PHYSICAL REVIEW E VOLUME 48, NUMBER 3 SEPTEMBER 1993 Vanishing thermal damping of Davydov's solitons Aurea R. Vasconcellos and Roberto Luzzi Caixa Postal 6165, 13083-970 Campinas, SB'0 Paulo, Brazil (Received 24 May 1993) We consider Davydov s biophysical model in the context of nonequilibrium statistical thermodynam- ics. We show that excitations of the Davydov-soliton type that can propagate in the system, which are strongly damped in near-equilibrium conditions, become near dissipation-free in the Frohlich-Bose- Einstein-like condensate and that this occurs after a certain threshold of pumped metabolic energy is reached. This implies the propagation of excitations at long distances in such biosystems. PACS number(s): 87.10.+e, 87.22.As, 05.70.Ln In the 1970s Davydov showed that, due to particular choose the basic variables deemed appropriate for the nonlinear interactions in biophysical systems, e.g., the a- description of the macroscopic state of the system. We helix protein, a novel mechanism for the localization and take the energy of the thermal bath, transport of vibrational energy is expected to arise in the (btb form of a solitary wave [1]. The subject was taken up by Eb(t)=Tr Q AQ + ')p(r, 0)— (la) a number of contributors, and a long list of results pub- q lished up to the first half of 1992 are discussed in the ex- the population of the polar vibrations, cellent review by Scott [2], who has also provided exten- sive = research on the subject [3]. As pointed out in that v (t) Tr [a za&P(t, 0)], (lb) review, one important open question concerning Davydov s soliton is that of its stability at normal physio- and the amplitudes of oscillation, logical conditions, or in other words, the possibility of = & r & the excitation to transport energy (and so information) at a, I Tr [a,p(r, o) ], (lc) long distances in the living organism, in spite of the relax- (a ~t)"=TrIazp(t, 0)], ation mechanisms that are expected to damp it out at very short (micrometer) distances. We address this ques- where a and b (a and b ) are annihilation (creation) tion here. First we note that there is equivalence between operators in the corresponding vibrational states of wave Davydov's model and the one used by Frohlich as [4,5], vector q, Qq is the frequency-dispersion relation of the shown et s work by Tuszynski al. [6]. In Frohlich it is modes of the bath, and, finally, shown that, as a consequence of the nonlinear interac- tions to which we have already referred, under appropri- p(t, 0)=exp P(t) PDHI, —Q—F~(t)a—a ate conditions a phenomenon akin to a Bose-Einstein q condensation may occur in the system, provided there is — t a supply of metabolic energy exceeding a critical value. g [f (t)a +f*(t)a ] (2) We proceed to analyze both phenomena in the Frohlich- Davydov model under arbitrary nonequilibrium condi- is the auxiliary (or coarse-grained) nonequilibrium opera- tions. For that purpose we resort to an informational sta- tor in the NSOM, the whole (fine-grained) NSO being a tistical thermodynamics based on the nonequilibrium sta- functional of it [7]. In Eq. (2), Po= I lkTO, P(t) ensures tistical operator method (NSOM) [7]. The NSOM, which the normalization of p, and F and f are the other provides microscopic foundations for phenomenological Lagrange multipliers that the variational approach to the irreversible thermodynamics [8], also allows for the con- NSOM introduces [7]; Hb is the Hamiltonian of the bath. struction of a nonlinear generalized quantum transport Further, we note that, introducing the canonical transfor- theory [9], which describes the evolution of the system at mation (reminiscent of Glauber's transformation to the macroscopic level in arbitrary nonequilibrium situa- coherent states in laser theory [10]), tions, a formalism to be used in our analysis below. The model consists of a biosystem that can sustain lon- a, =a, —(a, ir &, (3) gitudinal polar vibrations in interaction with a thermal bath of acousticlike vibrations. This interaction involves one finds that all possible three-particle collisions. An external source r = — l ) *(r) (4a) of metabolic energy is coupled to the polar vibrations, (a, f, iF,(r), and the bath is assumed to be constantly kept at a fixed v (t)=v (r)+~&a ~t) ~ (4b) temperature To by an efficient homeostatic mechanism. Next, as the first step in the NSOM, it is necessary to where 1063-651X/93/48(3)/2246(4)/$06. 00 2246 1993 The American Physical Society 48 VANISHING THERMAL DAMPING OF DAVYDOV'S SOLITONS 2247 + (t) — a v (t)=[e ' 1] (4c) vq(t) =Iq —rq '[v (t) —vq] —g (t), is the polar-mode population in the absence of the excita- where Iq is the pumping intensity rate, w is a relaxation tion of amplitude ( aq ~ t ). time associated with the decay of a polar vibration in two Applying the NSOM transport theory in the so-called acoustic vibrations in the bath, v is the equilibrium second-order approximation in relation theory (SOART) value, and 8 is a lengthy collision operator associated [9] and using Zubarev's approach [11]to the NSOM, we with the collisions of two polar phonons with one acous- find the equations of evolution ticlike phonon. They are given by V'", ~2[7), e q'5(Q —Q, —a) —coq)], (6a) g2 0 g~ qq g +, +, )+qqr1q, q5(Qq q+Qq. pcs', +, ' =2K (2) 2 . — q' — — 4q(t)= g~ Vqq ~ ri + I[1+v (t)][1+v (r)] v.(t)v (r)e ]5(Q +, co, a) ) ' —,, ' 2' (2) 2 — ps~,qq — + ~ tv q(t)[1+v (t)] [1+vq(t)]vq(t)e ]5(Qq q. +coq. coq) g2 g~ Vqqqq gq q —pcs' — co +—co ~ " (6b) 2 g~ Vqq qq tqV (qt)[1+v (t)]e [1+v,(t)]v (t)]5(Q + ), q' choose = ( = co—= 10' ', and where ro r, r ), goÃ= g & r- 10, ~ sec (+)— b, =2X10' sec '. Q =sq with s =10 cm/sec and we =COq+ COq (6c) kqq~ write S =I0F=I,F for the pumping intensities. The re- A( .)=co —co ~ (6d) sult is shown in Fig. 1: One may clearly see the onset of qq' q q V'" and V' ' are the matrix elements of the interaction involving, respectively, collision of one polar phonon with two acoustic phonons and collision of two polar phonons with one acoustic phonon, q is the equilibrium Planckian distribution of acoustic phonons at tempera- ture T0, and coq is the polar-mode frequency-dispersion relation. Equation (5) is then of the type proposed by Frohlich, IO where the nonlinear terms in 8 account for the transfer of energy (through mediation of the acoustic phonons) from the high-frequency to the low-frequency modes, (A leading to Frohlich-Bose condensation. Let us show this in numerical calculations. Taking into account the men- tioned type of energy transfer, we introduced a quite CL simplified model, that is, we take a mode as representa- D tive of all high-frequency modes (index 0 below), and CL Ld another for the low-frequency modes (index 1 below). We C5 C) are then left with a coupled set of equations, namely, (p vo(t) =Io —ro '[vo(t) —vo] g, ye~" vo—(t) +g, gv, (t) —g2v, (t)vo(t), (7a) v, (t) =I, —r, '[v, (t) —v, ] gorjv, (t)— +g,g ~",(r)+g, ,(r),(t), (7b) — 2 I I i I I ) I where g0 and g, are coupling constants, 6=co0 co&, and IO IO lO g = [exp(PA'b, ) —1] 1NTENSlTY S After a certain transient has elapsed (estimated to be of FIG. 1. Population of the representative modes of high fre- the order of picoseconds (12), a steady state is attained, quency, vo, and that of the representative modes of low frequen- i.e., dv/dt =0 in Eqs. (7). For a numerical solution we cy, v&, as a function of the source power. 2248 AUREA R. VASCONCELLOS AND ROBERTO LUZZI 48 the Frohlich e+ect, that is, a large increase of population in the low-frequency modes, which follows at an intensity threshold (roughly given by the value of S where v, steeply increases) of 500. Figure 2 shows, for the case of an a-helix protein, the domain in frequency space in which the Frohlich condensate occurs [12]. We consider now the dynamic aspects associated with the propagation of signals in Frohlich condensate, that is, we t look for quantities (aq ~ ) that, in SOART-NSOM, satisfy the equations of evolution = c— — — *+ *+ ~ ~ t ~ ~ ~ ~ '+ a t i a t I a t i t I a j a t a t a t c.c. ( q ) oq( q ) q( q ) Wq ( aq ) q( q ) g [R q q ( q ) ( q ) ( q +q q ) ] (9) and the accompanying complex conjugated equation, where coq coq+ JYq, pcs,'+' — 2t =r '+ g~ V' '~ g + [(e " 1)v —l]5(to„+co —0 + ) V'q~ q + g~ ~ qq [(1+v )e v e ~][e q6(coq a)q 0 )+e q5(coq ~ +f1 )] (10) q with r of Eq. (6a) [we omit to write down the long expression for W(it is a term of renormalization of frequency of no relevance in what follows)], and — +(coq coq +Qq +q +le) +(cdq coq Aq +q +le) where e is to be taken in the limit +0. we obtain Equation (9) is of the Davydov-soliton type, but with the presence of the damping term with damping constant iR g(x, t)= A'coo+Ra t) I .