Random walks in the space of conformations of toy proteins

TL;DR

This study models the conformational space of a toy protein as a random walk and finds evidence that it has hyperbolic geometry, revealing complex structural properties of protein folding landscapes.

Contribution

It introduces a novel approach to analyze protein conformational spaces using random walk models and compares empirical data with theoretical hyperbolic geometries.

Findings

Conformational space exhibits non-trivial dimensions.

Evidence of negative curvature in the space.

Supports hyperbolic geometry in protein folding landscapes.

Abstract

Monte Carlo dynamics of the lattice 48 monomers toy protein is interpreted as a random walk in an abstract (discrete) space of conformations. To test the geometry of this space, we examine the return probability $P(T)$, which is the probability to find the polymer in the native state after $T$ Monte Carlo steps, provided that it starts from the native state at the initial moment. Comparing computational data with the theoretical expressions for $P(T)$ for random walks in a variety of different spaces, we show that conformational spaces of polymer loops may have non-trivial dimensions and exhibit negative curvature characteristic of Lobachevskii (hyperbolic) geometry.