Computing committors via Mahalanobis diffusion maps with enhanced sampling data

Abstract

The study of phenomena, such as protein folding and conformational changes in molecules, is a central theme in chemical physics. Molecular dynamics (MD) simulation is the primary tool for the study of transition processes in biomolecules, but it is hampered by a huge timescale gap between the processes of interest and atomic vibrations that dictate the time step size. Therefore, it is imperative to combine MD simulations with other techniques in order to quantify the transition processes taking place on large timescales. In this work, the diffusion map with Mahalanobis kernel, a meshless approach for approximating the Backward Kolmogorov Operator (BKO) in collective variables, is upgraded to incorporate standard enhanced sampling techniques, such as metadynamics. The resulting algorithm, which we call the target measure Mahalanobis diffusion map (tm-mmap), is suitable for a moderate number of collective variables in which one can approximate the diffusion tensor and free energy. Imposing appropriate boundary conditions allows use of the approximated BKO to solve for the committor function and utilization of transition path theory to find the reactive current delineating the transition channels and the transition rate. The proposed algorithm, tm-mmap, is tested on the two-dimensional Moro–Cardin two-well system with position-dependent diffusion coefficient and on alanine dipeptide in two collective variables where the committor, the reactive current, and the transition rate are compared to those computed by the finite element method (FEM). Finally, tm-mmap is applied to alanine dipeptide in four collective variables where the use of finite elements is infeasible.

Variational deep learning of equilibrium transition path ensembles

“…14,18 Significant advancements in methods to parameterize the time independent committor have been made by resolving this model along physically motivated, predetermined order parameters. 21,23,28,[62][63][64][65][66][67][68] As we show, choosing among a large number of internal coordinates without consideration of their correlation or coupling risks neglecting important aspects of the transition path ensemble. This is because internal coordinates do not form an orthogonal set of coordinates, and collective motions such as the rotations of a single dihedral angle can be coupled with the motions of angles and other dihedrals.…”

“…3,[14][15][16] Learning this high dimensional function has attracted interest from a diversity of fields, and significant advancements has been made through methods that employ importance sampling and machine learning. [16][17][18][19][20][21][22][23][24][25][26][27][28] Some notable approaches have leveraged the confinement of the transition region to compute it using string methods, 16,17,29 coarse-grained the phase-space to approximate it through diffusion maps, 19,28,30,31 and parameterized neural-networks by either fitting the committor directly 18,21 or solving the variational form of the steady-state backward Kolmogorov equation 22 by combining it with importance sampling methods. [23][24][25] While the learning procedures applied previously have been successful in fitting high dimensional representations of the reaction coordinate or committors, their nonlinearity has largely resulted in a difficulty in interpreting the relative importance of physically distinct descriptors and converting those descriptors into a robust measure of the rate.…”

Manifold learning in atomistic simulations: a conceptual review

“…Equation ( 31) is approximately the average time the systems spends to commute between x k and x l , and the associated commute map is given by: where we can see the difference between the diffusion and commute distances are in the coefficients λ n and t n /2, respectively. Various methods exploit the relation of the effective timescale with eigenvalues [9,61,94,137,138,[145][146][147][148][149].…”

Manifold Learning in Atomistic Simulations: A Conceptual Review

“…where we can see the difference between the diffusion and commute distances are in the coefficients λ n and t n /2, respectively. Various methods exploit the relation of the effective timescale with eigenvalues [9,[117][118][119][120][121][122][123].…”