Abstract
Spectral gap optimization of order parameters (SGOOP) [P. Tiwary and B. J. Berne, Proc. Natl. Acad. Sci. U. S. A. 113, 2839 (2016)] is a method for constructing the reaction coordinate (RC) in molecular systems, especially when they are plagued with hard to sample rare events, given a larger dictionary of order parameters or basis functions and limited static and dynamic information about the system. In its original formulation, SGOOP is designed to construct a 1-dimensional RC. Here we extend its scope by introducing a simple but powerful extension based on the notion of conditional probability factorization where known features are effectively washed out to learn additional and possibly hidden features of the energy landscape. We show how SGOOP can be used to proceed in a sequential and bottom-up manner to (i) systematically probe the need for extending the dimensionality of the RC and (ii) if such a need is identified, learn additional coordinates of the RC in a computationally efficient manner. We formulate the method and demonstrate its utility through three illustrative examples, including the challenging and important problem of calculating the kinetics of benzene unbinding from the protein T4L99A lysozyme, where we obtain excellent agreement in terms of dissociation pathway and kinetics with other sampling methods and experiments. In this last case, starting from a larger dictionary of 11 order parameters that are generic for ligand unbinding processes, we demonstrate how to automatically learn a 2-dimensional RC, which we then use in the infrequent metadynamics protocol to obtain 16 independent unbinding trajectories. We believe our method will be a big step in increasing the utility of SGOOP in performing intuition-free sampling of complex systems. Finally, we believe that the utility of our protocol is amplified by its applicability to not just SGOOP but also other generic methods for constructing the RC.
Highlights
Spectral Gap Optimization of Order Parameters (SGOOP) is one such method to construct a reaction coordinate (RC) as a function of candidate order parameters for a given molecular system.[5,6]
We first demonstrate our method on two illustrative simple model potentials through a combination of which it can be clearly seen why a second component to the RC might or might not be needed, and how Spectral gap optimization of order parameters (SGOOP) can be used to identify the various components
Using SGOOP we demonstrate how the process of RC selection for infrequent metadynamics can be made almost automatic, starting from a larger dictionary of fairly generic and arbitrarily chosen 11 order parameters
Summary
Finding reaction coordinates (RC) and mechanistic pathways in complex systems and processes is a problem of great theoretical and practical interest for which over the decades numerous theoretical and numerical schemes have been proposed.[1,2,3,4] The problem becomes especially complicated in rare event systems, aptly summarized by Chandler and co-workers in their review as the problem of “throwing ropes over rough mountain passes, in the dark”.2 Spectral Gap Optimization of Order Parameters (SGOOP) is one such method to construct a RC as a function of candidate order parameters for a given molecular system.[5,6] This RC encapsulates the most relevant degrees of freedom in the system whose fluctuations must be enhanced in order to accurately sample the thermodynamics and kinetics of metastable states during biased molecular dynamics (MD) simulations such as metadynamics or umbrella sampling.[7]. Been demonstrated to be useful for a range of systems such as small peptides and protein–ligand systems, and falls in the broad family of many such related methods that attempt to learn RC for enhanced sampling from sub-optimally biased simulations, such as Ref. 8 and 9. Instead of trying to make the 1-d RC more and more sophisticated, it might be computationally cheaper and physically more interpretable to add a second or even more components to the RC, while still keeping the final dimensionality of the RC much lower than the space of order parameters considered These other RC components could serve to lift the degeneracy in the first component, and could directly be interpreted in terms of the different pathways or metastable states that they correspond to. We start with a dictionary of 11 generic order parameters such as protein-ligand and protein-protein distances, and use our automatically learned two-dimensional RC in an infrequent metadynamics framework[7,18,19] to calculate its dissociation rate constant and dominant unbinding pathway, in excellent agreement with previous studies and experiments.[20–24] We believe our method should be of considerable use to the enhanced sampling and molecular simulation communities
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