Local topology and perestroikas in protein structure and folding dynamics

Methods of local topology are introduced to the field of protein physics. This is done by interpreting the folding and unfolding processes of a globular protein in terms of conformational bifurcations that alter the local topology of the proteins Cα backbone. The mathematical formulation extends Arnold's perestroikas to piecewise linear chains using the discrete Frenet frame formalism. In the low-temperature folded phase, the backbone geometry generalizes the concept of a Peano curve, with its modular building blocks modeled by soliton solutions of a discretized nonlinear Schrödinger equation. The onset of thermal unfolding begins when perestroikas change the flattening and branch points that determine the centers of solitons. When temperature increases, the perestroikas cascade, which leads to a progressive disintegration of the modular structures. The folding and unfolding processes are quantitatively characterized by a correlation function that describes the evolution of perestroikas under temperature changes. The approach provides a comprehensive framework for understanding the Physics of protein folding and unfolding transitions, contributing to the broader field of protein structure and dynamics.