Optimizing the hydrogen-bond network in Poisson-Boltzmann equation-based pK(a) calculations

pK(a) calculation methods that are based on finite difference solutions to the Poisson-Boltzmann equation (FDPB) require that energy calculations be performed for a large number of different protonation states of the protein. Normally, the differences between these protonation states are modeled by changing the charges on a few atoms, sometimes the differences are modeled by adding or removing hydrogens, and in a few cases the positions of these hydrogens are optimized locally. We present an FDPB-based pK(a) calculation method in which the hydrogen-bond network is globally optimized for every single protonation state used. This global optimization gives a significant improvement in the accuracy of calculated pK(a) values, especially for buried residues. It is also shown that large errors in calculated pK(a) values are often due to structural artifacts induced by crystal packing. Optimization of the force fields and parameters used in pK(a) calculations should therefore be performed with X-ray structures that are corrected for crystal artifacts.     

Calculating pKa values in the cAMP-dependent protein kinase: the effect of conformational change and ligand binding

The conformational change observed upon ligand binding and phosphorylation for the cAMP-dependent protein kinase (protein kinase A-PKA) is of high importance for the regulation of its activity. We calculate pKa values and net charges for 18 3D structures of PKA in various conformations and liganded states to examine the role of electrostatics in ligand binding and activation. We find that the conformational change of PKA takes place without any significant net proton uptake/release at all pH values, thus indicating that PKA has evolved to reduce any pH-dependent barriers to the conformational motion. We furthermore find that the binding of ligands induces large changes in the net charge of PKA at most pH values, but significantly, we find that the net charge difference at physiological pH is close to zero, thus indicating that the active-site pKa values have been preorganized for substrate binding. We are unable to unequivocally resolve the identity of the groups responsible for determining the pH-activity profile of PKA but speculate that the titration of Lys 168 or the titration of ATP itself could be responsible for the loss of activity at high pH values. Finally, we examine the effect of point mutations on the pKa values of the PKA catalytic residues and find these to be relatively insensitive to both noncharge-altering and charge-altering mutations.     

On the evaluation and optimization of protein X-ray structures for pKa calculations

The calculation of the physical properties of a protein from its X-ray structure is of importance in virtually every aspect of modern biology. Although computational algorithms have been developed for calculating everything from the dynamics of a protein to its binding specificity, only limited information is available on the ability of these methods to give accurate results when used with a particular X-ray structure. We examine the ability of a pKa calculation algorithm to predict the proton-donating residue in the catalytic mechanism of hen egg white lysozyme. We examine the correlation between the ability of the pKa calculation method to obtain the correct result and the overall characteristics of 41 X-ray structures such as crystallization conditions, resolution, and the output of structure validation software. We furthermore examine the ability of energy minimizations (EM), molecular dynamics (MD) simulations, and structure-perturbation methods to optimize the X-ray structures such that these give correct results with the pKa calculation algorithm. We propose a set of criteria for identifying the proton donor in a catalytic mechanism, and demonstrate that the application of these criteria give highly accurate prediction results when using unmodified X-ray structures. More specifically, we are able to successfully identify the proton donor in 85% of the X-ray structures when excluding structures with crystal contacts near the active site. Neither the use of the overall characteristics of the X-ray structures nor the optimization of the structure by EM, MD, or other methods improves the results of the pKa calculation algorithm. We discuss these results and their implications for the design of structure-based energy calculation algorithms in general.     

Internal dynamics and ionization states of the macrophage migration inhibitory factor: comparison between wild-type and mutant forms

The macrophage migration inhibitory factor (MIF) is a cytokine that shares a common structural architecture and catalytic strategy with three isomerases: 4-oxalocrotonate tautomerase, 5-carboxymethyl-2-hydroxymuconate isomerase, and D-dopachrome tautomerase. A highly conserved N-terminal proline acts as a base-acid during the proton transfer reaction catalyzed by these enzymes. Such unusual catalytic strategy appears to be possible only due to the N-terminal proline pK(a) shifted to 5.0-6.0 units. Mutations of this residue result in a significant decrease of the catalytic activity of MIF. Two hypotheses have been proposed to explain the catalytic inefficiency of MIF: the lower basicity of primary amines with regard to secondary ones and the increased flexibility resulting from the replacement of a proline by residues like glycine. To investigate that, we have performed molecular dynamics simulations of MIF wild-type and its mutant P1G, as well as calculated the protonation properties of several mutant forms. It was found that the N-terminal glycine does not show larger fluctuations compared to proline, but the former residue is more exposed to the solvent throughout the simulations. The apparent pK(a) of these residues displays very little change (as expected from the structural rigidity of MIF) and is not significantly affected by the surrounding ionizable residues. Instead, the hydrophobic character of the active site seems to be the main factor in determining the pKa of the N-terminal residue and the catalytic efficiency of MIF.     

Role of beta-lactam carboxyl group on binding of penicillins and cephalosporins to class C beta-lactamases

Molecular models for the Henry Michaelis complexes of Enterobacter cloacae, a class C beta-lactamase, with penicillin G and cephalotin have been constructed by using molecular mechanic calculations, based on the AMBER force field, to examine the molecular differentiation mechanisms between cephalosporins and penicillins in beta-lactamases. Ser318Ala and Thr316Ala mutations in both complexes and Asn346Ala and Thr316Ala/Asn346Ala double mutation in penicillin G complex have also been studied. Results confirm that Thr316, Ser318, and Asn346 play a crucial role in the substrate recognition, via their interactions with one of the oxygens of the antibiotic carboxyl group. Both mutation Ser318Ala and Thr316Ala strongly affect the correct binding of cephalotin to P99, the first mainly by precluding the discriminating salt bridge between carboxyl and serine OH groups, and the second one by the Ser318, Lys315, and Tyr150 spatial rearrangements. On the other hand, Ser318Ala mutation has little effect on penicillin G binding, but the Thr316Ala/Asn346Ala double mutation causes the departure of the antibiotic from the oxyanion hole. Molecular dynamic simulations allow us to interpret the experimental results of some class C and A beta-lactamases.     

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.     

pKa calculations for class A beta-lactamases: methodological and mechanistic implications.

Beta-lactamases are responsible for resistance to penicillins and related beta-lactam compounds. Despite numerous studies, the identity of the general base involved in the acylation step is still unclear. It has been proposed, on the basis of a previous pKa calculation and analysis of structural data, that the unprotonated Lys73 in the active site could act as the general base. Using a continuum electrostatic model with an improved treatment of the multiple titration site problem, we calculated the pKa values of all titratable residues in the substrate-free TEM-1 and Bacillus licheniformis class A beta-lactamases. The pKa of Lys73 in both enzymes was computed to be above 10, in good agreement with recent experimental data on the TEM-1 beta-lactamase, but inconsistent with the proposal that Lys73 acts as the general base. Even when the closest titratable residue, Glu166, is mutated to a neutral residue, the predicted downward shift of the pKa of Lys73 shows that it is unlikely to act as a proton abstractor in either enzyme. These results support a mechanism in which the proton of the active Ser70 is transferred to the carboxylate group of Glu166.     

pKa predictions for proteins,RNAs, and DNAs with the Gaussian dielectric function using DelPhi pKa

We developed a Poisson‐Boltzmann based approach to calculate the values of protein ionizable residues (Glu, Asp, His, Lys and Arg), nucleotides of RNA and single stranded DNA. Two novel features were utilized: the dielectric properties of the macromolecules and water phase were modeled via the smooth Gaussian‐based dielectric function in DelPhi and the corresponding electrostatic energies were calculated without defining the molecular surface. We tested the algorithm by calculating values for more than 300 residues from 32 proteins from the PPD dataset and achieved an overall RMSD of 0.77. Particularly, the RMSD of 0.55 was achieved for surface residues, while the RMSD of 1.1 for buried residues. The approach was also found capable of capturing the large shifts of various single point mutations in staphylococcal nuclease (SNase) from ‐cooperative dataset, resulting in an overall RMSD of 1.6 for this set of pKa's. Investigations showed that predictions for most of buried mutant residues of SNase could be improved by using higher dielectric constant values. Furthermore, an option to generate different hydrogen positions also improves predictions for buried carboxyl residues. Finally, the calculations on two RNAs demonstrated the capability of this approach for other types of biomolecules. Proteins 2015; 83:2186–2197. © 2015 Wiley Periodicals, Inc.     

Simplified methods for pKa and acid pH-dependent stability estimation in proteins: removing dielectric and counterion boundaries

Much computational research aimed at understanding ionizable group interactions in proteins has focused on numerical solutions of the Poisson-Boltzmann (PB) equation, incorporating protein exclusion zones for solvent and counterions in a continuum model. Poor agreement with measured pKas and pH-dependent stabilities for a (protein, solvent) relative dielectric boundary of (4,80) has lead to the adoption of an intermediate (20,80) boundary. It is now shown that a simple Debye-Huckel (DH) calculation, removing both the low dielectric and counterion exclusion regions associated with protein, is equally effective in general pKa calculations. However, a broad-based discrepancy to measured pH-dependent stabilities is maintained in the absence of ionizable group interactions in the unfolded state. A simple model is introduced for these interactions, with a significantly improved match to experiment that suggests a potential utility in predicting and analyzing the acid pH-dependence of protein stability. The methods are applied to the relative pH-dependent stabilities of the pore-forming domains of colicins A and N. The results relate generally to the well-known preponderance of surface ionizable groups with solvent-mediated interactions. Although numerical PB solutions do not currently have a significant advantage for overall pKa estimations, development based on consideration of microscopic solvation energetics in tandem with the continuum model could combine the large deltapKas of a subset of ionizable groups with the overall robustness of the DH model.     

Effect of nucleotide substrate binding on the pKa of catalytic residues in barnase

A thermodynamic cycle is used to describe barnase catalysis, which considers explicitly the presence of different ionic states of the catalytic residues Glu-73 and His-102 in barnase during the enzyme-substrate recognition process. Reinterpretation of published experimental data using rate equations derived from this cycle provides estimates of the ionization constants of these catalytic side chains, in the free enzyme and in the barnase-GpA complex. In addition, the electrostatic properties of the barnase-d(CGAC) crystal complex and of a barnase-5′3′(AAGAAp)-O-methyl ester modeled complex are investigated by means of a continuum approach to account for solvent polarization effects. Taking GpA as a reference substrate, it is shown that Increasing the length of the bound nucleotide induces pK_a shifts in the catalytic side chains, which modulate the fraction of enzyme in the correct ionic form for achieving the transesterification reaction. The computed results are in good agreement with the experimental variation of the optimum pH of barnase activity. The present analysis underscores the influence of pH effects on the k_cat and K_M kinetic constants of barnase and provides the basic formalism for linking the effective kinetic parameters, which usually depend on the pH, to the theoretical estimates of the true kinetic constants. © 1996 Wiley-Liss, Inc.     

Molecular dynamics study of the stabilities of consensus designed ankyrin repeat proteins

Two designed ankyrin repeat (AR) proteins (E3_5 and E3_19) are high homologous (with about 87% sequence identity) and their crystal structures have a Calpha atom-positional root-mean-square difference of about 0.14 nm. However, it was found that E3_5 is considerably more stable than E3_19 in guanidinium hydrochloride and thermal denaturation experiments. With the goal of providing insights into the various factors contributing to the stabilities of the designed AR proteins and suggesting possible mutations to enhance their stabilities, homology modeling and molecular dynamics (MD) simulations with explicit solvent have been performed. Because the crystal structure of E3_19 was solved later than that of E3_5, a homology model of E3_19 based on the crystal structure of E3_5 was also used in the simulations. E3_5 shows a very stable trajectory in both crystal and solution simulations. In contrast, the C-terminal repeat of E3_19 unfolds in the simulations starting from either the modeled structure or the crystal structure, although it has a sequence identical to that of E3_5. A continuum electrostatic model was used to estimate the effect of single mutations on protein stability and to study the interaction between the internal ARs and the C-terminal capping AR. Mutations involving charged residues were found to have large effects on stability. Due to the difference in charge distribution in the internal ARs of E3_19 and E3_5, their interaction with the C-terminal capping AR is less favorable in E3_19. The simulation trajectories suggest that the stability of the designed AR proteins can be increased by optimizing the electrostatic interactions within and between the different repeats.     

Free energy calculations for theophylline binding to an RNA aptamer: Comparison of MM-PBSA and thermodynamic integration methods

We have applied the molecular mechanics Poisson-Boltzmann surface area (MM-PBSA) method (J. Srinivasan, T. E. Cheatham, P. Cieplak, P. A. Kollman, and D. A. Case, Journal of the American Chemical Society, 1998, Vol. 120, pp. 9401-9409) to study the interaction of an RNA aptamer with theophylline and its analogs. The MM-PBSA free energy analysis provides a reasonable absolute binding free energy for the RNA aptamer-theophylline complex formation. Energetic analysis reveals that the van der Waals interaction and the nonpolar contribution to solvation provide the basis for the favorable absolute free energy of complex. This trend is similar to other protein-ligand interactions studied previously. The MM-PBSA method also ranks the relative binding energies of five theophylline analogs approximately correctly, but not as well as the more conventional thermodynamic integration calculations, which were carried out to convert theophylline into its analogs. The comparison of MM-PBSA with TI suggests that the MM-PBSA method has some difficulties with the first-solvation-shell energetics.     

Free energy calculations on binding and catalysis by alpha-lytic protease: the role of substrate size in the P1 pocket

We present free energy calculations using molecular dynamics on different substrates of alpha-lytic protease in the gas phase, in solution, while forming a noncovalent Michaelis complex with the enzyme, and in a tetrahedral structure representing a transition state/intermediate for acylation by the enzyme. Various P1 substrates were studied, with P1 = Gly, Ala, Val, and Leu. In qualitative agreement with experiment, the enzyme was calculated to bind and catalyze most effectively substrates with P1 = Ala over those with P1 = Gly, Val or Leu. Also, the calculated relative solvation free energies of Gly----Ala and Ala----Val were in qualitative agreement with experimental values in corresponding model systems. However, the level of quantitative agreement with experiment achieved in our earlier study of relative binding and catalysis of native subtilisin and an Asn-155----Ala mutant was not achieved. We surmise that this is due to the greater difficulty in quantitatively simulating effects that are predominantly van der Waals and hydrophobic compared to those that are hydrogen bonding/electrostatic.     

A helix-turn-strand structural motif common in alpha-beta proteins.

By exhaustive structural comparisons, we have found that about one-third of the alpha-helix-turn-beta-strand polypeptides in alpha-beta barrel domains share a common structural motif. The chief characteristics of this motif are that first, the geometry of the turn between the alpha-helix and the beta-strand is somewhat constrained, and second, the beta-strand contains a hydrophobic patch that fits into a hydrophobic pocket on the alpha-helix. The geometry of the turn does not seem to be a major determinant of the alpha-beta unit, because the turns vary in length from four to six residues. However, the motif does not occur when there are few constraints on the geometry of the turn-for instance, when the turns between the alpha-helix and the beta-strands are very long. It also occurs much less frequently in flat-sheet alpha-beta proteins, where the topology is much less regular and the amount of twist on the sheet varies considerably more than in the barrel proteins. The motif may be one of the basic building blocks from which alpha-beta barrels are constructed.     

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.     

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.     

Hydrogen bonding is the prime determinant of carboxyl pKa values at the N-termini of alpha-helices

Experimentally determined mean pK(a) values of carboxyl residues located at the N-termini of alpha-helices are lower than their overall mean values. Here, we perform three types of analyses to account for this phenomenon. We estimate the magnitude of the helix macrodipole to determine its potential role in lowering carboxyl pK(a) values at the N-termini. No correlation between the magnitude of the macrodipole and the pK(a) values is observed. Using the pK(a) program propKa we compare the molecular surroundings of 18 N-termini carboxyl residues versus 233 protein carboxyl groups from a previously studied database. Although pK(a) lowering interactions at the N-termini are similar in nature to those encountered in other protein regions, pK(a) lowering backbone and side-chain hydrogen bonds appear in greater number at the N-termini. For both Asp and Glu, there are about 0.5 more hydrogen bonds per residue at the N-termini than in other protein regions, which can be used to explain their lower than average pK(a) values. Using a QM-based pK(a) prediction model, we investigate the chemical environment of the two lowest Asp and the two lowest Glu pK(a) values at the N-termini so as to quantify the effect of various pK(a) determinants. We show that local interactions suffice to account for the acidity of carboxyl residues at the N-termini. The effect of the helix dipole on carboxyl pK(a) values, if any, is marginal. Backbone amide hydrogen bonds constitute the single biggest contributor to the lowest carboxyl pK(a) values at the N-termini. Their estimated pK(a) lowering effects range from about 1.0 to 1.9 pK(a) units.     

DAPI binding to the DNA minor groove: a continuum solvent analysis

A continuum solvent model based on the generalized Born (GB) or finite-difference Poisson-Boltzmann (FDPB) approaches has been employed to compare the binding of 4'-6-diamidine-2-phenyl indole (DAPI) to the minor groove of various DNA sequences. Qualitative agreement between the results of GB and FDPB approaches as well as between calculated and experimentally observed trends regarding the sequence specificity of DAPI binding to B-DNA was obtained. Calculated binding energies were decomposed into various contributions to solvation and DNA-ligand interaction. DNA conformational adaptation was found to make a favorable contribution to the calculated total interaction energy but did not change the DAPI binding affinity ranking of different DNA sequences. The calculations indicate that closed complex formation is mainly driven by nonpolar contributions and was found to be disfavored electrostatically due to a desolvation penalty that outbalances the attractive Coulomb interaction. The calculated penalty was larger for DAPI binding to GC-rich sequences compared with AT-rich target sequences and generally larger for the FDPB vs the GB continuum model. A radial interaction profile for DAPI at different distances from the DNA minor groove revealed an electrostatic energy minimum a few Angstroms farther away from the closed binding geometry. The calculated electrostatic interaction up to this distance is attractive and it may stabilize a nonspecific binding arrangement.     

Putative implication of alpha-amylase loop 7 in the mechanism of substrate binding and reaction products release

Alpha-amylases are widespread endo-enzymes involved in the hydrolysis of internal alpha-(1,4) glycosidic linkages of starch polymers. Molecular modeling of amylose-amylase interactions is a step toward enzymatic mechanism understanding and rational design of new enzymes. From the crystallographic complex of barley alpha-amylase AMY2-acarbose, the static aspects of amylose-amylase docking have been characterized with a model of maltododecaose (DP12) (G. André, A. Buléon, R. Haser, and V. Tran, Biopolymers 1999, Vol. 50, pp. 751-762; G. André and V. Tran, Special Publication no. 246 1999, The Royal Society of Chemistry, H. J. Gilbert, G. J. Davies, B. Henrissat, and B. Svensson, Eds., Cambridge, pp. 165-174). These studies, consistent with the experimental subsite mapping (K. Bak-Jensen, G. André, V. Tran, and B. Svensson, Journal of Biological Chemistry, to be published), propose a propagation scheme for an amylose chain in the active cleft of AMY2. The topographical overview of alpha-amylases identified loop 7 as a conserved segment flanking the active site. Since some crystallographic experiments suspected its high flexibility, its putative motion was explored through a robotic scheme, an alternate route to dynamics simulations that consume CPU time. The present article describes the characteristics of the flexibility of loop 7: location and motion in AMY2. A back-and-forth motion with a large amplitude of more than 0.6 nm was evaluated. This movement could be triggered by two hinge residues. It results in the loop flipping over the active site to enhance the docking of the native helical substrate through specific interactions, it positions the catalytic residues, it distorts the substrate towards its transition state geometry, and finally monitors the release of the products after hydrolysis. The residues involved in the process are now rational mutation points in the hands of molecular biologists.