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.