I present a physical model to calculate protein-protein interactions. General formulations to calculate the electrostatic and the van der Waals free energies are brought by the boundary element method of solving linearized Poission-Boltzmann equation in an electrolyte solution, then further expanded to the application of the Fast Multipole Method(FMM). We built an efficient solver to investigate how the mutations on the active site of the protein-protein interface affect changes in binding affinities of protein complexes. Calculated results in addition to the structural analysis help us to understand the protein-protein interaction energy and provide a model to the important applications such as protein crystallization. The osmotic second virial coefficient B₂ is directly related to the solubility of protein molecule in electrolyte solution and determined by molecular interactions involving both solvent and solute molecules. Calculations of interaction energies account for the electrostatic and the van der Waals interactions with the structural anisotropic properties of protein molecules. The orientation dependence of interaction energies between two proteins is determined by the crystal space operations and small number of protein-protein pair configurations according to the anisotropic patch model are required to calculate B₂. With the extended FMMs, double-tree and single-tree algorithms, the boundary element formulations of interaction energies can be applied with low computational cost to the proteins. B₂ Calculations of Bovine Pancreatic Trypsin Inhibitor are firstly performed to validate our model and the results of lysozyme protein under different salts, concentrations, pH and temperatures are correlated to the experimental B₂. The reduced number of pair interaction energies between two proteins are interpolated to predict all pair interaction energies in the patch model as a precursor of the protein phase diagram calculation.