Protein solvation from theory and simulation: Exact treatment of Coulomb interactions in three-dimensional theories

Solvation forces dominate protein structure and dynamics. Integral equation theories allow a rapid and accurate evaluation of the effect of solvent around a complex solute, without the sampling issues associated with simulations of explicit solvent molecules. Advances in integral equation theories make it possible to calculate the angle dependent average solvent structure around an irregular molecular solution. We consider two methodological problems here: the treatment of long-ranged forces without the use of artificial periodicity or truncations and the effect of closures. We derive a method for calculating the long-ranged Coulomb interaction contributions to three-dimensional distribution functions involving only a rewriting of the system of integral equations and introducing no new formal approximations. We show the comparison of the exact forms with those implied by the supercell method. The supercell method is shown to be a good approximation for neutral solutes whereas the new method does not exhibit the known problems of the supercell method for charged solutes. Our method appears more numerically stable with respect to thermodynamic starting state. We also compare closures including the Kovalenko-Hirata closure, the hypernetted-chain (HNC) and an approximate three-dimensional bridge function combined with the HNC closure. Comparisons to molecular dynamics results are made for water as well as for the protein solute bovine pancreatic trypsin inhibitor. The proposed equations have less severe approximations and often provide results which compare favorably to molecular dynamics simulation where other methods fail.