Electrostatic screening in molecular dynamics simulations.

The screened Coulombic potential has been shown to describe satisfactorily equilibrium properties like pK shifts, the effects of charged groups on redox potentials and binding constants of metal ions. To test how well the screening of the electrostatic potential describes the dynamical trajectory of a macromolecular system, a series of comparative simulations have been carried out on a protein system which explicitly included water molecules and a system in vacuo. For the system without solvent the results of using (i) the standard potential form were compared with results of (ii) the potential where the Coulomb term was modified by the inclusion of a distance dependent dielectric, epsilon (r), to model the screening effect of bulk water, and (iii) standard potential modified by reducing the charge on ionized residue side chains. All molecular dynamics simulations have been carried out on bovine pancreatic trypsin inhibitor. Comparisons between the resulting trajectories, averaged structures, hydrogen bonding patterns and properties such as solvent accessible surface area and radius of gyration are described. The results show that the dynamical behaviour of the protein calculated with a screened electrostatic term compares more favourably with the time-dependent structural changes of the full system with explicitly included water than the standard vacuum simulation.     

Electrostatic effects in hemoglobin: hydrogen ion equilibria in human deoxy- and oxyhemoglobin A.

The modified Tanford-Kirkwood theory of Shire et al. [Shire, S. J., Hanania, G.I.H., & Gurd, F.R.N. (1974) Biochemistry 13, 2967] for electrostatic interactions was applied to the hydrogen ion equilibria of human deoxyhemoglobin and oxyhemoglobin. Atomic coordinates for oxyhemoglobin were generated by the application of the appropriate rigid rotation function to alpha and beta chains of the deoxyhemoglobin structure [Fermi, G. (1975) J. Mol. Biol. 97, 237]. The model employs two sets of parameters derived from the crystalline protein structures, the atomic coordinates of charged amino acid residues and static solvent accessibility factors to reflect their individual degrees of exposure to solvent. Theoretical titration curves based on a consistent set of pKint values compared closely with experimental potentiometric curves. Theoretical pK values at half-titration for individual protein sites corresponded to available observed values for both quaternary states. The results bring out the cumulative effects of numerous electrostatic interactions in the tetrameric structures and the major effects of the quaternary transition that result from changes in static solvent accessibility of certain ionizable groups.     

High apparent dielectric constant inside a protein reflects structural reorganization coupled to the ionization of an internal Asp

The dielectric properties of proteins are poorly understood and difficult to describe quantitatively. This limits the accuracy of methods for structure-based calculation of electrostatic energies and pK(a) values. The pK(a) values of many internal groups report apparent protein dielectric constants of 10 or higher. These values are substantially higher than the dielectric constants of 2-4 measured experimentally with dry proteins. The structural origins of these high apparent dielectric constants are not well understood. Here we report on structural and equilibrium thermodynamic studies of the effects of pH on the V66D variant of staphylococcal nuclease. In a crystal structure of this protein the neutral side chain of Asp-66 is buried in the hydrophobic core of the protein and hydrated by internal water molecules. Asp-66 titrates with a pK(a) value near 9. A decrease in the far UV-CD signal was observed, concomitant with ionization of this aspartic acid, and consistent with the loss of 1.5 turns of alpha-helix. These data suggest that the protein dielectric constant needed to reproduce the pK(a) value of Asp-66 with continuum electrostatics calculations is high because the dielectric constant has to capture, implicitly, the energetic consequences of the structural reorganization that are not treated explicitly in continuum calculations with static structures.     

Improving the accuracy of protein pKa calculations: Conformational averaging versus the average structure

Several methods for including the conformational flexibility of proteins in the calculation of titration curves are compared. The methods use the linearized Poisson-Boltzmann equation to calculate the electrostatic free energies of solvation and are applied to bovine pancreatic trypsin inhibitor (BPTI) and hen egg-white lysozyme (HEWL). An ensemble of conformations is generated by a molecular dynamics simulation of the proteins with explicit solvent. The average titration curve of the ensemble is calculated in three different ways: an average structure is used for the pK_a calculation; the electrostatic interaction free energies are averaged and used for the pK_a calculation; and the titration curve for each structure is calculated and the curves are averaged. The three averaging methods give very similar results and improve the pK_a values to approximately the same degree. This suggests, in contrast to implications from other work, that the observed improvement of pK_a values in the present studies is due not to averaging over an ensemble of structures, but rather to the generation of a single properly averaged structure for the pK_a calculation. Proteins 33:145–158, 1998. © 1998 Wiley-Liss, Inc.     

Calculation of protein ionization equilibria with conformational sampling: pK(a) of a model leucine zipper, GCN4 and barnase.

The use of conformational ensembles provided by nuclear magnetic resonance (NMR) experiments or generated by molecular dynamics (MD) simulations has been regarded as a useful approach to account for protein motions in the context of pK(a) calculations, yet the idea has been tested occasionally. This is the first report of systematic comparison of pK(a) estimates computed from long multiple MD simulations and NMR ensembles. As model systems, a synthetic leucine zipper, the naturally occurring coiled coil GCN4, and barnase were used. A variety of conformational averaging and titration curve-averaging techniques, or combination thereof, was adopted and/or modified to investigate the effect of extensive global conformational sampling on the accuracy of pK(a) calculations. Clustering of coordinates is proposed as an approach to reduce the vast diversity of MD ensembles to a few structures representative of the average electrostatic properties of the system in solution. Remarkable improvement of the accuracy of pK(a) predictions was achieved by the use of multiple MD simulations. By using multiple trajectories the absolute error in pK(a) predictions for the model leucine zipper was reduced to as low as approximately 0.25 pK(a) units. The validity, advantages, and limitations of explicit conformational sampling by MD, compared with the use of an average structure and a high internal protein dielectric value as means to improve the accuracy of pK(a) calculations, are discussed.     

Molecular dynamics studies of a DNA-binding protein: 2. An evaluation of implicit and explicit solvent models for the molecular dynamics simulation of the Escherichia coli trp repressor.

Although aqueous simulations with periodic boundary conditions more accurately describe protein dynamics than in vacuo simulations, these are computationally intensive for most proteins. Trp repressor dynamic simulations with a small water shell surrounding the starting model yield protein trajectories that are markedly improved over gas phase, yet computationally efficient. Explicit water in molecular dynamics simulations maintains surface exposure of protein hydrophilic atoms and burial of hydrophobic atoms by opposing the otherwise asymmetric protein-protein forces. This properly orients protein surface side chains, reduces protein fluctuations, and lowers the overall root mean square deviation from the crystal structure. For simulations with crystallographic waters only, a linear or sigmoidal distance-dependent dielectric yields a much better trajectory than does a constant dielectric model. As more water is added to the starting model, the differences between using distance-dependent and constant dielectric models becomes smaller, although the linear distance-dependent dielectric yields an average structure closer to the crystal structure than does a constant dielectric model. Multiplicative constants greater than one, for the linear distance-dependent dielectric simulations, produced trajectories that are progressively worse in describing trp repressor dynamics. Simulations of bovine pancreatic trypsin were used to ensure that the trp repressor results were not protein dependent and to explore the effect of the nonbonded cutoff on the distance-dependent and constant dielectric simulation models. The nonbonded cutoff markedly affected the constant but not distance-dependent dielectric bovine pancreatic trypsin inhibitor simulations. As with trp repressor, the distance-dependent dielectric model with a shell of water surrounding the protein produced a trajectory in better agreement with the crystal structure than a constant dielectric model, and the physical properties of the trajectory average structure, both with and without a nonbonded cutoff, were comparable.     

Optimizing pKa computation in proteins with pH adapted conformations

pK(A) in proteins are determined by electrostatic energy computations using a small number of optimized protein conformations derived from crystal structures. In these protein conformations hydrogen positions and geometries of salt bridges on the protein surface were determined self-consistently with the protonation pattern at three pHs (low, ambient, and high). Considering salt bridges at protein surfaces is most relevant, since they open at low and high pH. In the absence of these conformational changes, computed pK(A)(comp) of acidic (basic) groups in salt bridges underestimate (overestimate) experimental pK(A)(exp), dramatically. The pK(A)(comp) for 15 different proteins with 185 known pK(A)(exp) yield an RMSD of 1.12, comparable with two other methods. One of these methods is fully empirical with many adjustable parameters. The other is also based on electrostatic energy computations using many non-optimized side chain conformers but employs larger dielectric constants at short distances of charge pairs that diminish their electrostatic interactions. These empirical corrections that account implicitly for additional conformational flexibility were needed to describe the energetics of salt bridges appropriately. This is not needed in the present approach. The RMSD of the present approach improves if one considers only strongly shifted pK(A)(exp) in contrast to the other methods under these conditions. Our method allows interpreting pK(A)(comp) in terms of pH dependent hydrogen bonding pattern and salt bridge geometries. A web service is provided to perform pK(A) computations.     

Improved pKa calculations through flexibility based sampling of a water-dominated interaction scheme

Ionizable groups play critical roles in biological processes. Computation of pK(a)s is complicated by model approximations and multiple conformations. Calculated and experimental pK(a)s are compared for relatively inflexible active-site side chains, to develop an empirical model for hydration entropy changes upon charge burial. The modification is found to be generally small, but large for cysteine, consistent with small molecule ionization data and with partial charge distributions in ionized and neutral forms. The hydration model predicts significant entropic contributions for ionizable residue burial, demonstrated for components in the pyruvate dehydrogenase complex. Conformational relaxation in a pH-titration is estimated with a mean-field assessment of maximal side chain solvent accessibility. All ionizable residues interact within a low protein dielectric finite difference (FD) scheme, and more flexible groups also access water-mediated Debye-Hückel (DH) interactions. The DH method tends to match overall pH-dependent stability, while FD can be more accurate for active-site groups. Tolerance for side chain rotamer packing is varied, defining access to DH interactions, and the best fit with experimental pK(a)s obtained. The new (FD/DH) method provides a fast computational framework for making the distinction between buried and solvent-accessible groups that has been qualitatively apparent from previous work, and pK(a) calculations are significantly improved for a mixed set of ionizable residues. Its effectiveness is also demonstrated with computation of the pH-dependence of electrostatic energy, recovering favorable contributions to folded state stability and, in relation to structural genomics, with substantial improvement (reduction of false positives) in active-site identification by electrostatic strain.     

Protein dynamics in solution and in a crystalline environment: a molecular dynamics study

The effect of a solvent and a crystalline environment on the dynamics of proteins is investigated by the method of computer simulation. Three 25-ps molecular dynamics simulations at 300 K of the bovine pancreatic trypsin inhibitor (BPTI), consisting of 454 heavy atoms, are compared: one of BPTI in vacuo, one of BPTI in a box with 2647 spherical nonpolar solvent atoms, and one of BPTI surrounded by fixed crystal image atoms. Both average and time-dependent molecular properties are examined to determine the effect of the environment on the behavior of the protein. The dynamics of BPTI in solution or in the crystal environment are found to be very similar to that found in the vacuum calculation. The primary difference in the average properties is that the equilibrium structure in the presence of solvent or the crystal field is significantly closer to the X-ray structure than is the vacuum result; concomitantly, the more realistic environment leads to a number density closer to experiment. The presence of solvent has a negligible effect on the overall magnitude of the positional or dihedral angle fluctuations in the interior of the protein; however, there are changes in the decay times of the fluctuations of interior atoms. For surface residues, both the magnitude and the time course of the motions are significantly altered by the solvent. There tends to be an increase in the displacements of long side chains and the flexible parts of the main chain that protrude into the solvent. Further, these motions tend to have a more diffusive character with longer relaxation times than in vacuo. The crystal environment has a specific effect on a number of side chains which are held in relatively fixed positions through hydrogen-bond and electric interactions with the neighboring protein atoms. Most of the effects of the solution environment seem to be sufficiently nonspecific that it may be possible to model them by applying a mean field and stochastic dynamic methods.     

Acidic-basic properties of three alanine-based peptides containing acidic and basic side chains: comparison between theory and experiment

The purpose of this work was to evaluate the effect of the nature of the ionizable end groups, and the solvent, on their acid-base properties in alanine-based peptides. Hence, the acid-base properties of three alanine-based peptides: Ac-KK-(A)(7)-KK-NH(2) (KAK), Ac-OO-(A)(7)-DD-NH(2) (OAD), Ac-KK-(A)(7)-EE-NH(2) (KAE), where A, D, E, K, and O denote alanine, aspartic acid, glutamic acid, lysine, and ornithine, respectively, were determined in water and in methanol by potentiometry. With the availability of these data, the ability of two theoretical methods to simulate pH-metric titration of those peptides was assessed: (i) the electrostatically driven Monte Carlo method with the ECEPP/3 force field and the Poisson-Boltzmann approach to compute solvation energy (EDMC/PB/pH), and (ii) the molecular dynamics method with the AMBER force field and the Generalized Born model (MD/GB/pH). For OAD and KAE, pK(a1) and pK(a2) correspond to the acidic side chains. For all three compounds in both solvents, the pK(a1) value is remarkably lower than the pK(a) of a compound modeling the respective isolated side chain, which can be explained by the influence of the electrostatic field from positively charged ornithine or lysine side chains. The experimental titration curves are reproduced well by the MD/GB/pH approach, the agreement being better if restraints derived from NMR measurements are incorporated in the conformational search. Poorer agreement is achieved by the EDMC/PB/pH method.     

Molecular dynamics simulations of peptides and proteins with a continuum electrostatic model based on screened Coulomb potentials

A continuum electrostatics approach for molecular dynamics (MD) simulations of macromolecules is presented and analyzed for its performance on a peptide and a globular protein. The approach incorporates the screened Coulomb potential (SCP) continuum model of electrostatics, which was reported earlier. The model was validated in a broad set of tests some of which were based on Monte Carlo simulations that included single amino acids, peptides, and proteins. The implementation for large-scale MD simulations presented in this article is based on a pairwise potential that makes the electrostatic model suitable for fast analytical calculation of forces. To assess the suitability of the approach, a preliminary validation is conducted, which consists of (i) a 3-ns MD simulation of the immunoglobulin-binding domain of streptococcal protein G, a 56-residue globular protein and (ii) a 3-ns simulation of Dynorphin, a biological peptide of 17 amino acids. In both cases, the results are compared with those obtained from MD simulations using explicit water (EW) molecules in an all-atom representation. The initial structure of Dynorphin was assumed to be an alpha-helix between residues 1 and 9 as suggested from NMR measurements in micelles. The results obtained in the MD simulations show that the helical structure collapses early in the simulation, a behavior observed in the EW simulation and consistent with spectroscopic data that suggest that the peptide may adopt mainly an extended conformation in water. The dynamics of protein G calculated with the SCP implicit solvent model (SCP-ISM) reveals a stable structure that conserves all the elements of secondary structure throughout the entire simulation time. The average structures calculated from the trajectories with the implicit and explicit solvent models had a cRMSD of 1.1 A, whereas each average structure had a cRMSD of about 0.8A with respect to the X-ray structure. The main conformational differences of the average structures with respect to the crystal structure occur in the loop involving residues 8-14. Despite the overall similarity of the simulated dynamics with EW and SCP models, fluctuations of side-chains are larger when the implicit solvent is used, especially in solvent exposed side-chains. The MD simulation of Dynorphin was extended to 40 ns to study its behavior in an aqueous environment. This long simulation showed that the peptide has a tendency to form an alpha-helical structure in water, but the stabilization free energy is too weak, resulting in frequent interconversions between random and helical conformations during the simulation time. The results reported here suggest that the SCP implicit solvent model is adequate to describe electrostatic effects in MD simulation of both peptides and proteins using the same set of parameters. It is suggested that the present approach could form the basis for the development of a reliable and general continuum approach for use in molecular biology, and directions are outlined for attaining this long-term goal.     

Electrostatic properties of bovine beta-lactoglobulin

Bovine beta-Lactoglobulin (BLG) has been studied for many decades, but only recently structural data have been obtained, making it possible to simulate its molecular properties. In the present study, electrostatic properties of BLG are investigated theoretically using Poisson-Boltzmann calculations and experimentally following pH titration via NMR. Electrostatic properties are determined for several structural models, including an ensemble of NMR structures obtained at low pH. The changes in electrostatic forces upon changes in ionic strength, solvent dielectric constant, and pH are calculated and compared with experiments. pK(a)s are computed for all titratable sites and compared with NMR titration data. The analysis of theoretical and experimental results suggests that (1) there may be more than one binding sites for negatively charged ligands; (2) at low pH the core of the molecule is more compact than observed in the structures obtained via restrained molecular dynamics from NMR data, but loop and terminal regions must be disordered.     

The relative order of helical propensity of amino acids changes with solvent environment

A model peptide of sequence Ac-Y-VAXAK-VAXAK-VAXAK-NH(2), where X is substituted with one of nineteen amino acids (P excluded), was synthesized and titrated with methanol to study helical propensity as a function of solvent environment. The CD spectra of these peptides are largely random coil in 2 mM sodium phosphate buffer (pH 5.5) and show a conformational change to alpha-helix with increasing methanol content. Singular value decomposition was used to correct the CD spectra for the absorbing side chains of W, Y, F, C, and M, and this correction can be substantial. With correction both W and F become good helix formers. The free energy for helix propagation was calculated using the Lifson-Roig statistical model for each of the nineteen amino acids at each point in their titration. The results show that the rank order of helical propensity for the nineteen amino acids changes with solvent environment. This result will be particularly important if proteins undergo hydrophobic collapse before secondary structures are formed, because amino acids can then see different solvent environments as the secondary structures are formed. Related amino acids are found to have interesting correlations in the shape of their titration curves. This finding provides one explanation for the limiting 70% accuracy in predicting secondary structure from sequence, since the helical propensities used are calculated for an average solvent environment. Proteins 2000;39:132-141.     

The Molecular Determinants of the Increased Reduction Potential of the Rubredoxin Domain of Rubrerythrin Relative to Rubredoxin

Based on the crystal structures, three possible sequence determinants have been suggested as the cause of a 285 mV increase in reduction potential of the rubredoxin domain of rubrerythrin over rubredoxin by modulating the polar environment around the redox site. Here, electrostatic calculations of crystal structures of rubredoxin and rubrerythrin and molecular dynamics simulations of rubredoxin wild-type and mutants are used to elucidate the contributions to the increased reduction potential. Asn¹⁶⁰ and His¹⁷⁹ in rubrerythrin versus valines in rubredoxins are predicted to be the major contributors, as the polar side chains contribute significantly to the electrostatic potential in the redox site region. The mutant simulations show both side chains rotating on a nanosecond timescale between two conformations with different electrostatic contributions. Reduction also causes a change in the reduction energy that is consistent with a linear response due to the interesting mechanism of shifting the relative populations of the two conformations. In addition to this, a simulation of a triple mutant indicates the side-chain rotations are approximately anticorrelated so whereas one is in the high potential conformation, the other is in the low potential conformation. However, Ala¹⁷⁶ in rubrerythrin versus a leucine in rubredoxin is not predicted to be a large contributor, because the solvent accessibility increases only slightly in mutant simulations and because it is buried in the interface of the rubrerythrin homodimer.     

Vicinal coupling constants and protein dynamics

The effects of motional averaging on the analysis of vicinal spin-spin coupling constants derived from proton NMR studies of proteins have been examined. Trajectories obtained from molecular dynamics simulations of bovine pancreatic trypsin inhibitor and of hen egg white lysozyme were used in conjunction with an expression for the dependence of the coupling constant on the intervening dihedral angle to calculate the time-dependent behavior of the coupling constants. Despite large fluctuations, the time-average values of the coupling constants are not far from those computed for the average structure in the cases where fluctuations occur about a single potential well. The calculated differences show a high correlation with the variation in the magnitude of the fluctuations of individual dihedral angles. For the cases where fluctuations involve multiple sites, large differences are found between the time-average values and the average structure values for the coupling constants. Comparison of the simulation results with the experimental trends suggests that side chains with more than one position are more common in proteins than is inferred from X-ray results. It is concluded that for the main chain, motional effects do not introduce significant errors where vicinal coupling constants are used in structure determinations; however, for side chains, the motional average can alter deductions about the structure. Accurately measured coupling constants are shown to provide information concerning the magnitude of dihedral angle fluctuations.     

MCCE analysis of the pKas of introduced buried acids and bases in staphylococcal nuclease

The pK(a)s of 96 acids and bases introduced into buried sites in the staphylococcal nuclease protein (SNase) were calculated using the multiconformation continuum electrostatics (MCCE) program and the results compared with experimental values. The pK(a)s are obtained by Monte Carlo sampling of coupled side chain protonation and position as a function of pH. The dependence of the results on the protein dielectric constant (ε(prot)) in the continuum electrostatics analysis and on the Lennard-Jones non-electrostatics parameters was evaluated. The pK(a)s of the introduced residues have a clear dependence on ε(prot,) whereas native ionizable residues do not. The native residues have electrostatic interactions with other residues in the protein favoring ionization, which are larger than the desolvation penalty favoring the neutral state. Increasing ε(prot) scales both terms, which for these residues leads to small changes in pK(a). The introduced residues have a larger desolvation penalty and negligible interactions with residues in the protein. For these residues, changing ε(prot) has a large influence on the calculated pK(a). An ε(prot) of 8-10 and a Lennard-Jones scaling of 0.25 is best here. The X-ray crystal structures of the mutated proteins are found to provide somewhat better results than calculations carried out on mutations made in silico. Initial relaxation of the in silico mutations by Gromacs and extensive side chain rotamer sampling within MCCE can significantly improve the match with experiment.     

Minimal ensembles of side chain conformers for modeling protein–protein interactions

The goal of this article is to reduce the complexity of the side chain search within docking problems. We apply six methods of generating side chain conformers to unbound protein structures and determine their ability of obtaining the bound conformation in small ensembles of conformers. Methods are evaluated in terms of the positions of side chain end groups. Results for 68 protein complexes yield two important observations. First, the end‐group positions change less than 1 Å on association for over 60% of interface side chains. Thus, the unbound protein structure carries substantial information about the side chains in the bound state, and the inclusion of the unbound conformation into the ensemble of conformers is very beneficial. Second, considering each surface side chain separately in its protein environment, small ensembles of low‐energy states include the bound conformation for a large fraction of side chains. In particular, the ensemble consisting of the unbound conformation and the two highest probability predicted conformers includes the bound conformer with an accuracy of 1 Å for 78% of interface side chains. As more than 60% of the interface side chains have only one conformer and many others only a few, these ensembles of low‐energy states substantially reduce the complexity of side chain search in docking problems. This approach was already used for finding pockets in protein–protein interfaces that can bind small molecules to potentially disrupt protein–protein interactions. Side‐chain search with the reduced search space will also be incorporated into protein docking algorithms. Proteins 2012. © 2011 Wiley Periodicals, Inc.     

Discrete charge calculations of potentiometric titrations for globular proteins: sperm whale myoglobin, hemoglobin alpha chain, cytochrome c.

The modified Tanford-Kirkwood theory of Shire et al. for intramolecular electrostatic interactions has been applied to hydrogen ion equilibria of sperm whale ferrimyoglobin, human hemoglobin α-chain and horse cytochrome c. The model employs two sets of parameters derived from the crystalline protein structures, first, the atomic coordinates of charged amino acid residues and, second, static accessibility factors to reflect their solvent exposure. In addition, a consistent set of intrinsic pK values (pK_int) for the individual groups is employed. The theoretical pK values at half-titration for individual groups in each protein correspond to the available observed pK values, and the theoretical titration curves compare closely with experimental potentiometric curves.