  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 which 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.
