g a t h r e s h o l d and e x c i t a b i l i t y . Furthermore, the e x t e r n a l f i e l d can lead to a complete c o l l a p s e of the small amplitude os c i l l a t i o n . The c l o s e r the e x t e r n a l frequency λ i s to the i n t e r n a l one, u)j, the smaller i s the c r i t i c a l f i e l d s t r e n g t h F , which i s necessary f o r the breakdown of the small o s c i l l a t i o n and the sub sequent t r a n s i t i o n to the other one. For F >F , the system can only e x i s t i n the l a r g e amplitude s t a t e . TRis°state i s s t a b l e w i t h respect to F, but there e x i s t s a r e g i o n , where p a r t i a l quenching of the o s c i l l a t i o n can occur. In the resonance r e g i o n the system o s c i l l a t e s w i t h the e x t e r n a l frequency λ and w i t h an increased amplitude (entrainment r e g i o n ) . Far away from resonance, the i n t e r n a l f r e e o s c i l l a t i o n s are present. This behaviour i s completely absent i n systems, where no s e l f - s u s t a i n e d o s c i l l a t i o n s can e x i s t . A t y p i c a l example of such a system i s a n o n l i n e a r c o n s e r v a t i v e system. The resonance diagram has been drawn i n F i g u r e 2 f o r both, the small and the l a r g e a m p l i tude o s c i l l a t i o n . qc



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::J

Figure 1. X(t) — t (amplitude as a function of time) and X(t) — X(tJ (phaseplane) computer diagram of the generalized Van der Pol oscillator with F == Ο = λ for 4 different values of initial conditions. Both the small and large amplitude oscil lations are shown. 0

Figure 2. Resonance diagram (steadystate amplitude ν as a function of the de tuning parameter (1 — λ )/(λμ)) for 3 different values F (F < F < F ). Note the instability region of the small limit cycle. 2

G

O1


O2

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The r e s u l t i n g frequency and the steady s t a t e amplitude f o r both o s c i l l a t i o n s are given i n Figure 3 as a f u n c t i o n of the ex t e r n a l f i e l d amplitude f o r a c e r t a i n frequency λ. The t r a n s i t i o n from one LC to the other one occurs below that p o i n t , where the l a t t e r LC gets entrained (F t r a n s i t i o n to a new LC w i t h f r e quency λ (entrainment) -> c o l l a p s e of the LC ( i n s t a b i l i t y ) and onset of t r a v e l l i n g waves(nonlinear response s i g n a l ) . Q


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Figure 6. Resonance diagram (steadystate amplitude ν as a function of the external frequency λ) for different values of Fa (F < F < F < F ). The dashed part of Curve 4 is unstable; all resonance curves are asymmetric. o1

ft2

o3

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A s i m i l a r scheme can be given f o r a v a r y i n g e x t e r n a l frequen cy. T y p i c a l examples of perturbed o s c i l l a t i o n s are shown i n F i g u r e 7. Figure7e demonstrates that a f t e r some o s c i l l a t i o n s w i t h i n c r e a s i n g amplitude ( t r a n s i e n t s ) the system gets u n s t a b l e . The d i f f e r e n t regions of behaviour have been drawn i n Figure 8. The c r i t i c a l s t r e n g t h F , which i s necessary f o r the LC to c o l l a p s e and to c r e a t e a response s i g n a l , i s given by the e x t e r n a l as w e l l as by the i n t e r n a l parameters. Predominant are the d i e l e c t r i c ones(c^, d^), s i n c e these parameters determine the frequency and amplitude of the i n t e r n a l o s c i l l a t i o n . We have been able to show that pulsed and modulated e x t e r n a l f i e l d s can only s h i f t the i n s t a b i l i t y point towards smaller or higher values of the c r i t i c a l f i e l d s t r e n g t h per c y c l e . Q u a l i t a t i v e changes i n the o v e r a l l be haviour of the d r i v e n LC system are i m p o s s i b l e . The relevance of the model i s a matter of c o n t r o v e r s a l d i s c u s s i o n s . Though the chemical r e a c t i o n i s a r a t h e r s p e c u l a t i v e one, one should r e a l i z e that the s p e c i a l type of n o n l i n e a r r e a c t i o n can be replaced by an other one. The important step i s the combined e x i s t e n c e of both, a s p e c i a l chemical k i n e t i c s and a r e l a t e d e l e c t r i c behaviour. Both terms can be m o d i f i e d , but they must be based on p h y s i c a l laws and the e x t r a o r d i n a r y d i e l e c t r i c p r o p e r t i e s of the material. Model 3: Extended H i g h - P o l a r i z a t i o n Model The p o s s i b l e e x i s t e n c e of h i g h l y p o l a r i z e d metastable s t a t e s i n b i o l o g i c a l model membranes has l e d us to a p r e l i m i n a r y model for s p e c i f i c i n t e r a c t i o n s i n membranes (14). We have assumed that biomolecules are capable of n o n l i n e a r p o l a r i z a t i o n o s c i l l a t i o n s , when they are s u f f i c i e n t l y e x c i t e d by metabolic energy or by ex t e r n a l means, e.g. e l e c t r i c f i e l d s . R e s t r i c t i n g to the case of two i n t e r a c t i n g molecules f o r s i m p l i c i t y , the dynamics of the system i s given by a set of n o n l i n e a r d i f f e r e n t i a l equations f o r the po l a r i z a t i o n P. of the molecule i : ι

y. P. has been introduced as a phenomenological damping term to prevent the system from becoming u n s t a b l e , when only a very small amount of energy i s s u p p l i e d and to account f o r always present l o s s processes. y.(R) P. r e s u l t s from the d i p o l e - d i p o l e i n t e r a c t i o n , i . e. Y-.(R) P,f>,;hign er c o u p l i n g terms have been neglected. We have r e s t r i c t e d è o ^ t h e f i r s t three odd powers i n P. , these terms c o r respond to a symmetric n o n l i n e a r p o t e n t i a l . The two molecule model may be viewed as a s t a r t i n g p o i n t to d i s c u s s the dynamics of h i g h l y p o l a r i z a b l e biomolecules i n mem-


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Figure 7.

Oscillation and phase-plane diagrams of the externally driven limit cycle of the coherent oscillation model.

(a) Xj < ω , F < F ; (b) λ = ω , F < F ; (c) X = ω , F„ -» F ; (d) λ > ω , F < F ; (e) λ > ω,„ F — F : collapse of oscillation (ω = frequency of unperturbed limit cycle; Figures (a)-(d) belong to Region III, Figure (e) to Region IV of Figure 8). 0

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Figure 7. Continued


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Figure 7. Continued


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branes. Depending on both, the i n i t i a l c o n d i t i o n ( t o t a l energy) and the e x t e r n a l l y supplied energy (given by the frequency and i n t e n s i t y of the e x t e r n a l e l e c t r i c f i e l d ) , the o s c i l l a t o r s can be s t a b i l i z e d i n a h i g h l y p o l a r o s c i l l a t i n g s t a t e . For each molecule three types of s t a b l e s t a t e s (P4 = 0 , ±P. j + 0 ) are p o s s i b l e , i f there i s no c o u p l i n g between tSe molecules. Consequently three types of s t a b l e o s c i l l a t i o n s can e x i s t : a. o s c i l l a t i o n s around P. w i t h small amplitudes (the u s u a l l y found l i n e a r b e h a v i o u r ) b. o s c i l l a t i o n s around P. w i t h l a r g e amplitudes (giant d i p o l e o s c i l l a t i o n s w i t h zero'mean value) c. n o n l i n e a r o s c i l l a t i o n s around ±P4 j (with nonzero mean v a l u e ) . The three types of o s c i l l a t o r y behaviour are shown i n F i g u r e 9, w i t h P. .and P. , w r i t t e n as Ρ and P., r e s p e c t i v e l y . P. . must l.o ... ι-1 ,, 0 i f . i,l not n e c e s s a r i l y be a s t a b l e s t a t e , a metastable s t a t e i s conceiv able and perhaps more probable. D e t a i l s depend on the parameters α. and β.. Energy can be exchanged from one o s c i l l a t o r to the other one, a t y p i c a l s i t u a t i o n i s given i n Figure 10. This behaviour can ex p l a i n f u n c t i o n a l changes of i r r a d i a t e d systems: the h i g h l y p o l a r i z e d molecule A absorbes a d d i t i o n a l energy, which i s t r a n s f e r r e d p a r t i a l l y to molecule B. A f t e r some t r a n s i e n t p e r i o d s , Β i s h i g h l y e x c i t e d and i t can e x h i b i t a new f u n c t i o n . A d i r e c t e x c i t a t i o n of molecule Β may be forbidden, whereas A i s already s t a b i l i z e d i n the h i g h l y p o l a r s t a t e by i n t e r n a l means. Furthermore, a t y p i c a l n o n l i n e a r resonance behaviour of nonself-sustained o s c i l l a t o r s r e s u l t s , i t includes hysteresis e f f e c t s ( b i s t a b i l i t y ) and t r a n s i t i o n s to a new s t a t e (upper curve i n F i gure 11). This new s t a t e can e i t h e r be a small or a l a r g e ampli tude o s c i l l a t i o n ; i n both cases a s t a b l e s t a t e of non-zero p o l a r i z a t i o n (P. .) i s included. Provided that the system i s prepared i n a c e r t a i n s t a t e , small e x t e r n a l i n f l u e n c e s can d r i v e i t i n t o a s t a t e w i t h a completely d i f f e r e n t v i b r a t i o n . A p o s s i b l e r e a l i z a t i o n i s given i n Figure 11, the system can jump from one resonance curve to the other one. However, since we are d e a l i n g w i t h a l i n e a r l y damped, but otherwise conservative system, the response of the sys tem i s predominantly governed by the i n i t i a l c o n d i t i o n s . No compe t i t i o n between i n t e r n a l and e x t e r n a l o s c i l l a t i o n s i s p o s s i b l e , the r e s u l t i n g o s c i l l a t i n g s t a t e i s completely determined by the e x t e r nal constraints. 1


0

Model 4: L o c a l E x c i t a t i o n Model The p o s s i b l e existence of strong e x c i t a t i o n s of some modes of behaviour might a l l o w f o r energy storage mechanism i n c e r t a i n b i o l o g i c a l subunits. Some years ago we discussed the p o s s i b i l i t y of e x c i t i n g some or at l e a s t one a d d i t i o n a l degree of freedom by a f l u x of energy through the system. For a simple model system (a l i n e a r chain of molecules to which a s i n g u l a r degree of freedom i s coupled), we have found that the s i n g l e mode gets s t r o n g l y e x c i t e d ,



236


Figure 8. Regions of different behavior of the externally driven limit cycle of Model 2 (see Figures 5-7): (I) free oscilla tion (limit cycle with ω ) (II) quasi-peri odic oscillation; (III) entrainment (limit cycle with λ); (IV) unstable region -» on set of traveling waves. 0

;

Figure 9. Schematic of possible types of oscillations for Model 3 (potential VfPj as a function of polarization P); (I) har monic oscillations (linear regime); (II) nonlinear oscillations with a large ampli tude, < Ρ > = 0; (III) nonlinear oscil lations around the stable states ±P with < Ρ >Ty40. r

lf


237



KAISER

Figure 11. Resonance diagram (computer plot of the steady state amplitude of one oscillator as a function of the external frequency λ) for 4 different values of the damping term μ(μ = 0.5; 1.2; 1.3; 2.0) ((I) oscillations around P , (II) oscillations around Pj ) 0



238


Figure 12. Excitation of the singular mode N as a function of the external pump parameter (phononfluxΦ) of the local excitation model (Model 4). Both the steadystate excitation (hysteresis) and the possible oscillations on the hysteresis are shown. s


13.

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when the f l u x exceeds a c r i t i c a l v a l u e . I n a d d i t i o n , there i s a h y s t e r e s i s i n the dependence of the number of e x c i t a t i o n s Ν on the pump power or f l u x Φ (15). The steady s t a t e r e s u l t s are shown i n Figure 12. For Φ 4 * s i n g l e mode gets s t r o n g l y e x c i t e d , i . e . the energy i s p r e f e r e n t i a l l y channeled i n t o t h i s mode (Bose c o n d e n s a t i o n - l i k e e x c i t a t i o n ) . I f , by i n t e r n a l means, the system i s near Φ 4 only a small amount of energy i s necessary to d r i v e the system i n t o the h i g h l y e x c i t e d s t a t e . Furthermore, we have been able to show that o s c i l l a t i o n s on the h y s t e r e s i s are p o s s i b l e f o r Φ 4 ύ Φ < Φ 4· A d e t a i l e d i n s p e c t i o n of the t r a n s p o r t equations (generalizeS n o n l i n e a r P e i e r l s Boltzmann equations f o r phonons) shows that n o n l i n e a r k i n e t i c s , d i s s i p a t i o n and energy supply v i a t r a n s p o r t are i n d i s p e n s a b l e f o r such a behaviour t o occur. A d e s c r i p t i o n of the r a t h e r complicated t r a n s p o r t formalism i s beyond the scope of t h i s a r t i c l e and w i l l be omitted. F o r our pur poses the important r e s u l t i s the e x c i t a t i o n and s t a b i l i z a t i o n of a s i n g l e mode. Contrary t o the r e s u l t s of the other models w i t h a Bose c o n d e n s a t i o n - l i k e e x c i t a t i o n , the c o h e r e n t l y e x c i t e d mode i s the mode w i t h the highest and not w i t h the lowest frequency.


t

n

e

Conclusions I n t e r a c t i o n w i t h E x t e r n a l F i e l d s . The models considered ex h i b i t cooperative behaviour through n o n l i n e a r i n t e r n a l o s c i l l a t i o n s (models 1, 2, 4) o r through n o n l i n e a r resonances (model 3). This makes p l a u s i b l e the e x i s t e n c e of e f f e c t s , when the system i s d r i v e n by weak e x t e r n a l f i e l d s of appropriate frequency. On the b a s i s of F r o h l i c h ' s suggestions and the p r o p e r t i e s o f b i o l o g i c a l systems ( v i d . models 2, 3) one may take i n t o account the f o l l o w i n g p o s s i b l e i n t e r a c t i o n s of an electromagnetic f i e l d w i t h b i o l o g i c a l systems (16, 17): 4 1. the membrane i n t r i n s i c e l e c t r i c f i e l d ( 10 V/cm), which i s r e s p o n s i b l e f o r the metastable h i g h l y p o l a r i z e d s t a t e , 2. the coherent e l e c t r i c o s c i l l a t i o n s ( 1 0 - 1 0 Hz), l e a d i n g to c o l l e c t i v e biochemical r e a c t i o n s through long range i n t e r actions, 3. the r e s u l t i n g ELF branch (10 - 100 Hz r e g i o n ) , which may des c r i b e EEG a c t i v i t i e s . I t should be s t r e s s e d that the d i f f e r e n t regions are not indepen dent from each o t h e r , i n t e r a c t i o n s are not r e s t r i c t e d t o the above frequency regions and t o membranes. l 0

1 2

General Our models may be viewed as a r a t h e r s p e c u l a t i v e and very sim p l i f i e d v e r s i o n of any a c t u a l s i t u a t i o n , nevertheless they may serve as a s t a r t i n g p o i n t t o get more i n s i g h t i n t o the f u n c t i o n of b i o l o g i c a l systems and t h e i r i n t e r a c t i o n s w i t h e x t e r n a l f i e l d s . As a r e s u l t of these models, the l i m i t c y c l e concept seems t o be a relevant basis for theoretical investigations.


BIOLOGICAL


240

E F F E C T S O F NONIONIZING

RADIATION

We would l i k e to comment p a r t i c u l a r l y on four remarkable a s pects of t h i s concept: 1. n o n l i n e a r o s c i l l a t i o n s are the r e s u l t of c o o p e r a t i v i t y i n com p l e x systems; they are i n t i m a t e l y l i n k e d w i t h n o n e q u i l i b r i u m c o n s t r a i n t s and i r r e v e r s i b i l i t y , 2. l i m i t c y c l e s e x h i b i t a pronounced behaviour w i t h respect t o the frequency and i n t e n s i t y of an e x t e r n a l s t i m u l u s ; they are capable of e x p l a i n i n g extreme s e n s i t i v i t i e s by a mere t r i g g e r a c t i o n o f the e x t e r n a l s i g n a l , 3. n o n l i n e a r systems ( i n an o s c i l l a t i n g or b i s t a b l e s t a t e ) can be d r i v e n t o a new s t a t e o f order o r f u n c t i o n ; i n c e r t a i n c a ses, extremely small changes of the e x t e r n a l c o n s t r a i n t s are s u f f i c i e n t f o r the t r a n s i t i o n s , i . e . very weak p e r t u r b a t i o n s can create d r a s t i c changes, i f one a p p l i e s the appropriate frequencies, 4. both, the general l i m i t c y c l e concept and the more s p e c i f i c concept of coherent o s c i l l a t i o n s r e q u i r e the supply o f energy (metabolic energy, b i o l o g i c a l pump) to create and s t a b i l i z e the a c t i v e s t a t e . I t should be emphasized that the relevance o f the concept can only be checked by very accurate experimental i n v e s t i g a t i o n s . To e s t a b l i s h a c l e a r p i c t u r e , i t i s a b s o l u t e l y necessary t o measure both, the frequency and the i n t e n s i t y dependence of the e f f e c t s . I n ad d i t i o n , the measurements should i n c l u d e the d u r a t i o n of i r r a d i a t i o n and both, an increase and decrease i n the a p p l i e d f i e l d i n t e n s i t y to detect p o s s i b l e h y s t e r e s i s e f f e c t s . From a t h e o r e t i c a l p o i n t of view, one has t o solve the pro blem o f the onset o f a propagating pulse a f t e r the LC c o l l a p s e . This n e c e s s i t a t e s the i n t r o d u c t i o n of space dependent v a r i a b l e s and of appropriate boundary c o n d i t i o n s . This problem has already been discussed f o r model 2 (13). Literature Cited 1. 2. 3.

4. 5. 6. 7. 8.

Kaiser, F . , Z. Naturforsch., 1979, 34a, 134 Adey, W.R., Bawin, S.M., Neurosci. Res. Progr. Bull., 1977, 67 Fröhlich, Η., "The Biological Effects of Microwaves and Re lated Questions"; Advances in Electronics and Electron Phy sics. Vol. 53, 85-152, 1980 Fröhlich, H . , in "Theoretical Physics and Biology", Versailles 1967, Marois, M., Ed., North-Holland: Amsterdam, 1969 Fröhlich, H . , Int. J . Quant. Chem., 1968, 2, 641 Haken, Η., "Synergetics, an Introduction", Springer: Berlin, 1978 Nicolis, G., Prigogine, I., "Self-Organization in Nonequilibrium-Systems", J . Wiley: New York, 1977 Kaiser, F . , Nonlinear Oscillations (Limit Cycles) in Physical and Biological Systems p. 343-389, in "Nonlinear Electro-


13.

9. 10. 11. 12. 13. 14.


15. 16. 17.

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magnetics", Uslenghi, P.L.E., Ed., Academic Press: New York, 1980 Kaiser, F., Workshop on the "Mechanism of Microwave Biologi cal Effects", University of Maryland, May, 1979 Adey, W.R., Bio Systems, 1977, 8, 163 Kaiser, F., Biol. Cybernetics, 1977, 27, 55 Kaiser, F., Z. Naturforsch., 1978, 33a, 294 Kaiser, F., Z. Naturforsch., 1978, 33a, 414 Kaiser. F., Szabo, Z., (to be published) Szabo, Ζ., diploma thesis, University of Stuttgart, 1980 Kaiser, F., Z. Naturforsch., 1977, 32a, 697 Illinger, Κ., Workshop on the "Physical Basis of Electromagne tic Interactions with Biological Systems", University of Mary land, June, 1977 Kaiser, F., Symposium on the "Biological Effects of Electro magnetic Waves", Helsinki, Finland, 1978 Radio Science (in print 1980)

RECEIVED

December 9, 1980.


Coherent Modes in Biological Systems - American Chemical Society

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