  Methods of local topology are introduced to the field of protein physics.
This is achieved by explaining how the folding and unfolding processes of a
globular protein alter the local topology of the protein's C-alpha backbone
through conformational bifurcations. The mathematical formulation builds on the
concept of Arnol'd's perestroikas, by extending it 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
Schroedinger 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.
