Macromolecular and biomolecular folding landscapes typically contain high free energy barriers that impede efficient sampling of configurational space by standard molecular dynamics simulation. Biased sampling can artificially drive the simulation along prespecified collective variables (CVs), but success depends critically on the availability of good CVs associated with the important collective dynamical motions. Nonlinear machine learning techniques can identify such CVs but typically do not furnish an explicit relationship with the atomic coordinates necessary to perform biased sampling. In this work, we employ autoassociative artificial neural networks ("autoencoders") to learn nonlinear CVs that are explicit and differentiable functions of the atomic coordinates. Our approach offers substantial speedups in exploration of configurational space, and is distinguished from existing approaches by its capacity to simultaneously discover and directly accelerate along data-driven CVs. We demonstrate the approach in simulations of alanine dipeptide and Trp-cage, and have developed an open-source and freely available implementation within OpenMM.

The success of enhanced sampling molecular simulations that accelerate along collective variables (CVs) is predicated on the availability of variables coincident with the slow collective motions governing the long-time conformational dynamics of a system. It is challenging to intuit these slow CVs for all but the simplest molecular systems, and their data-driven discovery directly from molecular simulation trajectories has been a central focus of the molecular simulation community to both unveil the important physical mechanisms and to drive enhanced sampling. In this work, we introduce state-free reversible VAMPnets (SRV) as a deep learning architecture that learns nonlinear CV approximants to the leading slow eigenfunctions of the spectral decomposition of the transfer operator that evolves equilibrium-scaled probability distributions through time. Orthogonality of the learned CVs is naturally imposed within network training without added regularization. The CVs are inherently explicit and differentiable functions of the input coordinates making them wellsuited to use in enhanced sampling calculations. We demonstrate the utility of SRVs in capturing parsimonious nonlinear representations of complex system dynamics in applications to 1D and 2D toy systems where the true eigenfunctions are exactly calculable and to molecular dynamics simulations of alanine dipeptide and the WW domain protein.

Auto-associative neural networks ("autoencoders") present a powerful nonlinear dimensionality reduction technique to mine data-driven collective variables from molecular simulation trajectories. This technique furnishes explicit and differentiable expressions for the nonlinear collective variables, making it ideally suited for integration with enhanced sampling techniques for accelerated exploration of configurational space. In this work, we describe a number of sophistications of the neural network architectures to improve and generalize the process of interleaved collective variable discovery and enhanced sampling. We employ circular network nodes to accommodate periodicities in the collective variables, hierarchical network architectures to rank-order the collective variables, and generalized encoder-decoder architectures to support bespoke error functions for network training to incorporate prior knowledge. We demonstrate our approach in blind collective variable discovery and enhanced sampling of the configurational free energy landscapes of alanine dipeptide and Trp-cage using an open-source plugin developed for the OpenMM molecular simulation package.