Including Side Chain Flexibility in Continuum Electrostatic Calculations of Protein Titration

The electrostatic potential inside and around α-lactalbumin: Fluctuations and mean-field models

The electrostatic potential (EP) generated by the protein α-lactoalbumin in the presence of added salt is computed as a thermal average at a given point in space. With this aim, constant pH Monte Carlo (MC) simulations are performed within the primitive model, namely, the solvent is treated as a continuum dielectric. The study of the thermal and spatial fluctuations of the EP reveals that they are remarkably high inside the protein. The calculations indicate that fluctuations inside the protein are mainly due to the asymmetric distribution of the charge groups, while the charge fluctuations of the titratable groups play a minor role. The computed EP matches very well with the one obtained from the Poisson equation for the average charge density in spherical symmetry. The Tanford–Kirkwood multipole expansion reproduces the simulated angular-averaged potential rather accurately. Surprisingly, two of the simplest mean-field models, the linear Poisson–Boltzmann (PB) equation and Donnan potential, provide good estimations of the average EP in the effective protein surface (surface EP). The linear PB equation predicts a linear relationship between charge and surface EP, which is numerically reproduced only if the small ions within the protein are taken into account. On the other hand, the partition coefficients of the small ions inside and outside the protein predicted by Donnan theory reproduce reasonably well the simulation results.

A Fast and Interpretable Deep Learning Approach for Accurate Electrostatics-Driven pKa Predictions in Proteins

Existing computational methods for estimating pK_a values in proteins rely on theoretical approximations and lengthy computations. In this work, we use a data set of 6 million theoretically determined pK_a shifts to train deep learning models, which are shown to rival the physics-based predictors. These neural networks managed to infer the electrostatic contributions of different chemical groups and learned the importance of solvent exposure and close interactions, including hydrogen bonds. Although trained only using theoretical data, our pKAI+ model displayed the best accuracy in a test set of ∼750 experimental values. Inference times allow speedups of more than 1000× compared to physics-based methods. By combining speed, accuracy, and a reasonable understanding of the underlying physics, our models provide a game-changing solution for fast estimations of macroscopic pK_a values from ensembles of microscopic values as well as for many downstream applications such as molecular docking and constant-pH molecular dynamics simulations.

Prediction of protein pK a with representation learning

The behavior of proteins is closely related to the protonation states of the residues. Therefore, prediction and measurement of pK a are essential to understand the basic functions of proteins. In this work, we develop a new empirical scheme for protein pK a prediction that is based on deep representation learning. It combines machine learning with atomic environment vector (AEV) and learned quantum mechanical representation from ANI-2x neural network potential (J. Chem. Theory Comput. 2020, 16, 4192). The scheme requires only the coordinate information of a protein as the input and separately estimates the pK a for all five titratable amino acid types. The accuracy of the approach was analyzed with both cross-validation and an external test set of proteins. Obtained results were compared with the widely used empirical approach PROPKA. The new empirical model provides accuracy with MAEs below 0.5 for all amino acid types. It surpasses the accuracy of PROPKA and performs significantly better than the null model. Our model is also sensitive to the local conformational changes and molecular interactions.

Hybrid MC/MD for protein design

Computational protein design relies on simulations of a protein structure, where selected amino acids can mutate randomly, and mutations are selected to enhance a target property, such as stability. Often, the protein backbone is held fixed and its degrees of freedom are modeled implicitly to reduce the complexity of the conformational space. We present a hybrid method where short molecular dynamics (MD) segments are used to explore conformations and alternate with Monte Carlo (MC) moves that apply mutations to side chains. The backbone is fully flexible during MD. As a test, we computed side chain acid/base constants or pK_a's in five proteins. This problem can be considered a special case of protein design, with protonation/deprotonation playing the role of mutations. The solvent was modeled as a dielectric continuum. Due to cost, in each protein we allowed just one side chain position to change its protonation state and the other position to change its type or mutate. The pK_a's were computed with a standard method that scans a range of pH values and with a new method that uses adaptive landscape flattening (ALF) to sample all protonation states in a single simulation. The hybrid method gave notably better accuracy than standard, fixed-backbone MC. ALF decreased the computational cost a factor of 13.

Assessing the Conformational Equilibrium of Carboxylic Acid via Quantum Mechanical and Molecular Dynamics Studies on Acetic Acid

Accurate hydrogen placement in molecular modeling is crucial for studying the interactions and dynamics of biomolecular systems. The carboxyl functional group is a prototypical example of a functional group that requires protonation during structure preparation. To our knowledge, when in their neutral form, carboxylic acids are typically protonated in the syn conformation by default in classical molecular modeling packages, with no consideration of alternative conformations, though we are not aware of any careful examination of this topic. Here, we investigate the general belief that carboxylic acids should always be protonated in the syn conformation. We calculate and compare the relative energetic stabilities of syn and anti acetic acid using ab initio quantum mechanical calculations and atomistic molecular dynamics simulations. We focus on the carboxyl torsional potential and configurations of microhydrated acetic acid from molecular dynamics simulations, probing the effects of solvent, force field (GAFF vs GAFF2), and partial charge assignment of acetic acid. We show that while the syn conformation is the preferred state, the anti state may in some cases also be present under normal NPT conditions in solution.

PKA17-A Coarse-Grain Grid-Based Methodology and Web-Based Software for Predicting Protein pK a Shifts

We have developed and tested PKA17, a coarse-grain grid-based model for predicting protein pK _a shifts. Our pK _a predictor is currently deployed via a website interface. We have carried out parameter fitting using 442 Asp, Glu, His, Lys, and Arg residues for which experimental results are available in the literature. PROPKA software has been used for benchmarking. The average unsigned error and root-mean-square deviation (RMSD) have been found to be 0.628 and 0.831 pH units, respectively, for PKA17. The corresponding results with PROPKA are 0.761 and 1.063 units. We have assessed the robustness of the developed PKA17 methodology with a number of tests and have also explored the possibility of using a combination of PROPKA and PKA17 calculations in order to improve the accuracy of predicted pK _a values for protein residues. We have also once again confirmed that protein acidity constants are influenced almost entirely by residues in the immediate spatial proximity of the ionizable amino acids. The resulting PKA17 software has been deployed online with a web-based interface at http://users.wpi.edu/~jpcvitkovic/pka_calc.html. © 2019 Wiley Periodicals, Inc.

Light-Responsive Size of Self-Assembled Spiropyran-Lysozyme Nanoparticles with Enzymatic Function

Novel light-responsive nanoassemblies with switchable size and enzymatic activity are built from a protein and a water-soluble spiropyran. Assemblies are created by electrostatic self-assembly in aqueous solution such that the photochromic property of the spiropyran enables light responsiveness. Upon visible light exposure, the aggregate size increases from 200 to 400 nm. The enzyme retains its activity upon aggregation into the assembly, while it decreases through visible light irradiation. Fundamentally, we show how the two different spiropyran isomers, the open-ring merocyanine form and the closed-ring spiropyran form, bind differently to the protein, which triggers the assembly size and use of thermodynamic data to understand the binding process and the size response. Thus, as a proof of concept, a self-assembly driven light-tunable enzyme activity in conjunction with a triggerable assembly size is demonstrated for a model system. The concept bears future potential for various possible biological applications ranging from genetic control over vaccine applications to the detection of certain proteins.

DelPhiPKa: Including salt in the calculations and enabling polar residues to titrate

DelPhiPKa is a widely used and unique approach to compute pK_a 's of ionizable groups that does not require molecular surface to be defined. Instead, it uses smooth Gaussian-based dielectric function to treat computational space via Poisson-Boltzmann equation (PBE). Here, we report an expansion of DelPhiPKa functionality to enable inclusion of salt in the modeling protocol. The method considers the salt mobile ions in solvent phase without defining solute-solvent boundary. Instead, the ions are penalized to enter solute interior via a desolvation penalty term in the Boltzmann factor in the framework of PBE. Hence, the concentration of ions near the protein is balanced by the desolvation penalty and electrostatic interactions. The study reveals that correlation between experimental and calculated pK_a 's is improved significantly by taking into consideration the presence of salt. Furthermore, it is demonstrated that DelphiPKa reproduces the salt sensitivity of experimentally measured pK_a 's. Another new development of DelPhiPKa allows for computing the pK_a 's of polar residues such as cysteine, serine, threonine and tyrosine. With this regard, DelPhiPKa is benchmarked against experimentally measured cysteine and tyrosine pK_a 's and for cysteine it is shown to outperform other existing methods (DelPhiPKa RMSD of 1.73 vs RMSD between 2.40 and 4.72 obtained by other existing pK_a prediction methods).

Comparing pairwise-additive and many-body generalized Born models for acid/base calculations and protein design

Generalized Born (GB) solvent models are common in acid/base calculations and protein design. With GB, the interaction between a pair of solute atoms depends on the shape of the protein/solvent boundary and, therefore, the positions of all solute atoms, so that GB is a many-body potential. For compute-intensive applications, the model is often simplified further, by introducing a mean, native-like protein/solvent boundary, which removes the many-body property. We investigate a method for both acid/base calculations and protein design that uses Monte Carlo simulations in which side chains can explore rotamers, bind/release protons, or mutate. The fluctuating protein/solvent dielectric boundary is treated in a way that is numerically exact (within the GB framework), in contrast to a mean boundary. Its originality is that it captures the many-body character while retaining the residue-pairwise complexity given by a fixed boundary. The method is implemented in the Proteus protein design software. It yields a slight but systematic improvement for acid/base constants in nine proteins and a significant improvement for the computational design of three PDZ domains. It eliminates a source of model uncertainty, which will facilitate the analysis of other model limitations. © 2017 Wiley Periodicals, Inc.

Intrinsic Relative Stabilities of the Neutral Tautomers of Arginine Side-Chain Models

The specific protonation state of amino acids is crucial for the physicochemical properties of proteins and their biological functions. These protonation states influence, for instance, properties related to hydrogen bonding, solubility, and folding. pKa calculations for proteins are, therefore, important and require, in principle, a specification of the most stable protonated and deprotonated forms of each titratable group. This is complicated by the existence of multiple tautomers, like the five neutral tautomers of the guanidine moiety in arginine. In this study, the compounds N-methyl-guanidine and N-ethyl-guanidine were used to model the charged and all neutral protonation states of the arginine side chain. The relative stabilities of all five neutral tautomers were investigated systematically for the first time, using quantum-mechanical calculations. These relative stabilities were obtained in vacuo, water and chloroform, by combining the quantum-mechanical calculations with a continuum solvation model. The water model was used to represent arginines exposed to an aqueous solution, whereas the chloroform model has a polarity representative of a protein core or a membrane. This allowed determining the relative pKa's associated with each neutral tautomer in these environments. A key result is that significant differences in stability are found between the neutral tautomers, in both water and chloroform. Some tautomers are consistently found to be the most stable. These findings will be helpful to refine pKa calculations in proteins.

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 pKa 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 pKa 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 pKa shifts of various single point mutations in staphylococcal nuclease (SNase) from pKa-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 pKa predictions for buried carboxyl residues. Finally, the pKa calculations on two RNAs demonstrated the capability of this approach for other types of biomolecules.

Monte Carlo simulations of proteins at constant pH with generalized Born solvent, flexible sidechains, and an effective dielectric boundary

Titratable residues determine the acid/base behavior of proteins, strongly influencing their function; in addition, proton binding is a valuable reporter on electrostatic interactions. We describe a method for pK(a) calculations, using constant-pH Monte Carlo (MC) simulations to explore the space of sidechain conformations and protonation states, with an efficient and accurate generalized Born model (GB) for the solvent effects. To overcome the many-body dependency of the GB model, we use a "Native Environment" approximation, whose accuracy is shown to be good. It allows the precalculation and storage of interactions between all sidechain pairs, a strategy borrowed from computational protein design, which makes the MC simulations themselves very fast. The method is tested for 12 proteins and 167 titratable sidechains. It gives an rms error of 1.1 pH units, similar to the trivial "Null" model. The only adjustable parameter is the protein dielectric constant. The best accuracy is achieved for values between 4 and 8, a range that is physically plausible for a protein interior. For sidechains with large pKa shifts, ≥2, the rms error is 1.6, compared to 2.5 with the Null model and 1.5 with the empirical PROPKA method.

Rapid calculation of protein pKa values using Rosetta

We developed a Rosetta-based Monte Carlo method to calculate the pK(a) values of protein residues that commonly exhibit variable protonation states (Asp, Glu, Lys, His, and Tyr). We tested the technique by calculating pK(a) values for 264 residues from 34 proteins. The standard Rosetta score function, which is independent of any environmental conditions, failed to capture pK(a) shifts. After incorporating a Coulomb electrostatic potential and optimizing the solvation reference energies for pK(a) calculations, we employed a method that allowed side-chain flexibility and achieved a root mean-square deviation (RMSD) of 0.83 from experimental values (0.68 after discounting 11 predictions with an error over 2 pH units). Additional degrees of side-chain conformational freedom for the proximal residues facilitated the capture of charge-charge interactions in a few cases, resulting in an overall RMSD of 0.85 pH units. The addition of backbone flexibility increased the overall RMSD to 0.93 pH units but improved relative pK(a) predictions for proximal catalytic residues. The method also captures large pK(a) shifts of lysine and some glutamate point mutations in staphylococcal nuclease. Thus, a simple and fast method based on the Rosetta score function and limited conformational sampling produces pK(a) values that will be useful when rapid estimation is essential, such as in docking, design, and folding.

Progress in the prediction of pKa values in proteins

The pK(a) -cooperative aims to provide a forum for experimental and theoretical researchers interested in protein pK(a) values and protein electrostatics in general. The first round of the pK(a) -cooperative, which challenged computational labs to carry out blind predictions against pK(a) s experimentally determined in the laboratory of Bertrand Garcia-Moreno, was completed and results discussed at the Telluride meeting (July 6-10, 2009). This article serves as an introduction to the reports submitted by the blind prediction participants that will be published in a special issue of PROTEINS: Structure, Function and Bioinformatics. Here, we briefly outline existing approaches for pK(a) calculations, emphasizing methods that were used by the participants in calculating the blind pK(a) values in the first round of the cooperative. We then point out some of the difficulties encountered by the participating groups in making their blind predictions, and finally try to provide some insights for future developments aimed at improving the accuracy of pK(a) calculations.

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.

Histidine in continuum electrostatics protonation state calculations

A modification to the standard continuum electrostatics approach to calculate protein pK(a)s, which allows for the decoupling of histidine tautomers within a two-state model, is presented. Histidine with four intrinsically coupled protonation states cannot be easily incorporated into a two-state formalism, because the interaction between the two protonatable sites of the imidazole ring is not purely electrostatic. The presented treatment, based on a single approximation of the interrelation between histidine's charge states, allows for a natural separation of the two protonatable sites associated with the imidazole ring as well as the inclusion of all protonation states within the calculation.

Empirical prediction of protein pKa values with residue mutation

A fast, empirical method, Mut-pKa, is presented for predicting the pKa values of ionizable residues in proteins based on mutation. The method compares the effect of mutating each residue that may act as a hydrogen bond donor or acceptor for the ionizable residue. The energetic effect of each type of mutation, along with a desolvation measure and the overall background charge, is fit against pKa data for histidine and carboxyl residues. A total of 214 residues from 35 different proteins were used in the dataset. Using 11 parameters for each type of ionizable residue, a root mean squared error (RMSE) of 0.78 and 1.12 pH units were obtained for carboxyl and histidines residues, respectively, using leave one out cross validation (LOOCV). The results were particularly promising for buried residues, which had RMSE values of 0.99 and 1.13 for carboxyl and histidine residues, respectively. A number of desolvation measures were tested. The simplest measure, the number of atoms surrounding the residue, was found to work best. The effect of using dynamics was also studied using short molecular dynamics runs, followed by minimization of the structures. Mut-pKa has significantly fewer parameters than, but similar performance to, other empirical methods. Because of this and the LOOCV results, we believe the model is robust and that overfitting is not a problem.

Coupling Constant pH Molecular Dynamics with Accelerated Molecular Dynamics

An extension of the constant pH method originally implemented by Mongan et al. (J. Comput. Chem.2004, 25, 2038−2048) is proposed in this study. This adapted version of the method couples the constant pH methodology with the enhanced sampling technique of accelerated molecular dynamics, in an attempt to overcome the sampling issues encountered with current standard constant pH molecular dynamics methods. Although good results were reported by Mongan et al. on application of the standard method to the hen egg-white lysozyme (HEWL) system, residues which possess strong interactions with neighboring groups tend to converge slowly, resulting in the reported inconsistencies for predicted pK_a values, as highlighted by the authors. The application of the coupled method described in this study to the HEWL system displays improvements over the standard version of the method, with the improved sampling leading to faster convergence and producing pK_a values in closer agreement to those obtained experimentally for the more slowly converging residues.

MCCE2: improving protein pKa calculations with extensive side chain rotamer sampling

Multiconformation continuum electrostatics (MCCE) explores different conformational degrees of freedom in Monte Carlo calculations of protein residue and ligand pK(a)s. Explicit changes in side chain conformations throughout a titration create a position dependent, heterogeneous dielectric response giving a more accurate picture of coupled ionization and position changes. The MCCE2 methods for choosing a group of input heavy atom and proton positions are described. The pK(a)s calculated with different isosteric conformers, heavy atom rotamers and proton positions, with different degrees of optimization are tested against a curated group of 305 experimental pK(a)s in 33 proteins. QUICK calculations, with rotation around Asn and Gln termini, sampling His tautomers and torsion minimum hydroxyls yield an RMSD of 1.34 with 84% of the errors being <1.5 pH units. FULL calculations adding heavy atom rotamers and side chain optimization yield an RMSD of 0.90 with 90% of the errors <1.5 pH unit. Good results are also found for pK(a)s in the membrane protein bacteriorhodopsin. The inclusion of extra side chain positions distorts the dielectric boundary and also biases the calculated pK(a)s by creating more neutral than ionized conformers. Methods for correcting these errors are introduced. Calculations are compared with multiple X-ray and NMR derived structures in 36 soluble proteins. Calculations with X-ray structures give significantly better pK(a)s. Results with the default protein dielectric constant of 4 are as good as those using a value of 8. The MCCE2 program can be downloaded from http://www.sci.ccny.cuny.edu/~mcce.

Dielectric relaxation of cytochrome c oxidase: Comparison of the microscopic and continuum models

We have studied a charge-insertion process that models the deprotonation of a histidine side chain in the active site of cytochrome c oxidase (CcO) using both the continuum electrostatic calculations and the microscopic simulations. The group of interest is a ligand to Cu(B) center of CcO, which has been previously suggested to play the role of the proton pumping element in the enzyme; the group is located near a large internal water cavity in the protein. Using the nonpolarizable Amber-99 force field in molecular dynamics (MD) simulations, we have calculated the nuclear part of the reaction-field energy of charging of the His group and combined it with the electronic part, which we estimated in terms of the electronic continuum (EC) model, to obtain the total reaction-field energy of charging. The total free energy obtained in this MDEC approach was then compared with that calculated using pure continuum electrostatic model with variable dielectric parameters. The dielectric constant for the "dry" protein and that of the internal water cavity of CcO were determined as those parameters that provide best agreement between the continuum and microscopic MDEC model. The nuclear (MD) polarization alone (without electronic part) of a dry protein was found to correspond to an unphysically low dielectric constant of only about 1.3, whereas the inclusion of electronic polarizability increases the protein dielectric constant to 2.6-2.8. A detailed analysis is presented as to how the protein structure should be selected for the continuum calculations, as well as which probe and atomic radii should be used for cavity definition. The dielectric constant of the internal water cavity was found to be 80 or even higher using "standard" parameters of water probe radius, 1.4 A, and protein atomic radii from the MD force field for cavity description; such high values are ascribed to the fact that the standard procedure produces unphysically small cavities. Using x-ray data for internal water in CcO, we have explored optimization of the parameters and the algorithm of cavity description. For Amber radii, the optimal probe size was found to be 1.25 A; the dielectric of water cavity in this case is in the range of 10-16. The most satisfactory cavity description, however, was achieved with ProtOr atomic radii, while keeping the probe radius to be standard 1.4 A. In this case, the value of cavity dielectric constant was found to be in the range of 3-6. The obtained results are discussed in the context of recent calculations and experimental measurements of dielectric properties of proteins.

Defining the role of salt bridges in protein stability

Although the energetic balance of forces stabilizing proteins has been established qualitatively over the last decades, quantification of the energetic contribution of particular interactions still poses serious problems. The reasons are the strong cooperativity and the interdependence ofnoncovalent interactions. Salt bridges are a typical example. One expects that ionizable side chains frequently form ion pairs in innumerable crystal structures. Since electrostatic attraction between opposite charges is strong per se, salt bridges can intuitively be regarded as an important factor stabilizing the native structure. Is that really so? In this chapter we critically reassess the available methods to delineate the role ofelectrostatic interactions and salt bridges to protein stability, and discuss the progress and the obstacles in this endeavor. The basic problem is that formation of salt bridges depends on the ionization properties of the participating groups, which is significantly influenced by the protein environment. Furthermore, salt bridges experience thermal fluctuations, continuously break and re-form, and their lifespan in solution is governed by the flexibility of the protein. Finally, electrostatic interactions are long-range and might be significant in the unfolded state, thus seriously influencing the energetic profile. Elimination of salt bridges by protonation/deprotonation at extreme pH or by mutation provides only rough energetic estimates, since there is no way to account for the nonadditive response of the protein moiety. From what we know so far, the strength of electrostatic interactions is strongly context-dependent, yet it is unlikely that salt bridges are dominant factors governing protein stability. Nevertheless, proteins from thermophiles and hyperthermophiles exhibit more, and frequently networked, salt bridges than proteins from the mesophilic counterparts. Increasing the thermal (not the thermodynamic) stability of proteins by optimization of charge-charge interactions is a good example for an evolutionary solution utilizing physical factors.

Improved pK(a) prediction: combining empirical and semimicroscopic methods

Using three different methods we tried to compute 171 experimentally known pK(a) values of ionizable residues from 15 different proteins and compared the accuracies of computed pK(a) values in terms of the root mean square deviation (RMSD) from experiment. One method is based on a continuum electrostatic model of the protein including conformational flexibility (KBPLUS). The others are empirical approaches with PROPKA deploying physically motivated energy terms with adjustable parameters and PKAcal using an empirical function with no physical basis. PROPKA reproduced the pK(a) values with highest overall accuracy. Differentiating the data set into weakly and strongly shifted experimental pK(a) values, however, we found that PROPKA's accuracy is better if the pK(a) values are weakly shifted but on equal footing with that of KBPLUS for more strongly shifted values. On the other hand, PKAcal reproduces strongly shifted pK(a) values badly but weakly shifted values with the same accuracy as PROPKA. We tested different consensus approaches combining data from all three methods to find a general procedure for most accurate pK(a) predictions. In most of the cases we found that the consensus approach reproduced experimental data with better accuracy than any of the individual methods alone.

Modification of quinone electrochemistry by the proteins in the biological electron transfer chains: examples from photosynthetic reaction centers

Quinones such as ubiquinone are the lipid soluble electron and proton carriers in the membranes of mitochondria, chloroplasts and oxygenic bacteria. Quinones undergo controlled redox reactions bound to specific sites in integral membrane proteins such as the cytochrome bc(1) oxidoreductase. The quinone reactions in bacterial photosynthesis are amongst the best characterized, presenting a model to understand how proteins modulate cofactor chemistry. The free energy of ubiquinone redox reactions in aqueous solution and in the Q(A) and Q(B) sites of the bacterial photosynthetic reaction centers (RCs) are compared. In the primary Q(A) site ubiquinone is reduced only to the anionic semiquinone (Q(*-)) while in the secondary Q(B) site the product is the doubly reduced, doubly protonated quinol (QH(2)). The ways in which the protein modifies the relative energy of each reduced and protonated intermediate are described. For example, the protein stabilizes Q(*-) while destabilizing Q(=) relative to aqueous solution through electrostatic interactions. In addition, kinetic and thermodynamic mechanisms for stabilizing the intermediate semiquinones are compared. Evidence for the protein sequestering anionic compounds by slowing both on and off rates as well as by binding the anion more tightly is reviewed.

A fast and accurate computational approach to protein ionization

We report a very fast and accurate physics-based method to calculate pH-dependent electrostatic effects in protein molecules and to predict the pK values of individual sites of titration. In addition, a CHARMm-based algorithm is included to construct and refine the spatial coordinates of all hydrogen atoms at a given pH. The present method combines electrostatic energy calculations based on the Generalized Born approximation with an iterative mobile clustering approach to calculate the equilibria of proton binding to multiple titration sites in protein molecules. The use of the GBIM (Generalized Born with Implicit Membrane) CHARMm module makes it possible to model not only water-soluble proteins but membrane proteins as well. The method includes a novel algorithm for preliminary refinement of hydrogen coordinates. Another difference from existing approaches is that, instead of monopeptides, a set of relaxed pentapeptide structures are used as model compounds. Tests on a set of 24 proteins demonstrate the high accuracy of the method. On average, the RMSD between predicted and experimental pK values is close to 0.5 pK units on this data set, and the accuracy is achieved at very low computational cost. The pH-dependent assignment of hydrogen atoms also shows very good agreement with protonation states and hydrogen-bond network observed in neutron-diffraction structures. The method is implemented as a computational protocol in Accelrys Discovery Studio and provides a fast and easy way to study the effect of pH on many important mechanisms such as enzyme catalysis, ligand binding, protein-protein interactions, and protein stability.

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.

Structure-based pKa calculations using continuum electrostatics methods

Electrostatic free energy is useful for correlating structure with function in proteins in which ionizable groups play essential functional roles. To this end, the pK(a) values of ionizable groups must be known and their molecular determinants must be understood. Structure-based calculations of electrostatic energies and pK(a) values are necessary for this purpose. This unit describes protocols for pK(a) calculations with continuum electrostatics methods based on the numerical solution of the linearized Poisson-Boltzmann equation by the method of finite differences. Critical discussion of key parameters, approximations, and shortcomings of these methods is included. Two protocols are described for calculations with methods modified empirically to maximize agreement between measured and calculated pK(a) values. Applied judiciously, these methods can contribute useful and novel insight into properties of surface ionizable groups in proteins.

Identifying functional sites based on prediction of charged group behavior

This protocol describes the implementation and interpretation of THEMATICS, a simple computational predictor of functional information for proteins from the three-dimensional structure. This method is based on the computation of the electrical potential function for the protein and the calculation of the predicted titration curves for each of the titratable groups in the protein. While most of the titratable residues in a protein have predicted titration behavior that fits the Henderson-Hasselbalch equation, the ionizable residues in the active site generally deviate dramatically from the typical behavior. From the calculated titration curves, one identifies those residues that deviate significantly from Henderson-Hasselbalch behavior. A cluster of two or more of such deviant titratable residues in physical proximity is a reliable predictor of active-site location.

Electrostatic contribution to the thermodynamic and kinetic stability of the homotrimeric coiled coil Lpp-56: A computational study

The protein moiety of the Braun's E. coli outer membrane lipoprotein (Lpp-56) is an attractive object of biophysical investigation in several aspects. It is a homotrimeric, parallel coiled coil, a class of coiled coils whose stability and folding have been studied only occasionally. Lpp-56 possesses unique structural properties and exhibits extremely low rates of folding and unfolding. It is natural to ask how the specificity of the structure determines the extraordinary physical chemical properties of this protein. Recently, a seemingly controversial data on the stability and unfolding rate of Lpp-56 have been published (Dragan et al., Biochemistry 2004;43: 14891-14900; Bjelic et al., Biochemistry 2006;45:8931-8939). The unfolding rate constant measured using GdmCl as the denaturing agent, though extremely low, was substantially higher than that obtained on the basis of thermal unfolding. If this large difference arises from the effect of screening of electrostatic interactions induced by GdmCl, electrostatic interactions would appear to be an important factor determining the unusual properties of Lpp-56. We present here a computational analysis of the electrostatic properties of Lpp-56 combining molecular dynamics simulations and continuum pK calculations. The pH-dependence of the unfolding free energy is predicted in good agreement with the experimental data: the change in DeltaG between pH 3 and pH 7 is approximately 60 kJ mol(-1). The results suggest that the difference in the stability of the protein observed using different experimental methods is mainly because of the effect of the reduction of electrostatic interactions when the salt (GdmCl) concentration increases. We also find that the occupancy of the interhelical salt bridges is unusually high. We hypothesize that electrostatic interactions, and the interhelical salt bridges in particular, are an important factor determining the low unfolding rate of Lpp-56.

Electrostatic effects in a network of polar and ionizable groups in staphylococcal nuclease

His121 and His124 are embedded in a network of polar and ionizable groups on the surface of staphylococcal nuclease. To examine how membership in a network affects the electrostatic properties of ionizable groups, the tautomeric state and the pK(a) values of these histidines were measured with NMR spectroscopy in the wild-type nuclease and in 13 variants designed to disrupt the network. In the background protein, His121 and His124 titrate with pK(a) values of 5.2 and 5.6, respectively. In the variants, where the network was disrupted, the pK(a) values range from 4.03 to 6.46 for His121, and 5.04 to 5.99 for His124. The largest decrease in a pK(a) was observed when the favorable Coulomb interaction between His121 and Glu75 was eliminated; the largest increase was observed when Tyr91 or Tyr93 was substituted with Ala or Phe. In all variants, the dominant tautomeric state at neutral pH was the N(epsilon2) state. At one level the network behaves as a rigid unit that does not readily reorganize when disrupted: crystal structures of the E75A or E75Q variants show that even when the pivotal Glu75 is removed, the overall configuration of the network was unaffected. On the other hand, a few key hydrogen bonds appear to govern the conformation of the network, and when these bonds are disrupted the network reorganizes. Coulomb interactions within the network report an effective dielectric constant of 20, whereas a dielectric constant of 80 is more consistent with the magnitude of medium to long-range Coulomb interactions in this protein. The data demonstrate that when structures are treated as static, rigid bodies, structure-based pK(a) calculations with continuum electrostatics method are not useful to treat ionizable groups in cases where pK(a) values are governed by short-range polar and Coulomb interactions.

Why are proteins charged? Networks of charge-charge interactions in proteins measured by charge ladders and capillary electrophoresis

Almost all proteins contain charged amino acids. While the function in catalysis or binding of individual charges in the active site can often be identified, it is less clear how to assign function to charges beyond this region. Are they necessary for solubility? For reasons other than solubility? Can manipulating these charges change the properties of proteins? A combination of capillary electrophoresis (CE) and protein charge ladders makes it possible to study the roles of charged residues on the surface of proteins outside the active site. This method involves chemical modification of those residues to generate a large number of derivatives of the protein that differ in charge. CE separates those derivatives into groups with the same number of modified charged groups. By studying the influence of charge on the properties of proteins using charge ladders, it is possible to estimate the net charge and hydrodynamic radius and to infer the role of charged residues in ligand binding and protein folding.

DFT/electrostatic calculations of pK(a) values in cytochrome c oxidase

Using classical electrostatic calculations, earlier we examined the dependence of the protonation state of bovine cytochrome c oxidase (CcO) on its redox state. Based on these calculations, we have proposed a model of CcO proton pumping that involves His291, one of the Cu(B) histidine ligands, which was found to respond to redox changes of the enzyme Fe(a)(3)-Cu(B) catalytic center. In this work, we employ combined density functional and continuum electrostatic calculations to evaluate the pK(a)() values of His291 and Glu242, two key residues of the model. The pK(a) values are calculated for different redox states of the enzyme, and the influence of different factors on the pK(a)'s is analyzed in detail. The calculated pK(a)() values of Glu242 are between 9.4 and 12.0, depending on the redox state of the protein, which is in excellent agreement with recent experimental measurements. Assuming the reduced state of heme a(3), His291 of the oxidized Cu(B) center possesses a pK(a)() between 2.1 and 4.0, while His291 of the reduced Cu(B) center has a pK(a) above 17. The obtained results support the proposal that the His291 ligand of the Cu(B) center in CcO is a proton pump element.

Linear-scaling molecular orbital calculations for the pKa values of ionizable residues in proteins

In this report, we present a computational methodology for the pKa prediction of proteins, based on linear-scaling molecular orbital calculations for their solution-conformations obtained from NMR measurements. The method is used to predict the pKa values of five carboxylic acids (Asp7, Glu10, Glu19, Asp27, and Glu43) in turkey ovomucoid third domain (OMTKY3), and six aspartates residues (Asp 22, Asp 44, Asp 54, Asp 75, Asp 83, and Asp 93) in barnase. For OMTKY3, all the predicted pKa values are within 1 pH units from the available experimental ones, except for the case of Glu 43. For barnase, the root-mean-square deviation from experiment is 1.46 pH units. As a result, the proposed pKa calculation method correctly reproduces the relative order of the pKa values among the carboxylic acids located in different sites of the proteins. The calculated pKa values are decomposed into the contributions of short- and long-range structural difference effects. The results indicate that in both proteins the pKa value of the given carboxylic acid is partially influenced by long-range interactions with distant charged residues, which significantly contribute to determining the relative order of the pKa values. The current methodology based on LSMO provides us useful information about the titration behavior in a protein.

The pH-dependent conformational states of kyotorphin: a constant-pH molecular dynamics study

An extensive conformational study of the analgesic dipeptide kyotorphin (L-Tyr-L-Arg) at different pH values was performed using a constant-pH molecular dynamics method. This dipeptide showed a remarkable pH-dependent conformational variety. The protonation of the N-terminal amine was identified as a key element in the transition between the more extended and the more packed conformational states, as monitored by the dihedral angle defined by the atoms 1Cbeta-1Calpha-2Calpha-2Cbeta. The principal-component analysis of kyotorphin identified two major conformational populations (the extended trans and the packed cis) together with conformations that occur exclusively at extreme pH values. Other, less stable conformations were also identified, which help us to understand the transitions between the predominant populations. The fitting of kyotorphin's conformational space to the structure of morphine resulted in a set of conformers that were able to fulfill most of the constraints for the mu-receptor. These results suggest that there may be strong similarities between the kyotorphin receptor and the structural family of opioid receptors.

Exploring the evaluation of net charge, hydrodynamic size and shape of peptides through experimental electrophoretic mobilities obtained from CZE

This work explores the validity of simple CZE models to analyze the electrophoretic mobilities of 102 peptides reported in literature. These models are based mainly on fundamental physicochemical theories providing analytical expressions amenable to relatively simple numerical analysis. Thus, the Linderstrøm-Lang capillary electrophoresis model (LLCEM) and its perturbed version (PLLCEM), proposed and applied previously to the CZE of globular proteins, are adapted and used here for peptides. Also the effects of pK-shifts on net charge, hydration and hydrodynamic size and shape of peptides are analyzed and discussed. Emphasis is placed on the fact that these parameters are physically coupled, and thus a variation in the net charge may produce an appreciable change in the hydrodynamic size of peptides. Within the framework of CZE, peptides may be assumed as having a hydrodynamic volume associated with either spherical or spheroidal particles. The effects on peptide net charge and hydrodynamic size, of electrostatic interaction between a pair of charged groups in the chain and electrical permitivitty around the peptide domain are studied. The predictions of the PLLCEM and LLCEM are in good agreement with results reported previously in the literature. Several limitations concerning these models and some needs for further research are also described.

A simple clustering algorithm can be accurate enough for use in calculations of pKs in macromolecules

Structure and function of macromolecules depend critically on the ionization states of their acidic and basic groups. Most current structure-based theoretical methods that predict pK of ionizable groups in macromolecules include, as one of the key steps, a computation of the partition sum (Boltzmann average) over all possible protonation microstates. As the number of these microstates depends exponentially on the number of ionizable groups present in the molecule, direct computation of the sum is not realistically feasible for many typical proteins that may have tens or even hundreds of ionizable groups. We have tested a simple and robust approximate algorithm for computing these partition sums for macromolecules. The method subdivides the interacting sites into independent clusters, based upon the strength of site-site electrostatic interaction. The resulting partition function is factorizable into computationally manageable components. Two variants of the approach are presented and validated on a representative test set of 602 proteins, by comparing the pK(1/2) values computed by the proposed method with those obtained by the standard Monte Carlo approach used as a reference. With 95% confidence, the relative error introduced by the more accurate of the two methods is less than 0.25 pK units. The algorithms are one to two orders of magnitude faster than the Monte Carlo method, with the typical settings. A graphical representation is introduced that visualizes the clusters of strong site-site interactions in the context of the three-dimensional (3D) structure of the macromolecule, facilitating identification of functionally important clusters of ionizable groups; the approach is exemplified on two proteins, bacteriorhodopsin and myoglobin.

Factors influencing the energetics of electron and proton transfers in proteins. What can be learned from calculations

A protein structure should provide the information needed to understand its observed properties. Significant progress has been made in developing accurate calculations of acid/base and oxidation/reduction reactions in proteins. Current methods and their strengths and weaknesses are discussed. The distribution and calculated ionization states in a survey of proteins is described, showing that a significant minority of acidic and basic residues are buried in the protein and that most of these remain ionized. The electrochemistry of heme and quinones are considered. Proton transfers in bacteriorhodopsin and coupled electron and proton transfers in photosynthetic reaction centers, 5-coordinate heme binding proteins and cytochrome c oxidase are highlighted as systems where calculations have provided insight into the reaction mechanism.

Multiconformation continuum electrostatics analysis of the NhaA Na+/H+ antiporter of Escherichia coli with functional implications

Sodium proton antiporters are essential enzymes that catalyze the exchange of sodium ions for protons across biological membranes. Protonations and deprotonations of individual amino acid residues and of clusters formed by these residues play an important role in activating these enzymes and in the mechanism of transport. We have used multiconformation continuum electrostatics method to investigate the protonation states of residues in the sodium proton exchanger NhaA from Escherichia coli, the structure of which has been determined recently by x-ray crystallography. Our calculations identify four clusters of electrostatically tightly interacting residues as well as long-range interactions between residues required for activation. The importance of many of these residues has been demonstrated by the characterization of site-directed mutants. A number of residues with extreme pKa values, including several of the "pH sensor," can only undergo protonation/deprotonation reactions subsequent to conformational changes. The results of the calculations provide valuable information on the activation of the antiporter and the role of individual amino acid residues, and provide a solid framework for further experiments.

Electrostatic calculations for assignment of infrared difference bands to carboxyl groups getting protonated during protein reactions

Fourier transform infrared (FTIR) difference spectroscopy is predestined to monitor the protonation of carboxyl groups during protein reactions, making glutamic and aspartic amino acids unique to follow proton pathways. The absorption of the corresponding vibrations are clearly distinguishable from the absorption of other amino acids. However, the assignment to specific groups within the protein needs additional information, e.g., from induced spectral changes due to isotopic labeling or mutation. Here, the capability of electrostatic calculations to assign IR difference bands to specific carboxyl groups getting protonated is demonstrated by the ion pump mechanism of the sarcoplasmic reticulum Ca(2+)-ATPase. Active Ca(2+) transport is coupled to the hydrolysis of ATP. Two Ca(2+) ions are transported per ATP hydrolysed and two or three H(+) ions are countertransported. FTIR difference spectra show that during the Ca(2+) release step, carboxyl groups become protonated. Multiconformation continuum electrostatic calculations (MCCE) have been carried out to determine the equilibrium distribution of residue ionization and side chain conformation in dependence of pH. Available structural X-ray data from the calcium-bound and the calcium-free state allows us to simulate the transition between the two states monitored in the IR difference spectra. Exemplarily for Asp 800, ligand of both calcium ions, it is shown that MCCE calculations can identify this specific Asp to contribute to the IR bands and therefore to take part in the proton countertransport of the Ca(2+)-ATPase. In addition, an energy analysis can be performed to understand what interactions shift the pK(a).