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Coarse-graining conformational dynamics with multi-dimensional generalized Langevin equation: how, when, and why

Pinchen Xie, Yunrui Qiu, Weinan E

Abstract

A data-driven ab initio generalized Langevin equation (AIGLE) approach is developed to learn and simulate high-dimensional, heterogeneous, coarse-grained conformational dynamics. Constrained by the fluctuation-dissipation theorem, the approach can build coarse-grained models in dynamical consistency with all-atom molecular dynamics. We also propose practical criteria for AIGLE to enforce long-term dynamical consistency. Case studies of a toy polymer, with 20 coarse-grained sites, and the alanine dipeptide, with two dihedral angles, elucidate why one should adopt AIGLE or its Markovian limit for modeling coarse-grained conformational dynamics in practice.

Coarse-graining conformational dynamics with multi-dimensional generalized Langevin equation: how, when, and why

Abstract

A data-driven ab initio generalized Langevin equation (AIGLE) approach is developed to learn and simulate high-dimensional, heterogeneous, coarse-grained conformational dynamics. Constrained by the fluctuation-dissipation theorem, the approach can build coarse-grained models in dynamical consistency with all-atom molecular dynamics. We also propose practical criteria for AIGLE to enforce long-term dynamical consistency. Case studies of a toy polymer, with 20 coarse-grained sites, and the alanine dipeptide, with two dihedral angles, elucidate why one should adopt AIGLE or its Markovian limit for modeling coarse-grained conformational dynamics in practice.
Paper Structure (14 sections, 24 equations, 2 figures)

This paper contains 14 sections, 24 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Results
  3. Construction of AIGLE and AILE
  4. When is AIGLE appropriate for enforcing dynamical consistency
  5. Particle-based coarse-graining
  6. Collective variable-based coarse-graining
  7. Conclusion
  8. Second fluctuation-dissipation theorem on decay-Fourier basis
  9. Variational principle for orthogonality condition
  10. GLE Simulation with Markovian integrator
  11. Derivation of second fluctuation-dissipation theorem on decay-Fourier basis
  12. Objective function
  13. The friction constant in ab initio Langevin equation
  14. Details of training

Figures (2)

  • Figure 1: (a) Sketches of the all-atom model and the CG models of the toy polymer. The grey shades in the all-atom model represent the dangling particles attached to the backbone atoms. (b) Left: The dynamical diffusivity as a function of time. Right: The memory kernel as a function of time. The inset plots the effective Markovian friction $\eta_i$ for all backbone atoms. (c) MSD of $d$ as a function of time. (d) MFPT as a function of $d$ for expanding (upper) and contracting (lower) processes.
  • Figure 2: (a) Schematic representations of all-atom MD, AIGLE, and AILE simulation of alanine dipeptide. (b) The heatmap of the free energy surface $G(\phi, \psi)$. Colored solid circles mark five representative states. (c) Memory kernels of the orthogonally transformed CVs. The inset shows the integrated memory kernels. (d) RMSD of $\phi$ as a function of time. (e) Graph representation of $\{\mathrm{S_1, \cdots, S_5}\}$, matching panel (b) by color. The size of the nodes is arranged in ascending order of the equilibrium probability, reported as node labels. The MFPTs are reported as labels of bidirectional edges. We highlight the edge label with grey shade when $\frac{\tau_{\mathrm{GLE}}}{\tau_{\mathrm{MD}}}$ or $\frac{\tau_{\mathrm{LE}}}{\tau_{\mathrm{MD}}}$ is larger than 2 or smaller than 0.5.