APBS apolar calculations follow the very generic framework described in Wagoner JA, Baker NA. Assessing implicit models for nonpolar mean solvation forces: the importance of dispersion and volume terms. Proc Natl Acad Sci USA, 103, 8331-8336, 2006. (doi:10.1073/pnas.0600118103).
In particular, nonpolar solvation potentials of mean force (energies) are calculated according to:
and mean nonpolar solvation forces are calculated according to:
In these equations, gamma is the repulsive (hard sphere) solvent surface tension, A is the conformation-dependent solute surface area (see srad and srfm keywords), p (see press keyword) is the repulsive (hard sphere) solvent pressure, V is the conformation-dependent solute volume (see srad and srfm keywords), rho (see bconc keywords) is the bulk solvent density, and the integral involves the attractive portion (defined in a Weeks-Chandler-Andersen sense) of the Lennard-Jones interactions between the solute and the solvent integrated over the region of the problem domain outside the solute volume V. Lennard-Jones parameters are taken from APBS parameter files as read in through an APBS input file READ statement.
Note that the above expressions can easily be reduced to simpler apolar solvation formalisms by setting one or more of the coefficients to zero through the keywords.
Important: all apolar calculations require a parameter file which contains Lennard-Jones radius and well-depth parameters for all the atoms in the solute PDB. This parameter file must also contain radius and well-depth parameters for water (specifically: residue "WAT" and atom "OW"). Complete parameter files for protein and nucleic acid parameters are not currently available and are actively under development as a research project. Please contact Nathan Baker for additional information about the state of this research, particularly if you are interested in helping.