Perturbative study of classical Ablowitz-Ladik type soliton dynamics in relation to energy transport in $\alpha$-helical proteins

Classical Ablowitz-Ladik type soliton dynamics from three closely related classical nonlinear equations is studied using a perturbative method. Model nonintegrable equations are derived by assuming nearest neighbor hopping of an exciton(vibron) in the presence of a full exciton(vibron)-phonon interaction in soft molecular chains in general and spines of $\alpha$-helices in particular. In all cases, both trapped and moving solitons are found implying activation energy barrier for propagating solitons. Analysis further shows that staggered and nearly staggered trapped solitons will have a negative effective mass. In some models the exciton(vibron)-phonon coupling affects the hopping. For these models, when the conservation of probability is taken into account, only propagating solitons with a broad profile are found to be acceptable solutions. Of course, for the soliton to be a physically meaningful entity, total nonlinear coupling strength should exceed a critical value. On the basis of the result, a plausible modification in the mechanism for biological energy transport involving conformational change in $\alpha$-helix is proposed. Future directions of the work are also mentioned.