The capabilities of molecular simulations have been greatly extended by a number of widely used enhanced sampling methods that facilitate escaping from metastable states and crossing large barriers. Despite these developments there are still many problems which remain out of reach for these methods which has led to a vigorous effort in this area. One of the most important problems that remains unsolved is sampling high-dimensional free-energy landscapes and systems that are not easily described by a small number of collective variables. In this work we demonstrate a new way to compute freeenergy landscapes of high dimensionality based on the previously introduced variationally enhanced sampling, and we apply it to the miniprotein chignolin.enhanced sampling | protein folding | free-energy calculation | biomolecular simulation M olecular simulation has become an increasingly useful tool for studying complex transformations in chemistry, biology, and materials science. Such simulations provide extremely high spatial and temporal resolution but suffer from inherent time-scale limitations. Molecular dynamics simulations of biomolecules performed on standard hardware struggle to exceed the microsecond time scale. Modern, purpose-built hardware can reach the millisecond time scale (1), but these tools are not generally available and they still prove insufficient for many important problems (2). Widely used enhanced sampling methods also allow one to simulate processes with a time scale of milliseconds (3) and there is significant interest in extending these methods. One important class of enhanced sampling methods involves applying an external bias to one or more collective variables (CVs) of the system. Umbrella sampling and metadynamics fall into this category (4, 5), but there are many other examples (6-13). These methods are most effective when the transformations of interest are in some sense collective and can be described by a very small number of CVs.When the choice of CV is not obvious or the free-energy landscape cannot be projected meaningfully on a low-dimensional manifold, new strategies are needed. Here we describe an enhanced sampling strategy that allows one to bias high-dimensional freeenergy landscapes by exploiting the flexibility inherent in the recently introduced variationally enhanced sampling (VES) method (14). This method casts free-energy computation as a functional minimization problem, and it solves two important problems associated with high-dimensional free-energy computation. The first problem is that when exploring a high-dimensional space a system might never visit the regions of interest, but the variational principle introduced in ref. 14 permits us to specifically target the region we are interested in through the use of a so-called "target distribution." The second problem is that one must adopt an approximation for the high-dimensional bias and the variational principle allows us to do this in an optimal way. Furthermore, as we show here, for the first time to our knowledge, choos...

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In recent work, we demonstrated that it is possible to obtain approximate representations of high-dimensional free energy surfaces with variationally enhanced sampling ( Shaffer, P.; Valsson, O.; Parrinello, M. Proc. Natl. Acad. Sci. , 2016 , 113 , 17 ). The high-dimensional spaces considered in that work were the set of backbone dihedral angles of a small peptide, Chignolin, and the high-dimensional free energy surface was approximated as the sum of many two-dimensional terms plus an additional term which represents an initial estimate. In this paper, we build on that work and demonstrate that we can calculate high-dimensional free energy surfaces of very high accuracy by incorporating additional terms. The additional terms apply to a set of collective variables which are more coarse than the base set of collective variables. In this way, it is possible to build hierarchical free energy surfaces, which are composed of terms that act on different length scales. We test the accuracy of these free energy landscapes for the proteins Chignolin and Trp-cage by constructing simple coarse-grained models and comparing results from the coarse-grained model to results from atomistic simulations. The approach described in this paper is ideally suited for problems in which the free energy surface has important features on different length scales or in which there is some natural hierarchy.

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