Mobile polaron solutions and nonlinear electron transfer in helical protein models

We consider the electron transfer along helical forms of proteins. The spatial structure of the protein helices is modeled by three-dimensional oscillator networks whose constituents represent peptide groups. Covalent and hydrogen bonds between the peptide units are modeled by point-point interaction potentials. The electronic degree of freedom is described by a tight-binding system including besides the nearest-neighbor exchange interactions between covalently connected units also third- or fourth-nearest neighbor interactions between hydrogen-bonded sites. In addition each peptide unit possesses an internal vibrational degree of freedom. The various dynamical degrees of freedom are coupled to each other making the exchange of electronic, intramolecular, and bond-vibrational energy possible. In the first part of the paper we investigate the static polaron formation resulting from strong interactions between the electron and the intramolecular vibrations. The 3-10 helix and the $\alpha$ helix are investigated. Polaron states are constructed analytically on the basis of a variational approach. Compared to the $\alpha$ helix the 3-10 helix supports stronger localized polarons. In the second part of the paper we take the coupling of the polaron with the vibrations of the three-dimensional protein matrix into account focusing interest on the bond-assisted initiation of polaron motion. In detail it is demonstrated that the interplay of the protein matrix and the polaron dynamics conspire to activate not only the polaron motion but also to maintain a long-lived coherently traveling localized pattern along the lattice of peptide units. Starting from a nonequilibrium state it is shown that coexisting electron and bond-vibration breathers assist the relaxation dynamics towards energy equilibration and the attainment of a stationary regime.