Different microscopic and semimicroscopic approaches for calculations of electrostatic energies in macromolecules are examined. This includes the Protein Dipoles Langevin Dipoles (PDLD) method, the semimicroscopic PDLD (PDLD/S) method, and a free energy perturbation (FEP) method. The incorporation of these approaches in the POLARIS and ENZYME modules of the MOLARIS package is described in detail. The PDLD electrostatic calculations are augmented by estimates of the relevant hydrophobic and steric contributions, as well as the effects of the ionic strength and external pH. Determination of the hydrophobic energy involves an approach that considers the modification of the effective surface area of the solute by local field effects. The steric contributions are analyzed in terms of the corresponding reorganization energies. Ionic strength effects are studied by modeling the ionic environment around the given system using a grid of residual charges and evaluating the relevant interaction using Coulomb's law with the dielectric constant of water. The performance of the FEP calculations is significantly enhanced by using special boundary conditions and evaluating the long-range electrostatic contributions using the Local Reaction Field (LRF) model. A diverse set of electrostatic effects are examined, including the solvation energies of charges in proteins and solutions, energetics of ion pairs in proteins and solutions, interaction between surface charges in proteins, and effect of ionic strength on such interactions, as well as electrostatic contributions to binding and catalysis in solvated proteins. Encouraging results are obtained by the microscopic and semimicroscopic approaches and the problems associated with some macroscopic models are illustrated. The PDLD and PDLD/S methods appear to be much faster than the FEP approach and still give reasonable results. In particular, the speed and simplicity of the PDLD/S method make it an effective strategy for calculations of electrostatic free energies in interactive docking studies. Nevertheless, comparing the results of the three approaches can provide a useful estimate of the accuracy of the calculated energies. 0 1993 by John Wiley & Sons, Inc.

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 so-called “protein dielectric constant”, εp. 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 εp 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 εp 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 εp reduces the overestimate of charge−charge interactions relative to models that use small εp while not considering the protein relaxation explicitly. Unfortunately, the use of large εp 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 calculations is addressed. It is pointed out that running MD over protein configurations will not by itself lead to a more consistent value of εp. It is clarified that a smaller value of εp, 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 solvat...

The solution structures of the reduced and oxidized forms of the cytochrome c are used to reevaluate the reorganization energy for oxidation of cytochrome c. This is achieved by using the linear response approximation in concert with the NMR structures as pseudo energy constraints. Alternative estimates, obtained using a free energy perturbation approach employing umbrella sampling and a continuum dielectric approach, are also provided. The reorganization energy obtained is larger than that previously estimated using crystal structures of the protein. Nevertheless, the present estimate remains significantly smaller than the corresponding reorganization energy in water (9−15 kcal mol-1 as compared to ≈37 kcal mol-1 in water) and the protein contribution to the reorganization energy is only 8−10 kcal mol-1. This provides further support for the proposal that proteins assist in electron transfer reactions by reducing the relevant reorganization energies. The solution structures are also used to estimate the redox potential of cytochrome c. Several strategies are employed including a newly formulated scaled linear response approximation. The calculations agree reasonably well with the observed redox potential. Analysis of the group contributions to the reorganization energy and redox potential reveals a clear energetic linkage between these fundamental parameters of electron transfer and a redox-dependent surface feature likely to influence recognition of cytochrome c by its redox partners. Specifically, the rearrangement of Ile81 and other residues at the heme edge upon a change in oxidation state gives rise to a large contribution to both the redox potential and the reorganization energy. Finally this work is used to explore and illustrate the meaning of macroscopic dielectric models. It is shown that the “proper” dielectric constant depends strongly on the model used since it basically represents the implicit contributions of the given model rather than a fundamental physics. Thus we obtain different effective dielectric constants for different treatments of redox potential and reorganization energy.

Several strategies for evaluation of the protein-ligand binding free energies are examined. Particular emphasis is placed on the Linear Response Approximation (LRA) (Lee et. al., Prot Eng 1992;5:215-228) and the Linear Interaction Energy (LIE) method (Aqvist et. al., Prot Eng 1994;7:385-391). The performance of the Protein Dipoles Langevin Dipoles (PDLD) method and its semi-microscopic version (the PDLD/S method) is also considered. The examination is done by using these methods in the evaluating of the binding free energies of neutral C2-symmetric cyclic urea-based molecules to Human Immunodeficiency Virus (HIV) protease. Our starting point is the introduction of a thermodynamic cycle that decomposes the total binding free energy to electrostatic and non-electrostatic contributions. This cycle is closely related to the cycle introduced in our original LRA study (Lee et. al., Prot Eng 1992;5:215-228). The electrostatic contribution is evaluated within the LRA formulation by averaging the protein-ligand (and/or solvent-ligand) electrostatic energy over trajectories that are propagated on the potentials of both the polar and non-polar (where all residual charges are set to zero) states of the ligand. This average involves a scaling factor of 0.5 for the contributions from each state and this factor is being used in both the LRA and LIE methods. The difference is, however, that the LIE method neglects the contribution from trajectories over the potential of the non-polar state. This approximation is entirely valid in studies of ligands in water but not necessarily in active sites of proteins. It is found in the present case that the contribution from the non-polar states to the protein-ligand binding energy is rather small. Nevertheless, it is clearly expected that this term is not negligible in cases where the protein provides preorganized environment to stabilize the residual charges of the ligand. This contribution can be particularly important in cases of charged ligands. The analysis of the non-electrostatic term is much more complex. It is concluded that within the LRA method one has to complete the relevant thermodynamic cycle by evaluating the binding free energy of the "non-polar" ligand, l;, where all the residual charges are set to zero. It is shown that the LIE term, which involves the scaling of the van der Waals interaction by a constant beta (usually in the order of 0.15 to 0.25), corresponds to this part of the cycle. In order to elucidate the nature of this non-electrostatic term and the origin of the scaling constant beta, it is important to evaluate explicitly the different contributions to the binding energy of the non-polar ligand, DeltaG(bind,l;). Since this cannot be done at present (for relatively large ligands) by rigorous free energy perturbation approaches, we evaluate DeltaG(bind,l;) by the PDLD approach, augmented by microscopic calculations of the change in configurational entropy upon binding. This evaluation takes into account the van der Waals, hydrophobic, water penetration and entr...