Electrostatic pKa computations in proteins: role of internal cavities

The solvent accessible surface area (SASA) algorithm is conventionally used to characterize protein surfaces in electrostatic energy computations of proteins. Unfortunately, it often fails to find narrow cavities inside a protein. As a consequence pK(a) computations based on this algorithm perform badly. In this study a new cavity-algorithm is introduced, which solves this problem and provides improved pK(a) values. The procedure is applied to 20 pK(a) values of titratable groups introduced as point mutations in SNase variants, where crystal structures are available. The computations of these pK(a)s are particular challenging, since they are placed in a rather hydrophobic environment. For nine mutants, where the titratable residue is in contact with a large cavity, the RMSD(pKa) between computed and measured pK(a) values is 2.04, which is a considerable improvement as compared to the original results obtained with Karlsberg(+) (http://agknapp.chemie.fu-berlin.de/karlsberg/) that yielded an RMSD(pKa) of 8.8. However, for 11 titratable residues the agreement with experiments remains poor (RMSD(pKa) = 6.01). Considering 15 pK(a)s of SNase, which are in a more conventional less hydrophobic protein environment, the RMSD(pKa) is 2.1 using the SASA-algorithm and 1.7 using the new cavity-algorithm. The agreement is reasonable but less good than what one would expect from the general performance of Karlsberg(+) indicating that SNase belongs to the more difficult proteins with respect to pK(a) computations. We discuss the possible reasons for the remaining discrepancies between computed and measured pK(a)s.     

Calculation of pK(a) in proteins with the microenvironment modulated-screened coulomb potential

The MM-SCP has been applied to predict pK(a) values of titratable residues in wild type and mutants of staphylococcal nuclease (SNase). The calculations were based on crystal structures made available by the Garcia-Moreno Laboratory. In the mutants, mostly deeply buried hydrophobic residues were replaced with ionizable residues, and thus their pK(a) values could be measured and calculated using various methods. The data set used here consisted of a set of WT SNase for which His pK(a) at several ionic strengths had been measured, a set of mutants for which measured pK(a) were available and a set of 11 mutants for which the measured pK(a) were not known at the time of calculation. For this latter set, blind predictions were submitted to the protein pK(a) cooperative, 2009 workshop at Telluride, where the results of the blind predictions were discussed (the RMSD of the submitted set was 1.10 pH units). The calculations on the structures with known pK(a) indicated that in addition to weaknesses of the method, structural issues were observed that led to larger errors (>1) in pK(a) predictions. For example, different crystallographic conditions or steric clashes can lead to differences in the local environment around the titratable residue, which can produce large differences in the calculated pK(a) . To gain further insight into the reliability of the MM-SCP, pK(a) of an extended set of 54 proteins belonging to several structural classes were carried out. Here some initial results from this study are reported to help place the SNase results in the appropriate context.     

A buried lysine that titrates with a normal pKa: role of conformational flexibility at the protein-water interface as a determinant of pKa values

Previously we reported that Lys, Asp, and Glu residues at positions 66 and 92 in staphylococcal nuclease (SNase) titrate with pK(a) values shifted by up to 5 pK(a) units in the direction that promotes the neutral state. In contrast, the internal Lys-38 in SNase titrates with a normal pK(a). The crystal structure of the L38K variant shows that the side chain of Lys-38 is buried. The ionizable moiety is approximately 7 A from solvent and ion paired with Glu-122. This suggests that the pK(a) value of Lys-38 is normal because the energetic penalty for dehydration is offset by a favorable Coulomb interaction. However, the pK(a) of Lys-38 was also normal when Glu-122 was replaced with Gln or with Ala. Continuum electrostatics calculations were unable to reproduce the pK(a) of Lys-38 unless the protein was treated with an artificially high dielectric constant, consistent with structural reorganization being responsible for the normal pK(a) value of Lys-38. This reorganization must be local because circular dichroism and NMR spectroscopy indicate that the L38K protein is native-like under all conditions studied. In molecular dynamics simulations, the ion pair between Lys-38 and Glu-122 is unstable. The simulations show that a minor rearrangement of a loop is sufficient to allow penetration of water to the amino moiety of Lys-38. This illustrates both the important roles of local flexibility and water penetration as determinants of pK(a) values of ionizable groups buried near the protein-water interface, and the challenges faced by structure-based pK(a) calculations in reproducing these effects.     

Measuring the successes and deficiencies of constant pH molecular dynamics: a blind prediction study

A constant pH molecular dynamics method has been used in the blind prediction of pK(a) values of titratable residues in wild type and mutated structures of the Staphylococcal nuclease (SNase) protein. The predicted values have been subsequently compared to experimental values provided by the laboratory of García-Moreno. CpHMD performs well in predicting the pK(a) of solvent-exposed residues. For residues in the protein interior, the CpHMD method encounters some difficulties in reaching convergence and predicting the pK(a) values for residues having strong interactions with neighboring residues. These results show the need to accurately and sufficiently sample conformational space in order to obtain pK(a) values consistent with experimental results.     

A parameterized, continuum electrostatic model for predicting protein pKa values

Recognizing the limits of trying to achieve chemical accuracy for pK(a) calculations with a purely electrostatic model, we include empirical corrections into the Poisson-Boltzmann solver macroscopic electrostatics with atomic detail (Bashford, Biochemistry 1990;29:10219-10225), to improve the reliability and accuracy of the model. The total number of parameters is kept to a minimum to maximize the robustness of the model for compounds outside of the fitting dataset. The parameters are based on: (a) the electrostatic interaction between functional groups close to the titratable site, (b) the electrostatic work required to desolvate the residue, and (c) the site-to-site interactions. These interactions are straightforward to calculate once the electrostatic field has been solved for each residue using the linearized Poisson-Boltzmann equation and are assumed to be linearly related to the intrinsic pK(a). Two hundred and eighty-six residues from 30 proteins are used to determine the empirical parameters, which result in a root mean square error (RMSE) of 0.70 for the entire set. Eight proteins with 46 experimentally known values were excluded from the parameterization to test the model. This test set had a RMSE of 1.08. We show that the parameterized model improves the results over other models, although like other models the error is strongly correlated with the degree to which a residue is buried. The parameters themselves indicate that local effects are most important for determining the pK(a), whereas site-to-site interactions are found to be less significant.     

Toward accurate prediction of pKa values for internal protein residues: the importance of conformational relaxation and desolvation energy

Proton uptake or release controls many important biological processes, such as energy transduction, virus replication, and catalysis. Accurate pK(a) prediction informs about proton pathways, thereby revealing detailed acid-base mechanisms. Physics-based methods in the framework of molecular dynamics simulations not only offer pK(a) predictions but also inform about the physical origins of pK(a) shifts and provide details of ionization-induced conformational relaxation and large-scale transitions. One such method is the recently developed continuous constant pH molecular dynamics (CPHMD) method, which has been shown to be an accurate and robust pK(a) prediction tool for naturally occurring titratable residues. To further examine the accuracy and limitations of CPHMD, we blindly predicted the pK(a) values for 87 titratable residues introduced in various hydrophobic regions of staphylococcal nuclease and variants. The predictions gave a root-mean-square deviation of 1.69 pK units from experiment, and there were only two pK(a)'s with errors greater than 3.5 pK units. Analysis of the conformational fluctuation of titrating side-chains in the context of the errors of calculated pK(a) values indicate that explicit treatment of conformational flexibility and the associated dielectric relaxation gives CPHMD a distinct advantage. Analysis of the sources of errors suggests that more accurate pK(a) predictions can be obtained for the most deeply buried residues by improving the accuracy in calculating desolvation energies. Furthermore, it is found that the generalized Born implicit-solvent model underlying the current CPHMD implementation slightly distorts the local conformational environment such that the inclusion of an explicit-solvent representation may offer improvement of accuracy.     

Charge-charge interactions are key determinants of the pK values of ionizable groups in ribonuclease Sa (pI=3.5) and a basic variant (pI=10.2)

The pK values of the titratable groups in ribonuclease Sa (RNase Sa) (pI=3.5), and a charge-reversed variant with five carboxyl to lysine substitutions, 5K RNase Sa (pI=10.2), have been determined by NMR at 20 degrees C in 0.1M NaCl. In RNase Sa, 18 pK values and in 5K, 11 pK values were measured. The carboxyl group of Asp33, which is buried and forms three intramolecular hydrogen bonds in RNase Sa, has the lowest pK (2.4), whereas Asp79, which is also buried but does not form hydrogen bonds, has the most elevated pK (7.4). These results highlight the importance of desolvation and charge-dipole interactions in perturbing pK values of buried groups. Alkaline titration revealed that the terminal amine of RNase Sa and all eight tyrosine residues have significantly increased pK values relative to model compounds.A primary objective in this study was to investigate the influence of charge-charge interactions on the pK values by comparing results from RNase Sa with those from the 5K variant. The solution structures of the two proteins are very similar as revealed by NMR and other spectroscopic data, with only small changes at the N terminus and in the alpha-helix. Consequently, the ionizable groups will have similar environments in the two variants and desolvation and charge-dipole interactions will have comparable effects on the pK values of both. Their pK differences, therefore, are expected to be chiefly due to the different charge-charge interactions. As anticipated from its higher net charge, all measured pK values in 5K RNase are lowered relative to wild-type RNase Sa, with the largest decrease being 2.2 pH units for Glu14. The pK differences (pK(Sa)-pK(5K)) calculated using a simple model based on Coulomb's Law and a dielectric constant of 45 agree well with the experimental values. This demonstrates that the pK differences between wild-type and 5K RNase Sa are mainly due to changes in the electrostatic interactions between the ionizable groups. pK values calculated using Coulomb's Law also showed a good correlation (R=0.83) with experimental values. The more complex model based on a finite-difference solution to the Poisson-Boltzmann equation, which considers desolvation and charge-dipole interactions in addition to charge-charge interactions, was also used to calculate pK values. Surprisingly, these values are more poorly correlated (R=0.65) with the values from experiment. Taken together, the results are evidence that charge-charge interactions are the chief perturbant of the pK values of ionizable groups on the protein surface, which is where the majority of the ionizable groups are positioned in proteins.     

Predicting extreme pKa shifts in staphylococcal nuclease mutants with constant pH molecular dynamics

Accurate computational methods of determining protein and nucleic acid pK(a) values are vital to understanding pH-dependent processes in biological systems. In this article, we use the recently developed method constant pH molecular dynamics (CPHMD) to explore the calculation of highly perturbed pK(a) values in variants of staphylococcal nuclease (SNase). Simulations were performed using the replica exchange (REX) protocol for improved conformational sampling with eight temperature windows, and yielded converged proton populations in a total sampling time of 4 ns. Our REX-CPHMD simulations resulted in calculated pK(a) values with an average unsigned error (AUE) of 0.75 pK units for the acidic residues in Δ + PHS, a hyperstable variant of SNase. For highly pK(a)-perturbed SNase mutants with known crystal structures, our calculations yielded an AUE of 1.5 pK units and for those mutants based on modeled structures an AUE of 1.4 pK units was found. Although a systematic underestimate of pK shifts was observed in most of the cases for the highly perturbed pK mutants, correlations between conformational rearrangement and plasticity associated with the mutation and error in pK(a) prediction was not evident in the data. This study further extends the scope of electrostatic environments explored using the REX-CPHMD methodology and suggests that it is a reliable tool for rapidly characterizing ionizable amino acids within proteins even when modeled structures are employed.     

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.     

Variability in the pKa of histidine side-chains correlates with burial within proteins

Acidic pKas of histidines buried within the protein interior are frequently rationalized on the contradictory basis of either polar interactions within the protein or the effects of a hydrophobic environment. To examine these relationships, we surveyed the buried surface area, depth of burial, polar interactions, and crystallographic temperature factors of histidines of known pKa. It has been found that buried environments of histidines do not always result in acidic pKas. Instead, the variability of histidine pKas increases for residues where the majority of the side-chain is buried. Because buried histidines are always found in mixed polar/apolar environments, multiple environmental contributions to pKa values must be considered. However, the quantitative relationships between heterogeneous environments and pKa values are not immediately apparent from the available data.     

A self-consistent, microenvironment modulated screened coulomb potential approximation to calculate pH-dependent electrostatic effects in proteins.

An improved approach is presented for calculating pH-dependent electrostatic effects in proteins using sigmoidally screened Coulomb potentials (SCP). It is hypothesized that a key determinant of seemingly aberrant behavior in pKa shifts is due to the properties of the unique microenvironment around each residue. To help demonstrate this proposal, an approach is developed to characterize the microenvironments using the local hydrophobicity/hydrophilicity around each residue of the protein. The quantitative characterization of the microenvironments shows that the protein is a complex mosaic of differing dielectric regions that provides a physical basis for modifying the dielectric screening functions: in more hydrophobic microenvironments the screening decreases whereas the converse applies to more hydrophilic regions. The approach was applied to seven proteins providing more than 100 measured pKa values and yielded a root mean square deviation of 0.5 between calculated and experimental values. The incorporation of the local hydrophobicity characteristics into the algorithm allowed the resolution of some of the more intractable problems in the calculation of pKa. Thus, the divergent shifts of the pKa of Glu-35 and Asp-66 in hen egg white lysozyme, which are both about 90% buried, was correctly predicted. Mechanistically, the divergence occurs because Glu-35 is in a hydrophobic microenvironment, while Asp-66 is in a hydrophilic microenvironment. Furthermore, because the calculation of the microenvironmental effects takes very little CPU time, the computational speed of the SCP formulation is conserved. Finally, results from different crystal structures of a given protein were compared, and it is shown that the reliability of the calculated pKa values is sufficient to allow identification of conformations that may be more relevant for the solution structure.     

Redesigning protein pKa values

The ability to re-engineer enzymatic pH-activity profiles is of importance for industrial applications of enzymes. We theoretically explore the feasibility of re-engineering enzymatic pH-activity profiles by changing active site pK(a) values using point mutations. We calculate the maximum achievable DeltapK(a) values for 141 target titratable groups in seven enzymes by introducing conservative net-charge altering point mutations. We examine the importance of the number of mutations introduced, their distance from the target titratable group, and the characteristics of the target group itself. The results show that multiple mutations at 10A can change pK(a) values up to two units, but that the introduction of a requirement to keep other pK(a) values constant reduces the magnitude of the achievable DeltapK(a). The algorithm presented shows a good correlation with existing experimental data and is available for download and via a web server at http://enzyme.ucd.ie/pKD.     

Experimental pK(a) values of buried residues: analysis with continuum methods and role of water penetration

Lys-66 and Glu-66, buried in the hydrophobic interior of staphylococcal nuclease by mutagenesis, titrate with pK(a) values of 5.7 and 8.8, respectively (Dwyer et al., Biophys. J. 79:1610-1620; García-Moreno E. et al., Biophys. Chem. 64:211-224). Continuum calculations with static structures reproduced the pK(a) values when the protein interior was treated with a dielectric constant (epsilon(in)) of 10. This high apparent polarizability can be rationalized in the case of Glu-66 in terms of internal water molecules, visible in crystallographic structures, hydrogen bonded to Glu-66. The water molecules are absent in structures with Lys-66; the high polarizability cannot be reconciled with the hydrophobic environment surrounding Lys-66. Equilibrium thermodynamic experiments showed that the Lys-66 mutant remained folded and native-like after ionization of the buried lysine. The high polarizability must therefore reflect water penetration, minor local structural rearrangement, or both. When in pK(a) calculations with continuum methods, the internal water molecules were treated explicitly, and allowed to relax in the field of the buried charged group, the pK(a) values of buried residues were reproduced with epsilon(in) in the range 4-5. The calculations show that internal waters can modulate pK(a) values of buried residues effectively, and they support the hypothesis that the buried Lys-66 is in contact with internal waters even though these are not seen crystallographically. When only the one or two innermost water molecules were treated explicitly, epsilon(in) of 5-7 reproduced the pK(a) values. These values of epsilon(in) > 4 imply that some conformational reorganization occurs concomitant with the ionization of the buried groups.     

pKa predictions with a coupled finite difference Poisson-Boltzmann and Debye-Hückel method

Modeling charge interactions is important for understanding many aspects of biological structure and function, and continuum methods such as Finite Difference Poisson-Boltzmann (FDPB) are commonly employed. Calculations of pH-dependence have identified separate populations; surface groups that can be modeled with a simple Debye-Hückel (DH) model, and buried groups, with stronger resultant interactions that are dependent on detailed conformation. This observation led to the development of a combined FDPB and DH method for pK(a) prediction (termed FD/DH). This study reports application of this method to ionizable groups, including engineered buried charges, in staphylococcal nuclease. The data had been made available to interested research groups before publication of mutant structures and/or pK(a) values. Overall, FD/DH calculations perform as intended with low ΔpK(a) values for surface groups (RMSD between predicted and experimental pK(a) values of 0.74), and much larger ΔpK(a) values for the engineered internal groups, with RMSD = 1.64, where mutant structures were known and RMSD = 1.80, where they were modeled. The weaker resultant interactions of the surface groups are determined mostly by charge-charge interactions. For the buried groups, R(2) for correlation between predicted and measured ΔpK(a) values is 0.74, despite the high RMSDs. Charge-charge interactions are much less important, with the R(2) value for buried group ΔpK(a) values increasing to 0.80 when the term describing charge desolvation alone is used. Engineered charge burial delivers a relatively uniform, nonspecific effect, in terms of pK(a) . How the protein environment adapts in atomic detail to deliver this resultant effect is still an open question.     

Blind, one-eyed, or eagle-eyed? pKa calculations during blind predictions with staphylococcal nuclease

In the current contribution, the performance of Poisson-Boltzmann-based pK(a) calculations of SNase mutants as part of a blind prediction exercise facilitated by the pK(a) cooperative ("pK(a) _coop") is described. A one parameter setting ("quick&dirty" approach) is used to provide an industry perspective where strong time constraints are frequently encountered. On the one hand, results are analyzed in terms of root mean square deviation performance. Furthermore, the pK(a) calculations are assessed for their ability to properly assign protonation state. For this purpose, a new measure called BIPS (binary protonation state at physiological pH) is introduced. Significant differences were found with both comparison measures based on the class of residues examined. In addition, the performance of PROPKA3 as well as the NULL model is examined on the same data set. Finally, pK(a) calculations on SNase mutants with available structural information have been performed and provide support for our calculation methods. The performance on this subset is better than on the pK(a) cooperative mutation data. In the pK(a) _coop data, no structural information on the generated mutants is available. This suggests the occurrence of a substantial structural rearrangement on the insertion of additional charged groups into SNase, which leads to improved prediction quality.     

Molecular mechanisms of pH-driven conformational transitions of proteins: insights from continuum electrostatics calculations of acid unfolding

The acid unfolding of staphylococcal nuclease (SNase) is very cooperative (Whitten and García-Moreno, Biochemistry 2000;39:14292-14304). As many as seven hydrogen ions (H+) are bound preferentially by the acid-unfolded state relative to the native (N) state in the pH range 3.2-3.9. To investigate the mechanism of acid unfolding, structure-based pKa calculations were performed with a variety of continuum electrostatic methods. The calculations reproduced successfully the H+ binding properties of the N state between pH 5 and 9, but they systematically overestimated the number of H+ bound upon acid unfolding. The calculated pKa values of all carboxylic residues in the N state were more depressed than they should be. The discrepancy between the observed and the calculated H+ uptake upon acid unfolding was not improved by using high protein dielectric constants, structures relaxed with molecular dynamics, or other empirical modifications implemented previously by others to maximize agreement between measured and calculated pKa values. This suggests an important role for conformational fluctuations of the backbone as important determinants of pKa values of carboxylic groups. Because no global or subglobal conformational changes have been observed previously for SNase under acidic conditions above the acid-unfolding region, these fluctuations must be local. The acid unfolding of SNase does not seem to involve the disruption of the N state by accruement of intramolecular repulsive interactions, nor the protonation of key ion paired carboxylic residues. It is more consistent with modest contributions from many H+ binding groups, with an important role for local conformational fluctuations in the coupling between H+ binding and the global structural transition.     

Assessing the acid-base and conformational properties of histidine residues in human prion protein (125-228) by means of pK(a) calculations and molecular dynamics simulations

A thorough study of the acid-base behavior of the four histidines and the other titratable residues of the structured domain of human prion protein (125-228) is presented. By using multi-tautomer electrostatic calculations, average titration curves have been built for all titratable residues, using the whole bundles of NMR structures determined at pH 4.5 and 7.0. According to our results, (1) only histidine residues are likely to be involved in the first steps of the pH-driven conformational transition of prion protein; (2) the pK(a)'s of His140 and His177 are approximately 7.0, whereas those of His155 and His187 are < 5.5. 10-ns long molecular dynamics simulations have been performed on five different models, corresponding to the most significant combinations of histidine protonation states. A critical comparison between the available NMR structures and our computational results (1) confirms that His155 and His187 are the residues whose protonation is involved in the conformational rearrangement of huPrP in mildly acidic condition, and (2) shows how their protonation leads to the destructuration of the C-terminal part of HB and to the loss of the last turn of HA that represent the crucial microscopic steps of the rearrangement.     

Structural reorganization triggered by charging of Lys residues in the hydrophobic interior of a protein

Structural consequences of ionization of residues buried in the hydrophobic interior of proteins were examined systematically in 25 proteins with internal Lys residues. Crystal structures showed that the ionizable groups are buried. NMR spectroscopy showed that in 2 of 25 cases studied, the ionization of an internal Lys unfolded the protein globally. In five cases, the internal charge triggered localized changes in structure and dynamics, and in three cases, it promoted partial or local unfolding. Remarkably, in 15 proteins, the ionization of the internal Lys had no detectable structural consequences. Highly stable proteins appear to be inherently capable of withstanding the presence of charge in their hydrophobic interior, without the need for specialized structural adaptations. The extent of structural reorganization paralleled loosely with global thermodynamic stability, suggesting that structure-based pK(a) calculations for buried residues could be improved by calculation of thermodynamic stability and by enhanced conformational sampling.