Exact Topology of Dynamic Probability Surface of an Activated Process by Persistent Homology

TL;DR

This paper introduces a precise topological analysis method using persistent homology to identify key features of high-dimensional probability surfaces in activated processes, revealing the true transition states beyond traditional saddle-point models.

Contribution

The authors develop an exact approach leveraging persistent homology to analyze the topology of probability surfaces, providing new insights into transition states in high-dimensional spaces.

Findings

Transition states are identified as prominent probability peaks, not saddle points.

Principal component analysis can distort the topology and obscure dynamic features.

The method is general and applicable to other high-dimensional activated processes.

Abstract

To gain insight into reaction mechanism of activated processes, we introduce an exact approach for quantifying the topology of high-dimensional probability surfaces of the underlying dynamic processes. Instead of Morse indexes, we study the homology groups of a sequence of superlevel sets of the probability surface over high-dimensional configuration spaces using persistent homology. For alanine-dipeptide isomerization, a prototype of activated processes, we identify locations of probability peaks and connecting-ridges, along with measures of their global prominence. Instead of a saddle-point, the transition state ensemble (TSE) of conformations are at the most prominent probability peak after reactants/products, when proper reaction coordinates are included. Intuition-based models, even those exhibiting a double-well, fail to capture the dynamics of the activated process. Peak occurrence, prominence, and locations can be distorted upon subspace projection. While principal component analysis account for conformational variance, it inflates the complexity of the surface topology and destroy dynamic properties of the topological features. In contrast, TSE emerges naturally as the most prominent peak beyond the reactant/product basins, when projected to a subspace of minimum dimension containing the reaction coordinates. Our approach is general and can be applied to investigate the topology of high-dimensional probability surfaces of other activated process.