Spectral Map for\nSlow Collective Variables, Markovian\nDynamics, and Transition State Ensembles

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