Spectral Map for Slow Collective Variables, Markovian Dynamics, and Transition State Ensembles

TL;DR

This paper advances a spectral map technique to identify slow collective variables in complex molecular systems, enabling simplified, Markovian modeling of long-time dynamics and providing insights into protein folding mechanisms.

Contribution

The work introduces algorithmic improvements for spectral map analysis, including kinetic partitioning and transition state ensemble definition, applied to high-dimensional protein folding data.

Findings

Spectral map-derived slow CVs closely approach Markovian dynamics.

Coordinate-dependent diffusion coefficients have minimal impact on free-energy landscapes.

A single slow CV can effectively serve as a reaction coordinate in protein folding.

Abstract

Understanding the behavior of complex molecular systems is a fundamental problem in physical chemistry. To describe the long-time dynamics of such systems, which is responsible for their most informative characteristics, we can identify a few slow collective variables (CVs) while treating the remaining fast variables as thermal noise. This enables us to simplify the dynamics and treat it as diffusion in a free-energy landscape spanned by slow CVs, effectively rendering the dynamics Markovian. Our recent statistical learning technique, spectral map [Rydzewski, J. Phys. Chem. Lett. 2023, 14, 22, 5216-5220], explores this strategy to learn slow CVs by maximizing a spectral gap of a transition matrix. In this work, we introduce several advancements into our framework, using a high-dimensional reversible folding process of a protein as an example. We implement an algorithm for coarse-graining Markov transition matrices to partition the reduced space of slow CVs kinetically and use it to define a transition state ensemble. We show that slow CVs learned by spectral map closely approach the Markovian limit for an overdamped diffusion. We demonstrate that coordinate-dependent diffusion coefficients only slightly affect the constructed free-energy landscapes. Finally, we present how spectral map can be used to quantify the importance of features and compare slow CVs with structural descriptors commonly used in protein folding. Overall, we demonstrate that a single slow CV learned by spectral map can be used as a physical reaction coordinate to capture essential characteristics of protein folding.