Ball-and-Stick Local Elevation Umbrella Sampling: Molecular Simulations Involving Enhanced Sampling within Conformational or Alchemical Subspaces of Low Internal Dimensionalities, Minimal Irrelevant Volumes, and Problem-Adapted Geometries

A new method, ball-and-stick local elevation umbrella sampling (B&S-LEUS), is proposed to enhance the sampling in computer simulations of (bio)molecular systems. It enables the calculation of conformational free-energy differences between states (or alchemical free-energy differences between molecules), even in situations where the definition of these states relies on a conformational subspace involving more than a few degrees of freedom. The B&S-LEUS method consists of the following steps: (A) choice of a reduced conformational subspace; (B) representation of the relevant states by means of spheres ("balls"), each associated with a biasing potential involving a one-dimensional radial memory-based term and a radial confinement term; (C) definition of a set of lines ("sticks") connecting these spheres, each associated with a biasing potential involving a one-dimensional longitudinal memory-based term and a transverse confinement term; (D) unification of the biasing potentials corresponding to the union of all of the spheres and lines (active subspace) into a single biasing potential according to the enveloping distribution sampling (EDS) scheme; (E) build-up of the memory using the local elevation (LE) procedure, leading to a biasing potential enabling a nearly uniform sampling (radially within the spheres, longitudinally within the lines) of the active subspace; (F) generation of a biased ensemble of configurations using this preoptimized biasing potential, following an umbrella sampling (US) approach; and (G) calculation of the relative free energies of the states via reweighting and state assignment. The main characteristics of this approach are: (i) a low internal dimensionality, that is, the memory only involves one-dimensional grids (acceptable memory requirements); (ii) a minimal irrelevant volume, that is, the conformational volume opened to sampling includes a minimal fraction of irrelevant regions in terms of the free energy of the physical system or of user-specified metastable states (acceptable build-up duration requirements, high statistical efficiency); and (iii) a problem-adapted geometry (a priori specification of the conformational regions considered as relevant or irrelevant). In particular, the use of lines to connect the spheres ensures both a minimal irrelevant volume and a sufficient number of transitions between the states. As an illustration, the B&S-LEUS method is applied here to three test systems: (i) a solvated (blocked) alanine monopeptide (two-dimensional conformational subspace), used as a toy system to illustrate the versatility of the method in promoting the sampling of arbritrary regions of the Ramachandran map; (ii) a solvated polyalanine decapeptide (nine-dimensional conformational subspace), to evaluate the relative free energies of three different types of helices (π, α, and 310) based on a single simulation; and (iii) a solvated artifical hexopyranose, termed the "mother" of all d-hexopyranoses and constructed as a hybrid of all d-hexopyranose stereoisomers, where the method is applied (seven-dimensional mixed alchemical and conformational subspace) to calculate the relative free energies of the corresponding 32 isomers, anomers, and chair conformers, based on a single simulation.