Multi-level grid algorithms for faster molecular energetics

Bio-molecules reach their stable configuration in solvent which is primarily water with a small concentration of salt ions. One approximation of the total free energy of a bio-molecule includes the classical molecular mechanical energy E_MM (which is understood as the self intra-molecular energy in vacuum) and the solvation energy G_sol which is caused by the change of the environment of the molecule from vacuum to solvent (and hence also known as the molecule-solvent interaction energy). This total free energy is used to model and study the stability of bio-molecules in isolation or in their interactions with drugs. In this paper we present fast O(N logN) multi-level grid based approximation algorithms (where N is the number of atoms) for efficiently estimating the compute-intensive terms of E_MM and G_sol. The fast octree-based algorithm for G_sol is additionally dependent on an O(N) size computation of the biomolecular surface and its spatial derivatives (normals). We also provide several examples with timing results, and speed/accuracy tradeoffs, demonstrating the efficiency and scalability of our fast free energy estimation of bio-molecules, potentially with millions of atoms.