Variational tensor approach for approximating the rare-event kinetics of macromolecular systems

Abstract

Essential information about the stationary and slow kinetic properties of macromolecules is contained in the eigenvalues and eigenfunctions of the dynamical operator of the molecular dynamics. A recent variational formulation allows to optimally approximate these eigenvalues and eigenfunctions when a basis set for the eigenfunctions is provided. In this study, we propose that a suitable choice of basis functions is given by products of one-coordinate basis functions, which describe changes along internal molecular coordinates, such as dihedral angles or distances. A sparse tensor product approach is employed in order to avoid a combinatorial explosion of products, i.e., of the basis set size. Our results suggest that the high-dimensional eigenfunctions can be well approximated with relatively small basis set sizes.

Reliable Approximation of Long Relaxation Timescales in Molecular Dynamics

“…For the transfer operator this problem has been studied in [4,7,8] using trajectories of the original diffusion process given by (1). In contrast, we herein will consider trajectories of the effective dynamics (31) instead of the original diffusion process.…”

“…This approach has been considered in [4,7,8] to study the related eigenproblem of the transfer operator. Its main idea is to approximate the dominant eigenvalues of a self-adjoint transfer operator via an appropriate form of the Rayleigh variational principle instead via Galerkin discretization [7].…”

“…The domain of the operator L will be denoted as D(L) ⊂ H. It is also known that the process (x s ) s≥0 is a reversible process and that L is a self-adjoint operator with respect to the inner product (4). Whenever the potential V grows to infinity fast enough at infinity, its spectrum is discrete [9].…”

“…Often, these timescales are not accessible by direct numerical simulation. Therefore, different discrete coarse graining approaches for their approximation, like Markov state model (MSM) building [1,2] or time-lagged independent component analysis (TiCA) [3,4] have been introduced and successfully applied to various molecular systems [5,6]. These approaches are based on finite-dimensional Galerkin discretization [1] or variational approximation [7,8] of the transfer operator of the molecular dynamics process [9].…”

Analytical methods for structural ensembles and dynamics of intrinsically disordered proteins

“…The drawback is that if the protein system is large, the possible combination of basis functions becomes prohibitively large. Recently, a way of identifying a good way of combining basis functions has been proposed (Nüske et al 2016). This uses a sparse tensor product approach that allows description of high-dimensional eigenfunctions with a small set of eigenfunctions, making it possible to also tackle larger molecular systems.…”