Le formalisme du chapitre 3 permet d’obtenir l’hamiltonien effectif d´ecrivant la dynamique des polarons. Il est bas´e sur une th´eorie de champ moyen qui sup- pose les phonons de l’h´elice `a l’´equilibre thermodynamique et qui n´eglige tous les processus d’absorption et d’´emission de phonons. Cette th´eorie n’autorise que des transitions virtuelles au cours desquelles l’absorption (ou l’´emission) d’un ou plu- sieurs phonons est imm´ediatement suivie de l’´emission (ou de l’absorption) du mˆeme nombre de phonons. Au-del`a de l’approximation de champ moyen, il existe un cou- plage r´esiduel entre les polarons et les phonons qui, bien que conservant le nombre total de polarons, est `a l’origine de la relaxation de phase des ´etats bi-polaroniques qui, dans la litt´erature, est caract´eris´ee par le temps de d´ephasage T2. Le couplage polaron-phonon correspond `a une fluctuation stochastique de la constante de saut des vibrons habill´es `a travers la d´ependance des op´erateurs d’habillage par rapport aux coordonn´ees de phonons. Ce couplage favorise donc des transitions entre les ´
etats li´es et les ´etats libres via des processus r´eels d’absorption et d’´emission de phonons acoustiques.
Ainsi, pour comprendre la nature des m´ecanismes `a l’origine de la relaxation des ´
etats li´es, nous avons r´ealis´e une ´etude perturbative du couplage polaron-phonon de mani`ere `a caract´eriser les taux de transitions entre les diff´erents ´etats bi-polaroniques. En supposant la fr´equence de coupure des phonons grande devant l’intensit´e du cou- plage vibron-phonon (Ωc ), nous avons restreint notre analyse aux transitions r´esultant de l’´echange d’un seul phonon. De plus, notre ´etude repose sur l’utilisation du jeu de param`etres Eq. (3.29) si bien que l’´energie de liaison du petit polaron et le param`etre y sont donn´es par
= EB(1 + 4η)
y = −η (4.21)
o`u EB = χ2/W d´esigne la constante du petit polaron harmonique et o`u χ est d´efini par l’Eq. (3.25). De telles hypoth`eses semblent en effet tout `a fait appropri´ees avec le jeu de param`etres de l’amide-I. Le lecteur int´eress´e par cette ´etude pourra en trouver le d´etail complet `a la fin de ce chapitre. Cependant, nous allons en r´esumer les principales conclusions.
Ainsi, nous avons montr´e que le taux de relaxation associ´e `a la d´esint´egration des diff´erents ´etats bi-polaroniques est globalement ind´ependant de leur nature li´ee
ou libre. En particulier, `a temp´erature biologique, nos calculs ont montr´e que la dur´ee de vie correspondante ´etait de l’ordre de la pico-seconde. Par contre, le canal de relaxation d´epend fortement du type d’´etat bi-polaronique. En effet, nous avons montr´e que les ´etats EL-I transitent essentiellement vers les ´etats EL-II par absorp- tion de phonons. A l’inverse, les ´etats de type EL-II se d´esint`egrent `a la fois dans la bande des ´etats li´es EL-I et dans le continuum des ´etats libres. Enfin, sous l’action des phonons, les ´etats libres effectuent des transitions `a l’int´erieur du continuum qui leur est associ´e. Ces diff´erents processus conduisent globalement `a la d´esint´egration irr´eversible des ´etats li´es dans le continuum des ´etats libres.
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Relaxation channels of two-vibron bound states in␣-helix proteins
V. Pouthier*and C. Falvo
Laboratoire de Physique Mole´culaire, UMR CNRS 6624, Faculte´ des Sciences – La Bouloie, Universite´ de Franche-Comte´, 25030 Besanc¸on Cedex, France
共Received 16 October 2003; published 29 April 2004兲
Relaxation channels for two-vibron bound states in an anharmonic␣-helix protein are studied. According to a recently established small polaron model关V. Pouthier, Phys. Rev. E 68, 021909 共2003兲兴, it is shown that the relaxation originates in the interaction between the dressed anharmonic vibrons and the remaining phonons. This interaction is responsible for the occurrence of transitions between two-vibron eigenstates mediated by both phonon absorption and phonon emission. At biological temperature, the relaxation rate does not signifi- cantly depend on the nature of the two-vibron states involved in the process. The lifetime for both bound and free states is of the same order of magnitude and ranges between 0.1 and 1.0 ps for realistic parameter values. By contrast, the relaxation channels strongly depend on the nature of the two-vibron states which is a conse- quence of the breatherlike behavior of the two-vibron bound states.
DOI: 10.1103/PhysRevE.69.041906 PACS number共s兲: 87.15.⫺v, 03.65.Ge, 63.20.Ry, 63.22.⫹m
I. INTRODUCTION
Since the pioneer works of Davydov and co-workers关1兴, soliton mechanisms for bioenergy transport in proteins have received increasing attention during the last 25 years关2,3兴. The main idea is that the energy released by the hydrolysis of adenosine triphosphate can be stored in the CvO vibration
共amide-I兲 of a peptide group. The dipole-dipole coupling be-
tween peptide groups leads to the delocalization of these vi- brations and to the formation of vibrational excitons, i.e., vibrons. Therefore, the strong interaction between the vi- brons and the phonons of the protein yields a nonlinear dy- namics which favors the occurrence of the so-called Davy- dov’s soliton.
However, it has been pointed out that the solution of the Davydov’s problem is rather a small vibron polaron than a vibron soliton 关4–9兴. Indeed, the vibron bandwidth in pro- teins is smaller than the phonon cutoff frequency so that the nonadiabatic limit is reached. During its propagation, a vi- bron is dressed by a virtual cloud of phonons which yields a lattice distortion essentially located on a single site and which follows instantaneously the vibron 共small polaron兲. Nevertheless, the dressing effect leads to an attractive inter- action between vibrons mediated by virtual phonons. Such an interaction is responsible for the formation of bound states and it has been suggested that proteins can support solitons formed by bound states involving a large number of vibra- tional quanta关7–9兴.
Although this formalism gives a comprehensive schema for the formation of solitons in proteins, it assumes the har- monic approximation for the amide-I vibration. However, this approximation failed when several vibrons are excited because the intramolecular anharmonicity acts as an addi- tional nonlinear source. As the dressing effect, the anharmo- nicity is responsible for the formation of bound states关10– 15兴 and the fundamental question of the interplay between both sources of nonlinearity has been addressed in a recent
paper关16兴. In this work, we have restricted our attention to the formation of two-vibron bound states共TVBS兲 only. In- deed, although the influence of the anharmonicity in molecu- lar lattices has been the subject of intense research during the last decade, this research was essentially restricted to classi- cal lattices关17–19兴. In particular, the formation of discrete breathers, i.e., highly localized nonlinear vibrations, has been demonstrated. However, in spite of the great interest that these classical nonlinear objects have attracted, no clear evi- dence has been found for their existence in real lattices. By contrast, TVBS have been observed in several low- dimensional molecular lattices关20–28兴. These quantum ob- jects correspond to the first quantum states which experience the nonlinearity and can thus be viewed as the quantum counterpart of breathers or soliton excitations 关12兴. Their characterization is thus essential and appears as a first step to understand the formation of multivibron solitons.
In Ref.关16兴, we have shown that the anharmonicity modi- fies the vibron-phonon interaction which results in an en- hancement of the dressing effect. Anharmonic vibrons are thus more sensitive to the dressing than harmonic vibrons. Moreover, both nonlinear sources favor the occurrence of two kinds of bound states whose properties strongly depend on the anharmonicity. In the harmonic situation, the two bound states appear as combinations of states involving the trapping of the two vibrons onto the same amide-I mode and onto nearest neighbor amide-I modes. By contrast, the in- tramolecular anharmonicity reduces the hybridization be- tween these two kinds of trapping so that low frequency bound states refer to the trapping of the two vibrons onto the same amide-I mode whereas high frequency bound states characterize their trapping onto nearest neighbor amide-I vi- brations.
In this study, the dynamical coupling between the dressed anharmonic vibrons and the remaining phonons was disre- garded. Therefore, the present paper is devoted to the char- acterization of this coupling and to a detailed analysis of the relaxation pathways. The TVBS lifetime is determined with a special emphasis on the influence of the different nonlinear sources.
*Electronic address: vincent.pouthier@univ-comte.fr
PHYSICAL REVIEW E 69, 041906 共2004兲
At biological temperature, the lifetime of the Davydov’s soliton is still an open question. It has been shown that the amide-I excitation, in vivo, corresponds to a localized state
关29,30兴. Instead of traveling in a coherent manner, it follows
a stochastic, diffusional path along the lattice. In other words, the single-vibron Davydov soliton does not last long enough to be useful at biological temperatures and it has been shown that two-vibron solitons are more stable and appear as good candidates for bioenergy transport 关31,32兴. However, recent calculations performed by Ivic et al. 关9兴 clearly show that the multivibron soliton lifetime is of about a few picoseconds, i.e., the same order of magnitude as the single-vibron soliton lifetime found by Cottingham and Sch- weitzer关33,34兴.
The paper is organized as follows. In Sec. II, the dressed anharmonic vibron point of view described in details in Ref.
关16兴 is first summarized. Then, the coupling Hamiltonian be-
tween these anharmonic polarons and the remaining phonons is determined. The TVBS relaxation rate is expressed in Sec. III and studied numerically in Sec. IV. The results are finally discussed in Sec. V.
II. VIBRON-PHONON HAMILTONIANS AND TWO- VIBRON EIGENSTATES
A. The general vibron-phonon Hamiltonian
According to the original Davydov’s model, the collective dynamics of the amide-I modes is described by a one- dimensional lattice with N sites containing the CvO vibra- tions. The nth amide-I mode is assumed to behave as a high frequency anharmonic oscillator described by the standard operators bn
†
and bn. This oscillator is characterized by its harmonic frequency0and by the cubic and quartic anhar-
monic parameters ␥3 and ␥4, respectively. Finally, the
dipole-dipole coupling between nearest neighbor amide-I modes is introduced via the hopping constant J. These CvO vibrations interact with the phonons of the lattice which characterize the dynamics of the external motions of the pep- tide groups. Within the harmonic approximation, the phonons correspond to a set of N low frequency acoustic modes labeled兵q其 and described by the phonon operators aq† and aq. The frequency of the qth mode is defined as ⍀q
⫽⍀c兩sin(q/2)兩, where ⍀c denotes the phonon cutoff fre- quency.
As shown in Ref. 关16兴, a unitary transformation is per- formed to remove the intramolecular anharmonicity of each amide-I mode and a modified Lang-Firsov transformation is applied to renormalize the vibron-phonon interaction. As a result, the vibron-phonon Hamiltonian is defined as 共within
the conventionប⫽1兲 Hˆ⫽
兺
n ˆ0 bn†bn⫺Aˆbn †2 bn2⫺Bˆbn†⫹1bn†bn⫹1bn ⫺J1关⌰n †共N n⫺1兲⌰n⫹1共Nn⫹1兲bn † bn⫹1⫹H.c.兴 ⫺J2冋
⌰n †2冉
Nn⫺ 3 2冊
⌰n⫹1 2冉
Nn⫹1⫹ 1 2冊
bn †2 bn2⫹1⫹H.c.册
⫺J3关⌰n †共N n⫺1兲⌰n⫹1共Nn⫹1兲bn †关N n⫹Nn⫹1兴bn⫹1 ⫹H.c.兴⫹兺
q ⍀q aq † aq, 共1兲 where Nn⫽bn † bn and A⫽30␥3 2 /0⫺6␥4. In Eq. 共1兲, H.c.stands for the Hermitian conjugate and the different param- eters are expressed in terms of both the anharmonic param- eters and the small polaron binding energy EBas
ˆ0⫽0⫺2A⫺B⫺共1⫹4兲EB, Aˆ⫽A⫹共1⫹8兲EB, Bˆ⫽B⫹共1⫹4兲EB,
B⫽144J
冉
␥3 0冊
2 , J1⫽J冋
1⫹44冉
␥3 0冊
2 ⫺12␥4 0册
, J2⫽4J冉
␥3 0冊
2 , J3⫽J冋
22冉
␥3 0冊
2 ⫺12␥4 0册
, ⫽120冉
␥3 0冊
2 ⫺12␥4 0 . 共2兲Note that the small polaron binding energy EB and the an- harmonic parameter A appear as the relevant parameters to
characterize the nonlinearity of the system. In Eq. 共1兲,
⌰n(Nn) stands for the dressing operator expressed as
⌰n共Nn兲⫽exp共⫺Qn关1⫹2⫹2Nn兴兲, 共3兲 where Qnis defined as Qn⫽
兺
q冑
EB 2N⍀q sin共q兲 i兩sin共q/2兲兩e ⫺iqna q † ⫺H.c. 共4兲The Hamiltonian, Eq.共1兲, describes the dynamics of an- harmonic vibrons dressed by virtual phonons, i.e., anhar- monic small polarons. It takes into account the intramolecu- lar anharmonicity up to the second order and allows for a renormalization of the main part of the initial vibron-phonon coupling. However, this coupling remains through the dress- ing operators⌰n(Nn) which depend on the phonon coordi- nates in a highly nonlinear way. Therefore, to separate the vibron degrees of freedom from the phonon coordinates, a mean field procedure is applied关4–6兴. The full Hamiltonian
Hˆ is thus written as Hˆ⫽Hˆe f f⫹Hp⫹⌬H, where Hp is the
phonon Hamiltonian and where Hˆe f f⫽具(Hˆ⫺Hp)典 denotes the effective Hamiltonian of the dressed anharmonic vibrons.
⌬H⫽Hˆ⫺Hp⫺具(Hˆ⫺Hp)典 stands for the remaining part of the vibron-phonon interaction. The symbol具¯典 represents a thermal average over the phonon degrees of freedom at tem- perature T.
Finally, starting from a set of anharmonic vibrons strongly coupled to the phonons of the protein, the previous proce- dure yields a different point of view in which the system consists in a set of interacting small polarons weakly coupled to the remaining phonons. Due to this weak coupling, the small polarons are supposed to be in stationary states accord-
V. POUTHIER AND C. FALVO PHYSICAL REVIEW E 69, 041906 共2004兲
ing to the effective Hamiltonian Hˆe f fand the phonons can be viewed as a thermal bath responsible for their relaxation. It is thus assumed that the bath is not affected by the polarons since this effect is explicitly taken into account through the definition of the small polarons, i.e., vibrons dressed by a lattice distortion.
To conclude this section, let us mention that the phonon anharmonicity was disregarded in the present work. How- ever, it is expected to play an important role since the pres- ence of several amide-I excitations leads to a large lattice distortion when compared to the distortion induced by a single excitation. Nevertheless, such an effect needs a more appropriate theory and will be addressed in a forthcoming work.
B. The effective vibron Hamiltonian and the two-vibron eigenstates
The effective dressed anharmonic vibron Hamiltonian is written as Hˆe f f⫽
兺
n ˆ0 bn † bn⫺Aˆbn †2 bn 2 ⫺Bˆbn†⫹1bn † bn⫹1bn ⫺J1关⌽共Nn⫹Nn⫹1兲bn † bn⫹1⫹H.c.兴 ⫺J2关⌽共Nn⫹Nn⫹1兲4bn †2 bn⫹1 2 ⫹H.c.兴 ⫺J3关⌽共Nn⫹Nn⫹1兲bn †关N n⫹Nn⫹1兴bn⫹1⫹H.c.兴, 共5兲where⌽(X)⫽exp关⫺S(T)(1⫹2⫹2X)兴 and where S(T) is
the coupling constant introduced by Ivic and co-workers as (kBdenotes the Boltzmann constant兲
S共T兲⫽4EB N⍀c
兺
q冏
sin冉
q 2冊冏
cos冉
q 2冊
2 coth冉
⍀q 2kBT冊
. 共6兲In Ref.关16兴, a detailed analysis of the two-vibron energy spectrum of the Hamiltonian Hˆe f f is presented. Within the number state method关12–15兴, the two-vibron wave function is first expanded as兩⌿典⫽兺⌿(n1,n2)兩n1,n2) where兩n1,n2)
denotes a local basis vector characterizing two vibrons lo- cated onto the sites n1 and n2, respectively. Note that the
restriction n2⭓n1 is applied due to the indistinguishable
character of the vibrons so that the dimension of the two- vibron subspace is N(N⫹1)/2. Then, by taking advantage of the lattice periodicity, the wave function is expanded as a Bloch wave as ⌿共n1,n2⫽n1⫹m兲⫽ 1 冑N
兺
n1 eik(n1⫹m/2)⌿ k共m兲, 共7兲where the total momentum k is associated with the motion of the center of mass of the two vibrons. The wave function
⌿k(m) refers to the degree of freedom m which character- izes the distance between the two vibrons. Since k is a good quantum number, the Hamiltonian Hˆe f fappears as block di- agonal and the Schro¨dinger equation can be solved for each
k value. For a given k value, the protein exhibits (N⫹1)/2
eigenstates 兩⌿k典, where the index ⫽1, . . . ,(N⫹1)/2. Due to the nonlinear sources, there are two different eigen- states, i.e., two-vibron free states 共TVFS兲 and TVBS. The TVFS correspond to a delocalization of the separating dis- tance m. The wave function⌿k(m) behaves as a plane wave and the TVFS belong to an energy continuum. By contrast, the TVBS correspond to a localization of the separating dis- tance and characterize the trapping of the two quanta over only a few neighboring sites. We have shown that the protein supports two kinds of bound states, called TVBS-I and TVBS-II. The TVBS-I, denoted兩⌿k,⫽1典, are located below the TVFS continuum over the entire Brillouin zone whereas for TVBS-II, two situations occur depending on the strength of the small polaron binding energy. For small values of EB, the band disappears inside the continuum when兩k兩 is lower than a critical wave vector kcwhereas, for strong values of EB, the band is located below the continuum over the entire Brillouin zone. As a result, the notation兩⌿k,⫽2典refers ei-
ther to a free state or to TVBS-II, depending on the situation. In the harmonic situation, both TVBS-I and TVBS-II appear as combinations of states involving the trapping of the two vibrons onto the same amide-I mode and onto nearest neigh- bor amide-I modes. By contrast, the intramolecular anharmo- nicity reduces the hybridization between these two kinds of trapping so that TVBS-I refer to the trapping of the two vibrons onto the same amide-I mode whereas TVBS-II char- acterize the trapping onto nearest neighbor amide-I vibra- tions.
C. The vibron-phonon coupling Hamiltonian
By comparing Eqs. 共1兲 and 共5兲, it is straightforward to
show that the coupling Hamiltonian ⌬H corresponds to a
modulation of the different lateral contributions describing vibron hops, i.e., the terms proportional to J1, J2, and J3in
Eq.共1兲. However, in␣-helix proteins, it has been shown that
J2⬇J3⬇J1/0关16兴. As a result J2and J3are of about three
orders of magnitude lesser than J1and can be neglected. The
coupling Hamiltonian⌬H is thus written as
⌬H⫽⫺J1
兺
n,␦⫽⫾1关⌰n †共N n⫹1兲⌰n⫹␦共Nn⫹␦兲 ⫺具⌰n †共N n⫹1兲⌰n⫹␦共Nn⫹␦兲典兴bn † bn⫹␦. 共8兲In addition, the small polaron binding energy is about one order of magnitude smaller than the phonon cutoff frequency so that the dressing operator, Eq.共3兲, can be linearized 关8,9兴. As a consequence, by neglecting the rather small parameter
in Eq.共3兲 关16兴, the coupling between the anharmonic po- larons and the remaining phonons is finally expressed as
⌬H⬇⫺
兺
n,␦⫽⫾1⌬J共n,n⫹␦兲bn
†
bn⫹␦, 共9兲
where
RELAXATION CHANNELS OF TWO-VIBRON BOUND . . . PHYSICAL REVIEW E 69, 041906 共2004兲
⌬J共n,n⫹␦兲⫽J1
兺
q
冋
⫺i冑
EB
2N⍀q
sin共q兲e⫺iqn
兩sin共q/2兲兩 ⫻共1⫺e⫺iq␦兲a
q
†
⫺H.c.