Modeling Coil-Globule-Helix Transition in Polymers by Self-Interacting Random Walks

Random walks (RWs) have been important in statistical physics and can describe the statistical properties of various processes in physical, chemical, and biological systems. In this study, we have proposed a self-interacting random walk model in a continuous three-dimensional space, where the walker and its previous visits interact according to a realistic Lennard-Jones (LJ) potential uLJr=εr0/r12-2r0/r6. It is revealed that the model shows a novel globule-to-helix transition in addition to the well-known coil-to-globule collapse in its trajectory when the temperature decreases. The dependence of the structural transitions on the equilibrium distance r0 of the LJ potential and the temperature T were extensively investigated. The system showed many different structural properties, including globule-coil, helix-globule-coil, and line-coil transitions depending on the equilibrium distance r0 when the temperature T increases from low to high. We also obtained a correlation form of k_BT_c = λε for the relationship between the transition temperature T_c and the well depth ε, which is consistent with our numerical simulations. The implications of the random walk model on protein folding are also discussed. The present model provides a new way towards understanding the mechanism of helix formation in polymers like proteins.