Davydov soliton

Davydov Hamiltonian is formally similar to the Fröhlich-Holstein Hamiltonian for the interaction of electrons with a polarizable lattice. Thus the Hamiltonian of the energy operator $\backslash hat\{H\}$ is

:$$

\hat{H}=\hat{H}_{\text{ex}}+\hat{H}_{\text{ph}}+\hat{H}_{\text{int}}

where $\backslash hat\{H\}\_\{\backslash text\{ex\}\}$ is the exciton Hamiltonian, which describes the motion of the amide I excitations between adjacent sites; $\backslash hat\{H\}\_\{\backslash text\{ph\}\}$ is the phonon Hamiltonian, which describes

the vibrations of the lattice; and $\backslash hat\{H\}\_\{\backslash text\{int\}\}$ is the interaction Hamiltonian, which describes the interaction of the amide I excitation with the lattice.

The exciton Hamiltonian $\backslash hat\{H\}\_\{\backslash text\{ex\}\}$ is

:$\backslash begin\{align\}$

\hat{H}_{\text{ex}} =& \sum_{n,\alpha}E_{0}\hat{A}_{n,\alpha}^{\dagger}\hat{A}_{n,\alpha} \\

&-J_1\sum_{n,\alpha}\left(\hat{A}_{n,\alpha}^{\dagger}\hat{A}_{n+1,\alpha}+\hat{A}_{n,\alpha}^{\dagger}\hat{A}_{n-1,\alpha}\right) \\

&+J_2\sum_{n,\alpha}\left(\hat{A}_{n,\alpha}^{\dagger}\hat{A}_{n,\alpha+1}+\hat{A}_{n,\alpha}^{\dagger}\hat{A}_{n,\alpha-1}\right)

\end{align}

where the index $n=1,2,\backslash cdots,N$ counts the peptide groups along the α-helix spine, the index $\backslash alpha=1,2,3$ counts each α-helix spine, $E\_\{0\}=32.8$ zJ is the energy of the amide I

vibration (CO stretching), $J\_1=0.155$ zJ is the dipole-dipole coupling energy between a particular amide I bond and those ahead and behind along the same spine,{{cite journal

| last1 = Nevskaya

| first1 = N. A.

| last2 = Chirgadze

| first2 = Yuriy Nikolaevich

| title = Infrared spectra and resonance interactions of amide-I and II vibrations of α-helix

| journal = Biopolymers

| volume = 15

| issue = 4

| pages = 637–648

| year = 1976

| doi = 10.1002/bip.1976.360150404

| pmid = 1252599

| s2cid = 98650911

}} $J\_2=0.246$ zJ is the

dipole-dipole coupling energy between a particular amide I bond and those on adjacent spines in the same unit cell of the protein α-helix, $\backslash hat\{A\}\_\{n,\backslash alpha\}^\{\backslash dagger\}$ and $\backslash hat\{A\}\_\{n,\backslash alpha\}$ are respectively

the boson creation and annihilation operator for an amide I exciton at the peptide group $(n,\backslash alpha)$.

{{cite journal

| last1 = Hyman

| first1 = James M.

| last2 = McLaughlin

| first2 = David W.

| last3 = Scott

| first3 = Alwyn C.

| year = 1981

| title = On Davydov's alpha-helix solitons

| journal=Physica D: Nonlinear Phenomena

| volume=3

| issue=1

| pages = 23–44

| bibcode =1981PhyD....3...23H

| doi= 10.1016/0167-2789(81)90117-2

}}

{{cite journal

| last1 = Scott

| first1 = Alwyn C.

| title = Davydov's soliton

| journal = Physics Reports

| volume = 217

| issue = 1

| pages = 1–67

| year = 1992

| doi = 10.1016/0370-1573(92)90093-F

|bibcode = 1992PhR...217....1S }}

{{cite journal

| last1 = Cruzeiro-Hansson

| first1 = Leonor

| last2 = Takeno

| first2 = Shozo

| year = 1997

| title = Davydov model: the quantum, mixed quantum-classical, and full classical systems

| journal=Physical Review E

| volume=56 | issue=1 | pages = 894–906

| bibcode = 1997PhRvE..56..894C

| doi= 10.1103/PhysRevE.56.894

}}

The phonon Hamiltonian $\backslash hat\{H\}\_\{\backslash text\{ph\}\}$ is{{cite journal

| last1 = Davydov

| first1 = Alexander S.

| title = Solitons in quasi-one-dimensional molecular structures

| journal = Soviet Physics Uspekhi

| volume = 25

| issue = 12

| pages = 898–918

| year = 1982

| doi = 10.1070/pu1982v025n12abeh005012

}}{{cite journal

| last1 = Georgiev

| first1 = Danko D.

| last2 = Glazebrook

| first2 = James F.

| title = Thermal stability of solitons in protein α-helices

| journal = Chaos, Solitons and Fractals

| volume = 155

| pages = 111644

| year = 2022

| doi = 10.1016/j.chaos.2021.111644

| s2cid = 244693789

| mr = 4372713

| arxiv = 2202.00525

| bibcode = 2022CSF...15511644G

}}{{cite journal

| last1 = Zolotaryuk

| first1 = Alexander V.

| last2 = Christiansen

| first2 = P. L.

| last3 = Nordеn

| first3 = B.

| last4 = Savin

| first4 = Alexander V.

| title = Soliton and ratchet motions in helices

| journal = Condensed Matter Physics

| volume = 2

| issue = 2

| pages = 293–302

| year = 1999

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| bibcode = 1999CMPh....2..293Z

| doi-access = free

}}{{cite journal

| last1 = Brizhik

| first1 = Larissa S.

| last2 = Luo

| first2 = Jingxi

| last3 = Piette

| first3 = Bernard M. A. G.

| last4 = Zakrzewski

| first4 = Wojtek J.

| title = Long-range donor-acceptor electron transport mediated by alpha-helices

| journal = Physical Review E

| volume = 100

| issue = 6

| pages = 062205

| year = 2019

| doi = 10.1103/PhysRevE.100.062205

| pmid = 31962511

| arxiv = 1909.08266

| bibcode = 2019PhRvE.100f2205B

| s2cid = 202660869

}}

:$$

\hat{H}_{\text{ph}}=\frac{1}{2}\sum_{n,\alpha}\left[w_1(\hat{u}_{n+1,\alpha}-\hat{u}_{n,\alpha})^{2}+w_2(\hat{u}_{n,\alpha+1}-\hat{u}_{n,\alpha})^{2}+\frac{\hat{p}_{n,\alpha}^{2}}{M_{n,\alpha}}\right]

where $\backslash hat\{u\}\_\{n,\backslash alpha\}$ is the displacement operator from the equilibrium position of the peptide group $(n,\backslash alpha)$, $\backslash hat\{p\}\_\{n,\backslash alpha\}$ is the momentum operator of the peptide group $(n,\backslash alpha)$, $M\_\{n,\backslash alpha\}$ is the mass of the peptide group $(n,\backslash alpha)$, $w\_1=13-19.5$ N/m is an effective elasticity coefficient of the lattice (the spring constant of a hydrogen bond) and $w\_2=30.5$ N/m is the lateral coupling between the spines.{{cite journal

| last1 = Savin

| first1 = Alexander V.

| last2 = Zolotaryuk

| first2 = Alexander V.

| title = Dynamics of the amide-I excitation in a molecular chain with thermalized acoustic and optical modes

| journal = Physica D: Nonlinear Phenomena

| volume = 68

| issue = 1

| pages = 59–64

| year = 1993

| doi = 10.1016/0167-2789(93)90029-Z

| bibcode = 1993PhyD...68...59S

}}

Finally, the interaction Hamiltonian $\backslash hat\{H\}\_\{\backslash text\{int\}\}$ is

:$$

\hat{H}_{\text{int}}=\chi\sum_{n,\alpha}\left[(\hat{u}_{n+1,\alpha}-\hat{u}_{n,\alpha})\hat{A}_{n,\alpha}^{\dagger}\hat{A}_{n,\alpha}\right]

where $\backslash chi=35-62$ pN is an anharmonic parameter arising from the coupling between the exciton and the lattice displacements (phonon) and parameterizes the strength of the exciton-phonon interaction. The value of this parameter for α-helix has been determined via comparison of the theoretically calculated absorption line shapes with the experimentally measured ones.

There are three possible fundamental approaches for deriving equations of motion from Davydov Hamiltonian:

• quantum approach, in which both the amide I vibration (excitons) and the lattice site motion (phonons) are treated quantum mechanically;{{cite journal

| last1 = Kerr

| first1 = William C.

| last2 = Lomdahl

| first2 = Peter S.

| title = Quantum-mechanical derivation of the equations of motion for Davydov solitons

| journal = Physical Review B

| volume = 35

| issue = 7

| pages = 3629–3632

| year = 1987

| doi = 10.1103/PhysRevB.35.3629

| pmid = 9941870

| bibcode = 1987PhRvB..35.3629K

| hdl = 10339/15922

| hdl-access = free

}}

• mixed quantum-classical approach, in which the amide I vibration is treated quantum mechanically but the lattice is classical;
• classical approach, in which both the amide I and the lattice motions are treated classically.{{cite journal

| last1 = Škrinjar

| first1 = M. J.

| last2 = Kapor

| first2 = D. V.

| last3 = Stojanović

| first3 = S. D.

| title = Classical and quantum approach to Davydov's soliton theory

| journal = Physical Review A

| volume = 38

| issue = 12

| pages = 6402–6408

| year = 1988

| doi = 10.1103/PhysRevA.38.6402

| pmid = 9900400

| bibcode = 1988PhRvA..38.6402S

}}

The mathematical techniques that are used to analyze the Davydov soliton are similar to some that have been developed in polaron theory.{{cite journal

| last1 = Sun

| first1 = Jin

| last2 = Luo

| first2 = Bin

| last3 = Zhao

| first3 = Yang

| title = Dynamics of a one-dimensional Holstein polaron with the Davydov ansätze

| journal = Physical Review B

| volume = 82

| issue = 1

| pages = 014305

| year = 2010

| doi = 10.1103/PhysRevB.82.014305

| arxiv = 1001.3198

| bibcode = 2010PhRvB..82a4305S

| s2cid = 118564115

}} In this context, the Davydov soliton corresponds to a polaron that is:

• large so the continuum limit approximation is justified,
• acoustic because the self-localization arises from interactions with acoustic modes of the lattice,
• weakly coupled because the anharmonic energy is small compared with the phonon bandwidth.

The Davydov soliton is a quantum quasiparticle and it obeys Heisenberg's uncertainty principle. Thus any model that does not impose translational invariance is flawed by construction. Supposing that the Davydov soliton is localized to 5 turns of the α-helix results in significant uncertainty in the velocity of the soliton $\backslash Delta\; v=133$ m/s, a fact that is obscured if one models the Davydov soliton as a classical object.

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Category:Biological matter

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