Topology and geometry of molecular conformational spaces and energy landscapes

TL;DR

This paper explores the geometric and topological properties of molecular conformational spaces and energy landscapes, combining theoretical and computational methods to better understand their structure considering molecular symmetries.

Contribution

It introduces a mathematical framework using principal bundles and orbifolds to describe molecular configuration spaces, integrating geometric and topological data analysis tools.

Findings

Configuration spaces form principal bundles and orbifolds when symmetries are considered.

Geometric and topological tools reveal the structure of energy landscapes.

The approach advances understanding of molecular conformational complexity.

Abstract

Understanding the geometry and topology of configuration or conformational spaces of molecules has relevant applications in chemistry and biology such as the proteins folding problem, drug design and the structure activity relationship problem. Despite their relevance, configuration spaces of molecules are only partially understood. In this paper we discuss both theoretical and computational approaches to the configuration spaces of molecules and their associated energy landscapes. Our mathematical approach shows that when symmetries of the molecules are taken into account, configuration spaces of molecules give rise to certain principal bundles and orbifolds. We also make use of a variety of geometric and topological tools for data analysis to study the topology and geometry of these spaces.