Davydov soliton

In quantum biology, the Davydov soliton (after the Soviet Ukrainian physicist Alexander Davydov) is a quasiparticle representing an excitation propagating along the self-trapped amide I groups within the α-helices of proteins. It is a solution of the Davydov Hamiltonian.

The Davydov model describes the interaction of the amide I vibrations with the hydrogen bonds that stabilize the α-helices of proteins. The elementary excitations within the α-helix are given by the phonons which correspond to the deformational oscillations of the lattice, and the excitons which describe the internal amide I excitations of the peptide 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 the C=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 called self-localization or self-trapping.^[3]^[4]^[5] Solitons in which the energy is distributed in a fashion preserving the helical symmetry are dynamically unstable, and such symmetrical solitons once formed decay rapidly when they propagate. On the other hand, an asymmetric soliton which spontaneously breaks the local translational and helical symmetries possesses the lowest energy and is a robust localized entity.^[6]

Davydov Hamiltonian

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 ${\hat {H}}$ is

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

where ${\hat {H}}_{\text{ex}}$ is the exciton Hamiltonian, which describes the motion of the amide I excitations between adjacent sites; ${\hat {H}}_{\text{ph}}$ is the phonon Hamiltonian, which describes the vibrations of the lattice; and ${\hat {H}}_{\text{int}}$ is the interaction Hamiltonian, which describes the interaction of the amide I excitation with the lattice.^[3]^[4]^[5]

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

{\begin{aligned}{\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{aligned}}

where the index $n=1,2,\cdots ,N$ counts the peptide groups along the α-helix spine, the index $\alpha =1,2,3$ counts each α-helix spine, $E_{0}=32.8$ z J is the energy of the amide I vibration (CO stretching), $J_{1}=0.155$ z J is the dipole-dipole coupling energy between a particular amide I bond and those ahead and behind along the same spine,^[7] $J_{2}=0.246$ z J 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,^[7] ${\hat {A}}_{n,\alpha }^{\dagger }$ and ${\hat {A}}_{n,\alpha }$ are respectively the boson creation and annihilation operator for an amide I exciton at the peptide group $(n,\alpha )$ .^[8]^[9]^[10]

The phonon Hamiltonian ${\hat {H}}_{\text{ph}}$ is^[11]^[12]^[13]^[14]

{\hat {H}}_{\text{ph}}={\frac {1}{2}}\sum _{n,\alpha }\left

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