“ Vinča ” institute of nuclear sciences

1. “Vinča” institute of nuclear sciences On the possible role of small polarons in the charge and energy transport in the a-helix proteins D. Čevizović, S. Zeković, Z. Ivić

2. I Introduction:brief history of the problem

3. The mechanism of the long range intramolecular vibrational energy transfer in biological macromolecules (like DNA and a-helix) has been an attractive area of investigation over the last 30 years. But, in spite of the efforts, this problem remains unresolved! Very attractive explanation on the microscopic level has been proposed in mid seventies by Davydov. The essence of this theory is that the energy released in the hydrolysis of ATP can be captured by the protein molecules and then transported along the polypeptide chain in a soliton form which arises due to the self-trapping (ST) of the amide-I (CO stretching) quanta. However, due to the lack of direct experimental evidence of soliton existence in these substances, Davydov's idea for a long time has been considered just as an interesting theoretical concept of academic interest only.

4. However, the concept of self-trapping and its possible relevance for the transport processes in molecular systems cannot be excluded as a whole. Situation changed at the beginning of eighties when Careri and Scot suggested that appearance of the so-called unconventional amide-I band, observed in a previous year experiment by Careri in crystalline acetanilide (ACN) may be explained in terms of Davydov's soliton theory (DST). Their ideas have encouraged restored interest in DST. Alternative interpretation of Careri's experiments in terms of small-polaron (SP) theories [6] indicates that ST may arise in a-helix and ACN. These ST states should be identified as nonadiabatic small-polarons rather than solitons, since the values of basic energy parameters commonly used in the studies of vibrational energy ST fall into nonadiabatic weak coupling regime. The present condition in the field is that the original soliton idea cannot explain intramolecular energy transfer over the large distances.

5. Hydrogen -bonded molecular crystals ACN a -helix When foreign particle or excitation (electronic or vibronic) is injected or created into a solid, their presence induces a distortion in the surrounding medium. Such quasiparticle consisting of the original exciton together with the distortion it induces in the host medium is polaron. Many hydrogen -bonded molecular crystals have similar structure, so each of them can approximate by quasi 1D macromolecular chain.

6. w h K J is intersite transfer integral (it is propo-rtional to quasiparticle energy bandwidth) M is mass of molecular group is characteristic phonon energy F is vibron-phonon coupling parameter is vibron –phonon interaction constant Literature values of the basic physical parameters c[N] |F|[cm-1] J[cm-1] [cm-1] M [kg] C=O stretch 4 50 (opt) 5,6 (6,2)•10-11 14(16) 2,25•10-25 ACN N-H stretch 4 50 (opt) 33•10-11 83 a-helix 44-54 (opt) (*) 5,7•10-25 7,8 (3,5-6,2) •10-11 10 (*) calculated from k=(39-58,5) [N/m] Table 1. Basic physical parameters [6,7,8,11]

7. Polarons The character of ST states (if any arise!) is determined by values of basic energy parameters of the system, and more importantly their mutual relationship: characteristic phonon energy, ћwK, quasiparticle energy bandwidth 2J, and quasiparticle binding energy Eb: It is useful to determine the system parameter space by the following parame-ters: adiabatic parameter coupling constant In spite of orbitals overlap from neighboring lattice sites, these quasiparicles can move through crystal by successive hopping from one site to neighbour site.

8. Full theoretical description of the ST states is yet unknown. But, two limiting cases are well considered: nonadiabatic limes B<<1 adiabatic limes B>>1 lattice deformation is slow and can not follow quasiparticle motion (lattice defomation forms “frozen” structure). quasiparticle and lattice deformation form new entity: we have “dressed” quasiparticle, with new effective mass, and with reduced energy band. Polaron dimension depend on value of S parameter: S>>1 we have AdSP, S<<1 we have AdLP (soliton). In this case we have NaSP. Theoretical description of those systems is based on the methods that employ unitary transformation (Classical SP theories are based on the use LF transformation) Formation of nonadiabatic SP is possible in crystals with narrow energy band (in the case of small value of intersite transfer integral), and strong polarisation field(quasi-particle can not “esape”).

9. Our goal In this paper we investigate the possibility of the SP creation in the hydrogen -bonded molecular crystals (such as a-helix proteins and ACN) accounting of the self -trapping of the intramolecular vibrational energy quanta. We apply mean -field (MF) theory based upon the modified Lang -Firsov (MLF) transformation and Bogolyubov theorem. Optimization procedure was carried out in the region of parameter space which corresponds to the values of system parameters usually employed in the theoretical examination of energy transfer in polypeptides.

10. II Basis of theoretical analysis

11. Starting hamiltonian In order to set up theoretical model for investigating the quasiparticle dynamics in deformable media, we chose, as a theoretical basis of our analysis 1D Hamiltonian in the following form: D- vibron excitation energy, An+(An) describes presence (absence) of the vibron quanta on n-th lattice site, bq+(bq) creates (annihilates) phonon quanta, J denotes intersite transfer integral, Fq=F-q* denotes vibron-phonon coupling parameter, wq is phonon frequency, and R0 is lattice constant. Vibron-phonon coupling: acoustics phonons optics phonons

12. Modified LF transformation Due to the vibron –phonon interaction, our vibrons can form polaron states. In order to rewrite system Hamiltonian in therms of polarons and new phonons, in nonadiabatic case we must apply unitary transformation. On the other hand, if we have weakly dressed polarons, we must apply variational approach. System parameters that characterizes a-helix proteins suggest that vibron ST would result in the creation of the weakly dressed non adiabatic SP (vibron bandwidth is substantially less than maximal phonon frequency), so it is necessary to apply strategy that can intermediate between heavy dressed and weakly dressed quasiparticle (partial dressing strategy). Table 1. (ACN (C=O stretch): B=0,16; S=0,1) and (a-helix: B=0,3; S=0,04).

13. This situation may be successfully described by means of the MLF theory: variational parameter whose magnitude defines the degree of dressing and the cha-racetr of ST states new phonons in the chain with shifted equilibrium positions of the molecular group. dressed quasiparticle (polaron) Itis necessary to compare LF method and partially dressed method, and prove that partially dressed method is more favorable to describe our systems.

14. Transformed Hamiltonian SP band energy temperature dependent SP energy shift quasiparticle band narrowing factor In the wave number representation transformed Hamiltonian has a form: A=e-S(T) is a temperature –dependent modulation factor (for the dipole –dipole interaction) which measures the coherence of transfer of the vibrational energy of the intramolecular mode from site to site, and it is proportional to inversion value of polaron effective mass.

15. Variational parameters Optimal form of variational parameters is determined in accordance with the Bogolyubov theorem by means of the minimization of the model free energy of the system: where E0=Epol+Eph represents the free energy of the fully decoupled polaron-phonon system.

16. IIIResults

17. Part 1 a) vibron iteract with acoustic phonons

18. b) vibron interact with optic phonons

19. Part 2 Comparison with LFapproach Obtained results for variational parameters are very similar with parameters that appears in standard LF approach: Comparing fqand fqLF, one can see that actually, LF approach is equivalent with MLF approach in the case when S(T) tends to infinity. This is the case of heavy dressed quasiparticles. In order to compare partial dressing model (MLF) with LF approach it is necessary to compare ground state energy for both models. In the region of the parameter space where EgsLF is lower than EgsMLF, LF approach gives a better description of the system, while in the opposite case, the MLF is more suitable.

20. Comparison LF and MLF approaches in system parameter space (S,B) MLF a–helix (0.04,0.3) LF ACN (0.1,0.16)

21. Part 3 Investigation of the SP properties in dependence of system parameter space Iverse value of polaron effective mass is proportional to modulation factor. Therefore, behavior of A in system parameter space may give us information about polaron degree of dressing, and consequently, which type of polaron can exist for these system parameter values. Dependences of A versus S are graphically presented on the following graphs, where by full color lines are shown curves that corresponds to one value of adiabatic parameter (so, we call them adiabate curves). By red dashed line stability curve is presented , and by black dashed line critical adiabate curve is presented. A(S) is considered for several values of B, and for several temperatures of the system. Stability curve is the locus of the points where the first derivative of S(A) vanishes. Those corresponds to the set of points where the second derivative of Egs vanishes. Points below the curve correspond to stable states (minima) while points above the curve correspond to the maxima of energy (unstable states).

22. Interaction with acoustical phonons

23. Interaction with optical phonons

24. Our theory predict that: For small values of B, A is unique function of S. But, for higher B (higher than some critical value BC), there appears to be three values of A for given value of S. In the case when B<BC, with increasing S there occurs the continual transition of partially dressed polaron band states towards ST (immobile) small polarons of infinite mass. In the case when B>BC, for each adiabate there exists a metastable region bounded by two critical coupling constants where the coexistence is predicted and there is an abrupt transition from the free band (partially dressed) state into a ST state. This change is expressed stronger in the case of large B values (highly adiabatic region) where a slightly dressed, practically free band state transits into a ST state. With temperature increase critical value of adiabatic parameter decreases (for both cases).

25. IVConclusion

26. Based on results presented, we made the revision of SP concept of energy transfer in polypeptides. We found that traditional SP theories cannot be directly applied to the vibrational quanta transfer in these substances. Our analysis shows that the existing vibron transfer theories should be modified in accordance with partial dressing strategy: true eigenstates of system should correspond to a partialy dressed polarons, rather than to the fully dressed SP states.

27. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] A. S. Davydov, Teoriya tverdogo tela, Nauka, Moskva, (1976). I. G. Lang, Yu. A. Firsov, Zh. Eksp. Teor. Fiz.,43 (1962), pp. 1843. E. I. Rashba in: E. I. Rashba, M. Struge, (Eds.) "Excitons", North-Holland (1982), pp. 543. D. Emin, Adv. Phys.,22 (1973), pp. 57. Ž. Pržulj, D. Čevizović, S. Zeković, and Z. Ivić, Chem. Phys. Lett., 462 (2008), pp. 213. D. M. Alexander and J. A. Krumhansl, Phys. Rev. B,33 (1986), pp. 7172. A.C.Scott, Phys. Rep., 217 (1992), pp. 1. Peter Hamm, Julian Edler, Phys. Rev. B, 73 (2006), pp. 094302-1. V. Pouthier, Phys. Rev. E,78 (2008), pp. 061909-1. D. Emin, Phys. Rev. Lett.,28 (1972), pp. 604. D. W. Brown, Z. Ivić, Phys. Rev. B,40 (1989), pp. 9876.