| 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 EMM (which is understood as the self intra-molecular energy in vacuum) and the solvation energy Gsol 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 octree based approximation schemes for estimating the compute-intensive terms of EMM and Gsol. The algorithms run in O(M logM) time and use O(M) space, where M is the number of atoms in the molecule. Additionally, we show how to approximate the polarization force (i.e., derivatives of polarization energy) acting on all M atoms of the molecule within the same time and space bounds. The algorithms for Gsol and polarization forces are dependent on an O(M) size sampling of the biomolecular surface and its spatial derivatives (normals). We also present fast octree based algorithms for approximating interface areas (plain as well as hydrophobic and hydrophilic) of bio-molecular complexes. We include 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. |