Variational cross-validation of slow dynamical modes in molecular kinetics

Abstract

Markov state models (MSMs) are a widely used method for approximating the eigenspectrum of the molecular dynamics propagator, yielding insight into the long-timescale statistical kinetics and slow dynamical modes of biomolecular systems. However, the lack of a unified theoretical framework for choosing between alternative models has hampered progress, especially for non-experts applying these methods to novel biological systems. Here, we consider cross-validation with a new objective function for estimators of these slow dynamical modes, a generalized matrix Rayleigh quotient (GMRQ), which measures the ability of a rank-$m$ projection operator to capture the slow subspace of the system. It is shown that a variational theorem bounds the GMRQ from above by the sum of the first $m$ eigenvalues of the system's propagator, but that this bound can be violated when the requisite matrix elements are estimated subject to statistical uncertainty. This overfitting can be detected and avoided through cross-validation. These result make it possible to construct Markov state models for protein dynamics in a way that appropriately captures the tradeoff between systematic and statistical errors.

References in corpus (4)

• Canonical sampling through velocity-rescaling
• Identification of slow molecular order parameters for Markov model construction
• Improved coarse-graining of Markov state models via explicit consideration of statistical uncertainty
• Concentration inequalities of the cross-validation estimator for Empirical Risk Minimiser

Cited by in corpus (34)

• Machine learning for molecular simulation
• VAMPnets: Deep learning of molecular kinetics
• Time-lagged autoencoders: Deep learning of slow collective variables for molecular kinetics
• Data-driven model reduction and transfer operator approximation
• Variational Encoding of Complex Dynamics
• Variational Koopman models: slow collective variables and molecular kinetics from short off-equilibrium simulations
• Eigendecompositions of Transfer Operators in Reproducing Kernel Hilbert Spaces
• Identification of simple reaction coordinates from complex dynamics
• Coarse-Grained Modelling Out of Equilibrium
• Special Topic: Markov Models of Molecular Kinetics
• Kernel-based approximation of the Koopman generator and Schrödinger operator
• Variational Selection of Features for Molecular Kinetics
• GraphVAMPNet, using graph neural networks and variational approach to markov processes for dynamical modeling of biomolecules
• Adaptive Markov State Model estimation using short reseeding trajectories
• Correlation-based feature selection to identify functional dynamics in proteins
• Variational approach for learning Markov processes from time series data
• Integrated VAC: A robust strategy for identifying eigenfunctions of dynamical operators
• Identification of kinetic order parameters for non-equilibrium dynamics
• Deep learning Markov and Koopman models with physical constraints
• ivis Dimensionality Reduction Framework for Biomacromolecular Simulations
• Towards a Benchmark for Markov State Models: The Folding of HP35
• Manifold Learning in Atomistic Simulations: A Conceptual Review
• Deeptime: a Python library for machine learning dynamical models from time series data
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• Deflation reveals dynamical structure in nondominant reaction coordinates
• Binding Pathway of Opiates to $\mu$ Opioid Receptors Revealed by Unsupervised Machine Learning
• Theoretical restrictions on longest implicit timescales in Markov state models of biomolecular dynamics
• Note: Variational Encoding of Protein Dynamics Benefits from Maximizing Latent Autocorrelation
• Unveiling the hidden reaction kinetic network of carbon dioxide in supercritical aqueous solutions
• Exploring the Regulatory Function of the N-terminal Domain of SARS-CoV-2 Spike Protein Through Molecular Dynamics Simulation
• Accelerated estimation of long-timescale kinetics by combining weighted ensemble simulation with Markov model "microstates" using non-Markovian theory
• Progress in deep Markov State Modeling: Coarse graining and experimental data restraints
• Kernel Embedding based Variational Approach for Low-dimensional Approximation of Dynamical Systems
• Automatic Feature Selection in Markov State Models Using Genetic Algorithm