Locating landmarks on high-dimensional free energy surfaces

Abstract

Coarse graining of complex systems possessing many degrees of freedom can often be a useful approach for analyzing and understanding key features of these systems in terms of just a few variables. The relevant energy landscape in a coarse-grained description is the free energy surface as a function of the coarsegrained variables, which, despite the dimensional reduction, can still be an object of high dimension. Consequently, navigating and exploring this high-dimensional free energy surface is a nontrivial task. In this paper, we use techniques from multiscale modeling, stochastic optimization, and machine learning to devise a strategy for locating minima and saddle points (termed "landmarks") on a high-dimensional free energy surface "on the fly" and without requiring prior knowledge of or an explicit form for the surface. In addition, we propose a compact graph representation of the landmarks and connections between them, and we show that the graph nodes can be subsequently analyzed and clustered based on key attributes that elucidate important properties of the system. Finally, we show that knowledge of landmark locations allows for the efficient determination of their relative free energies via enhanced sampling techniques.free energy surface | stochastic optimization | activation-relaxation | machine learning | network representation U nderstanding the conformational equilibria of complex systems remains a significant challenge in the computational molecular sciences. Whether one is interested in predicting biomolecular structure, generating and thermodynamically ranking the polymorphs of molecular crystals, or studying the phase behavior of complex materials, the very large number of degrees of freedom renders such problems highly nontrivial. Often the most important conformational states in a system can be characterized in terms of a subset of collective degrees of freedom or "collective variables" (CVs), and the problem of mapping out the conformational equilibria amounts to generating the marginal probability distribution in these CVs, from which the associated free energy surface (FES) can be generated. Unfortunately, due to the existence of many minima on the potential energy surface separated by high barriers, transitions from basin to basin on this surface are rare, so that the FES cannot be generated on any reasonable timescale using standard molecular dynamics (MD) or Monte Carlo (MC) methods. Various enhanced sampling approaches have been devised to accelerate the exploration of such "rough" or "frustrated" energy landscapes to generate such FESs, either by elevating the temperature in the subspace of the CVs (1-3) or by applying a bias potential on the FES (4-7) as in the popular metadynamics approach (4, 5). Recently, we have shown that these two classes of methods can be effectively combined (8), and others have shown that various types of free energy dynamics are possible (9).Although it is often claimed that a low-dimensional manifold involving the selected CVs and embedded in the full phas...

Enhanced, targeted sampling of high-dimensional free-energy landscapes using variationally enhanced sampling, with an application to chignolin

“…done in reconnaissance metadynamics (19) or in ref. 20. Other methods do without an external bias altogether, like temperatureaccelerated molecular dynamics (MD) (21), in which an artificially large friction is applied to a set of CVs to ensure that they evolve slowly compared with the atomic motions.…”

Spectral gap optimization of order parameters for sampling complex molecular systems

“…Last but not the least, having an optimal low-dimensional CV can also help in the building of Brownian dynamics-type models (16,17). Indeed, given the importance of this problem, there exists a range of methods that have been proposed to solve it (18)(19)(20)(21)(22)(23)(24)(25).…”

Capturing the Role of Temperature and Entropy in Crystal Polymorphs Through Molecular Simulations

“…Attempts to define minima on an arbitrary free energy landscape generally invoke the use of order parameters [149,[180][181][182]. Such approaches assume that for an arbitrary free energy surface, there exists a set of order parameters such that the exact distribution of those parameters for a cluster of configurations will be both necessary and sufficient to precisely determine which minima the configurations corresponds with.…”

“…Because MD acts on ensembles, complex effects can be observed which influence the stability and would be missed in lattice energy-based studies [144][145][146][147][148][149][150][151][152][153][154]. As one specific example, Yu and Tuckerman modeled six polymorphs of benzene with a free energy sampling method and proposed that the experimentally observed high-pressure benzene crystal may actually be a mixture of the benzene II and benzene III structures typically identified as distinct polymorphs in energy minimization studies [147].…”

“…The inability of static lattice models to faithfully represent true crystal structures has led investigators in recent years to begin generating full configuration ensembles in polymorph prediction studies using free energy simulation methods [144][145][146][147][148][149][150][151][152][153][154]. The free energy of a crystal structure can be estimated in molecular simulations with either a harmonic approximation, or with a fully atomistic Monte Carlo or molecular dynamics sampling of the phase space around each lattice energy minima.…”