Inquantum biology, theDavydov soliton (after the Soviet Ukrainian physicistAlexander Davydov) is aquasiparticle representing anexcitation propagating along the self-trappedamide I groups within theα-helices ofproteins. It is a solution of the DavydovHamiltonian.

The Davydov model describes the interaction of the amide Ivibrations with thehydrogen bonds that stabilize the α-helices of proteins. The elementary excitations within the α-helix are given by thephonons which correspond to the deformational oscillations of the lattice, and theexcitons which describe the internal amide I excitations of thepeptide groups. Referring to the atomic structure of an α-helix region of protein the mechanism that creates the Davydov soliton (polaron, exciton) can be described as follows:vibrational energy of theC=O stretching (or amide I)oscillators that is localized on the α-helix acts through a phonon coupling effect to distort the structure of the α-helix, while the helical distortion reacts again through phonon coupling to trap the amide I oscillation energy and prevent its dispersion. This effect is calledself-localization orself-trapping.^[3][4][5]Solitons in which theenergy is distributed in a fashion preserving thehelicalsymmetry are dynamically unstable, and suchsymmetrical solitons once formed decay rapidly when they propagate. On the other hand, anasymmetric soliton whichspontaneously breaks the local translational and helical symmetries possesses the lowest energy and is a robust localized entity.^[6]

DavydovHamiltonian is formally similar to theFröhlich-Holstein Hamiltonian for the interaction of electrons with a polarizable lattice. Thus theHamiltonian of theenergy operator${\displaystyle \hat{H}}$ is

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

where${\displaystyle {\hat{H}}_{\text{ex}}}$ is theexcitonHamiltonian, which describes the motion of the amide I excitations between adjacent sites;${\displaystyle {\hat{H}}_{\text{ph}}}$ is thephononHamiltonian, which describesthevibrations of thelattice; and${\displaystyle {\hat{H}}_{\text{int}}}$ is theinteractionHamiltonian, which describes the interaction of the amide I excitation with the lattice.^[3][4][5]

TheexcitonHamiltonian${\displaystyle {\hat{H}}_{\text{ex}}}$ is

where the index${\displaystyle n=1,2,\cdots ,N}$ counts the peptide groups along the α-helix spine, the index${\displaystyle \alpha =1,2,3}$ counts each α-helix spine,${\displaystyle E_0=32.8}$zJ is the energy of the amide Ivibration (CO stretching),${\displaystyle J_1=0.155}$zJ is thedipole-dipole coupling energy between a particular amide I bond and those ahead and behind along the same spine,^[7]${\displaystyle J_2=0.246}$zJ is thedipole-dipole coupling energy between a particular amide I bond and those on adjacent spines in the same unit cell of theproteinα-helix,^[7]${\displaystyle {\hat{A}}_{n,\alpha }^{†}}$ and${\displaystyle {\hat{A}}_{n,\alpha }}$ are respectivelythebosoncreation and annihilation operator for an amide I exciton at thepeptide group${\displaystyle (n,\alpha )}$.^[8][9][10]

ThephononHamiltonian${\displaystyle {\hat{H}}_{\text{ph}}}$ is^{[11][12][13][14]}

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

where${\displaystyle {\hat{u}}_{n,\alpha }}$ is thedisplacement operator from the equilibrium position of thepeptide group${\displaystyle (n,\alpha )}$,${\displaystyle {\hat{p}}_{n,\alpha }}$ is themomentum operator of the peptide group${\displaystyle (n,\alpha )}$,${\displaystyle M_{n,\alpha }}$ is themass of the peptide group${\displaystyle (n,\alpha )}$,${\displaystyle w_1=13-19.5}$N/m is aneffective elasticity coefficient of the lattice (thespring constant of ahydrogen bond)^[9] and${\displaystyle w_2=30.5}$N/m is the lateral coupling between the spines.^[12][15]

Finally, theinteractionHamiltonian${\displaystyle {\hat{H}}_{\text{int}}}$ is

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

where${\displaystyle \chi =35-62}$pN is an anharmonic parameter arising from the coupling between the exciton and the lattice displacements (phonon) and parameterizes the strength of theexciton-phononinteraction.^[9] The value of this parameter forα-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;^[16]
• mixed quantum-classical approach, in which the amide I vibration is treated quantum mechanically but the lattice is classical;^[10]
• classical approach, in which both the amide I and the lattice motions are treated classically.^[17]

The mathematical techniques that are used to analyze the Davydov soliton are similar to some that have been developed in polaron theory.^[18] In this context, the Davydov soliton corresponds to apolaron that is:

• large so the continuum limit approximation is justified,^[9]
• acoustic because the self-localization arises from interactions with acoustic modes of the lattice,^[9]
• weakly coupled because the anharmonic energy is small compared with the phonon bandwidth.^[9]

The Davydov soliton is aquantum quasiparticle and it obeysHeisenberg's uncertainty principle. Thus any model that does not impose translational invariance is flawed by construction.^[9] Supposing that the Davydov soliton is localized to 5 turns of theα-helix results in significant uncertainty in thevelocity of thesoliton${\displaystyle \mathrm{\Delta }v=133}$ m/s, a fact that is obscured if one models the Davydov soliton as a classical object.

• Biological matter
• Biophysics
• Proteins
• Quantum biology

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