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Shape optimization of metastable states

Noé Blassel, Tony Lelièvre, Gabriel Stoltz

Abstract

The definition of metastable states is an ubiquitous task in the design and analysis of molecular simulation, and is a crucial input in a variety of acceleration methods for the sampling of long configurational trajectories. Although standard definitions based on local energy minimization procedures can sometimes be used, these definitions are typically suboptimal, or entirely inadequate when entropic effects are significant, or when the lowest energy barriers are quickly overcome by thermal fluctuations. In this work, we propose an approach to the definition of metastable states, based on the shape-optimization of a local separation of timescale metric directly linked to the efficiency of a class of accelerated molecular dynamics algorithms. To realize this approach, we derive analytic expressions for shape-variations of Dirichlet eigenvalues for a class of operators associated with reversible elliptic diffusions, and use them to construct a local ascent algorithm, explicitly treating the case of multiple eigenvalues. We propose two methods to make our method tractable in high-dimensional systems: one based on dynamical coarse-graining, the other on recently obtained low-temperature shape-sensitive spectral asymptotics. We validate our method on a benchmark biomolecular system, showcasing a significant improvement over conventional definitions of metastable states.

Shape optimization of metastable states

Abstract

The definition of metastable states is an ubiquitous task in the design and analysis of molecular simulation, and is a crucial input in a variety of acceleration methods for the sampling of long configurational trajectories. Although standard definitions based on local energy minimization procedures can sometimes be used, these definitions are typically suboptimal, or entirely inadequate when entropic effects are significant, or when the lowest energy barriers are quickly overcome by thermal fluctuations. In this work, we propose an approach to the definition of metastable states, based on the shape-optimization of a local separation of timescale metric directly linked to the efficiency of a class of accelerated molecular dynamics algorithms. To realize this approach, we derive analytic expressions for shape-variations of Dirichlet eigenvalues for a class of operators associated with reversible elliptic diffusions, and use them to construct a local ascent algorithm, explicitly treating the case of multiple eigenvalues. We propose two methods to make our method tractable in high-dimensional systems: one based on dynamical coarse-graining, the other on recently obtained low-temperature shape-sensitive spectral asymptotics. We validate our method on a benchmark biomolecular system, showcasing a significant improvement over conventional definitions of metastable states.

Paper Structure

This paper contains 63 sections, 8 theorems, 188 equations, 21 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Dynamical setting.
  3. Local metastability and quasi-stationary timescales.
  4. Main contributions of this work.
  5. Outline of the work.
  6. Main results
  7. Framework and notation
  8. Assumptions on $V$ and $a$.
  9. Functional spaces.
  10. Lipschitz shape perturbations.
  11. Spectral properties of the Dirichlet generator.
  12. Shape perturbation analysis.
  13. Shape perturbation formulas
  14. Revisiting eigenvalue derivatives as boundary integrals
  15. Numerical optimization
  16. ...and 48 more sections

Key Result

Proposition 1

Let $\Omega\subset \mathbb{R}^d$ be a bounded, open Lipschitz domain, and $k\geq 1$. There exists $h_0>0$ such that, for all $\theta\in B_{{{\mathcal{W}}^{1,\infty}}}(0,h_0)$, is still a bounded, open Lipschitz domain.

Figures (21)

  • Figure 1: The standard framework of the Hadamard shape derivative: a reference domain $\Omega$ is deformed into $\Omega_\theta$ defined in \ref{['eq:stable_perturbation']} following a perturbation field $\theta\in{{\mathcal{W}}^{1,\infty}}$. Regularity properties of a shape functional $J(\Omega)$ are studied via those of the map $\theta\mapsto J(\Omega_\theta)$.
  • Figure 2: Directional shape perturbation of the triple Dirichlet eigenvalue $\lambda_k(\Omega)$ in the direction $\theta$. The slopes of the Gateaux right-tangents (in black dashed lines) correspond to the eigenvalues of the matrix $M^{\Omega,k}(\theta)$ (counted with multiplicity). In this case, the bottom eigenvalue has multiplicity two, and two half-tangents coincide.
  • Figure 3: Two-dimensional potentials \ref{['eq:two_d_pot']}, for decreasing values of the parameter $\varepsilon$. In each case, the potential has a local minimum in each quadrant of the plane, and two saddles on each axis. The saddles on the $y$-axis separate two deep energy basins, while the saddle points on the $x$-axis form shallow energy barriers inside these basins. Some energy level sets, in thin white lines, highlight the well structure.
  • Figure 4: Free energy profiles and effective diffusion coefficients for the CVs $\xi_1$ and $\xi_2$ defined in \ref{['eq:reaction_coordinates']}, and for the potential \ref{['eq:two_d_pot']}. Various values of the parameter $\varepsilon$ are color-coded. Free energy profiles are depicted in solid lines and effective diffusion coefficients are plotted in dashed lines.
  • Figure 5: Domain-dependent eigenvalues (dotted lines) and their coarse-grained approximations (dashed lines), for parametric families of domains defined in CV space.
  • ...and 16 more figures

Theorems & Definitions (17)

  • Proposition 1
  • Theorem 1
  • Remark 1
  • Corollary 1
  • proof : Proof of Corollary \ref{['cor:boundary_expression']}.
  • Lemma 1
  • proof
  • Remark 2
  • Proposition 2
  • Proposition 3
  • ...and 7 more