Energy transport in α-helical protein models: One-strand versus three-strand systems

We study the transport of vibrational energy in $\alpha$-helix forms of proteins within the frame of a steric Davydov model. Our main objective is the comparison of the localization and transport properties of the widely used one-strand approximation of the $\alpha$ helix associated with a linear chain of hydrogen bonded peptide groups and the complete three-dimensional helical system consisting of three such strands being connected via covalent bonds. We construct polaron solutions of the set of discrete equations utilizing a nonlinear map method for finding the ground state. With respect to the polaron wave pattern, we found that while the excitonic wave function of the one-strand system is localized at a site and monotonically decaying apart from it on the three-strand system a multihump structure arises whose envelope exhibits monotonic decay. For the polaronic states the static deformations of the H bonds are stronger for an isolated strand than for a strand embedded in the three-dimensional protein cage. In the excited region the radius of the helix slightly decreases. The participation numbers show that the exciton localization is more strongly pronounced on the one-strand system than on its three-strand counterpart. In our dynamical studies we pay special attention to the mobility of the polarons when their internal pinning modes are initiated. The latter are identified as certain isolated Floquet eigenvalues obtained from the Floquet map, which is derived from the linearized equations in tangent space. It is shown that the polarons on the one-strand system move with higher velocity than on the three-strand system. In the latter case the preferred transport path proceeds across the hydrogen bonds and polaron motion along the covalent channel is suppressed.