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

arXiv: https://arxiv.org/abs/2409.06428 · HTML: https://mlsanity.com/paper/2409.06428/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 108 references that are in the map.
- First posted on arXiv 10 Sep 2024.

## Builds on (most cited first, 8 of 8)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Time-lagged autoencoders: Deep learning of slow collective variables for molecular kinetics](https://mlsanity.com/paper/1710.11239/) (2017)
- [Computing Committor Functions for the Study of Rare Events Using Deep Learning](https://mlsanity.com/paper/1906.06285/) (2019)
- [Variational Koopman models: slow collective variables and molecular kinetics from short off-equilibrium simulations](https://mlsanity.com/paper/1610.06773/) (2016)
- [Nonlinear Discovery of Slow Molecular Modes using State-Free Reversible VAMPnets](https://mlsanity.com/paper/1902.03336/) (2019)
- [Unfolding Hidden Barriers by Active Enhanced Sampling](https://mlsanity.com/paper/1705.07414/) (2017)
- [Characterizing metastable states with the help of machine learning](https://mlsanity.com/paper/2204.07391/) (2022)
- [Reweighted Manifold Learning of Collective Variables from Enhanced Sampling Simulations](https://mlsanity.com/paper/2207.14554/) (2022)

## Built on by (most cited first, 3 of 3)

- [Machine Learning of Slow Collective Variables and Enhanced Sampling via Spatial Techniques](https://mlsanity.com/paper/2412.20868/) (2024)
- [Learning collective variables that preserve transition rates](https://mlsanity.com/paper/2506.01222/) (2025)
- [Unraveling the Mechanism of Drug Binding to SARS-CoV-2 RNA Pseudoknot with Thermodynamics-Driven Machine Learning](https://mlsanity.com/paper/2604.14906/) (2026)

## Abstract (arXiv)

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.