Inherent structure versus geometric metric for state space discretization

Directed kinetic transition network model

Molecular dynamics simulations contain detailed kinetic information related to the functional states of proteins and macromolecules, but this information is obscured by the high dimensionality of configurational space. Markov state models and transition network models are widely applied to extract kinetic descriptors from equilibrium molecular dynamics simulations. In this study, we developed the Directed Kinetic Transition Network (DKTN)-a graph representation of a master equation which is appropriate for describing nonequilibrium kinetics. DKTN models the transition rate matrix among different states under detailed balance. Adopting the mixing time from the Markov chain, we use the half mixing time as the criterion to identify critical state transition regarding the protein conformational change. The similarity between the master equation and the Kolmogorov equation suggests that the DKTN model can be reformulated into the continuous-time Markov chain model, which is a general case of the Markov chain without a specific lag time. We selected a photo-sensitive protein, vivid, as a model system to illustrate the usage of the DKTN model. Overall, the DKTN model provides a graph representation of the master equation based on chemical kinetics to model the protein conformational change without the underlying assumption of the Markovian property.

Approximating dynamic proximity with a hybrid geometry energy-based kernel for diffusion maps

The diffusion map is a dimensionality reduction method. The reduction coordinates are associated with the leading eigenfunctions of the backward Fokker-Planck operator, providing a dynamic meaning for these coordinates. One of the key factors that affect the accuracy of diffusion map embedding is the dynamic measure implemented in the Gaussian kernel. A common practice in diffusion map study of molecular systems is to approximate dynamic proximity with RMSD (root-mean-square deviation). In this paper, we present a hybrid geometry-energy based kernel. Since high energy-barriers may exist between geometrically similar conformations, taking both RMSD and energy difference into account in the kernel can better describe conformational transitions between neighboring conformations and lead to accurate embedding. We applied our diffusion map method to the β-hairpin of the B1 domain of streptococcal protein G and to Trp-cage. Our results in β-hairpin show that the diffusion map embedding achieves better results with the hybrid kernel than that with the RMSD-based kernel in terms of free energy landscape characterization and a new correlation measure between the cluster center Euclidean distances in the reduced-dimension space and the reciprocals of the total net flow between these clusters. In addition, our diffusion map analysis of the ultralong molecular dynamics trajectory of Trp-cage has provided a unified view of its folding mechanism. These promising results demonstrate the effectiveness of our diffusion map approach in the analysis of the dynamics and thermodynamics of molecular systems. The hybrid geometry-energy criterion could be also useful as a general dynamic measure for other purposes.