Local topology and perestroikas in protein structure and folding dynamics

TL;DR

This paper introduces a novel mathematical framework using local topology and perestroikas to analyze protein folding and unfolding, linking geometric changes to thermal dynamics and soliton models.

Contribution

It extends Arnol'd's perestroikas to protein backbone chains using discrete Frenet frames, providing a new topological perspective on folding dynamics.

Findings

Perestroikas change during thermal unfolding.

Backbone geometry modeled by solitons of nonlinear Schrödinger equation.

Progressive disintegration of modular structures with temperature.

Abstract

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.