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 ${\displaystyle {\hat {H}}}$ is

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

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

The quasiparticle (exciton) Hamiltonian ${\displaystyle {\hat {H}}_{\text{qp}}}$ is:

${\displaystyle {\hat {H}}_{\text{qp}}=}$ ${\displaystyle \sum _{n,\alpha }E_{0}{\hat {A}}_{n,\alpha }^{\dagger }{\hat {A}}_{n,\alpha }}$ ${\displaystyle -J\sum _{n,\alpha }\left({\hat {A}}_{n,\alpha }^{\dagger }{\hat {A}}_{n+1,\alpha }+{\hat {A}}_{n,\alpha }^{\dagger }{\hat {A}}_{n-1,\alpha }\right)}$ ${\displaystyle +L\sum _{n,\alpha }\left({\hat {A}}_{n,\alpha }^{\dagger }{\hat {A}}_{n,\alpha +1}+{\hat {A}}_{n,\alpha }^{\dagger }{\hat {A}}_{n,\alpha -1}\right)}$

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 I vibration (CO stretching), ${\displaystyle J=0.155}$ zJ is the dipole-dipole coupling energy between a particular amide I bond and those ahead and behind along the same spine, ${\displaystyle L=0.246}$ 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, ${\displaystyle {\hat {A}}_{n,\alpha }^{\dagger }}$ and ${\displaystyle {\hat {A}}_{n,\alpha }}$ are respectively the boson creation and annihilation operator for a quasiparticle at the peptide group ${\displaystyle (n,\alpha )}$.[7][8][9]

The phonon Hamiltonian ${\displaystyle {\hat {H}}_{\text{ph}}}$ is

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

where ${\displaystyle {\hat {u}}_{n,\alpha }}$ is the displacement operator from the equilibrium position of the peptide group ${\displaystyle (n,\alpha )}$, ${\displaystyle {\hat {p}}_{n,\alpha }}$ is the momentum operator of the peptide group ${\displaystyle (n,\alpha )}$, ${\displaystyle M_{n,\alpha }}$ is the mass of the peptide group ${\displaystyle (n,\alpha )}$, and ${\displaystyle w=13-19.5}$ N/m is an effective elasticity coefficient of the lattice (the spring constant of a hydrogen bond).[8]

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

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

where ${\displaystyle \chi =35-62}$ pN is an anharmonic parameter arising from the coupling between the quasiparticle (exciton) and the lattice displacements (phonon) and parameterizes the strength of the exciton-phonon interaction.[8] 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;[10]
• mixed quantum-classical approach, in which the amide I vibration is treated quantum mechanically but the lattice is classical;[9]
• classical approach, in which both the amide I and the lattice motions are treated classically.[11]

The mathematical techniques that are used to analyze the Davydov soliton are similar to some that have been developed in polaron theory.[12] In this context, the Davydov soliton corresponds to a polaron that is:

• large so the continuum limit approximation is justified,[8]
• acoustic because the self-localization arises from interactions with acoustic modes of the lattice,[8]
• weakly coupled because the anharmonic energy is small compared with the phonon bandwidth.[8]

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.[8] Supposing that the Davydov soliton is localized to 5 turns of the α-helix results in significant uncertainty in the velocity of the soliton ${\displaystyle \Delta v=133}$ m/s, a fact that is obscured if one models the Davydov soliton as a classical object.