Download - A quantum field theoretical approach to the collective behaviour of biological systems

Nuclear Physics B251 [FSI3] (1985) 375-400 O North-Holland Publishing Company

A Q U A N T U M FIELD THEORETICAL A P P R O A C H TO THE COLLECTIVE BEHAVIOUR OF BIOLOGICAL SYSTEMS

E. DEL G I U D I C E I, S. DOGLIA 2, M. MILANI 3

Dipartimento di Fisica dell" Universita, Via Celoria, 16-20133 Milano, Italy

G. V1TIELLO 4

Dipartimento di Fisica dell" Universita, 84100 Salerno, Italy

Received 4 March 1983 (Revised 27 February 1984)

An analysis at the level of molecular excitations is proposed for describing the biological system collective dynamics. In the resulting quan tum field theory scheme the system is considered to oscillate between two different regimes: one characterized by a localized Bose condensat ion of quanta, the other one by an extended homogeneous condensat ion with a long-range correlation. The transition between these regimes is seen as the transition between different vacua.

Dissipativity, which is typical of the biological systems, is shown to be the macroscopic manifestation of a microscopic invariance law.

1. Introduction

Biological systems have so far been investigated along two complementary approaches. The first approach, which has been followed by a vast majority of people, splits the living systems into a wide number of their components and accumulates theoretical and experimental data about each of them; the problem of combining these into a working scheme is left to a later stage. The second approach attempts to formulate general macroscopic requirements about living matter which must be implemented by any microscopic realization. This second approach can indeed give an account of an important extra component of the living system which is usually out of reach of the first one, namely the timing of the different steps of biological processes.

A bridge between these seemingly opposite approaches could be built following the same philosophy as in many-body physics, successfully employed for understanding the properties of condensed matter.

According to this point of view, a crystal for instance cannot be described in terms of the constituent atoms alone. One must take into account also the existence

Istituto Nazionale di Fisica Nucleare, Sez. di Milano. 2 Gruppo Nazionale di Struttura della Materia del CNR, Milano. 3 also: Istituto Nazionale di Fisica Nucleate, Sez. di Milano. 4 also: Istituto Nazionale di Fisica Nucleate, Sez. di Napoli.

375

376 E. Del Giudice et al. / Collective behaviour o f biological systems

of other particles, namely phonons, plasmons and so on, which describe the collective modes of the system. When the crystal is broken into its elementary components, namely the original atoms, those peculiar particles disappear: a particle corresponding to a collective mode cannot emerge from a. consideration of the system's elementary components taken one by one; it is built up from the cooperation of many individual degrees of freedom.

Systems with collective modes are naturally described by field theories. Further- more, quantum theory has proven to be the only successful tool for describing atoms, molecules and their interactions.

On the other hand, as it has been stressed [i], "since all living organisms are made up of molecules and atoms it is only possible to explain the mechanism of biological processes at the molecular level by using quantum theory". Quantum field theory (QFT) appears then a useful tool for the microscopic analysis of living matter.

Living matter could be schematized as an assembly of water molecules and electric dipoles tied together into macromolecules. Our approach can be summarized as follows:

(i) fields are introduced to describe the molecule excitations; (ii) collective properties of the set of dipoles are derived in a QFT framework; (iii) such properties have to be compared with the macroscopic requirements on

living matter, derived in the framework of the global point of view mentioned at the beginning, in order to connect them in a consistent scheme;

(iv) finally such a scheme is compared with the observed behaviour. The lagrangian of a biological system is expected to be very complicated so that we concentrate our attention on symmetry considerations. The basic symmetry of the dipole interaction is the rotational symmetry. On the other hand, the existence of a preferred direction in the ground state (vacuum) of the system would break the symmetry and - not surprisingly in a QFT approach - induce the appearance of massless bosons (the Goldstone bosons) which play the role of carriers of a long-range correlation among the constituents. These Goidstone bosons are collective modes and we identify them with Fr6hlich's coherent waves. We attribute the emergence of the preferred direction in the ground state to Davydov's solitons that occur because of metabolic activity on the macromolecule chains in the absence of any interaction among the macromolecules. In this way we link together the spontaneous appearance of order and self-organization in living, matter with the microscopic symmetry properties of the dynamics.

The bridging between microscopic and macroscopic levels in living matter has been already proposed by integrating the conservative and dissipative mechanisms, as described by Davydov [ i ,2 ] and Fr/Ahlich [3] respectively [4]. The Raman spectroscopic evidence [5] has been the basis for such an integration. This scheme could be summarized as follows. The metabolic reaction energy output is supplied to the macromolecular chains (for instance proteins, DNA) in detinite small-sized

E. Del Giudice et al. / Collective behaviour o f biological systems 377

sites. The induced local deformation propagates along the chain as a Davydov soliton. These solitons simultaneously induce a displacement of the chain vibrational levels and a rearrangement of the surrounding water structure. This process goes on until a resonance between a couple of levels of the macromolecule and water is achieved. At this point the transition between the "charge" and "discharge" regimes takes place. The energy previously stored in soliton form can be dissipated into the water as a Fr6hlich polarization wave, which, increasingly enhanced by the ongoing metabolic reaction, travels through the cell contributing to its organization. The main features of the above model rely on a global consideration of the phenomenological evidence.

Let us reconsider now the problem from a more general point of view. The most important general result is that a living system has to be a dissipative

one, namely it must be an open system able to dissipate outwards all the incoming energy [7]. This general requirement however does not describe the actual story of the energy while flowing inside the system. A consideration of the phenomenology shows that the energy uptake occurs under very different conditions than its outflow. Actually, biological systems are endowed with energy by means of chemical reactions occurring at definite times and sites on the macromolecular material; this energy is transported a long distance and may be released in a quite large region. The dynamical regime of the energy uptake looks then different from the dynamical one governing the energy release, so that the general principle of dissipativity has to be articulated into a principle of energy charge and discharge.

Vibrational solitons have been shown to be the long-range energy carriers along the molecule chains. Davydov has worked out the quantum theory of such classically behaving objects.

On the other hand the discharge of energy and the subsequent creation of order inside living matter is described in terms of Fr/Shlich electric dipole waves. In the dipole set considered by Fr6hlich, an externally supplied flow of energy exceeding a given threshold populates just one vibrational mode. A giant electric polarization wave sweeps then through the living matter producing a long-range correlation and ordering inside the system.

The QFT approach presented here relies on symmetry considerations and is not dependent on particular dynamical assumptions [8]. It puts in more general terms those mechanisms of Bose condensation and coherence, which are the essential elements of the Davydov solitons and Fr/Shlich waves.

In this approach solitons are considered as localized Bose condensations of quanta [9, 10]. Then we could understand the phenomenological model discussed above as an oscillation between two different collective regimes, one characterized by a localized condensation and the other characterized by a homogeneous condensation.

We will give an account of this transition in terms of a transition between two different "vacua" ; the first "vacuum" being the water at low organization

378 E. Del Giudice et al. / Collective behaviour of biological systems

surrounding the macromolecule in the soliton regime, while the second "vacuum" is the highly organized water involved in the FriShlich regime.

The QFT scheme naturally brings us to understand some features recently detected

such as zero frequency modes. In sect. 2 we will give a general outline of the mathematical framework of the

QFT model. Sects. 3 and 4 are devoted to a detailed illustration of Davydov and Fr6hlich regimes respectively. Sect. 5 contains a discussion and conclusions. We present in the appendices the formulae and the mathematical proofs which are not

essential for understanding the line of thought only.

2. The quantum field theory framework

We schematize the living system under investigation as one or more long chains of macromolecules made of weakly bound monomers. These biochains are embedded in water. The model of the living system and its dynamics we consider are the ones presented in ref. [4] and which have been summarized in the introduction. The molecules of the chains and of water carry electric dipole moment. The field theory approach consists of an investigation of the molecular excitat ions dynamics.

In this section we sketch the quantum field theory framework where we embed both the Davydov and the Fr/Shlich regimes of the cell collective dynamics. We will show that the same mechanism of Bose condensation controls both regimes although

with different forms due to different geometrical setups. Let us start from the Davydov soliton regime. We consider biomolecules that can

be described as one-dimensional chains. The basic Heisenberg complex boson field @(¢, t) is introduced to describe the intramolecular excitation with an electric dipole moment directed along the one-dimensional chain. The field ~b obeys the Heisenberg

equation A (¢9)O(s ¢, t) = J[qJ(~:, t ) ] , (2.1)

where the kinetic operator A(0) and source terms J[qJ(s ¢, t)] are chosen in such a way that the dynamical equation (2.1) is a nonlinear Schr6dinger equation. We also

introduce the associated free-field equation for the boson field ~b:

a (c3)~b(s ¢, t ) = 0 . (2.2)

A solution of the dynamics (2.1) is obtained when the dynamical mapping

(al~,(~: ' t ) lb)=(alqt(cb(~, t))lb ) (2.3)

is found [1 1]. la) and Ib> are elements of the Fock space of the physical states of

the (I + I )-dimensional system under investigation. Any invariant transformation h for eq. (2.2) is dynamically rearranged by the

dynamical mapping (2.3) into an invariant transformation g for (2.1)

(alg~b(~ ¢, t ) lb)= (alqt(hdP(~, t ) ) lb) , (2.4)

E. Del Giudice et al. / Collective behaviour of biological systems 379

where

A(c3)qJ'(s¢ , t )= J[~ ' (s ¢, t )] , ~/,'(~¢, t) = g~b(s ¢, t ) , (2.5)

A(cg)~'(s¢ , t ) = 0 , ~b'(s ¢, t) = h~b(s ¢, t ) . (2.6)

In order to match the Davydov approach [2, 12], we are looking to the ground state expectation value of ~'(~¢, t) in the semi-classical (c-number) limit. Let us write

~b'(s ¢, t) = ~b(~ ¢, t) + f ( s c, t ) , (2.7)

where the c-number function f(~¢, t) is a solution of (2.2)

A(O) f (~ , t) = 0. (2.8)

In eq. (2.7) ~b is a field with a zero vacuum expectation value so that the solution f(~:, t) o feq . (2.8) is the vacuum expectation value of ~'(~ ¢, t) and defines a localized exciton condensation in the vacuum [9, 10].

As shown in the next section, eqs. (2.7) and (2.5) lead to a soliton solution only in 1 + I dimensions and we recover in this way the Davydov result about the intrinsic one-dimensionality of his soliton. When the system cannot be considered (I + 1 )-dimensional any longer because of the interplay between biomolecules and water, as discussed in the introduction, a soliton solution cannot be obtained from eq. (2.5) but the dynamics can still have the form (2.1) (written of course in 3+1 dimensions and in matrix form due to the different nature of the qJ-field, see sect. 4). The Fr6hlich regime could then emerge. The dynamics will be governed by the same invariant transformation (2.7) with f(~, t) now being a constant:

P(s ¢, t )= P(s ¢, t )+ const. (2.9)

P(s ¢, t) is the dipole wave field to which we will refer also as the polaron field in the following. In general, however, due to the complexity of the biological system the dynamics could be quite different from (2.1) in 3 + 1 dimensions. For this reason we are going to study in sect. 4 the Fr6hlich regime without specifying the details of the dynamics. We will only assume that the lagrangian of the system is SU(2) invariant and spontaneous breakdown of symmetry occurs. Both assumptions rest on reasonable phenomenological grounds since experiments show that the excited dipole moment (which in 3 + I dimensions is no longer frozen) plays a relevant role and polarization can be seen as an order parameter indicating a spontaneous breakdown of symmetry.

By using functional integration techniques we will show that, as expected from the Goldstone theorem [13], a massless collective mode P(s ¢, t), the dipole-wave quantum (i.e. the Goldstone boson) exists, while the dynamical rearrangement of the symmetry leads again to a condensation mechanism [14-16]. Thus the macroscopic coherence of the Fr6hlich regime is a consequence of the invariant transformation (2.9) under the economic assumption that Davydov and Fr6hlich regimes are governed just by the same dynamics in I + I and 3 + 1 dimensions, respectively.


Eq. (2.9) (invariant transformation for the P(~:, t) field equation) shows that a homogeneous condensation of the gapless mode P(~, t) occurs in 3 + 1 dimensions, whereas in the (1 + l)-dimensional case a localized condensation of excitons with nonzero effective mass is induced.

In sect. 4 the original SU(2) symmetry group is shown to be dynamically rearranged into the E(2) group (to which the transformation (2.9) belongs) through the dynamical map:

(a[~(¢, t)lb)=(alF[,~(~, t), P(~, t)]lb>. (2.10)

In eq. (2.10) the symbol F has been used in the place of ~ (see eq. (2.3)) to stress the differences between the (3 + I )- and the ( 1 + 1 )-dimensional case. Furthermore, the scattering matrix is shown to be independent of zero momentum polaron interactions due to its invariance under the transformation (2.9). This is analogous to the low-energy theorems in particle physics and in many-body theories and gives an account of the stability of the Fr6hlich regime under small external perturbations which excite "'soft" dipole-wave quanta.

Our considerations are in the infinite volume limit. Of course, realistic systems are of finite volume in both regimes. This fact, among other effects, influences the stability of the soliton in the Davydov regime and will result in a nonzero effective mass for the polaron in the Fr/Shlich regime. The finiteness of the soliton lifetime is relevant in the Davydov-Fr6hl ich transition. On the other hand, the not exactly gapless nature of the polaron turns into a finite range force in realistic system. Recent experiments indeed give evidence for large but finite range correlations.

In the following sections we discuss these finite volume effects also in connection with the occurrence of a threshold energy in the Fr6hlich regime. Let us also mention that, although in our treatment of the Davydov soliton the inclusion of a phonon field is not required since the beginning, nevertheless we will be led to consider a phonon equation in a very natural way, the phonon field being related to the density probability of the exciton field in the soliton state. We thus recover as a result the full Davydov description, with the further advantage of having an unified (microscopic) formalism for both the Davydov and the Fr6hlich regimes. Then it becomes clear how the same mechanism of boson condensation (see eqs. (2.7) and (2.9)) controls both regimes although with different effects due to different geometrical setups.

3. The Davydov regime

In this section we will give the details of the field theoretical description of the Davydov soliton as a localized Bose condensation of excitation quanta. Let us begin with an infinitely long one-dimensional chain of weakly bound monomers of mass M, each of them endowed with a dipole moment directed along the chain. At the end we will switch to the real case of a finite length. A quantum complex boson

E. Del Giudice et al. / Collective behaviour of biological systems 38 I

field operator 0(~, t) describes the excitation produced on this one-dimensional structure by any possible reason (chemical reactions, thermal or radiative supply, and so on). The field 0 is made dependent on the "cont inuum" coordinate ~: = x/a, where x is the position on the chain and a is the lattice equilibrium molecular distance. Notice that we have assumed the field as a complex boson one since the dipole degree of freedom is frozen in I + 1 dimensions. Following the Davydov line of thought, we introduce dynamics described by the nonlinear Schr6dinger equation

/ 0 2 |ih ~ - A+ J_--~.~|O(~, t ) = - G~'+(~ :, t)~(~ r, t)~b(~ r, t ) . (3.1)

\

k 0t 0 C /

A normal ordering is understood in the r.h.s., while in the l.h.s, the constant A is an energy term, which Davydov takes as

A = e - D + W-2J. (3.2)

e is the free exciton energy, D the change in the static interaction energy, W the chain deformation energy, J the resonance interaction energy between neighbouring monomers. We follow in this paper the approach outlined in refs. [9, 10], which corresponds to the LSZ asymptotic field formalism in relativistic QFT. Consequently the excitation ~ is expressed in terms of quasiparticles, whose fields are solutions of the free-field equation for the complex boson field <b:

0 2

Examples of quasi-particles are quasi-electrons and magnons in ferromagnets, phonons in crystals, and so on. In our case, the above statement, i.e. all observable phenomena are to be described in terms of quasi-particles, is embodied in the dynamical map [ I 1 ]

(alq,(~, t)]b)= (al q'[,/,(~, t)]lb), (3.4)

where ~ is a series of normal ordered products of ~. Usually the explicit form of the functional qt consistent with the dynamics (3.1) is obtained by means of perturbative techniques.

Following the procedure of refs. [9, 10] for the ,~b 4 model and the sine-Gordon model, we relate the c-number Davydov soliton

( E t ) h v - v ~ , Xc(~,t)=(½~)~/2expi ~-~ (~ - ~:o) [cosh ( ~ ( ~ - ~o- vt))]- ' (3.5a)

f lx,.(~, 0[ 2 d~' = 1, (3.5b)

to the fundamental state (vacuum) expectation value (v.e.v.) of the field 0(~:, t). The soliton Xc(~, t) in eq. (3.5) is a solution of the classical nonlinear Schr6dinger

382 E. Del Giudice et aL / Collective behaviour o f biological systems

equation similar to eq. (3.1), but with the quantum operator field 4,(s c, t) replaced by the c-number function Xc(~:, t). Moreover in eq. (3.5)

h2o 2 E~ = A + T ~ - . - JI.~ 2 ,

G = - - . /.t 4J (3.6)

Now we assume that a t ransformation B exists such that

Xc(~:, t) = lim (0[B0(~, t)10), (3.7) ta~0

where h is the Planck constant. B is assumed to be an invariant t ransformation for the Heisenberg equation (3.1)

( i h - - - A + J O ' (~ , t )=-GO'+(~ , t )O ' (~ , t )O ' (~ , t ) , Ot

O'(s r, t) = B~/,(s r, t ) . (3.8)

The dynamical map (3.4) suggests that B corresponds to a t ransformat ion/3 for the quasi-particle field ~b such that (see also eq. (2.4))

(o1 n¢,(~:, t)[0) -- (01 ~'[/3,b (~:, t)][0). (3.9)

Requiring that /3 be a canonical t ransformation, we get

[~b(~, t), q~(~:', t)] = [~b'(~:, t), ~'(~:', t)] = i8(~:- ~ ' ) ,

[~b(~, t), ~b(~:', t)] = [d/(~, t), ~b'(~:', t ) ] = 0 , (3.10)

etc., where

~b'(¢, t) =/3~b(s r, t ) . (3.11)

Since B is an invariant t ransformation for the t/,-field equation, /3 has to be an invariant t ransformation for the ~b-field, too, i.e.

j / h ~ - A + ~b (st, t) = 0.

Eqs. (3.10) and (3.12) are satisfied when d~ transforms as follows:

~b'(~, t) = fld~(s c, t) = #b(~:, t) +f(~:, t ) ,

where f ( s e, t) is a solution of the free field equation (3.3), i.e.

at 82 ( ih O--- A + J-d-~)f(~, t) =O ,

(3.12)

(3.13)

(3.14)


such as

f_ (¢ , t) = C e - " ~ - ~ " e "~¢/2J-EJ/~) , (3.15a)

f+(~, t) = C e ~'(~-~" e i ( ~ v ~ / 2 J - E d / ~ ) , (3.15b)

where Eo and tz are given in eq. (3.6). A superposition of solutions (3.15) would be divergent both for x - ~ + m and

x ~ - o o ( x = ¢ - v t ) . We will choose only one of the solutions (3.15), say (3.15a), so that f(¢, t) is limited at least at one side of the chain.

This problem will be discussed again at the end of this section, and in appendix B, where we consider the realistic, finite length situation.

All the above arguments rest on the validity of the condition (3.7) that has been actually shown to hold in refs. [9, 10] where the canonical transformation (3.13) is implemented in eq. (3.9)*. In ref. [9] a tree approximation computat ion is presented which explicitly shows how the dynamical rearrangement (3.9) occurs. The dynamical rearrangement B ~ / 3 is induced by the nonlinearity of the dynamics (3.1). Actually the factor (cosh x) -t in eq. (3.5) can be expressed as a power series:

i - 2 ~. ( - I ) ~ e -<2"+1)~ . (3.16)

cosh x n=o

A perturbative analysis would show how the f(~:, t) in the argument of ~[~b(~, t )+ f ( s c, t)] combine to reproduce eqs. (3.5a) and (3.16)) as an effect of the nonlinear dynamics (3.1). The exponential factor in eq. (3.15) makes it consistent with the value (3.6) of Eo so that the soliton is energetically preferred when compared with the usual (plane wave) exciton. This last point has been discussed in details in refs. [1,2, 12].

We will express the soliton solution in terms of a coherent state representation [ 10]. The generator D of the transformation (3.13) in the v = 0 frame of reference is

o=-f d¢[g(¢, t)cb(¢, t ) - g , ( ¢ , t)c~(¢, t ) ] , (3.17)

where

g(~:, t ) = 0(~:)fo(~:, t ) ,

fo(~:, t ) : f (~ : , t)l~=o, (3.18)

and 0(~) is the step function. Using the canonical commutat ion relation (3.10), it can be shown that

~b'(~:, t ) = e-~n4~(~:, t) e ~D . (3.19)

" It has been shown in ref. [17] that in order that B preserve the canonical commutation relation for O(s c, t) it is necessary to include the bound state modes in the Fock space and introduce "quantum coordinates" (collective coordinates).


Then we have

(01 e-'°4~(~, t) e ' ° [0 )= (fo[OS(~:, t ) l fo)= fo(~:, t ) , (3.20)

having in t roduced the state [fo) as

[fo) -- e '°[0) • (3.21)

]fo) is a coherent state for ~b(¢, t); if ak is the annihilat ion opera tor for ~b(~, t) and the c -number function a(k) is related (in a suitable ~:-domain [10]) to the fo(s c, t) Fourier ampli tude, we have

ak I fo) = a ( k)lfo) , (3.22)

(folfo) = 1. (3.23)

Moreover , the dynamical map (3.4) together with eq. (3.20) gives

(f,,lto(~, t)lf, ,)= ~[fo(~ ¢, t ) ] , (3.24)

(fold[to(~ ¢, t)]]fo) = J[ ~[fo(~: , / ) ] ] . (3.25)

Use o f eqs. (3.24) and (3.25) leads to the classical nonl inear Schr6dinger equat ion

i ±-A ÷J e[f] = -GI (3.26) 0t

once a boost to the original frame of reference has been performed. Eq. (3.26) coincides with the Davydov equat ion (2.10) of ref. [2] and is actually solved by the soliton (3.5)

~ [ f ( ~ , t)] = gc(~:, t ) . (3.27)

This equat ion fixes the constant factor C in eq. (3.15)

C = ( 2 / z ) I /2 e ~'e,, e -~°~,~/2s .

The relat ionship between the coherent state representat ion (3.27) (see also eqs. (3.24) and (3.21)) in terms of the ~b-fieid and the more convent ional representat ion in terms of Heisenberg field to is discussed in ref. [10].

In conclusion the above procedure tells us that the B- t ransformat ion defined by eq. (3. i 3) induces a coherent dependen t condensat ion of excitons 4~ (s c, t), control led by the funct ion f ( s ¢, t), whose square modulus is related to the density of condensed excitons.

The soliton A,,(s c, t) emerges from the local condensat ion o f the const i tuent field ~b as a consequence of the nonl inear self-coupling of the Heisenberg exciton field ~/,(s c, t). The nonl inear origin o f the soliton solution is shown by its d isappearance when the nonl inear coupling constant G goes to zero. Notice also that eq. (3.26) exhibits a nontrivial self-potential ]tO[: due to the (cosh x) -~ factor in (3.5). The Davydov soliton appears then as a dynamical ly self-sufficient entity. Scott has been the first to stress this point o f view [18], a l though in a different framework.


The analysis could look at first different from the original approach proposed by

Davydov since we have introduced only one excitation field whereas Davydov introduces from the beginning a phonon field on the same ground as the excitation one. In appendix A we will show that a phonon equation for a deformation propagating with a velocity v of the soliton can be deduced. The economy of the present approach is then apparent: in a natural way the theory introduces a deformation field (the phonon) localized around the soliton center and propagating with the soliton velocity. This result can be understood intuitively: we may expect that a self-interacting intramolecular excitation field ~ could produce conformational changes in the chain lattice which are just described by a phonon field. The conventional approach separates from the very beginning the vibrational excitation and the corresponding longitudinal sound wave; this requires additional assumptions such as a coupling between the phonon and the vibrational excitation. These two excitations belong to the same dynamical self-sufficient entity because the longitudinal sound wave is induced by a localized vibrational excitation and at the same

time acts as a potential well for it. The soliton can be then considered as a response of a one-dimensional chain to

any external, although weak, localized perturbation (for a given class of parameters). In this way an incoherent energy supply is transformed by the nonlinear dynamics (3.1) into a first level coherent form via the localized Bose condensation of excitons (or, if one prefers, phonons) which is the actual Davydov soliton. In appendix A we discuss the equivalence between our approach and the Davydov approach. Thanks to the equivalence of the two approaches we are able to recover within our framework all the results of Davydov, such as those concerning the soliton stability, the sol i ton-photon interaction and the attraction of the electrons by the self- generated potential wells, leading to quasi-superconducting properties of the

molecular chain.

We want here to make a final point on the effect of the finite length of the molecular chain on the stability properties of the soliton. The stability of the soliton (3.5) implies the stability of the chain localized deformation and vice versa (see appendix A, eqs. (A.7) and (A.10)). The stability of the deformation is analyzed in appendix B where it is shown that for a chain of a finite length the local deformation is not stable against decay in a number of constituents (phonons); consequently the Davydov soliton Xc will also decay into a number of free excitons (electric dipole vibrations).

4. The FrGhlich regime

While Davydov solitons travel along a one-dimensional chain, an electric polarization is slowly induced in the surrounding water. The physical basis of such a mechanism will be discussed elsewhere. Here we limit ourselves to quote the experimental evidence that the surrounding water can be brought into an electret


state by very low electric fields at low frequencies, once an activation energy of approximately the same amount required for the soliton formation has been supplied to the biomolecule [19, 20]. This fact suggests that biomolecules with solitons on them create almost spontaneously an electrical polarization into the water. Due to such an interplay between biomolecules and water dipoles the "'vacuum" of our system changes, so that the confinement in 1 + l dimensions is increasingly lost and a (3 + 1 )-dimensional system of dipoles becomes the subject of our story. Since the dipole degrees of freedom have been unfrozen, let the molecular vibrational field be denoted by ~kj(x, t) where the suffix j specifies the dipole quantum number. Without loss of generality we assume for simplicity that the suffix j can only run over two values. The Heisenberg field ~,(x) is thus introduced as a (spinor-like) doublet field

~#(x) = ( ~#~ (x)'~ (4. I) \ ~ ( x ) / '

where the notation x-= (~, t) is introduced and arrows denote the dipole internal quantum number: j = 1', ,I,. Use of a representation of higher dimensionality (more than two values for j ) requires only small changes in the following discussion, cf. refs. [14, 16]. We will describe the system by a lagrangian invariant under the group of dipole rotations, i.e. the SU(2) group of rotations for our doublet field (4.1)

~(x) ~ e'~,",/2~(x) , (4.2)

where )t~ are real parameters and o', i = I, 2, 3, are the Pauli matrices. In the following we will concentrate on the symmetry structure of the problem and use the path- integral formalism [14-16], so that we are not compelled to introduce an explicit form of the lagrangian. Then, our conclusion will be independent of the details of the dynamics. Again we start with an infinite volume; this assumption will be dropped at the end.

We introduce the generating functional

W [ J , j , ~ ] = N f [dO][dO*]exp(i f d'x(~[q'(x)]

+ J* ( x ) O ( x ) + ~b ~ ( x ) J ( x ) + j r ( x ) Dt~fl(x)

+ D~(x)j(x) + D ~ ( x ) n ( x ) - leDge'(x))) . (4.3)

(u) D~ (x) (a = + , - , 3 ) is the dipole density operator, whose explicit form is not requ!red in the following.

Under infinitesimal SU(2) transformation (4.2)

(i) D~, ( x ) ~ (i~ ~k~ D, (x)+ Aje,~kD~, (x) , (4.4)

with e,jk the completely antisymmetric tensor; as usual

~ ~ iD~:'( x ) D~, (x)=D~I~(x)+


In eq. (4.3) N is given by

N=f [d4,][dq,+]exp(i f d'x(Le[q,(x)]-ieD~'(x))). (4.5)

tO and 4 + and the sources J+, J anticommute, whereas the sources j a n d j commute. The polarized surrounding water, which acts as an external field, is described by the e-term in eqs. (4.3) and (4.5). This term is actually a symmetry breaking one and the limit e -~ 0 should be performed at the end of the computations. As expected from the Goldstone theorem [13] it can be proved (see appendix C) that gapless modes exist when a spontaneous breakdown of SU(2) symmetry occurs, i.e. when the fundamental state (vacuum) expectation value (D~31(x)) is different from zero:

( D ~ 3 ' ( x ) ) = P ~ 0. (4.6)

The existence of such a polarization P # 0 is the basic point of the Fr6hlich regime [3]. The Davydov soliton and the induced electret in the water, actually producing the condition (4.6), switch on such a regime. Eq. (4.6) is consistent with the original SU(2) invariance of the lagrangian only if massless modes (Goldstone particles [ 13]) exist. They appear as collective excitations describing the long-range correlation forces which align the individual molecular dipoles in the third direction.

Once the breakdown of SU(2) symmetry (eq. (4.6)) is introduced as an effect of the water polarization, the dynamics by itself creates Goldstone modes which are bound states of the 4,(x) field. The quasi-particle set includes thus a quasi-molecular field d,(x) and a gapless boson field P ( x ) which is a bound state of 0-fields. The field equations for the quasi-particle doublet field ~b and the polaron P ( x ) are

with

A ( ~ ) ~ ( x ) = 0 , (4.7)

K(`9)P(x) = 0 , (4.8)

f d3k P ( x ) = J (27r)3/2 Pk eit~ x-i°'d , (4.9)

K(`9) = - ( i ~ t + to(`9)) . (4.10)

The P ( x ) commutation relations are:

[P(x), P+(y)],=,, = 8(x-y), [ P(x) , P(y)] = [ P+(x) , P+ (y)] = 0. (4.1 1 )

In eqs. (4.7) and (4.10) the differential operators A (`9) and K (,9) are determined once the dynamics of the system is specified. Use of the Lehmann-Symanzik-Zimmerman

F~(i)( x (LSZ) formula gives the S-matrix and the dipole field ~ , , ) as a function of the

388 E. Del Giudice et aL / Collective behaviour of biological systems

quasi-particle fields ~b, ~ , P and P÷:

S(~b, ~b', P, P * ) = ( : exp [-ia(~b, ~b +, P, P+)]:) , (4.12)

S D ° ) ( x , ~b, da' P, P+) ~;) • + , = (D~, (x). exp [ - ia (cb , d~ , P, P*)] : ) , (4.13)

where

a(dp, dp', P, P+) = f d 4 x [ p - t / 2 p ( x ) K ( 5 ) D ~ )(x)

+ p - ' / 2 D ~ + ) ( x ) K ( - ' ~ ) P + ( x ) + Z ' / 2 ¢b* (x )a (~ )q j ( x )

+ Z - ' / 2 ~ O + ( x ) A ( - ~ ) 4 ~ ( x ) ] , i = 1 , 2 , 3 . (4.14)

Z is the molecular wave function renormalization constant. The dynamical mappings (4.12) and (4.13) [see eq. (2.10)] give us an insight into the dynamical rearrangement of the symmetry by showing how the quasi-particle fields transform when the Heisenberg field qJ(x) is t ransformed according to eq. (4.2). In appendix C we show that the quasi-particle field transformations are

P'(x , a ) = P ( x ) + iA ,(2t P) '/2 ,

P " (x, A) : P * ( x ) - iA,(½P) '/2 ,

~'(x, a ) = ,~(x) ,

~b'+ (x, A) = ~b-(x), (4.15)

for A2 = A3 = O;

for A, = A 3 = O;

P ' ( x , A ) P ( x ) - ' ,/2 = ; t : ( ~ P ) ,

P'*(x , A) = P + ( x ) - A 2 ( ~ P ) '/2 ,

,~'(x, ~) = ~ ( x ) ,

4,'+(x, ,~) = ¢ + ( x ) , (4.16)

P'(x , A) = e w2A~p(x) ,

P'~ (x, A ) = e + " / 2 ~ P + ( x ) ,

qb'(x, A ) = e~i/2~'h"~c~(x) ,

~b'~ (x, A) = ~+(x) e -~i/2)~'~ , (4.17)

for A, = A3=0.

The transformations (4.15)-(4.17) belong to the E(2) group which is a contraction of SU(2) 116]. The dipole system does not appear as SU(2) rotational invariant but exhibits an (unbroken) U(I ) symmetry around the third direction (the total polariz-


ation direction) (see eqs. (4.17)) and two translational symmetries (see eqs. (4.15) and (4.16)) which govern the spatially homogeneous boson condensation of P(x) (compare it with the localized condensation of the ~(s ~, t) field in the Davydov regime, eq. (3.13)). To better understand the role of the P(x) condensation, we look at the generators of the E(2) transformations: from eqs. (4.15)-(4.17) they are

f d3x[P(x)f(x)+ P+(x)f*(x)], (4.18a) D~I) (½p),/2 f - -

I

D} 2)= -i(~P) '/2 f d3 x[ P ( x ) f ( x ) - P+(x)f*(x)] , (4.18b)

I d3x[~+(x)½°'3tb(x) - P+(x)P(x)]. (4.18c) D(3)=

f (x ) is a square-integrable function which regularizes the integrals (4.18). The translations in (4.15), (4.16) should be indeed understood as the limit f o r f ( x ) ~ I of

P(x)-~ P'(x, ;~) =/im [P(x)+ i,~,f(x)(½P)'/:], (4.19)

etc. The invariance of (4.8) under (4.19) implies that f must be a solution of the

equation

g (3)f(x) = 0. (4.20)

D~" in eqs. (4.18) are then time-independent so that no energy supply is needed to condensate the P(x) quanta. Notice that D ~) have the form (4.18) when written in terms of quasi-particles, otherwise they are bilinear:

= f d3x ~+(x)~o'i~b(x), (4.21) D (,)

and satisfy the original SU(2) algebra

[D (i), D (j)] = ieok D~k) . (4.22)

The generators D; ~) in (4.18) obey, on the contrary, the E(2) algebra

( I ) [D r , D~ 2)]=const~,

[D~ ' ) , /~q) l - "~(~) _j. j - + l u j . (4.23)

The fields 4~(x) and ~b+(x) in (4.15) and (4.16) remain unchanged. Rotations around the first and the second axis would now require an energy supply. The dipole quantum number is carried away from the ~b-field by the Goldstone field P(x) which does undergo an invariant transformation. Since invariant transformations (4.15)- (4.17) do not involve expenditure of energy (the generators (4.18) are in fact time independent), no supply of energy is required for the dipole ordering which is


controlled by the condensation of the P(x) quanta (eqs. (4.15), (4.16)). The system can thus dissipate outward all the incoming energy since no energy is required to reach its ordered state. Of course, we are not considering here temperature effects, finite volume effects and other effects which are usually disregarded in the description of a living system as a dissipative one (see sect. 1 ). When such effects are disregarded, the above arguments show that what appears macroscopically as dissipativity is the macroscopic manifestation of a microscopic fundamental invariance law. To better understand how the ordered state of polarization P (see eqs. (4.16), (4.18c)) is described through eqs. (4.15) and (4.16) as a condensation of the gapless mode P(x) in the ground state 10), observe that

(01 e a'?" e ,D;,.p, (x)P(x) e ~°','' e'° ' ," '10) = Plf(x) l ~- , P, (4.24)

with

Eq. (4.24) can also be written as

PklO) = 0. (4.25)

( f l P ' ( x ) P ( x ) l y ) - - Plf(x)] 2 , P, (4.26) f • I

I f ) -- e " ' " " e't '""'10) - (4 .27)

We thus see that a coherent state arises from the condensation transformation:

P k l f ) = P ' / 9 ; , I f ) , ( f ! f ) = t . (4.281

J~, is the Fourier amplitude o f f ( x ) introduced in (4.19) and (4.20). Since D} ~ and D} -'~ are time independent, no energy is involved in the transformation (4.27) and

thus lf), like 10), is a minimum energy state, f ( x ) acts as a volume cut-off in (4.18) so that contributions of the order I / V, as the volume V tends to infinity, are missing in D} ~, i = i ,2. Here is the origin of the dynamical rearrangement SU(2)-+ E(2) (see eqs. (4.22) and (4.23)). When these missing infrared contributions are considered, the SU(2) algebra is recovered, as it is shown in detail in refs. [14-16]. The suppression of the infrared terms has deep phenomenological consequences; the

D ~ and r~2~ generators r ~ r , dynamically introduced by the rearrangement SU(2)--, E(2), are actually at the origin of the coherent structure of the ground state (vacuum) in the Fr6hlich regime. On the other hand, the S-matrix is invariant under the D} ~

D t2~ induced translations of the P(x) field in the limit f (x ) -~ 1 (see eqs. (4.15), and r (4.16)). The invariance of the S-matrix (see appendix C) implies its independence of a~, i = 1,2, 3. The independence of &, i = i, 2, implies that

S(4)(x) ,4) ' (x) ,P(x) ,P+(x))=S(OS(x),4) ' (x) ,;~P(x),aP*(x)) . (4.29)

Consequently S is independent of P(x) in the zero momentum limit. This statement is called the low-energy theorem by analogy with many-body theories (e.g. in

E. Del Giudice et al./ Collective behaviour of biological systems 391

ferromagnetism [21]) and elementary particle physics [22, 23]). By this low-energy theorem (4.29) the Fr6hlich regime is stable against external perturbations inducing

soft ( low-momentum) modes. Long-range correlations are then present in the dipole system corresponding to the long-range forces mediated by the quanta P(x). In the infinite volume limit (f(x)--> I ) these quanta are gapless as shown above. However, in a realistic system the volume is finite and surface effects are to be considered. The condition (4.6) will be approximately valid only in a convenient internal domain with possible "distortions" near the boundary surfaces. Let us write

(D~'(x)).--ei f '/" d3x f d3p • J-I/• (2~) 3e ipxA(e 'p 'p°=O)

= ei I d3p $,7( p)A(e, p, Po = 0) , (4.30)

where 77 = V ~/3 and /~ , (p ) is a function which approaches/~(p) as r/-~ 0. Eq. (4.30)

reproduces eq. (C.4) of appendix C in the r/-+0 limit, having performed the time and the Po integrations.

Since l im. ~o f `5,(P)f(P) d3p =f(O) - lim,~o f $( p - r / ) f (p) d3p we approximate `5~(p) by ,5( p - 77) in eq. (4.30) for small 77. Then we obtain (see eq. (C.7) of appendix C)

( D ~ 3 ) ( x ) ) ~ = eid(e, 7, Po = O)

( ' , ) =eip(~7) ~o(p=~7)+iea to(p=T1)_iea at r / 0. (4.31)

In eq. (4.31) w(p = 7/) is a sort of effective mass for the bound state mode arising as a finite volume effect. As r/--, 0, or, alternatively, "far from the boundaries", to(p = r/)--, 0 and the condition (4.6) is satisfied.

Due to the nonzero effective mass of the quantum P(x) , finite (but long) range correlation forces are expected in realistic (finite volume) systems. An independent confirmation for the dressing of P(x) soft modes is provided by eq. (4.19) which exhibits a polaron condensation, i.e. the coherence domain extending over a range r /~ when the appropriate cut-off function fn (x ) is defined.

As the coherence domain size and the effective polaron mass are related, the domain volume is controlled by the polaron effective mass: a polaron of zero effective mass would correspond to an unbounded homogeneous Fr6hlich regime.

We will discuss elsewhere the mechanism which prevents the polaron from having a zero effective mass, thus creating (realistic) finite volume systems. When the polaron has nonzero effective mass its field equation is

K, UOP(x) = 0, (4.32)

where 71 denotes the dependence of K(a) on the effective mass. The requirement that

P(x) ~ P(x )+ f , ( x ) × const (4.33)


be an invariant transformation (see eq. (4.19)) compels f , ( x ) to be a solution of

eq. (4.32). The S-matrix independence of (4.33) leads again to eq. (4.29) where

c~P(x) is now replaced by Ko(O)P(x). This means that an energy threshold (the effective mass) is introduced in the system in agreement with the prediction of the original Fr6hlich model. It is interesting to note that the threshold is lower for larger

systems since the domain size goes as l/-q.

When P(x) has the same frequency in two finite and contiguous systems dipole

waves can propagate through the boundary without expenditure of energy since eq. (4.33) is an invariant transformation in both systems; their separation would increase

the energy and therefore an attractive force develops. In the opposite case of different

frequencies eq. (4.33) is not an invariant transformation in the global system. Eq. (4.32) is different in the two systems separately and thus different J~ functions must be used in the two cases. The propagation of a dipole wave through the boundary

will then require expenditure of energy and a repulsive force will develop.

5. Discussion and conclusions

The dynamical scheme discussed above can be summarized as follows. Energy

released by the chemical metabolic reactions can be stored on the macromolecule

chains in soliton form. Solitons induce an electret state in the surrounding water.

The dipole rotational symmetry is consequently broken and coherent electric dipole

waves (Goldstone modes) propagate in the system. Let us now discuss the phenomenological aspects related to the above picture. It

has been proven that a localized excitation - such as that induced by an interaction with small metabolites - produce solitons on a chain with a positive anharmonic

coupling among its vibrational modes [1,2]. Spectroscopic evidence for such a

positive anharmonicity is available for different chains such as a-helix proteins [24-26], DNA in B-forms [4, 27], and other unidimensional systems [28, 29].

Moreover, an activation energy of about some tenth of an eV has been calculated

for a soliton on a-helix proteins. An activation energy of just the same order of magnitude has been detected for the water electret around almost all the biopolymers [19]. Again, this electret is characterized by very long relaxation times [20] which correspond to the existence of zero frequency oscillations, namely the Goldstone

modes discussed in sect. 3 [30]. The Raman investigation [5] of bacterial cells at room temperature under particular nutritional conditions [31] shows that Raman

lines appear when cells are metabolically active, i.e. when the system is open to an

external energy flow. Frequencies and intensities of the Raman lines are strongly time dependent and

an important point is that no lines at all appear in the first part of the life cycle.

A correspondence between spectroscopic and biochemical trends has been recently pointed out [32], suggesting that the Raman pattern and the biochemical one both

arise from a unique clock powered by the cell dynamics.


Furthermore the anomalous antistokes ratios measured for the low Raman frequency lines (below 200cm -~) [5] prove that the system is far from thermal equilibrium. This pattern suggests that a cycle based on alternating Davydov and Fr/Shlich regimes, as discussed in previous sections, is at work.

First of all, notice that absence of metabolic activity does not allow any detectable electric signal. We have seen above that a soliton can be the consequence of a metabolic reaction at a macromolecule site. Then the formation of solitons is a prerequisite for the appearance of Raman lines.

The empty spectrum in the initial part of the life cycle is the manifestation of the charge phase, governed by the Davydov regime.

The subsequent appearance of lines marks the transition to the Fr6hlich regime, after an electret has been built up and the energy threshold corresponding to the system size has been reached.

More recently a course of experiments on rouleau formation in blood [33]* has

proven that the interaction among erythrocytes (red blood cells) behaves according to the Fr/Shlich theory requirements. Actually, long-range attractive forces have been observed to be effective between cells at distances greater by several orders of magnitude than the range of chemical forces. Such an interaction leads to the formation of rouleaux of stacked erythrocytes, which is responsible for the process of blood coagulation and increases anomalously in a number of blood diseases.

This long-range interaction disappears when: (i) the cells are depleted by the source of energy but is restored when the cell metabolic energy is supplied again; (ii) the cell membrane potential is reduced to zero, eliminating the high electric field (up to 108 V/m) associated and the consequent effects on the dynamics of the system; (iii) the membrane of the cell is disorganized by addition of poisons, i.e. organization of the molecules of the membrane is ruined. Moreover the interaction has been found to be specific, since a preferential rouleau formation has been observed among cells of the same species when a mixture of different types of cells have been examined. Again this specificity, which could explain a number of yet unexplained cellular processes (embryo cell differentiation and separation) is expected if a difference in the frequencies of Fr6hlich waves exists in different cells. Considerable experimental evidence suggests then, among many other important things, that the Fr6hlich regime starts when the energy flow exceeds a threshold [3, 30]. This is a strong experimental support for the validity of the low-energy theorem.

In conclusion we have worked out a QFT framework to integrate the opposite and complementary functions of charge and discharge of energy in biological systems.

We have discussed the transition between the regime characterized by the pre- dominance of solitons on the molecular one-dimensional chains and the regime characterized by a homogeneous condensation of polarons in the cell medium. A

* A model based on the ideas proposed in this paper is presented by Del Giudice et al. in this reference.


preliminary discussion of the present theoretical scheme has been presented in refs. [37]. We have yet to tackle the major problem of closing the cycle, namely the transition between the second regime and the first one, or the end of the discharge

phase and the onset of the charge again. The solution of this problem is connected with the definition of the cell size which

is fixed by the end of the expansion of the correlated region powered by the Fr6hlich

condensation. We have then to modify the simplified theory given in sect. 4 to this more realistic

case, getting in this way the conditions for the end of the process and his confinement inside the cell volume. A mechanism for this transition is under investigation and

will be discussed in a forthcoming paper.

It is our pleasure to thank A.S. l)avydov, H. Fr6hlich, G. Parisi and H. Umezawa for useful discussions and comments.

Note added in proof

Experimental evidence about the role of the solvent-water on the dynamics of a biopolymer, agarose, has been recently given by the study of Rayleigh scattering of M6ssbauer radiation by Albanese, Deriu and Ugozzoli [38]. Their results show that the influence of water on the dynamics of the polymer chain is a function of the percentage weight 6 of water content and that two different dynamical regimes exist below and above ,8 = 20%. Above ~ = 20% a privileged direction with highest mean displacements of the chain atoms is observed perpendicular to the polymer chain, indicating a large anisotropy in the dynamics of the systems.

Appendix A

The phonon equation for the chain deformation and the Davydov equations (2.5a)

and (2.5b) of ref. [2] are obtained as follows. Consider the density probability of ~b in the soliton state

pJ(~, t ) = -/(fl~0+(~, t)q,(~, t ) [ f ) = ylg, (~ ¢, 012 , (A.I)

where the constant 3' is related to the other parameters of the theory. From eq.

(3.5a) we get 1 (A.2)

pl(~, t) = 7 ~ cosh ~ [~(~: _ ~,,_ vt)]"

It can be easily checked that pJ(~, t) satisfies the phonon equation

_ v 2 p t ( ~ , t) = 0 . (A.3)

To explicitly recover in a formal way both the original Davydov equations one more


step is required. Let vac be the sound wave velocity on the chain when solitons are absent.

v,c=(wlM) '/2 , (A.4)

with w the chain elasticity coefficient and

V 2 = V~c- V~c(l - s2), (A.5)

13 s = - - 4 I . ( A . 6 )

/)at: Eq. (A.3) can be rewritten as

( ~2 ~2) 2K ~ixc(~,t),2= 0 (A.7)

which is the Davydov equation (2.5b) of ref. [2]. In eq. (A.7) the parameters are

defined by K =½w(I - s2)y, (A.8)

3 -<3-~/3f(~' t) = p i (s ¢, t ) . (A.9)

Furthermore, by combining eqs. (A.I), (A.8), (3.6), (3.26) and (3.27), we get

where

32 L _ A + - 2 K - - ih <3 t J-~-52

aCi/(s~> t)] @ x<(s~, t) = °,

K 2

(A.IO)

# = w(l - s 2 ) j " (A.I 1 )

Eq. (A. 10) is just the Davydov eq. (2.5a) of ref. [2]. In this way the equivalence between our approach and the original one becomes apparent. Eqs. (A.7) and (A. 10) are c-number equations describing the coherent collective behaviour of the underly- ing quantum system of excitons which is governed by the nonlinear dynamics (3.1), in the presence of the local boson condensation (3.13).

A p p e n d i x B

The stability of the soliton (3.5) implies the stability of the local deformation/3 j (or pl) and vice versa (see eqs. (A.7) and (A. 10)). For this reason we analyze the stability of the deformation /3Y:

K flf(~, t ) - w ( l ) - s 2------~ tgh[/z(sc-s¢°- vt)]+cOnst ' (B.I)

as obtained from eqs. (A.9) and (A.2). For the sake of simplicity we choose the value zero for the constant in eq. (B.I). The kink /3s(s ~, t) can be also described in


terms of a local condensation of quantum elementary constituents (phonons) [9]. Thus, the stability of the deformation /3t(¢, t) is studied by evaluating the overlap between the kink state 1/3) at t = : ~ and a state of a finite number of constituent phonons [10, 34]. We compute (see ref. [10]).

<01/3>1 . . . . = exp [ - ~ f dk'~" '2wk] , (B.2) L 3 _1

where we have used

I/3> = e ' ( ~ 1 0 ) , B.3)

G = - I dY/3J(Y)~(Y' t ) . (B.4)

In eq. (B.2) /~i is the Fourier amplitude of/3J(~ :, t) and w k = ( m 2 + k 2 ) j/e. In eq.

(B.3) the state [0) is the free field vacuum state and G is the generator of the kink state; in eq. (B.4) y = ~:- ~:o- vt and/3 is the Heisenberg deformation field operator whose asymptotic field has an effective mass m.

Since

~c : / I/3'12 d), : f I/~'l-~ dk < / o~kl~'12 dk, (B.5)

the resulting transition probability is zero. Of course this is related to the existence of a conserved topological charge K = / 3 ' ( y = + ~ ) - / 3 ' ( y = - ~ ) ~ 0, i.e. to the fact that the tratlsformation (B.3) would require an infinite amount of energy to be implemented. Observe however that the above argument fails if the integrations in (B.5) are not over an infinite range. Thus, for a chain of finite length the transition probability is not zero and the local deformation is not stable against decay in a number of constituents (phonons); and this is true also for the Davydov soliton X,

which will decay in a number of free excitons (electric dipole vibrations).

Appendix C

In this appendix we show the existence of the Goldstone mode [13], i.e. the dipole-wave quantum or polaron, when spontaneous breakdown of SU(2) symmetry, eq. (4.6), occurs. The main steps of the proof are similar to those followed in ref. [15] where spontaneous breakdown of SU(2) symmetry in ferromagnets is studied by means of path integral techniques. The same problem is handled in a perturbative approach in ref. [35].

We choose J = n = 0 in eq. (4.3) and then we apply the transformation (4.2) to the numerator of eq. (4.3). This change of variables should not influence the integration, so that

- 0 . (c.1)

E. Del Giudice et al. / Collective behaviour of biological .systems 397

By operat ing on eq. (C. ! ) the functional derivatives 6/6j (y) and 6/~j+(y), we obtain a t j = j " = 0

(I) (2) ( D , (y))~ = ( 0 . D , (y)). = (C.2)

f 4 (2) (I e d x ( D , ( x ) D , ) ( y ) ) , = 0 , (C.3)

(3) _I 4 11) (I) ( D , (y)), = e d x(D~, ( x ) D , (y))~, (C.4)

I 4 (21 (2) ( D ~ ( y ) ) , = e d x ( D , (x)D~, (y))~ . (C.5)

Moreover, by applying the operators (6 /6 j* ( z ) ) (6 /~ j (y ) ) and (6 /6 j ( z ) ) (6 /6 j+(y) ) to eq. (C.I) , putting j = 0 and subtracting we also have

(I) (I) -- (2) (2) (D,,, ( x ) D , (y))~ (C.6) -(D, ( x ) D , (y)) , .

In eqs. (C.2)-(C.6) the symbol ( . . . ) , . . j j denotes a functional average and a suffix is unders tood to be zero when missing.

Let us now write

f d~'p e_,px_~., ( [ 1 1 ) (D~ ' (x )D~ ' (y ) ) , .= i ~ p, p)[po_to~+ie a Po+tov - i e a

+ c o n t i n u u m contribution i = 1 ,2 , (C.7) with

p(x - y) =- - p . (x - y ) + po(tx - t,')"

In eq. (C.7) top is the energy of a quasi-particle which is a bound state o f the

molecular field ~,. Eq. (C.6) explicitly shows the symmetry l ,~,2 in the dipole fields (~)

D+ (x) through the equat ions

P,(P)= P2(P) , (C.8)

and a, = a2 where p, are the spectral functions. Moreover

P( e ) =- ( D~'(x)),. = ieA(e, 0) , (C.9)

where

Since

j ( e , p ) = p ( p ) ( p l 1 ) o - tOp + iea Po + top - iea "

(C.IO)

P = (D~3'(x)) = lira ( D~j~(x)),, (C. I 1)

we get P # 0 in the limit e --, 0 only when to o = 0 at p = 0 in eq. (C.9). We have then

P = 2 p / a . (C.12)


If N is the number of the dipole sites, the total polarization in the third direction is NP, since P is the local polarization (see eq. (4.6)). Then

(olp210) = N P ( N P + 1). (C.13)

From (C.7) and (C.12) we have

(OIP'"P'"Io)=oN, i= 1,2, (C.14)

and finally

(olp2}o) = 2oN + (Np) 2 , (c.15)

which compared with (C.13) gives

p =~P ,

and a = I (from eq. (C. 12)).

The above procedure has shown that a gapless boson mode exists because of nonzero polarization. The role of the external field e-term, namely the water electret, is now clear; eqs. (C.4) and (C.5) could not be derived from the generating functional without this term. The quasi-particle set includes thus a quasi-molecular field ~b(x) and a gapless boson field P(x) which is a bound state of ~b-fields.

Let us now sketch the proof of eqs. (4.15)-(4.17) (see also refs. [14-16]. Let

4~(x) --, ¢'(x, x) ,

P(x)--, P'(x, A) , (C.16)

when 6 (x ) --, 4/(x) = exp (~iA~o'i)~,(x) (see eqs. (2.10), (4.12), (4.13)):

(alo'(x)lb) = (a[F[ ¢'(x, A), P'(x, a)]lb>.

Since the original invariance of the theory cannot be lost, the free field equations (4.7) and (4.8) are invariant under (C.16):

A (a) ¢ ' (x, ,~ ) = 0, (C. 17a)

K(a)P'(x, A ) = 0 . (C.17b)

The SU(2) invariance of S implies its invariance under the quasi-particle transformations (C.16). Thus

0 - - S(qb'(x, A ), &'+(x, A ), P'(x, A ), P'+(x, A )) = O, (C.18) ~At

and (see eq. (4.4))

a D,~(x, ¢ ' (x , A), ¢ '+(x, A), P'(x, A), P'~(x, A))

=--e , kD~k~(x ,¢ ' ( x ,A) ,¢ '÷(x ,A) ,P ' ( x ,A) ,P ' ( x ,A) . (C.19)


cb', '~ P' P'~ ~b , and are then solutions of eqs. (C.18) and (C.19) under the constraints

(C.17). Using eqs. (4.12)-(4.14) eqs. (4.15)-(4.17) are obtained.

References

[1] A.S. Davydov, Biology and quantum mechanics (Pergamon, O~tford, 1982) [2] A.S. Davydov, Phys. Scripta 20 (1979) 387 [3] H. Fr6hlich, Rivista del Nuovo Cim. 7 (1977) 399; Advances in Electronics and Electron Physics,

ed. L. Marion and C. Matron, vol. 53 (1980) p. 85 [4] E. Del Giudice, S. Doglia and M. Milani, Phys. Scripta 26 (1982) 232 [5] S.J. Webb, Phys. Reports 60 (1980) 201 [6] E. Del Giudice, S. Doglia and M. Milani, Phys. Left. 90A (1982) 104 [7] I. Prigogine and G. Nicolis, Self-organization in non-equilibrium systems; from dissipative structures

to order through fluctuations (Wiley, New York, 1977) [8] H. Matsumoto, M. Tachiki and H. Umezawa, Thermo-field dynamics and condensed states ( North-

Holland, Amsterdam, 1982) [9] H. Matsumoto, P. Sodano and H. Umezawa, Phys. Rev. 19D (1979) 511

[10] L. Mercaldo, I. Rabulto and G. Vitiello, Nucl. Phys. 188B (1981) 193 Il l] H. Umezawa, NuovoCim. 40(1965) 450;

L. Leplae, R.N. Sen and H. Umezawa, Nuovo Cim. 49 (1967) 1; H. Umezawa, Renormalization and invariance in quantum field theory, ed. E.R. Caianello (Plenum, New York, 1974) p. 275

[12] A.S. Davydov and N.I. Kislukha, Soy. Phys. JEPT 44 (1976) [13] J. Goldstone, Nuovo Cim. 19 (1961) 154;

J. Goldstone, A. Salam and S. Weinberg, Phys. Rev. 127 (1962) 965 [14] H. Matsumoto, H. Umezawa, G. Vitiello and J.K. Wyly, Phys. Rev. 9D (1974) 2806 [15] M.N. Shah, H. UmezawaandG. Vitiello, Phys. Rev. 10B (1974) 4724 [16] C. De Concini and G. Vitiello, Nucl. Phys. 116B (1976) 141;

C. De Concini and G. Vitiello, Phys. Left. 70B (1977) 355 [17] H. Matsumoto, G. Oberlechner, M. Umezawa and H. Umezawa, J. Math. Phys. 20 (1979) 208 [18] A.C. Scott, Phys. Rev. 26A (1982) 578 [19] J.B. Hasted, H.M. Millany and D. Rosen, J. Chem. Soc., Faraday Trans. 77 (1981) 2289 [20] S. Celaschi and S. Mascarenhas, Biophys. J. 20 (1977) 273 [21] F.J. Dyson, Phys. Rev. 102 (1956) 1217 [22] S.L. Adler, Phys. Rev. 137B (1965) 1022; 139B (1965) 1638 [23] Y. Fujimoto and J. Papastamatiou, Nuovo Cim. 40A (1977) 468 [24] N.A. Nevskaya and Yu. N. Chirgadze, Biopolymers 15 (1976) 637 [25] J.M. Hyman, D.W. McLaughlin and A.C. Scott, Physica 3D (1981) 23;

A.C. Scott, Phys. Rev. 26A (1982) 578 [26] P.S. Lomdahl, L. MacNeil, A.C. Scott, M.E. Stoneham and S.J. Webb, Phys. Left. 92A (1982) 207 [27] J.A. Kruhmansl and D.V. Alexander, in Structure and dynamics: nucleic acids and proteins, ed. E.

Clementi and R.H. Sarma (Adenine, 1983) [28] E. Del Giudice, S. Doglia and M. Milani, Phys. Scripta 23 (1981) 307 [29] G. Careri, U. Buontempo, F. Carla, E. Gratton and A.C. Scott, Phys. Rev. Left. 51 (1983) 304 [30] S. Mascarenhas, personal communication (August 1982) [31] S.J. Webb, Nutrition, time and motion in metabolism and genetics (G.C. Thomas, Springfield, Ill.,

1975) [32] E. Del Giudice, S. Doglia, M. Milani and M.P. Fontana, Cell Biophys. 6 (1984) no. 2 [33] L.S. Sewchand, D" Roberts and S. Rowlands, Cell Biophys. 4 (1982) 253;

S. Rowlands, C.P. Eisenberg and L.S. Sewchand, J. Biol. Phys. I l (1983) I E. Del Guidice, S. Doglia and M. Milani, J. Biol. Phys., in press

[34] J.C. Taylor, Ann. of Phys. 115 (1978) 153


[35] M.N. Shah and G. Vitiello, Nuovo Cim. 30B (1975) 21 [36] J.H. Heller, in Biological etiects and health implications of microwave radiation (Richmond, 1969")

p. 116: N.D. Devyatkov, Sov. Phys. Usp. 16 (1974) 568; S.J. Webb, Ann. N.Y Acad. Sci. 247 (1975) 237

[37] E. Del Giudice, S. Doglia, M. Milani and G. Vitiello, Phys. Lett. A95 (1983) 508; in Nonlinear electrodynamics in biological systems, ed. W. Ross Adey and A. F. Lawrence (Plenum 1984), p. 459; Solitons and self-organization in biological systems, in Proc. Second Int. Workshop on Nonlinear and turbulent processes, Kiev, USSR, October 1983

[38] G. Albanese, A. Deriu and F. Ugozzoli, in Proc. Int. Conf. on the Application of M6ssbauer effect, Alma Ata, USSR, September 1983, in press