1. D. Čevizović1,2, S. Galović1,2, A. Reshetnyak3, and Z. Ivić1
Vibron Self-trapped States in Biological Macromolecules: Comparison of Different Theoretical Approaches
D. Čevizović1,2, S. Galović1,2, A. Reshetnyak3, and Z. Ivić1
1 University of Belgrade, "Vinča" Institute of Nuclear Sciences, Laboratory of theoretical and condensed matter physics, Belgrade, Serbia
2 Joint Institute for Nuclear Research, Bogoliubov Laboratory of Theoretical Physics, Dubna, Russia
3 Institute of Strength Physics and Material Science, Tomsk, Russia
Dubna-Nano2012, Dubna, 12 July, 2012

2. abstract
• We present a study of the applicability of the variational treatments based on using of the modified Lang-Firsov unitary transformation (MLF method) in the investigation of the vibron self-trapped states in biological macromolecular chains.
• Here we compare the values of the ground state energy predicted by MLF methods with the values of the ground state energy predicted by the standard small-polaron theory, for various values of the basic energy parameters of the system.
• We obtain regions in system parameter space where MLF approach gives better description of the vibron states.

3. Basic aims to solve:
To apply modified Lang-Firsov unitary transformation (MLF method) to investigate the vibron self-trapped states in biological macromolecular chains
To calculate and compare the values of the ground state energy predicted by MLF methods with the values of the ground state energy predicted by the standard small-polaron theory (for various values of the basic energy parameters of the system).
To chose the most adequate approach among standard SP approach, and variational: fq approach and d approach for description behaviour of the ST vibron states and then to use them to solve quantum transport problem in macromolecule chains.

4. Outline: Motivations. Basics of the general theory of ST phenomena.
Partial dressing concept
Theoretical models (Hamiltonian, Vibron-phonon interaction, Mean field Hamiltonian, Variational energy)
Derivation of the basic system parameter
Obtained results for optical and acoustic phonon cases
Conclusions & Outlook

5. motivation
• Energy released by the hydrolysis of adenosine triphosphate is a universal energy source allowing many biological processes.
• However, it is yet not clearly understood how this energy can be transported along macromolecule at long distances, without being dissipated or dispersed.
• Earlier explanation based on quantum mechanics: Davydov soliton theory (possibility of nondissipative long range transport) [1. A.S. Davydov, Phys. D, 3,1,1981;Solitons in Molecular Systems. Kluwer Academic Publishers, (1991)]

6. Basics of the general theory of ST phenomena
• basic energy parameters that determine system properties [2]:
(SP bending energy)
(characteristic phonon energy)
(vibron bandwidth)
• asymptotic solutions of the ST problem:
adiabatic limit
ASP
ALP (soliton)
nonadiabatic limit
NaSP
exciton-phonon strong coupling •
LF unitary transf.
exciton-phonon weak coupling ?
[2] E.I. Rashba in: E.I. Rashba, M. Struge (Eds.), Excitons, Nort-Holland, Amsterdam (1982)

7. Polypeptide macromolecular chains: energy parameters
(nonadiabatic limit)
SP formalism of energy transport [3,4] ✓
According to [4,5], soliton concept cannot resolve energy transport in polypeptide chain since the basic energy parameters of these media do not satisfy the conditions for soliton formation!
Peculiariry! ✓
Theory of nonadiabatic SP states is applicable in exciton-phonon strong coupling limit only!
[3] D.M. Alexander, J.A. Krumhansl, PRB, 33, 7172 (1986)
[4] D.W. Brown, Z. Ivic, PRB, 40, 9876 (1989)

8. polypeptide macromolecular chains: energy parameters
According to [5, Pouthier]: weak to intermediate coupling
According to [6, Hamm & Tsironis]: it seems that (depending of the values of the system parameters and temperature) the abrupt transition from partially dressed to self-trapped quasiparticle may occur
Instead of standard SP approach (LF), we need an approach which involve the concept of partial dressing
[5] V. Pouthier, JCP, 132, (2010)
[6] P. Hamm and G. Tsironis, PRB, 78, (2008)

9. Partial dressing concept
partial dressing concept presents as variational method , which was introduced to investigate the problems related to properties of ST states in some crystal structures.
in order to investigate under which conditions ST states may occur in some crystal structures, it was formulated “fq approach” [Y. Toyozawa, Prog. Theor. Phys., 26, 29, (1961)].
Latter, similar approach of partial dressing (“d approach” [ D. Emin, Adv. in Phys., 22, 57, (1973)]) was used to investigate the intermediate region between adiabatic and nonadiabatic limits of ST states [Brown, Z. Ivic, PRB, 40, 9876 (1989), Z.Ivic, Phys. D, 113, 218 (1998) ].
[7] Y. Toyozawa, Prog. Theor. Phys., 26, 29, (1961)
[8] D. Emin, Adv. in Phys., 22, 57, (1973)
[9] Z.Ivic, Phys. D, 113, 218 (1998)

10. Theoretical models system under consideration includes:
single vibron (excited on the n-th structural element (SE) of the macromolecule);
phonons
we suppose that:
vibron excitation can move from n-th to its neighbouring SE;
vibron excitation interacts with phonons.
Fig. b) α-Helix with peptide groups & 3 types hydrogen-bonds
Fig. a) α-Helix & 3 types hydrogen-bonds

11. Step I Theoretical models: Hamiltonian
where:
D is the vibron excitation energy,
an+(an) -- presence (absence) of the vibron quanta on n-th SE,
bq+(bq) --- creates (annihilates) phonon quanta,
wq , J - the phonon frequency, and intersite overlap integral (characterize the vibron transfer between neighbouring SE in the chain),
Fq*=F-q is the vibron-phonon coupling parameter (it governs the character of ST states).

12. Theoretical models: Vibron-phonon interaction
1) interaction with dispersionless optical phonon modes (MC model) [11,12]:
2) interaction with acoustical phonon modes (ADP model) [12]:
where c is the vibron-phonon coupling constant, M is the mass of the molecular group, k is the stiffness of the chain
[10] T. Holstein, Ann. Phys., 8, 325 (1959)
[11] A.S. Davydov, Teoriya tverdogo tela, Moskva (1976); G.D. Mahan, Many-Particle Physics, New York (1986)

13. Step II Theoretical models: Transition to Small Polaron picture
1. standard SP picture (LF transformation)
2. partial dressed SP picture (MLF transformation) “fq -variational” approach
3. partial dressed SP picture (MLF transformation) “d -variational” approach

14. Step III Theoretical models: Mean field Hamiltonian
mean field Hamiltonian -> better upper bound of the system free energy, and system ground state energy [13]
denotes the average over the new phonon ensemble
in k representation
[12] David Yakorny and Robert Silbey, The Journ. of Chem. Phys., 65, 1042 (1976)

15. Theoretical models: Variational energy
variational energy of the SP band states
vibron-band narrowing factor (characterize the degree of the reduction of the overlap integral, or equivalently, the enhancement of the polaron effective mass)
With
phonon average number

16. Step IV Theoretical models: Vibron ground state energy: determination of the variational parameter(s)
phonon vacuum
ground state of
determination of the variational parameter(s):
“fq -variational” approach
“d -variational” approach
k0=p/R0 (J<0), or k0=0 (J>0)

17. Derivation of the basic system parameter (D. Čevizović, S. Galović, A
Derivation of the basic system parameter (D. Čevizović, S. Galović, A. R. and Z. Ivić, arXiv: )
1) first, we perform the summation over q by the rule
2) second, inroduce the (S,B) system parameter space:
SP binding energy
SP band narrowing factor (LF)

18. Obtained results: eGS for optical phonon case
fq approach
d approach

19. Obtained results: W for optical phonon case
fq approach
d approach

20. Obtained results: eGS for acoustic phonon case
fq approach
d approach

21. Obtained results: W for acoustic phonon case
fq approach
d approach

22. Conclusions & Outlook
1. both variational approaches gives similar qualitative and quantitative predictions!
2. In comparison with standard SP theories, both variational approaches predicts for certain values of the system parameters, an abrupt junction of vibron ST properties from light, (practically free) to heavy (practically immobile) vibron ST state). This transition is followed by abrupt enhancement of vibron effective mass m*. Obtained results are in good agreement with some results obtained by numerical approach in [P. Hamm and G. Tsironis, PRB, 78, (2008)].
3. in both cases (interaction with optical and acoustical phonons) “fq approach” gives something lower values of the ground state energy eGS, compared with ones obtained by “d approach”
4. in non adiabatic region the difference in eGS is too small (practically it may be neglected)

23. Thank you for attention!
Conclusions & Outlook
Because of many bio-macromolecular structures behave as them are in nonadiabatic, weak (or intermediate) coupling limit, it seems that “d approach” may be more adequated method for study of ST vibron states in such structures.
Now, we should to pass to study of the vibron transport along macromolecular chain to describe energy transfer in it!
Thank you for attention!