Investigating Molecular Kinetics by Variationally Optimized Diffusion Maps

Abstract

Identification of the collective coordinates that describe rare events in complex molecular transitions such as protein folding has been a key challenge in the theoretical molecular sciences. In the Diffusion Map approach, one assumes that the molecular configurations sampled have been generated by a diffusion process, and one uses the eigenfunctions of the corresponding diffusion operator as reaction coordinates. While diffusion coordinates (DCs) appear to provide a good approximation to the true dynamical reaction coordinates, they are not parametrized using dynamical information. Thus, their approximation quality could not, as yet, be validated, nor could the diffusion map eigenvalues be used to compute relaxation rate constants of the system. Here we combine the Diffusion Map approach with the recently proposed Variational Approach for Conformation Dynamics (VAC). Diffusion Map coordinates are used as a basis set, and their optimal linear combination is sought using the VAC, which employs time-correlation information on the molecular dynamics (MD) trajectories. We have applied this approach to ultra-long MD simulations of the Fip35 WW domain and found that the first DCs are indeed a good approximation to the true reaction coordinates of the system, but they could be further improved using the VAC. Using the Diffusion Map basis, excellent approximations to the relaxation rates of the system are obtained. Finally, we evaluate the quality of different metric spaces and find that pairwise minimal root-mean-square deviation performs poorly, while operating in the recently introduced kinetic maps based on the time-lagged independent component analysis gives the best performance.

Galerkin approximation of dynamical quantities using trajectory data

“…They can also be used as a basis set for DOEA and similar analyses. 57,59,87 Here we extend this line of research, showing that diffusion maps can also be used to construct basis functions that obey homogeneous boundary conditions on arbitrary sets as required for use in DGA. We note that the diffusion process represented by P DMAP is not intended as an approximation of the dynamics, but rather as a tool for building the basis functions φ i .…”

“…Various approaches have been used to construct basis sets for describing dynamics in DOEA schemes. 39,40,52,59,60 However, if D c is nonempty these functions cannot be used in DGA. In particular, the linear basis in TICA cannot be used.…”

Markov models of the apo-MDM2 lid region reveal diffuse yet two-state binding dynamics and receptor poses for computational docking

“…The sign structure of the slowest relaxation eigenmode φ 1 shows population flux from unbound to bound states of the lid regions, indicated by two diffuse basins aligned with tIC 1 , the degree of freedom representing the slowest conformational motions (Figure 2a). Interestingly, compared to the tICA landscapes reported for many protein folding systems, [24][25][26] the lid 2/18 landscape is remarkably diffuse, even in the secondary eigenmodes ( Figure S4), reflecting the lack of residual structure. Similar landscapes have been found in other MSM studies of disordered proteins.…”

Bridging microscopic and macroscopic mechanisms of p53-MDM2 binding using molecular simulations and kinetic network models

“…Recent methodological advances have exploited the variational approach to conformational dynamics 27 to enable the construction of optimal MSMs given the available trajectory data 28,29 . To implement this approach, we used time-structure-based Independent Component Analysis (tICA) 30,31 to project the trajectory data to a low-dimensional subspace that best preserves the slowest conformational transitions.…”