toward a theory of living state—part ii

Toward a Theory of Living State-Part II

R. K. Mishra and Satish Kumar

Institute of Self Organizing Systems and Biophysics

Bijni Campus

North Eastern Hill University

Shillong 793 003 India

ABSTRACT

It was suggested earlier that matter capable of initiating and supporting phenomena exhibited by living organisms must have structural principles not encountered elsewhere. A pumped, open three boson field was suggested as necessary. Conse- quences on the requirement of time and temperature for any biological phenomena are examined in the present article.

1. INTRODUCTION

In an earlier paper [7], and in a more detail in Part I of this article [9] it was shown that a boson sea far from equilibrium, upon pumping of energy, is expected to propagate energy in solitonic beams. In this article, we examine the consequences of this propagation. We recall the proposition made by Friihlich referring to pumps on dipolar assemblies, leading to Bose-Einstein ‘condensation’ (ordering) in the lowest mode which is subsequently followed by biological processes [2, 31. We had previously [6, 81 dealt with linear aspects in the general scheme of condensation in mixed bosonic assembly, i.e., boson sea comprised of diverse bosons. A temperature dependence was noted [8]. However, the condensation is due to nonlinear processes between the system and the bath. That aspect is now examined as the dominant factor to account for temperature dependence of ‘condensation’ effect.

2. FRijHLICH’S MODEL [2, 31

The model conceives of isolating the three subsystems of a model biosystem:

APPLIED MATHEMATZCS AND COMPUTATION 56:167-176 (1993)

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168 R. K. MISHRA AND SATISH KUMAR

(1) a set of coupled periodic oscillators along the length of the macro- molecule. These oscillators, as per Friihlich, may be, for example, the low frequency modes associated with the hydrogen bonds in the macromolecules;

(2) the rest of the system constituting a heat bath maintained at a constant temperature T; and

(3) an energy pump that couples to the oscillating units.

These oscillating units, which occur more or less periodically along the chain, are expected through interaction among themselves, to split into a narrow band of frequencies and spread over the lowest frequency wO to the high- est frequency u; n . These modes represent the normal modes (longitudinal electric) of collective motion in the molecule as a whole. These coupled oscillators kept in a thermalizing heat bath would ultimately follow a Planck distribution. However, since these are also fed with energy from an energy pump with rate S (where S = Zsi, and si is the rate of pumping of energy to the ith mode), after a certain time, a dynamical equilibrium would be set up that may be adequately represented by a Pauli master equation. Thus, the rate of change of occupancy (hi> of the ith mode would be given by

rii = si + &Li + 1) - n,e@Y]

+ xc {(fq + l)nje% - ni(nj + l)e%} (1)

where the first term on the right-hand side gives the rate of inflow of quanta to the ith level by the pump, and the second term gives the inflow and outflow of quanta to and from the ith level due to the interaction of the oscillators with the heat bath. The coupling constant +J is weakly dependent on the mode frequency. Here fi = (l/k ,T), where k, is the Boltzman constant. The third term on the right-hand side represents the second order process of transference of quanta from one normal mode to the other via the heat bath. Thus, the term x0 is essentially a nonlinear interaction. When the dynamical equilibrium is reached, the occupancy would attain stationarity and the right-hand side of (1) is zero, giving lzi, a Bose-type distribution, e.g.,

n, = 1 + $(l - g-P@) [eP(hW,-pL) _ l]-l, ( 1 (2)

Theory of the Living State 169

where I_L is the chemical potential given by

4 + XCCni + l) e-PP =

4 + X~n,ew (3)

and

s = cs, = c$~{[n,eW - (ni + l)]}. (4)

The crux of the problem depends on the existence of chemical potential /.A It can be shown from (3) that we set 0 < f_~ < hw, and ,z + fiw,, when we have S * S,, where S, is some critical pumping rate of energy. Thus, when the pump exceeds the critical rate, almost all the quanta go to the lower energy level and Bose-Einstein condensation becomes a distinct possibility. It should be remembered that, in the absence of the pump (si = 01, or in absence of a nonlinear-type interaction, the chemical potential vanishes.

By making use of (4) in (31, the latter can be reduced to

-PP - 1 = -xs

e C#I~ + X+CniePhwi

In the high temperature limit, e -s fLw - 1 g - /3 fiw and (5) in this limit can be written as

xs ‘CL = 4” + X$C {ni( 1 + Phwi)}

Equation (6) can be translated in the following form:

(6)

Making use of the high temperature limit and dropping terms quadratic in p, we can write (7) as

PP+2 + PFX+Cni = Xs. (8)


Following FrShlich [31, one can write ni as

ni = ny + m, (9)

where nT is the number of quanta at thermal equilibrium and is given by

1 nT =

ePhw. 2-1’

and mi is the excess quanta over the thermal equilibrium distribution. Now in the second term of the left-hand side of (B), we can write

cni = En: + Emi = NT + N 2 t i

where C,nT = NT

and &mi = N.

Using these notations we can write (8) as

PPP~ + PPX+(N + NT) = xs. (10)

Differentiating (10) with respect to (T) temperature, assuming 4 and C,(n,> = N + NT to be independent of temperature, we get (remembering that p = l/K,T)

P dP 1 --

l-CT2 + %- = 4’ + ,~#J(N + NT) 1 - ,yS( N + NT)+?

1

dT [ +2 + ,yc#~( N + NT)]’ (11)

Now, invoking the conditions of temperature independence of Al. and high temperature limit, the left-hand side of (11) reduces to zero. We, therefore,

[ 42 + x4( N + NT)] 1

- XS$( N + NT) = 0. (12)


Upon simplification, we get from (12)

4sg +x[4+X(N+N’.)]g =o.

We rewrite (13) in the following form

1 dS -4 dX -- = S dT x[4 + X(N + NT)] 27’

Integrating (14) on both sides, we get

s=c N+t [ 1 X

(13)

(14)

(15)

where in (15) C is an undetermined constant of integration. Following a microscopic theory [6], one can estimate the linear and

nonlinear couplings 4 and X. An outline of these calculations is given in Appendix A. From here, one can find that

- = Ipfi. X

42 2 (16)

Further, identifying the coupling 4 with the thermal broadening (I3 of the level we can approximate

4=T/h and X=$r2/h.

Now substituting these values of 4 and X from (17) into (151, we get

S=C IV+:,. [ 1

(17)

(18)

The threshold time ( [ > is taken from the beginning of the irradiation to the onset of the “condensation.” One should note that a biological effect cannot take place without condensation to start first. The latter delay is due to the mechanics of the process initiated. A microscopic approach for the calculation


of l was initiated by Wu and Austin [12-141. Here we just hint at it for the sake of completeness.

The microscopic Hamiltonian for the model is given by the following Hamiltonian:

+ energy pump (1%

ai is the annihilation (creation) operator for the ith mode of the coupled harmonic oscillation, bP(bL> is the same operation for the pth mode of the bath oscillation of frequency R,, and g is the coupling constant between the oscillators and the bath. The term that is responsible for the occurrence of the chemical potential is the nonlinear term in the Frohlich model. It can be shown that when we evaluate the self-energy of the phonons using the Hamiltonian in (19), it is the fourth order term in the coupling constant g that will give the appropriate contributions to self-energy. From Appendix A, one can see that it is the x term that is proportional to g4; on the other hand, from (171, x is proportional to P.

This gives us the idea that self-energy [ C( p, to>] is proportional to P. Time 5 is related to C( p, w ) by

h

l= IzmC(p,q

From this relation, one can easily infer that

ia;.

(20)

(21)

This gives us that LaT. This perhaps explain why fever sets a limit to biological processes and the

phenomenon of life.

3. BISTABILITY

We have found from the above discussion that if a system of oscillators are kept in contact with a heat bath and are continuously being pumped energy, then after a finite time, the system would obtain a dynamical equilibrium as


long as the occupancy is constant. Under such a condition (dn/dt = O), the energy distribution of the quanta of the oscillators become a Bose-Einstein distribution condensation becomes feasible.

Let us now investigate the potential energy profile of the system. Before and after the attainment of dynamical before the equilibrium is reached, the oscillator quanta would naturally obey the Planck distribution.

equilibrium. So, the system has two types of stability, one of which can be achieved only after condensation.

SIGNIFICANCE OF THIS STUDY

Bose-Einstein condensation condensation and specify coherence as a

necessary and sufficient condition to reduce energy barriers prohibiting migration, homing, biological and biochemical action, conforma- tional change, shortening of structures, or flip-flop bistable states of molecules. We ask if even this would be possible at body temperature. It seems that strength of the pump is the crucial feature, emphasizing a selection of three-dimensional

.

Metastable State Zi (Bose-Condensed State)

-L 02 2w

2 2

The Most Stable State (Thermal Equilibrium: Chaotic State)

FIG. 1


words, we signify molecules with specific oscillators, with necessary radiant power, for a possible encounter, or action at given temperature to bring about coherence. This transforms a somewhat more general pump to a much more specific one.

One of the impediments to connecting macroscopic properties to sub- molecular quantum behavior is furnished by the time of the process. Here again, the nature of the pump dominates the macroscopic behavior.

Experimental work by various researchers in support of these ideas is listed in references [l, 4, 5, 10, 111.

APPENDIX A

Let us start from a microscopic description of the system. The typical Hamiltonian describing the system is

CL pL,v,i

(22)

where ui<u+> is the annihilation (creation) operator for the ith mode of the coupled harmonic oscillators of frequency wi; b,(b,+) is the annihilation (creation) operator for the pth mode of the bath oscillators of frequency a,, and g is the coupling constant between the oscillators and the bath. First order exchange (one-quantum process) would give &type terms, whereas second order exchange (two-quantum process) would give x-type terms. Thus, using the “Fermi Golden Rule,” we get

4 =

Xij =

P2(ap)6(wj

Here (b,fb,) heat bath. Hence, we have

x (Wi + cl” - ‘R/J cm, cm”,

Gw3 ,-g41jl[cblbp>p(n,)(~~~“)~(~~)(~~~~)zx

+ fl, - ,n,)S( wi + R, - wj - 0,)] X&In, dR, dR,. (24)

is the average number of oscillators in the +h mode in the

(23)

(b,+b,) = l/(ea”o~ - 1).


p(R,) is the density f 1 o evels that may be taken as proportional to 0:. Let us now integrate

I, = lI e P’ph

(e p”“p - l)( epnnu - 1) a; do& dR”6(fl” - R, + q).

(25)

In the high temperature limits, @hoi -K 1 (since oi = IO” HZ), thereby yielding

1 z+ = yI(5) 1 + $ + J$ + ...

(Ph) [ I

4! = ~

( W)” .

Thus, +i becomes essentially mode independent and may be taken as 4. Similar arguments hold for xij so that we may write it as X. Also, regard the integral

/

m x’e’dx r(9) z-

0 (ex - 1)4 3g

Using the above approximations and integrals (25) and (26), we get

A - = Ipfi. 42 2

(26)

(27)

Recognize that in the high temperature limit, 4 would be given by the thermal broadening (I) so that

c$= I/h and x= ipI”/fi. (28)

Assuming, for simplicity, that the width I is very weakly dependent on temperature (at high temperature limit), we find that C#J is independent and x is dependent on temperature.

In this calculation, we have taken 4 with a dimension of s-i. To have its correspondence with the #J of (3), our relationship (7) would be modified as

1 4=E and x=,p12m. (29)


REFERENCES

1

2

3

4

5

6

7

8

9

10

11

12

13

14

N. D. Devyatkov, Influences of Millimeter band electromagnetic radiation on

biological objects, in Scientific Session of Division of Gene& Physics and Astronomies, U.S.S.R. Academic Sciences, 17-18 Jan, 1973; Soviet Phys. Uspekhi 16:568 (1977). H. Frohlich, The biological effects of microwaves and related questions, Interna- tional Journal of Quantum Chemistry 2:641 (1968). H. Friihlich, The biological effects of microwaves and related questions, Adv. Electronics & Electron Phys. 69:85-152 (1980). W. Grundler, F. Keilman, and H. Friihlich, Resonant growth rate response of

yeast cells irradiated by weak microwaves, Phys. Lett. A 62:463-466 (1977). N. Kolios and W. R. Melander, Laser induced stimulation of chymotrypsis activity, Phys. Leti. A 57:102-104 (1976).

R. K. Mishra, Excitons and Bose-Einstein condensation in living systems, Inter- national JournaZ of Quantum Chemistry 10:691-706 (1976). R. K. Mishra and S. K. Dubey, The Living State X. A system of interacting bosons, Living State II, (R. K. Mishra, Ed.) World Scientific Publication, Singapore, 1985, pp. 492-509. R. K. Mishra, Theory of Living State VII Bose-Einstein-like two localised orders in temperature and time domain, International Journal of Quantum Chemistry 23:1579-1587 (1983). R. K. Mishra, G. Sen, Towards a theory of living state, Appl. Math. Comput. (this issue). S. J. Webb and M. E. Stoneham, Resonances between lOi and 10” Hs in active bacterial cells as seen in Laser Raman spectroscopy, Phys. Lett A 60:

267-270 (1977). S. J. Webb, M. E. Stoneham, and H. Frohlich, Evidence of non-thermal excitation of energy levels in attinobiological systems, Phys. Lett. A 63:407-408 (1977). T. M. Wu and S. Austin, Biological Bose-condensation with time threshold for biological effects, Phys. Lett. A 73:266-268 (1979). T. M. Wu and S. Austin, Bose-condensation in biosystems and the possibility of a

metastable coherenced state, Phys. Lett. A 64:151 (1977). T. M. Wu and S. Austin, Cooperative behaviour in biological systems, Phys. Lett. A 65:74 (1978).

toward a theory of living state—part ii

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