Macroscopic frameworks model the entire system, protein and solvent, as either a regularly shaped or irregularly shaped object situated within a dielectric medium. From this, the energy terms can be resolved using the Poisson-Boltzmann equation (PBE). For a regularly shaped protein (e.g., idealized sphere), the PBE can be solved analytically ; however, for a more realistic, irregularly shaped protein, the PBE must be solved numerically. The numerical PB-approach for computing pK a values was pioneered by Bashford and Karplus and has since been continually refined .