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$π$-Girsanov: A Generalized Method to Construct Markov State Models from Non-Equilibrium and Multiensemble Biased Simulations

Mingyuan Zhang, Yong Wang, Bettina G. Keller, Hao Wu

Abstract

We introduce $π$-Girsanov, a new method for constructing Markov state models from biased enhanced-sampling molecular dynamics simulations based on Girsanov reweighting. The key idea behind this new method is to separate the reweighting stationary density from the reweighting of the correlation function. We evaluate the effectiveness of this approach on several analytical potentials and on a model biomolecular system, comparing its performance with the original method. Our results show that $π$-Girsanov not only improves the estimation in a single-ensemble setting, but also resolves key challenges in estimating transition matrices from multiensemble and non-equilibrium biased trajectories. Overall, $π$-Girsanov represents a substantial advance in kinetic reweighting, strengthening the connection between enhanced sampling techniques and Markov state modeling.

$π$-Girsanov: A Generalized Method to Construct Markov State Models from Non-Equilibrium and Multiensemble Biased Simulations

Abstract

We introduce -Girsanov, a new method for constructing Markov state models from biased enhanced-sampling molecular dynamics simulations based on Girsanov reweighting. The key idea behind this new method is to separate the reweighting stationary density from the reweighting of the correlation function. We evaluate the effectiveness of this approach on several analytical potentials and on a model biomolecular system, comparing its performance with the original method. Our results show that -Girsanov not only improves the estimation in a single-ensemble setting, but also resolves key challenges in estimating transition matrices from multiensemble and non-equilibrium biased trajectories. Overall, -Girsanov represents a substantial advance in kinetic reweighting, strengthening the connection between enhanced sampling techniques and Markov state modeling.
Paper Structure (20 sections, 36 equations, 7 figures, 1 table)

This paper contains 20 sections, 36 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Theory
  3. Molecular Dynamics
  4. Markov State Modeling
  5. Girsanov Reweighting
  6. The $\pi$-Girsanov Method
  7. Methods
  8. Enhanced Sampling Simulations
  9. Computation of the Stationary Distribution from Enhanced Sampling Simulations
  10. MSM Construction and Evaluation
  11. Details in Simulation, Modeling and Software
  12. Results
  13. 1D Double Well Potential
  14. 1D Prinz Potential
  15. 2D Müller-Brown Potential
  16. ...and 5 more sections

Figures (7)

  • Figure 1: Summary of results for 1D double well potential simulations: (A) The ITS and the first eigenvector $\psi_1$ of $\mathbf T(\tau)$ reweighted with different methods from the rerun metadynamics trajectories. (B) The ITS and the first eigenvector $\psi_1$ of $\mathbf T(\tau)$ reweighted with different methods from the build-up metadynamics trajectories. (C) The energy landscape of the system. (D) The ITS and the first eigenvector $\psi_1$ of $\mathbf T(\tau)$ reweighted with different methods from the umbrella sampling trajectories. (E) The ITS and the first eigenvector $\psi_1$ of $\mathbf T(\tau)$ reweighted with different methods from the steered MD trajectories. For each column of each subplots, the y-axis scale is set to uniform so comparison can be made.
  • Figure 2: Summary of results for 1D Prinz potential rerun and build-up metadynamics simulations: (A) The energy landscape of the system. (B) The ITS and the leading eigenvectors of $\mathbf T(\tau)$ reweighted with different methods from the rerun metadynamics trajectories. (C) The ITS and the leading eigenvectors of $\mathbf T(\tau)$ reweighted with different methods from the build-up metadynamics trajectories. For each column of each subplots, the y-axis scale is set to uniform so comparison can be made.
  • Figure 3: Summary of results for 1D Prinz potential umbrella sampling and steered MD simulations: (A) The ITS and the leading eigenvectors of $\mathbf T(\tau)$ reweighted with different methods from the umbrella sampling trajectories. (C) The ITS and the leading eigenvectors of $\mathbf T(\tau)$ reweighted with different methods from the steered MD trajectories. For each column of each subplots, the y-axis scale is set to uniform so comparison can be made.
  • Figure 4: Dimensionality reduction of the 2D Müller-Brown potential. (A) The energy landscape of the potential. (B) The scatterplot of the sample coordinates used in FMRC training. The resulting RC is shown as colorbar and its isolines are plotted as black dashed lines. (C) The projected free energy surface computed from the training data along the RC.
  • Figure 5: Summary of results for 2D Müller-Brown potential rerun and build-up simulations: (A) The ITS and the leading eigenvectors of $\mathbf T(\tau)$ reweighted with different methods from the rerun metadynamics trajectories. (B) The ITS and the leading eigenvectors of $\mathbf T(\tau)$ reweighted with different methods from the build-up metadynamics trajectories. For each column of each subplots, the y-axis scale is set to uniform so comparison can be made.
  • ...and 2 more figures