Consistent calculations of pK_a's of ionizable residues in proteins: Semi-microscopic and microscopic approaches

One of the most direct benchmarks for electrostatic models of macromolecules is provided by the pK_a's of ionizable groups in proteins. Obtaining accurate results for such a benchmark presents a major challenge. Microscopic models involve very large opposing contributions and suffer from convergence problems. Continuum models that consider the protein permanent dipoles as a part of the dielectric constant cannot reproduce the correct self-energy. Continuum models that treat the local environment in a semi-microscopic way do not take into account consistently the protein relaxation during the charging process. This work describes calculations of pK_a's in protein in an accurate yet consistent way, using the semi-microscopic version of the protein dipoles Langevin dipoles (PDLD) model, which treats the protein relaxation in the microscopic framework of the linear response approximation. This approach allows one to take into account the protein structural reorganization during formation of charges, thus reducing the problems with the use of the socalled "protein dielectric constant", ∈_p. The model is used in calculations of pK_a's of the acidic groups of lysozyme, and the calculated results are compared to the corresponding results of discretized continuum (DC) studies. It is found that the present approach is more consistent than current DC models and also provides improved accuracies. Significant emphasis is given to the self-energy term, which has been pointed out in our early works but has been sometimes overlooked or presented as a small effect. The meaning of the dielectric constant ∈_P used in DC models is clarified and illustrated, establishing the finding (e.g. King et. al., J. Phys. Chem. 1991, 95, 4366) that this parameter represents the contributions that are not treated explicitly in the given model, rather than the "true" dielectric constant. It is pointed out that recent suggestions to use large ∈_p to obtain improved DC results might not be much different than our earlier suggestion to use a large effective dielectric for charge-charge interactions. This ∈_p reduces the overestimate of charge-charge interactions relative to models that use small ∈_p while not considering the protein relaxation explicitly. Unfortunately, the use of large ∈_p does not reproduce consistently the self-energies of isolated ionized groups in protein interiors. The recent interest in taking protein flexibility into account in pK_a's calculations is addressed. It is pointed out that running MD over protein configurations will not by itself lead to a more consistent value of ∈_p. It is clarified that a smaller value of ∈_p, which is not really more (or less) consistent with the physics of the proteins, will be obtained if one uses our LRA (linear response approximation) formulation, generating configurations of both neutral and ionized states of the protein. It is also stated that such studies have been a standard part of our approach for some time. The present model involves a consecutive running of all-atom MD simulations of solvated proteins and an automated used of the electrostatic PDLD model. This allows one to move consistently to any level of explicit solvent model, keeping an arbitrary number of solvent molecules in an explicit all-atom representation, while treating the rest as dipoles. This capacity is used in examining the microscopic basis of the PDLD models by comparing its free energy contributions to those obtained by the all-atom linear response approximation treatment. The agreement appears to be quite encouraging, thus further verifying the microscopic character of the PDLD model. Finally it is reclarified that real continuum models cannot provide proper descriptions of charges in protein and that current DC models are becoming more and more microscopic in nature.