- Email: [email protected]
Quantum mechanical concepts of coherent states in biological systems
ELSEVIER
Bioelectrochemistry
and Bioenergetics
41( 1996) 19-26
Quantum mechanical concepts of coherent states in biological systems T.M. Wu Department
of Physics,
State University
of New York at Binghamton,
Binghamton,
NY 13902,
USA
Received9 November 1995;revised 29 January 19%
Abstract In order to demonstrate the possibility of coherent behaviour in biological systems, FrGhlich wrote down the rate equations and showed that if energy is supplied above a critical rate to the branch or branches of electromagnetic modes, Bose-Einstein condensation into the lowest energy state occurs. The general forms of the rate equations were dictated by requiring a Bose-Einstein distribution for thermal equilibrium when there is no energy supplied. Using microscopic theory and perturbation calculations, we were able to obtain the rate equations of Friihlich from the Hamiltonian of the biological system under study. Two simple cases were presented to demonstrate the enhancement of excitations in biological systems by external stimulation. Friihlich’s model has been particularly useful in understanding recent experiments revealing the effects of low level microwaves on various biological systems. One important general characteristic reported in these experiments is the existence of a time threshold for the initiation of the differing biological effects. We obtained an approximation of this time threshold. Keywords:
Coherent
states; Bose-Einstein
condensation;
Perturbation
calculations;
1. Introduction Although the structure, function and detailed biochemi-
cal content of biological micro-organisms have been studied extensively, very little work has been done on the effects of millimeter electromagnetic radiation on biological systems. Such work has recently taken on a new importance with the development of new methods of measurement and their theoretical interpretation in terms of possible collective excitations of biological systems (e.g. membranes, nucleic acids, proteins and cells). Large non-biological, physical systems which are built from more fundamental constituents (crystals, atomic nuclei, spin arrays, etc.> possess vibrational modes; these modes are characterized by the coherent motion of many of the constituent parts of the large system. Thus, it is more than likely that biological systems such as cell membranes, large biological macromolecules, or intact cells, also possess vibrational modes which will couple strongly or weakly to electromagnetic radiation. To illustrate this, let us consider a simple cell membrane consisting of a bilayer of phospholipid molecules with protein interspersed. These macromolecular assemblies contain
Frijhlich
model;
irradiation
dipolar molecules that are arranged in such a way as to give rise to an array of dipoles. The charge groups which comprise the dipoles are displaced relative to each other with some frequency f. Each dipole is imbedded in a complex structure, and it is possible that the interaction between the dipole and this underlying superstructure results in the manifestation of a vibrational excitation. The detailed nature of this excitation is very complex, involving deformations of the underlying structure. In addition, each dipole also interacts with all other dipoles via electromagnetic forces. Other types of interactions mediated by the structure are also possible. The net result of these mutual interactions is to spread the frequency f into a narrow band and to provide for the sharing of energy between the individual dipoles. If a particular dipole is perturbed, the perturbation propagates to other dipoles until the whole array of dipoles is excited to some collective quantum state which we will call a dipole wave or electromagnetic oscillation. The excitation energy of this electromagnetiti oscillation is expected to lie in a narrow band close to hf, where h is Planck’s constant. Based on the relation
f = v/h, where u is the velocity
of the sound wave in organic material
0302-4598/%/$15.00 Copyright 0 1996Publishedby Elsevier ScienceS.A. All rights reserved. PII SO302-4598(96)01921-6
Microwave
and A is the
20
T.M. Wu /Bioelecfrochemistry
wavelength, Frohlich [l-3] has estimated the frequency of the oscillation. If we take u = lo5 ems-‘, which is the approximate velocity of sound in water and many organic liquids, and A = 10m6 cm, which is a typical dimension of a large biological molecule, then the frequency is equal to 10” Hz. In general, the expected frequencies are lOlo10” Hz for membranes, 10’2-10’4 Hz for proteins or more general for certain bond-stretching groups, and lo9 Hz for DNA or RNA molecules. There is plenty of experimental evidence verifying the existence of these waves in biological systems. From theoretical considerations, Frijhlich [4-S] suggested that if the energy is supplied to the vibrational modes at a rate greater than a certain critical value, then the phenomenon of Bose Condensation to the lowest excitation of a single mode can occur. Such a condensation to the lowest excitation mode might serve not only as a method for energy storage but also as a channel for specific biological processes such as cell division or protein synthesis. The Frijhlich model has been of special interest because recent experimental [g-16] and theoretical [ 17-261 reports conform in many respects to its predictions. The experiments are concerned primarily with the irradiation of biosystems with low level microwaves. It has been demonstrated that microwaves between 40 and 14OGHz can increase or decrease the division rate of bacterial and yeast cells, change the oncogenic properties of certain tumor cells, and alter the in vivo rates of synthesis of macromolecules. Webb [14-161 has reported the induction of lambda prophages in cells of Escherichiu cali after lysogenie cells were exposed to microwave fields. In theoretical work, Kaiser [27-301 has applied a modified PearlsBoltzmann transport equation formalism to Frijhlich’s theory, and Mills 131-341 has analysed the asymptotic solutions of FrBhlich’s equations.
2. FrShlich model
In this section, we consider a simple model suggested by Friihlich. It is presumed that the biological system consists of three parts: (1) the main oscillating units of giant dipoles (e.g. sections of the membrane of cells or parts of H-bands within biological macromolecules), (2) the remainder of the biological system which forms the heat bath, and (3) an external energy source which couples to the vibrating units. Long-range Coulomb interaction between the dipoles will produce a narrow band of energy, >, which corresponds to the normal wj (oJoo+Iom,, modes of the electromagnetic vibrations. The heat bath is, of course, a very complex system. Interaction with the heat bath will lead to emission and absorption of quanta by these oscillating electromagnetic modes. Here we consider processes involving one and two quanta only. The interac-
and Bioenergetics
41 (1996)
19-26
tion involves several factors: dipoles of water and other molecules, mobile ions, certain electronic degrees of freedom, and, to some extent, elastic displacements. The assumption that the heat bath is in a thermal equilibrium enforces the principle of detailed balance at its temperature. Therefore, the equation for the thermal average rate of change of the ith mode quanta is (rii)=si-~i[(ni>eao,-(l -
+(Q]
CAjj[(ni>(l
+
(nj>)ePW’
j
-(nj>(l where si and Aij quantum and Aij
+ (n,))&]
(1)
is the rate of energy supplied to the ith mode, +i are coupling constants to the heat bath for oneand two-quanta processes respectively. Both & will in general depend on temperature. We use
p = fi/kT.
For the stationary solutions of this rate equation, one requires that ( Ai) = 0. As a result, the ith quanta is 1 eS(Wt-wf) - 1
‘i (ni> = 1+ cpi + C AiinjepoJ
j
[
(2)
I
where c Aij( epoJ - 1) eP”=l+
++‘xAij(l
+)
”
(3)
and pi is the chemical potential of the ith mode. The inequality in Eq. (31, together with the requirement that at thermal equilibrium ( iri> = 0, dictates that wi 2 pi r 0. If there is no energy supplied (si = O>, then pi = 0 and Eq. (2) becomes the thermal equilibrium distribution as required. From Eq. (2) and (31, one notices that as si increases, pi will increase. When si exceeds a critical value s,, pi approaches wo. the lowest energy in the excitation band. Therefore a large number of quanta are condensed into the lowest energy state. This is exactly the Bose-Einstein condensation in a Bose gas system when the temperature is lower than a certain critical value. In our case, the corresponding phase transition is not induced by lowering the temperature; instead it occurs by keeping the temperature constant and increasing the energy supply beyond the critical value so. From Eq. (3), it should be noted that when Aij = 0, pi = 0. Thus, it is the two-quanta processes that are responsible for Bose-Einstein condensation; the larger the A,,, the greater the effect. Subsequently, other investigators approached the theoretical problem from differing standpoints. Lifshits [34] concluded that the Frijhlich model was erroneous because Friihlich did not include in the two-quanta term the contributions from the simultaneous creation or destruction of
TM.
Wu / Bioelectrochmtistry
and Bioenergetics
the quanta (i) and (j). Thus, according to Lifshits, the average rate of change of the ith mode quanta should be (hi> = si - c$i[(ni)ei~~~ - Cnij[(n,>(l j +zj>(l -(l
- (1 + (Q)] + (nj>)eP”’
+ (n&!@Q + (n,>)(l
+ (?zj>) + (ni)(frj)e+‘+Q] (4)
Eq. (4) can easily be reduced to the form (iti> =si fB,[(r+‘~“~-
(1 + +ri>)]
(5)
with Bi=
xA,,[l j
+ (nj)
+ (nj)&]
+ &
(6)
theory
Using rate equation, Friihlich was able to demonstrate that the condensation occurs when the external energy supply attains a certain critical value. The general forms of the rate equation were dictated by requiring a Bose-Einstein distribution for thermal equilibrium when there is no power supplied. Beyond this, the forms were somewhat arbitrary. To avoid this arbitrariness, we will study this
19-26
21
problem from a microscopic point of view using the finite temperature Green’s Function method including all orders of interaction strength. To each oscillating mode, wi, we assign a creation operator, a,?, and a destruction operator, ai. The normal modes of oscillation will interact with the remainder of the biological system (the heat bath) which is represented by a set of independent excitation Ok, associated with creation and destruction operators, b: and b, respectively. Furthermore, the external energy supply associated with creation operator, p,? and destruction operator pj, and excitation energy, Of, feeds into the electromagnetic oscillatory units and acts as impetus for the initiation of the biological effects. The Hamiltonian of the biosystem can then be written as A?= &+a+~,+ i
Therefore, the contribution from two-quanta terms including the terms suggested by Lifshits has the same form as the one-quantum term. Thus, Lifshits concluded that no Bose-Einstein condensation will occur in the biological system. It was later shown [18] that the addition of the Lifshits terms actually does not cancel the Bose-Einstein condensation phenomenon predicted by Frohlich.
3. Microscopic
41 (1996)
C,R,b;b,+
c6,p;pf
f
k
+~C(ubjb;ai+a*b,~b,a;) d‘ +;
&( @fa,b: 11“
+ C(YPf4 if
+ /3 *aja+bk)
+ Y*P;ai)
(7)
where CX, p and y are the coupling constants for the one quantum process, the two quanta process, and the energy source with the vibration modes respectively. Strictly speaking, we should consider higher order and possibly inharmonic terms; however, it can be shown that accounting for those terms have only minor effects and, within certain limits, they may be ignored altogether. The electromagnetic oscillations are Bosons. The excitations within the heat bath and the energy source can be either Bosons,
iGc1i.w)
Fig. 1. Diagrams
represent
the rate of change of the number
of phonon
iG:li,w)
with energy
wi.
22
T.M. Wu / Bioekctrochemistry
and Bioenergetics
41 (1996)
19-26
or Fermions, e.g. phonons or electrons. The operators a,?, ai, b:, b,, pf’ and pf satisfy the commutation relations
dwdw’do” PI3
a,a,+-a,+a, = Sij a,:,,?-a;at b,b:
X{[G,>(i,o)G,<(
= aiaj - ajai = 0
+G:
+ b:b, = S,,
(i,w)G,>
(j,w’)GL
(k,d’)]
X8(,--d-d’)+[G~(i,w)G,<(j,w’)
b,b, & b,b, = b:b; PfP,’ * P,‘P,=
j,d)Gb<(k,w”)
f b;b;
= 0
xG,>(k,o”)
Sf,
(8)
+G,<(
i,w)G,>
(j,w’)G:
(k,w”)]
xqw-o’+w”)}{
(+ sign for fermions and - sign for Bosons). The rate of change of the number of quanta in the ith mode is given by +G:(i,w)G,>(I,o’)]S(w-
with n, = a,?~~; the number of quanta in the ith mode and fi is Planck’s constant divided by 21r. The expectation value of the rate of change of the number of phonons with energy wi in the biological system is then given by
= fIm(
Gb(i,t)= G,(i,t)=
~~*(~(f)la:(f)bj(f)b:(f)lcp(f)) j,k
= -i<‘P(f)IT{ai(f)a+(0)}l~(f))
-i(‘P(t)IT{bi(t)b+(O)}I40(t)) -i<~(f)IT{P~(f)P~(O)}(f))
(13)
+ CP((P(t)l~+(t)~j(t)b:(t)l~(f)) j.k
where T is the time ordering operator. The retarded and advanced Fourier transforms of the above defined Green’s Functions can be written as
+ CP * (~P(f)la,+(f)aj(f)bk(f)l(P(f)) j,k
G,‘(i,w) = l’n(wi) o-q+is
+ Cr(‘P(t)la+(t)Pl(t)l~o(t))}
(10)
where lcp(t)) is the state vector of the whole system, and the angular bracket is the ensemble average. The ensemble average (6) for any operator 6 is obtained by 8 = Tr( ~6) where p is the density operator defined by
The retarded and advanced functions for the heat bath and the 2 energy pump have a similar form Gz (i,o)
=
G;(i,u)
=
G; (i,o)
=
G;(i,o)
=
(11) Using finite temperature Green’s Function techniques [35], one can easily obtain the rate of change of the number of phonons with energy wi as shown in Fig. 1. It is given by the equation (&)=
(12)
where G,‘(i,w), (G,<(i,w)), Gj’(i,~), (G, < (i,w)), and GY’ (i, w), (G,< (i, 0)) are the retarded and advanced Fourier transforms of the finite temperature Green’s Functions defined by G,(i,f)
=(‘P(t)lrii(f)l~(f))
w’)
--
2cY-d;’ f.&2
dodw’dw” Im
c,/, ( j.k
PI3
X[G,>(i,w)G~(j,o’)G,>(k,o”) +G,< (i,u)Gb>
(j,d)G:
(k,d’)]
1 +Nb( ai) o-fii+s -Nb(
‘i)
w-Q-it3 1 +N,(fi;) co- it& +iS -NY(@) o-fii;-is
(15)
where n(Zi), Nb(ai), and N,( fij) are the quantum distributions of the biological system, heat bath, and energy pump respectively. Wi = wi f Z&i) is the modified excitation of the biological system phonon with energy oi when there is no interaction between the biological system and heat reservoir, and ZO,(i) is the proper self-energy of the excitation. Owing to the nature of the weak dipolar interac-
TM.
Wu /Bioelectrochemistry
tion, one expects the self-energy, .Zfa(i), to be small. In general s=,(i) depends on oi, but the energy band is very narrow; it can be approximated to be a constant. Similarly, fii = ai + S,(i) and fii = ai + z,,(i) where s’,(i) and &(i) are the proper self-energies for the heat bath and energy supply excitations respectively. The vertex corrections G, p, 7 consist of all possible proper corrections to all orders of interaction. The actual detail forms of the vertex corrections 5, p can be obtained from the coupling equations as shown diagrammatically in Fig. 2. The coupling strengths cr, p and y are very complicated quantities and in general depend on oi. Since the energy band of the biological system under this study is very narrow, the coupling strengths and their vertex corrections will be considered to be constants. Substituting Eq. (14) and Eq. (15) into Eq. (12) and integrating over o, w’, and w”, one gets
(1
+n(Wi))Nb(‘j)(l
n(oi)(l
Summing over j and taking the imaginary part, Eq. (16) becomes Crii)=
(l
+Nb(zi+Bj))iVb(fij)
-$Z(n(zi)(l i
-(l
+n(Wi))Nh(oi+~j)(l
-$x(n(zi)(l i
+n(Sjj))(l
‘(‘i)(’
+Nb(wi-7jj))
- (1 + n( q>)n( Oj)Nb( wi - Oj)
+n(zJ)(1
+Nb(3ik))
-(l
+n(izj))&(Tjj-oi)
+“(oJ)n(oj)(l
- g{n(q)(
+n(zi)jn(zj>Nb(ak)
Wj+32,--Wi+i6 +
+Nb(~j)))
‘k)
Gj + jjz, - Oi - iS +
+.(zi))n(‘j)(l+Nb(ak))
(16)
I +n(‘j))(’
(1
19-26
Oj-fik--Z++iS
+Nb(ak))
jjj--a,--Oi+iS
41 (1996)
+
Ilj-Bk-Zi-iS
j.k +
+ Nb( 'j))Nb(
4'i)('
(iz,) = - $IrnC
and Bioenergetics
1 -t-N&))
+Nb(zj-zi))} - (1 +n(wi))N(wi)} (17)
Because the heat bath is large compared with the biological system under consideration, it is a good approximation to assume the distribution of the excitation in the heat
+n(Tjj))Nb(ak)
iZj-fik-Gi-it3
Fig. 2. The vertex
corrections
for the coupling
constants
a and p.
24
T.M. Wu /Bioelectrochemistry
bath is in equilibrium. Therefore, substituting N(Zi) = (ep7j! -t 11-l in Eq. (17) yields
Nb(Wi) =
and Bioenergetics
41 (1996)
19-26
where si, +ij, and Aij are the same as (201, Eq. (20, and
inEq. (191,Eq.
(iti)=S,-C$i(T,wi)[n(Wi)e~G~-(l+.(zX))] - CAij(T,Wi,Sjr)[n(wi)(l -n;Gj)(l
+n(Gj))ePe~
+ n( Oi))ePol]
(‘8)
s; = ~(,,,;)
- n( Z;))
(19)
c&~(T,C+) = T(l
+N(Wi))epa,
(20)
where
and 1 ‘N(OjGii) Aij(T,7iJi,iiij)
= $empzl -
N( 0; - Zj),
i
Oj>Oi oj < w; (21)
One notices that Si given in Eq. (19) is the net power absorbed by the ith mode. Eq. (18) is exactly the same rate Eq. (1) suggested by Frijhlich. To confirm the weakness of Lifshits’ conclusion, the problem must be treated microscopically. The Hamiltonian in Eq. (7) should be modified to take care of the contributions from creating two quanta or destroying two quanta simultaneously. Thus, SF’= cwia,Tai
+ xOkb:b,
i
+ ~O,p~p,
k
+i
f
&( pafa,b: ZJk
+ig(6ajaib: Ilk
+j? *aja+bk)
+6*afa,?b,)
+ C(ypf4 if
+ y*pf+a,)
(22)
where S is the coupling constant for the simultaneous creation of two quanta. Using this modified Hamiltonian and repeating the same calculation for (A), we get (hi> = Si-
&[(ni)epW~--
If Aij = rij as Lifshits presumes, then Eq. (23) can be reduced to Lifshits’ form. However, as seen in Eq. (21) and Eq. (241, allowing Aij = rij is incorrect. Nevertheless, presupposing that the coupling constants are of the same order of magnitude for each type of two-quanta interaction, the coefficient, Ajj is much larger than the coefficient, rij. This can be demonstrated readily for biological systems where oi is of the order of 10” s-‘. At room temperature hw,/kT = lo-*, thus rij and Aij can be approximately written as cj = (2r/612kT/h2)( Aij=
(24P12kT/h2)(lo,-
For narrow the Aij quency
c.j = (&12/h)( Aij = (24
p [*/h.)(
tni>)(l
kT/hZ)
(27)
kT/?iA w)
(28)
Since the range of normal frequencies of the oscillations is very small, Aw -zz W. Therefore, the terms in Eq. (27) are much smaller than those in Eq. (28). The simultaneous creation and destruction of quanta proposed by Lifshits can be regarded as negligible. Thus, the predominant “twoquanta” process is precisely the one postulated by Frijhlich, and the addition of the Lifshits term does not cancel out the Bose condensation phenomenon predicted by Frohlich. As Frijhlich has pointed out to us, oscillations as introduced in Refs. [ 17,181 and [341 form only a part of the degrees of freedom of the heat bath. Very important for the relevant energy transfer are non-oscillating systems, e.g. free ions, mobile ionic groups (dipoles), and certain low-lying electronic levels. For all of these it can be shown that Lifshits’ contention is incorrect. In fact some of them may permit excitations in the energy range (0, ( w,,, - w,)> but not in the range (20,, 2 o,,, ), thus making two-phonon absorption or emission impossible.
for biological effects
+ (nj))eP(Wg-WJ
+ (ni>)]
+
(26)
the rij term, since wi = oj owing to the very range of normal frequencies, let wi = wj = 0. For term, let 1wj - wil = A w, the width of the freband. Thus Eq. (25) and (26) become
4. Time threshold
-(l
oil)-’
(25)
(1 +)]
- CAij[(n,>(l i -(nj>(l
w; + co,)-’
+
- C~ij[(ni)(nj)eP(W,+W,)
<“j>)]
(23)
Frijlich’s model has been standing recent experiments level microwaves on various portant general characteristic experiments is the existence
particularly useful in underrevealing the effects of low biological systems. One imthat is reported in these of a time threshold for the
T.M. Wu /Bioelectrochemistty
initiation of the differing biological effects. Sevastyanova and Vilenskaya [l 11 performed experiments involving the irradiation of mice bone marrow cells by microwaves and X-rays simultaneously; they observed a time threshold of 30 min of microwave exposure before a protective effect could be induced by radiation. Smolyanskaya and Vilenskaya [lo] observed the effects of low level millimeter waves on the induction coefficient of colicin synthesis. They also found that a minimum irradiation time of 30 min was needed to produce measurable biological effects. In addition, one must keep in mind that when considering these time courses, the threshold will depend on the particular biological activity being monitored as well as temperature. An example of the latter is clearly demonstrated in Smolyanskaya and Vilenskaya’s experiment. Irradiation for 30 min at 20°C produced no change in the rate of colicin synthesis, whereas at 37”C, colicin synthesis increased after 30 min of irradiation time. Therefore, it is expected that the analysis of the time threshold will not be a trivial matter. Both the detailed microscopic structure of the biological system and its thermodynamic behavior must be considered to provide a full understanding of the specific time thresholds encountered for different biosysterns. Using the Frahlich model, we intend to obtain an approximation for a segment of this time threshold. [35] In applying the Frijhlich hypothesis, one sees that the biological effects cannot begin until condensed phase occurs. It has not been determined how long this condensation has to be maintained in order to produce biological effects. Generally, the time needed to initiate biological effects will consist of two parts, 7, and T*. r1 represents the time from the beginning of irradiation to the onset of condensation; r2 represents the time elapsed from the onset of the
and Bioenergetics
41 (1996)
19-26
25
condensed phase to the manifestation of the actual biological effects. Thus, the total time, T, elapsed before biological effects occur from the initiation of irradiation, is given by T= 7, + TV. In the following analysis, we will evaluate T] using the Hamiltonian given in Eq. (7). T, represents the minimum time required for the biosystem to exhibit effects after irradiation and is the lifetime of the collective excitations in the biological system. An explicit form of T* requires an indepth biochemical analysis of the particular system under scrutiny and is beyond the scope of the present study. However r2 represents an important part and should be studied in the future. As mentioned in the previous section, the heat bath is a very complex system. The excitation modes in the heat bath can be either Bosons or Fermions. For simplicity, we will only consider the case that the excitations in the heat bath are also Bosons and only to the first order in interaction. However, this method can also easily be extended to other kinds of excitations. The lifetime of T, is given by the relation T, = Zi/lIm~I, where Im 2 is the imaginary part of the self-energy of vibration mode in the energy band. It should be noted that the lifetime (and thus the self-energy) of interest is due to terms with coupling constants /3 and p * because these terms are responsible for creating the condensed phase. The rest of the terms in the Hamiltonian do not contribute to the condensation in the lowest order of coupling constant and therefore are not relevant to the present calculation. The self-energy for excitation with momentum p in the biological system is shown in Fig. 3 and is given by -q P&J . -* =~Cjdo’[G,‘(p-p’,o,-w’)c:(p’,o’) d
+G,<(p-p’,Z,-o’)G,‘:(p’,w’) + G,’ (p +p’&,
+ d)Gb(: ( p’.o’)
+G~(p+p’,~,,+o’)Gb>(p’.~‘)]
(29)
Since we are only interested in a rough estimate of the lifetime of the phonon in the biological system, it will be calculated to lowest order in coupling constant. Thus, replacing p by /3, WP by o,,, and the full Green’s Functions by the non-interaction ones and performing the energy integral yields
1+ n(“J + 4 %-p,) -q Pi,Wi)= Ip12C j [ wP,- LJp,- w,,-,,+i6 + Fig. 3. The self-energy
for the excitation
with momentum
p.
4%,)
- 4 wPi-P,)
up, + Opj - w~,-~ i + i6
1
(30)
26
TM.
Wu / Bioelecrrochemistry
Therefore, the self-energy can be approximately written as Z( P+J = !E!J~,dx~$!L PI
1 X i
cop - 0, - wp-pl + it3
+
1 cop + 0, - wpmpl + ia i
where the summation over heat bath momenta has been converted to an integration, and x = cos 6; 8 is the angle between p and p’. Using the Debye model for the excitations in the heat bath, ,n;l = u’p’, where u’ the velocity of sound in the heat bath. The form of the dispersion relation for the biological system is unknown at present and depends in an intricate manner upon the biological structure. To obtain an approximate result, however, it is presumed that op-op-pl= + -yA where A is the width of the energy band which is very narrow and y is a positive quantity less than 1. The value of y depends on the form of the dispersion relation for the biological system. The real part of self-energy in Eq. (31) is impossible to evaluate without knowing the form of wP. Fortunately we are only interested in the imaginary part which can be obtained easily from EQ. (31) as Id=
-IPl
2 ykTA m
(32)
Therefore, the lifetime r, becomes 7, = di* u3/l j31’ykTA
(33)
which is inversely proportional to the temperature T and to the square of the coupling constant. Thus, as the temperature is raised, the condensed phase should occur sooner, in turn causing the biological effects to begin sooner. This agrees, at least qualitatively, with Smolyanskaya and Vilenskaya’s experiment where changing temperature from 20°C to 37°C brought the biological effects into play while holding the irradiation time fixed at 30 min. 7, given in Eq. (33) represents the irradiation time needed to produce the condensation and is the minimum time needed to produce biological effects. In order to apply this equation to specific circumstances, model studies and more experiments need to be performed to obtain expressions for the coupling constants and the dispersion relations in the energy bands. Given this information, a more
and Bioenergetics
41 (1996)
19-26
complete understanding of the Frijhlich model and all its implications can be obtained. There are three common characters. (1) They depend very strongly on frequency (exhibiting resonance behavior). (2) There is a critical power supply blow of which no effect is observed. (3) There is a time threshold for initiating biological effects. References [l] [2] [3] [4] [5] [6] [7] [S] [9] [lo] [l 11 [ 121 [13] 1141 [15] [16] [17] [IS] [I91 [20] [21] [22] [23] [24] 1251 [26] 1271 [28] [29] [30] [31] 1321 [33] [34] [35]
H. Friihlich, Riv. Nuovo Cimento. 7 (1977) 399. H. Friilich, Biosystems, 8 (1977) 193. H. Friihlich, Adv. Electronics and Electron Phys., 53 (1980) 85. H. Frohlich, Phys. Lett., 26A (1968) 402. H. Friihlich, Int. J. Quant. Chem., 2 (1968) 641. H. Friihlich, Nature, 228 (1970) 1093. H. Frohlich, Phys. I..&. A, 39 (1972) 153. H. Friihlich, Phys. Lett. A, 51 (1975) 21. RI. Kiselev and N.P. Zalyubovskaya, Sov. Phys. USPEKHI, 16 (1974) 576. A.Z. Smolyanskaya and R.L. Vilenskaya, Sov. Phys. USPEKHI, 16 (1974) 571. L.A. Sevastyanova and R.L. Vilenskaya, Sov. Phys. USPEKHI, 16 (1974) 570. W. Grundler and F. Keilmann, Z. Naturforsch. Teil C, 33 (1978) 15. W. Grundler, F. Keilmann and H. Frohlich. Phys. Lett. A, 62 (1977) 463. S.J. Webb and M.E. Stoneham, Phys. Lett. A, 60 (1977) 267. S.J. Webb, Phys. Lett. A, 73 (1979) 145. S.J. Webb, R. Lee and M.E. Stoneham, Int. J. Quant. Chem. Quant. BioI. Symp., 4 (1977) 277. T.M. Wu and S. Austin, Phys. Lett. A, 64(1977) 151. T.M. Wu and S. Austin, Phys. Len. A, 65 (1978) 74. T.M. Wu and S. Austin, J. Theor. Biol., 71 (1978) 209. T.M. Wu, The Living State II, 1985, p. 318. T.M. Wu. Molecular and Biological Physics of Living Systems, 1990, p. 19. T.M. Wu, Bioelectrodynamics and Biocommunication, World Scientific, 1994, p. 387. L. Lauck. J. Theor. Biol., 158 (1992) 1. L.M. Ristorsk, Z. Phys. B, 88 (1992) 145. J. Pokomy and J. Fiala, Czech. 1. Phys., 44 (1994) 67. C.J.S. Clarke, J. Phys. A, 27 (1994) 5495. F. Kaiser, Phys. Lett A, 62 (1977) 63. F. Kaiser, Biol. Cybemet, 27 (1977) 155. F. Kaiser, Z. Naturforsch. Teil A, 33 (1978) 294. F. Kaiser, Z. Naturforsch. Teil A, 33 (1978) 418. R.E. Mills, Phys. Lett. A, 74 (1979) 278. R.E. Mills, Phys. Rev. A, 28 (1983) 379. R.E. Mills, Phys. Rev. A, 43 (1991) 3176. M.A. Lifshits, Biofiz.. 17 (1972) 694. A.A. Abrikosov, L.P. Gor’kov and I.Y. Dzyaloshinskii, Quantum Field Theoretical Method in Statistical Physics, Pergamon, Oxford.
Bioelectrochemistry
and Bioenergetics
41( 1996) 19-26
Quantum mechanical concepts of coherent states in biological systems T.M. Wu Department
of Physics,
State University
of New York at Binghamton,
Binghamton,
NY 13902,
USA
Received9 November 1995;revised 29 January 19%
Abstract In order to demonstrate the possibility of coherent behaviour in biological systems, FrGhlich wrote down the rate equations and showed that if energy is supplied above a critical rate to the branch or branches of electromagnetic modes, Bose-Einstein condensation into the lowest energy state occurs. The general forms of the rate equations were dictated by requiring a Bose-Einstein distribution for thermal equilibrium when there is no energy supplied. Using microscopic theory and perturbation calculations, we were able to obtain the rate equations of Friihlich from the Hamiltonian of the biological system under study. Two simple cases were presented to demonstrate the enhancement of excitations in biological systems by external stimulation. Friihlich’s model has been particularly useful in understanding recent experiments revealing the effects of low level microwaves on various biological systems. One important general characteristic reported in these experiments is the existence of a time threshold for the initiation of the differing biological effects. We obtained an approximation of this time threshold. Keywords:
Coherent
states; Bose-Einstein
condensation;
Perturbation
calculations;
1. Introduction Although the structure, function and detailed biochemi-
cal content of biological micro-organisms have been studied extensively, very little work has been done on the effects of millimeter electromagnetic radiation on biological systems. Such work has recently taken on a new importance with the development of new methods of measurement and their theoretical interpretation in terms of possible collective excitations of biological systems (e.g. membranes, nucleic acids, proteins and cells). Large non-biological, physical systems which are built from more fundamental constituents (crystals, atomic nuclei, spin arrays, etc.> possess vibrational modes; these modes are characterized by the coherent motion of many of the constituent parts of the large system. Thus, it is more than likely that biological systems such as cell membranes, large biological macromolecules, or intact cells, also possess vibrational modes which will couple strongly or weakly to electromagnetic radiation. To illustrate this, let us consider a simple cell membrane consisting of a bilayer of phospholipid molecules with protein interspersed. These macromolecular assemblies contain
Frijhlich
model;
irradiation
dipolar molecules that are arranged in such a way as to give rise to an array of dipoles. The charge groups which comprise the dipoles are displaced relative to each other with some frequency f. Each dipole is imbedded in a complex structure, and it is possible that the interaction between the dipole and this underlying superstructure results in the manifestation of a vibrational excitation. The detailed nature of this excitation is very complex, involving deformations of the underlying structure. In addition, each dipole also interacts with all other dipoles via electromagnetic forces. Other types of interactions mediated by the structure are also possible. The net result of these mutual interactions is to spread the frequency f into a narrow band and to provide for the sharing of energy between the individual dipoles. If a particular dipole is perturbed, the perturbation propagates to other dipoles until the whole array of dipoles is excited to some collective quantum state which we will call a dipole wave or electromagnetic oscillation. The excitation energy of this electromagnetiti oscillation is expected to lie in a narrow band close to hf, where h is Planck’s constant. Based on the relation
f = v/h, where u is the velocity
of the sound wave in organic material
0302-4598/%/$15.00 Copyright 0 1996Publishedby Elsevier ScienceS.A. All rights reserved. PII SO302-4598(96)01921-6
Microwave
and A is the
20
T.M. Wu /Bioelecfrochemistry
wavelength, Frohlich [l-3] has estimated the frequency of the oscillation. If we take u = lo5 ems-‘, which is the approximate velocity of sound in water and many organic liquids, and A = 10m6 cm, which is a typical dimension of a large biological molecule, then the frequency is equal to 10” Hz. In general, the expected frequencies are lOlo10” Hz for membranes, 10’2-10’4 Hz for proteins or more general for certain bond-stretching groups, and lo9 Hz for DNA or RNA molecules. There is plenty of experimental evidence verifying the existence of these waves in biological systems. From theoretical considerations, Frijhlich [4-S] suggested that if the energy is supplied to the vibrational modes at a rate greater than a certain critical value, then the phenomenon of Bose Condensation to the lowest excitation of a single mode can occur. Such a condensation to the lowest excitation mode might serve not only as a method for energy storage but also as a channel for specific biological processes such as cell division or protein synthesis. The Frijhlich model has been of special interest because recent experimental [g-16] and theoretical [ 17-261 reports conform in many respects to its predictions. The experiments are concerned primarily with the irradiation of biosystems with low level microwaves. It has been demonstrated that microwaves between 40 and 14OGHz can increase or decrease the division rate of bacterial and yeast cells, change the oncogenic properties of certain tumor cells, and alter the in vivo rates of synthesis of macromolecules. Webb [14-161 has reported the induction of lambda prophages in cells of Escherichiu cali after lysogenie cells were exposed to microwave fields. In theoretical work, Kaiser [27-301 has applied a modified PearlsBoltzmann transport equation formalism to Frijhlich’s theory, and Mills 131-341 has analysed the asymptotic solutions of FrBhlich’s equations.
2. FrShlich model
In this section, we consider a simple model suggested by Friihlich. It is presumed that the biological system consists of three parts: (1) the main oscillating units of giant dipoles (e.g. sections of the membrane of cells or parts of H-bands within biological macromolecules), (2) the remainder of the biological system which forms the heat bath, and (3) an external energy source which couples to the vibrating units. Long-range Coulomb interaction between the dipoles will produce a narrow band of energy, >, which corresponds to the normal wj (oJoo+Iom,, modes of the electromagnetic vibrations. The heat bath is, of course, a very complex system. Interaction with the heat bath will lead to emission and absorption of quanta by these oscillating electromagnetic modes. Here we consider processes involving one and two quanta only. The interac-
and Bioenergetics
41 (1996)
19-26
tion involves several factors: dipoles of water and other molecules, mobile ions, certain electronic degrees of freedom, and, to some extent, elastic displacements. The assumption that the heat bath is in a thermal equilibrium enforces the principle of detailed balance at its temperature. Therefore, the equation for the thermal average rate of change of the ith mode quanta is (rii)=si-~i[(ni>eao,-(l -
+(Q]
CAjj[(ni>(l
+
(nj>)ePW’
j
-(nj>(l where si and Aij quantum and Aij
+ (n,))&]
(1)
is the rate of energy supplied to the ith mode, +i are coupling constants to the heat bath for oneand two-quanta processes respectively. Both & will in general depend on temperature. We use
p = fi/kT.
For the stationary solutions of this rate equation, one requires that ( Ai) = 0. As a result, the ith quanta is 1 eS(Wt-wf) - 1
‘i (ni> = 1+ cpi + C AiinjepoJ
j
[
(2)
I
where c Aij( epoJ - 1) eP”=l+
++‘xAij(l
+
”
(3)
and pi is the chemical potential of the ith mode. The inequality in Eq. (31, together with the requirement that at thermal equilibrium ( iri> = 0, dictates that wi 2 pi r 0. If there is no energy supplied (si = O>, then pi = 0 and Eq. (2) becomes the thermal equilibrium distribution as required. From Eq. (2) and (31, one notices that as si increases, pi will increase. When si exceeds a critical value s,, pi approaches wo. the lowest energy in the excitation band. Therefore a large number of quanta are condensed into the lowest energy state. This is exactly the Bose-Einstein condensation in a Bose gas system when the temperature is lower than a certain critical value. In our case, the corresponding phase transition is not induced by lowering the temperature; instead it occurs by keeping the temperature constant and increasing the energy supply beyond the critical value so. From Eq. (3), it should be noted that when Aij = 0, pi = 0. Thus, it is the two-quanta processes that are responsible for Bose-Einstein condensation; the larger the A,,, the greater the effect. Subsequently, other investigators approached the theoretical problem from differing standpoints. Lifshits [34] concluded that the Frijhlich model was erroneous because Friihlich did not include in the two-quanta term the contributions from the simultaneous creation or destruction of
TM.
Wu / Bioelectrochmtistry
and Bioenergetics
the quanta (i) and (j). Thus, according to Lifshits, the average rate of change of the ith mode quanta should be (hi> = si - c$i[(ni)ei~~~ - Cnij[(n,>(l j +zj>(l -(l
- (1 + (Q)] + (nj>)eP”’
+ (n&!@Q + (n,>)(l
+ (?zj>) + (ni)(frj)e+‘+Q] (4)
Eq. (4) can easily be reduced to the form (iti> =si fB,[(r+‘~“~-
(1 + +ri>)]
(5)
with Bi=
xA,,[l j
+ (nj)
+ (nj)&]
+ &
(6)
theory
Using rate equation, Friihlich was able to demonstrate that the condensation occurs when the external energy supply attains a certain critical value. The general forms of the rate equation were dictated by requiring a Bose-Einstein distribution for thermal equilibrium when there is no power supplied. Beyond this, the forms were somewhat arbitrary. To avoid this arbitrariness, we will study this
19-26
21
problem from a microscopic point of view using the finite temperature Green’s Function method including all orders of interaction strength. To each oscillating mode, wi, we assign a creation operator, a,?, and a destruction operator, ai. The normal modes of oscillation will interact with the remainder of the biological system (the heat bath) which is represented by a set of independent excitation Ok, associated with creation and destruction operators, b: and b, respectively. Furthermore, the external energy supply associated with creation operator, p,? and destruction operator pj, and excitation energy, Of, feeds into the electromagnetic oscillatory units and acts as impetus for the initiation of the biological effects. The Hamiltonian of the biosystem can then be written as A?= &+a+~,+ i
Therefore, the contribution from two-quanta terms including the terms suggested by Lifshits has the same form as the one-quantum term. Thus, Lifshits concluded that no Bose-Einstein condensation will occur in the biological system. It was later shown [18] that the addition of the Lifshits terms actually does not cancel the Bose-Einstein condensation phenomenon predicted by Frohlich.
3. Microscopic
41 (1996)
C,R,b;b,+
c6,p;pf
f
k
+~C(ubjb;ai+a*b,~b,a;) d‘ +;
&( @fa,b: 11“
+ C(YPf4 if
+ /3 *aja+bk)
+ Y*P;ai)
(7)
where CX, p and y are the coupling constants for the one quantum process, the two quanta process, and the energy source with the vibration modes respectively. Strictly speaking, we should consider higher order and possibly inharmonic terms; however, it can be shown that accounting for those terms have only minor effects and, within certain limits, they may be ignored altogether. The electromagnetic oscillations are Bosons. The excitations within the heat bath and the energy source can be either Bosons,
iGc1i.w)
Fig. 1. Diagrams
represent
the rate of change of the number
of phonon
iG:li,w)
with energy
wi.
22
T.M. Wu / Bioekctrochemistry
and Bioenergetics
41 (1996)
19-26
or Fermions, e.g. phonons or electrons. The operators a,?, ai, b:, b,, pf’ and pf satisfy the commutation relations
dwdw’do” PI3
a,a,+-a,+a, = Sij a,:,,?-a;at b,b:
X{[G,>(i,o)G,<(
= aiaj - ajai = 0
+G:
+ b:b, = S,,
(i,w)G,>
(j,w’)GL
(k,d’)]
X8(,--d-d’)+[G~(i,w)G,<(j,w’)
b,b, & b,b, = b:b; PfP,’ * P,‘P,=
j,d)Gb<(k,w”)
f b;b;
= 0
xG,>(k,o”)
Sf,
(8)
+G,<(
i,w)G,>
(j,w’)G:
(k,w”)]
xqw-o’+w”)}{
(+ sign for fermions and - sign for Bosons). The rate of change of the number of quanta in the ith mode is given by +G:(i,w)G,>(I,o’)]S(w-
with n, = a,?~~; the number of quanta in the ith mode and fi is Planck’s constant divided by 21r. The expectation value of the rate of change of the number of phonons with energy wi in the biological system is then given by
= fIm(
Gb(i,t)= G,(i,t)=
~~*(~(f)la:(f)bj(f)b:(f)lcp(f)) j,k
= -i<‘P(f)IT{ai(f)a+(0)}l~(f))
-i(‘P(t)IT{bi(t)b+(O)}I40(t)) -i<~(f)IT{P~(f)P~(O)}(f))
(13)
+ CP((P(t)l~+(t)~j(t)b:(t)l~(f)) j.k
where T is the time ordering operator. The retarded and advanced Fourier transforms of the above defined Green’s Functions can be written as
+ CP * (~P(f)la,+(f)aj(f)bk(f)l(P(f)) j,k
G,‘(i,w) = l’n(wi) o-q+is
+ Cr(‘P(t)la+(t)Pl(t)l~o(t))}
(10)
where lcp(t)) is the state vector of the whole system, and the angular bracket is the ensemble average. The ensemble average (6) for any operator 6 is obtained by 8 = Tr( ~6) where p is the density operator defined by
The retarded and advanced functions for the heat bath and the 2 energy pump have a similar form Gz (i,o)
=
G;(i,u)
=
G; (i,o)
=
G;(i,o)
=
(11) Using finite temperature Green’s Function techniques [35], one can easily obtain the rate of change of the number of phonons with energy wi as shown in Fig. 1. It is given by the equation (&)=
(12)
where G,‘(i,w), (G,<(i,w)), Gj’(i,~), (G, < (i,w)), and GY’ (i, w), (G,< (i, 0)) are the retarded and advanced Fourier transforms of the finite temperature Green’s Functions defined by G,(i,f)
w’)
--
2cY-d;’ f.&2
dodw’dw” Im
c,/, ( j.k
PI3
X[G,>(i,w)G~(j,o’)G,>(k,o”) +G,< (i,u)Gb>
(j,d)G:
(k,d’)]
1 +Nb( ai) o-fii+s -Nb(
‘i)
w-Q-it3 1 +N,(fi;) co- it& +iS -NY(@) o-fii;-is
(15)
where n(Zi), Nb(ai), and N,( fij) are the quantum distributions of the biological system, heat bath, and energy pump respectively. Wi = wi f Z&i) is the modified excitation of the biological system phonon with energy oi when there is no interaction between the biological system and heat reservoir, and ZO,(i) is the proper self-energy of the excitation. Owing to the nature of the weak dipolar interac-
TM.
Wu /Bioelectrochemistry
tion, one expects the self-energy, .Zfa(i), to be small. In general s=,(i) depends on oi, but the energy band is very narrow; it can be approximated to be a constant. Similarly, fii = ai + S,(i) and fii = ai + z,,(i) where s’,(i) and &(i) are the proper self-energies for the heat bath and energy supply excitations respectively. The vertex corrections G, p, 7 consist of all possible proper corrections to all orders of interaction. The actual detail forms of the vertex corrections 5, p can be obtained from the coupling equations as shown diagrammatically in Fig. 2. The coupling strengths cr, p and y are very complicated quantities and in general depend on oi. Since the energy band of the biological system under this study is very narrow, the coupling strengths and their vertex corrections will be considered to be constants. Substituting Eq. (14) and Eq. (15) into Eq. (12) and integrating over o, w’, and w”, one gets
(1
+n(Wi))Nb(‘j)(l
n(oi)(l
Summing over j and taking the imaginary part, Eq. (16) becomes Crii)=
(l
+Nb(zi+Bj))iVb(fij)
-$Z(n(zi)(l i
-(l
+n(Wi))Nh(oi+~j)(l
-$x(n(zi)(l i
+n(Sjj))(l
‘(‘i)(’
+Nb(wi-7jj))
- (1 + n( q>)n( Oj)Nb( wi - Oj)
+n(zJ)(1
+Nb(3ik))
-(l
+n(izj))&(Tjj-oi)
+“(oJ)n(oj)(l
- g{n(q)(
+n(zi)jn(zj>Nb(ak)
Wj+32,--Wi+i6 +
+Nb(~j)))
‘k)
Gj + jjz, - Oi - iS +
+.(zi))n(‘j)(l+Nb(ak))
(16)
I +n(‘j))(’
(1
19-26
Oj-fik--Z++iS
+Nb(ak))
jjj--a,--Oi+iS
41 (1996)
+
Ilj-Bk-Zi-iS
j.k +
+ Nb( 'j))Nb(
4'i)('
(iz,) = - $IrnC
and Bioenergetics
1 -t-N&))
+Nb(zj-zi))} - (1 +n(wi))N(wi)} (17)
Because the heat bath is large compared with the biological system under consideration, it is a good approximation to assume the distribution of the excitation in the heat
+n(Tjj))Nb(ak)
iZj-fik-Gi-it3
Fig. 2. The vertex
corrections
for the coupling
constants
a and p.
24
T.M. Wu /Bioelectrochemistry
bath is in equilibrium. Therefore, substituting N(Zi) = (ep7j! -t 11-l in Eq. (17) yields
Nb(Wi) =
and Bioenergetics
41 (1996)
19-26
where si, +ij, and Aij are the same as (201, Eq. (20, and
inEq. (191,Eq.
(iti)=S,-C$i(T,wi)[n(Wi)e~G~-(l+.(zX))] - CAij(T,Wi,Sjr)[n(wi)(l -n;Gj)(l
+n(Gj))ePe~
+ n( Oi))ePol]
(‘8)
s; = ~(,,,;)
- n( Z;))
(19)
c&~(T,C+) = T(l
+N(Wi))epa,
(20)
where
and 1 ‘N(OjGii) Aij(T,7iJi,iiij)
= $empzl -
N( 0; - Zj),
i
Oj>Oi oj < w; (21)
One notices that Si given in Eq. (19) is the net power absorbed by the ith mode. Eq. (18) is exactly the same rate Eq. (1) suggested by Frijhlich. To confirm the weakness of Lifshits’ conclusion, the problem must be treated microscopically. The Hamiltonian in Eq. (7) should be modified to take care of the contributions from creating two quanta or destroying two quanta simultaneously. Thus, SF’= cwia,Tai
+ xOkb:b,
i
+ ~O,p~p,
k
+i
f
&( pafa,b: ZJk
+ig(6ajaib: Ilk
+j? *aja+bk)
+6*afa,?b,)
+ C(ypf4 if
+ y*pf+a,)
(22)
where S is the coupling constant for the simultaneous creation of two quanta. Using this modified Hamiltonian and repeating the same calculation for (A), we get (hi> = Si-
&[(ni)epW~--
If Aij = rij as Lifshits presumes, then Eq. (23) can be reduced to Lifshits’ form. However, as seen in Eq. (21) and Eq. (241, allowing Aij = rij is incorrect. Nevertheless, presupposing that the coupling constants are of the same order of magnitude for each type of two-quanta interaction, the coefficient, Ajj is much larger than the coefficient, rij. This can be demonstrated readily for biological systems where oi is of the order of 10” s-‘. At room temperature hw,/kT = lo-*, thus rij and Aij can be approximately written as cj = (2r/612kT/h2)( Aij=
(24P12kT/h2)(lo,-
For narrow the Aij quency
c.j = (&12/h)( Aij = (24
p [*/h.)(
tni>)(l
kT/hZ)
(27)
kT/?iA w)
(28)
Since the range of normal frequencies of the oscillations is very small, Aw -zz W. Therefore, the terms in Eq. (27) are much smaller than those in Eq. (28). The simultaneous creation and destruction of quanta proposed by Lifshits can be regarded as negligible. Thus, the predominant “twoquanta” process is precisely the one postulated by Frijhlich, and the addition of the Lifshits term does not cancel out the Bose condensation phenomenon predicted by Frohlich. As Frijhlich has pointed out to us, oscillations as introduced in Refs. [ 17,181 and [341 form only a part of the degrees of freedom of the heat bath. Very important for the relevant energy transfer are non-oscillating systems, e.g. free ions, mobile ionic groups (dipoles), and certain low-lying electronic levels. For all of these it can be shown that Lifshits’ contention is incorrect. In fact some of them may permit excitations in the energy range (0, ( w,,, - w,)> but not in the range (20,, 2 o,,, ), thus making two-phonon absorption or emission impossible.
for biological effects
+ (nj))eP(Wg-WJ
+ (ni>)]
+
(26)
the rij term, since wi = oj owing to the very range of normal frequencies, let wi = wj = 0. For term, let 1wj - wil = A w, the width of the freband. Thus Eq. (25) and (26) become
4. Time threshold
-(l
oil)-’
(25)
(1 +
- CAij[(n,>(l i -(nj>(l
w; + co,)-’
+
- C~ij[(ni)(nj)eP(W,+W,)
<“j>)]
(23)
Frijlich’s model has been standing recent experiments level microwaves on various portant general characteristic experiments is the existence
particularly useful in underrevealing the effects of low biological systems. One imthat is reported in these of a time threshold for the
T.M. Wu /Bioelectrochemistty
initiation of the differing biological effects. Sevastyanova and Vilenskaya [l 11 performed experiments involving the irradiation of mice bone marrow cells by microwaves and X-rays simultaneously; they observed a time threshold of 30 min of microwave exposure before a protective effect could be induced by radiation. Smolyanskaya and Vilenskaya [lo] observed the effects of low level millimeter waves on the induction coefficient of colicin synthesis. They also found that a minimum irradiation time of 30 min was needed to produce measurable biological effects. In addition, one must keep in mind that when considering these time courses, the threshold will depend on the particular biological activity being monitored as well as temperature. An example of the latter is clearly demonstrated in Smolyanskaya and Vilenskaya’s experiment. Irradiation for 30 min at 20°C produced no change in the rate of colicin synthesis, whereas at 37”C, colicin synthesis increased after 30 min of irradiation time. Therefore, it is expected that the analysis of the time threshold will not be a trivial matter. Both the detailed microscopic structure of the biological system and its thermodynamic behavior must be considered to provide a full understanding of the specific time thresholds encountered for different biosysterns. Using the Frahlich model, we intend to obtain an approximation for a segment of this time threshold. [35] In applying the Frijhlich hypothesis, one sees that the biological effects cannot begin until condensed phase occurs. It has not been determined how long this condensation has to be maintained in order to produce biological effects. Generally, the time needed to initiate biological effects will consist of two parts, 7, and T*. r1 represents the time from the beginning of irradiation to the onset of condensation; r2 represents the time elapsed from the onset of the
and Bioenergetics
41 (1996)
19-26
25
condensed phase to the manifestation of the actual biological effects. Thus, the total time, T, elapsed before biological effects occur from the initiation of irradiation, is given by T= 7, + TV. In the following analysis, we will evaluate T] using the Hamiltonian given in Eq. (7). T, represents the minimum time required for the biosystem to exhibit effects after irradiation and is the lifetime of the collective excitations in the biological system. An explicit form of T* requires an indepth biochemical analysis of the particular system under scrutiny and is beyond the scope of the present study. However r2 represents an important part and should be studied in the future. As mentioned in the previous section, the heat bath is a very complex system. The excitation modes in the heat bath can be either Bosons or Fermions. For simplicity, we will only consider the case that the excitations in the heat bath are also Bosons and only to the first order in interaction. However, this method can also easily be extended to other kinds of excitations. The lifetime of T, is given by the relation T, = Zi/lIm~I, where Im 2 is the imaginary part of the self-energy of vibration mode in the energy band. It should be noted that the lifetime (and thus the self-energy) of interest is due to terms with coupling constants /3 and p * because these terms are responsible for creating the condensed phase. The rest of the terms in the Hamiltonian do not contribute to the condensation in the lowest order of coupling constant and therefore are not relevant to the present calculation. The self-energy for excitation with momentum p in the biological system is shown in Fig. 3 and is given by -q P&J . -* =~Cjdo’[G,‘(p-p’,o,-w’)c:(p’,o’) d
+G,<(p-p’,Z,-o’)G,‘:(p’,w’) + G,’ (p +p’&,
+ d)Gb(: ( p’.o’)
+G~(p+p’,~,,+o’)Gb>(p’.~‘)]
(29)
Since we are only interested in a rough estimate of the lifetime of the phonon in the biological system, it will be calculated to lowest order in coupling constant. Thus, replacing p by /3, WP by o,,, and the full Green’s Functions by the non-interaction ones and performing the energy integral yields
1+ n(“J + 4 %-p,) -q Pi,Wi)= Ip12C j [ wP,- LJp,- w,,-,,+i6 + Fig. 3. The self-energy
for the excitation
with momentum
p.
4%,)
- 4 wPi-P,)
up, + Opj - w~,-~ i + i6
1
(30)
26
TM.
Wu / Bioelecrrochemistry
Therefore, the self-energy can be approximately written as Z( P+J = !E!J~,dx~$!L PI
1 X i
cop - 0, - wp-pl + it3
+
1 cop + 0, - wpmpl + ia i
where the summation over heat bath momenta has been converted to an integration, and x = cos 6; 8 is the angle between p and p’. Using the Debye model for the excitations in the heat bath, ,n;l = u’p’, where u’ the velocity of sound in the heat bath. The form of the dispersion relation for the biological system is unknown at present and depends in an intricate manner upon the biological structure. To obtain an approximate result, however, it is presumed that op-op-pl= + -yA where A is the width of the energy band which is very narrow and y is a positive quantity less than 1. The value of y depends on the form of the dispersion relation for the biological system. The real part of self-energy in Eq. (31) is impossible to evaluate without knowing the form of wP. Fortunately we are only interested in the imaginary part which can be obtained easily from EQ. (31) as Id=
-IPl
2 ykTA m
(32)
Therefore, the lifetime r, becomes 7, = di* u3/l j31’ykTA
(33)
which is inversely proportional to the temperature T and to the square of the coupling constant. Thus, as the temperature is raised, the condensed phase should occur sooner, in turn causing the biological effects to begin sooner. This agrees, at least qualitatively, with Smolyanskaya and Vilenskaya’s experiment where changing temperature from 20°C to 37°C brought the biological effects into play while holding the irradiation time fixed at 30 min. 7, given in Eq. (33) represents the irradiation time needed to produce the condensation and is the minimum time needed to produce biological effects. In order to apply this equation to specific circumstances, model studies and more experiments need to be performed to obtain expressions for the coupling constants and the dispersion relations in the energy bands. Given this information, a more
and Bioenergetics
41 (1996)
19-26
complete understanding of the Frijhlich model and all its implications can be obtained. There are three common characters. (1) They depend very strongly on frequency (exhibiting resonance behavior). (2) There is a critical power supply blow of which no effect is observed. (3) There is a time threshold for initiating biological effects. References [l] [2] [3] [4] [5] [6] [7] [S] [9] [lo] [l 11 [ 121 [13] 1141 [15] [16] [17] [IS] [I91 [20] [21] [22] [23] [24] 1251 [26] 1271 [28] [29] [30] [31] 1321 [33] [34] [35]
H. Friihlich, Riv. Nuovo Cimento. 7 (1977) 399. H. Friilich, Biosystems, 8 (1977) 193. H. Friihlich, Adv. Electronics and Electron Phys., 53 (1980) 85. H. Frohlich, Phys. Lett., 26A (1968) 402. H. Friihlich, Int. J. Quant. Chem., 2 (1968) 641. H. Friihlich, Nature, 228 (1970) 1093. H. Frohlich, Phys. I..&. A, 39 (1972) 153. H. Friihlich, Phys. Lett. A, 51 (1975) 21. RI. Kiselev and N.P. Zalyubovskaya, Sov. Phys. USPEKHI, 16 (1974) 576. A.Z. Smolyanskaya and R.L. Vilenskaya, Sov. Phys. USPEKHI, 16 (1974) 571. L.A. Sevastyanova and R.L. Vilenskaya, Sov. Phys. USPEKHI, 16 (1974) 570. W. Grundler and F. Keilmann, Z. Naturforsch. Teil C, 33 (1978) 15. W. Grundler, F. Keilmann and H. Frohlich. Phys. Lett. A, 62 (1977) 463. S.J. Webb and M.E. Stoneham, Phys. Lett. A, 60 (1977) 267. S.J. Webb, Phys. Lett. A, 73 (1979) 145. S.J. Webb, R. Lee and M.E. Stoneham, Int. J. Quant. Chem. Quant. BioI. Symp., 4 (1977) 277. T.M. Wu and S. Austin, Phys. Lett. A, 64(1977) 151. T.M. Wu and S. Austin, Phys. Len. A, 65 (1978) 74. T.M. Wu and S. Austin, J. Theor. Biol., 71 (1978) 209. T.M. Wu, The Living State II, 1985, p. 318. T.M. Wu. Molecular and Biological Physics of Living Systems, 1990, p. 19. T.M. Wu, Bioelectrodynamics and Biocommunication, World Scientific, 1994, p. 387. L. Lauck. J. Theor. Biol., 158 (1992) 1. L.M. Ristorsk, Z. Phys. B, 88 (1992) 145. J. Pokomy and J. Fiala, Czech. 1. Phys., 44 (1994) 67. C.J.S. Clarke, J. Phys. A, 27 (1994) 5495. F. Kaiser, Phys. Lett A, 62 (1977) 63. F. Kaiser, Biol. Cybemet, 27 (1977) 155. F. Kaiser, Z. Naturforsch. Teil A, 33 (1978) 294. F. Kaiser, Z. Naturforsch. Teil A, 33 (1978) 418. R.E. Mills, Phys. Lett. A, 74 (1979) 278. R.E. Mills, Phys. Rev. A, 28 (1983) 379. R.E. Mills, Phys. Rev. A, 43 (1991) 3176. M.A. Lifshits, Biofiz.. 17 (1972) 694. A.A. Abrikosov, L.P. Gor’kov and I.Y. Dzyaloshinskii, Quantum Field Theoretical Method in Statistical Physics, Pergamon, Oxford.