# Topological metrics as evolutionary and dynamical descriptors of conformational landscapes within protein families

## TL;DR

This paper introduces topological metrics to link protein structure with dynamics and evolution, showing that static structures can predict dynamic behavior.

## Contribution

The novel contribution is using topological metrics like LTE to connect protein structure with dynamics and evolutionary trajectories.

## Key findings

- LTE strongly correlates with dynamical measures like DFI and tracks evolutionary changes in proteins.
- Topological metrics applied to static structures can predict conformational dynamics and functional evolution.
- Molecular dynamics simulations reveal how topological changes underpin functional evolution.

## Abstract

Identifying the key order parameters that connect a protein‘s native structure to its dynamical and evolutionary behavior remains a central challenge. We introduce topological and geometrical metrics—specifically, writhe and Local Topological Energy (LTE)—to investigate these connections. Applying these tools to both present-day and ancestral forms of thioredoxin and β-lactamase, we show that LTE strongly correlates with established dynamical measures such as the Dynamical Flexibility Index (DFI). Remarkably, LTE distributions also track the evolutionary trajectories of these proteins, suggesting that the topological geometry of the native state encodes key aspects of both dynamics and evolution. Through molecular dynamics simulations, we further reveal critical shifts in the topological landscape of proteins, providing a molecular mechanism by which functional evolution proceeds via alterations in conformational dynamics. Extending our analysis to over 100 proteins, we provide the first compelling evidence that topological descriptors derived from static structures can reliably predict dynamical behavior. In general, our findings demonstrate that simple geometrical metrics capture essential features of protein conformational landscapes, offering a powerful new approach to bridging protein structure, dynamics, and evolution.

Advances in protein science and artificial intelligence have resulted in a wealth of protein structures, yet a fundamental challenge remains: how does a protein‘s native structure encode its dynamical behavior? Addressing this question is crucial for understanding protein function and advancing protein engineering, particularly for designing nanomachines. We introduce a novel approach, rooted in knot theory, to bridge this gap. We demonstrate that topological metrics, even when applied to static protein structures, reveal key aspects of protein dynamics. This discovery provides a rigorous and quantitative framework for predicting protein flexibility and evolutionary adaptation directly from structure. Our approach has broad implications, from improving AI-based protein modeling to anticipating functional outcomes of mutations, with applications in protein engineering, biotechnology and medicine.