The Poisson-Boltzmann equation, PBE, the linearized PBE, LPBE, and generalized Born, GB, model are implicit solvent methods that accelerate biophysical calculations by eliminating the need to integrate across the solvent's degrees of freedom. This thesis compares the predictions of the Poisson-Boltzmann equation, PBE, to those of another electrostatic theory, the counterion condensation theory, CCT. It demonstrates that the CCT's predictions of the salt dependence of the electrostatic binding free energy agree with those of the PBE, but its predictions of the electrostatic binding free energy itself do not. This observation is explained by deriving a simple analytical expression for the salt dependence of the electrostatic binding free energy from the GB model. This expression indicates that essentially any electrostatic theory with the same long-distance predictions will predict the same salt dependence, it explains the observed correlations between this salt dependence and various empirical quantities, and it provides a rapid method for predicting its change upon mutation of the charged residues of the binding partners. Implicit solvent models do include several approximations whose validity should be evaluated to determine the reliability of their estimates of experimental quantities. In this thesis the effect of finite ion sizes upon PBE solutions is examined with the size-modified PBE, SMPBE. These findings show that the SMPBE predicts different dependences of electrostatic energies upon ion size than the PBE, but whether these changes lead to different predictions for observable quantities is not clear. Additionally, the ion-exclusion layer does not match the dependences on ion size given by the SMPBE, and combined with the superior physical basis of the SMPBE, the use of the ion-exclusion layer is called into question. Additionally, this thesis improves stochastic solvers of the LPBE and presents a new stochastic solver of the GB model. By optimizing the bias-generating parameters in a walk-on-spheres, WOS, LPBE solver, dividing the variance evenly across the atoms of the molecule, and using an approximate nearest-neighbor solver, WOS solvers can solve the LPBE in times comparable to deterministic solvers. The stochastic GB solver has the advantage over traditional analytical solvers of the GB in that analytical solutions to the GB model must approximate a set of parameters called the Born radii, and the error due to this approximation cannot usually be evaluated. The stochastic solver, on the other hand quantifies this error and converges to the exact GB model with additional computation time. This behavior allows the validity of the Born approximation itself to be evaluated, and the results presented here indicate that the GB model gives different estimates of the electrostatic binding free energy than the LPBE. More research will be required to refine the GB approximation to better compute the electrostatic binding free energy, and that the radii in the stochastic method are arbitrarily precise indicates that the stochastic solver will be useful in this endeavor.
