The Prediction of Protein pKa's Using QM/MM: The pKa of Lysine 55 in Turkey Ovomucoid Third Domain

Automated preparation of nanoscopic structures: Graph-based sequence analysis, mismatch detection, and pH-consistent protonation with uncertainty estimates

Structure and function in nanoscale atomistic assemblies are tightly coupled, and every atom with its specific position and even every electron will have a decisive effect on the electronic structure, and hence, on the molecular properties. Molecular simulations of nanoscopic atomistic structures therefore require accurately resolved three-dimensional input structures. If extracted from experiment, these structures often suffer from severe uncertainties, of which the lack of information on hydrogen atoms is a prominent example. Hence, experimental structures require careful review and curation, which is a time-consuming and error-prone process. Here, we present a fast and robust protocol for the automated structure analysis and pH-consistent protonation, in short, ASAP. For biomolecules as a target, the ASAP protocol integrates sequence analysis and error assessment of a given input structure. ASAP allows for pK a  prediction from reference data through Gaussian process regression including uncertainty estimation and connects to system-focused atomistic modeling described in Brunken and Reiher (J. Chem. Theory Comput. 16, 2020, 1646). Although focused on biomolecules, ASAP can be extended to other nanoscopic objects, because most of its design elements rely on a general graph-based foundation guaranteeing transferability. The modular character of the underlying pipeline supports different degrees of automation, which allows for (i) efficient feedback loops for human-machine interaction with a low entrance barrier and for (ii) integration into autonomous procedures such as automated force field parametrizations. This facilitates fast switching of the pH-state through on-the-fly system-focused reparametrization during a molecular simulation at virtually no extra computational cost.

Protein pKa Prediction by Tree-Based Machine Learning

Protonation states of ionizable protein residues modulate many essential biological processes. For correct modeling and understanding of these processes, it is crucial to accurately determine their pKa values. Here, we present four tree-based machine learning models for protein pKa prediction. The four models, Random Forest, Extra Trees, eXtreme Gradient Boosting (XGBoost), and Light Gradient Boosting Machine (LightGBM), were trained on three experimental PDB and pKa datasets, two of which included a notable portion of internal residues. We observed similar performance among the four machine learning algorithms. The best model trained on the largest dataset performs 37% better than the widely used empirical pKa prediction tool PROPKA and 15% better than the published result from the pKa prediction method DelPhiPKa. The overall root-mean-square error (RMSE) for this model is 0.69, with surface and buried RMSE values being 0.56 and 0.78, respectively, considering six residue types (Asp, Glu, His, Lys, Cys, and Tyr), and 0.63 when considering Asp, Glu, His, and Lys only. We provide pKa predictions for proteins in human proteome from the AlphaFold Protein Structure Database and observed that 1% of Asp/Glu/Lys residues have highly shifted pKa values close to the physiological pH.

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.

Fully Atomistic Multiscale Approach for pKa Prediction

The ionization state of titratable amino acids strongly affects proteins structure and functioning in a large number of biological processes. It is therefore essential to be able to characterize the pK_a of ionizable groups inside proteins and to understand its microscopic determinants in order to gain insights into many functional properties of proteins. A big effort has been devoted to the development of theoretical approaches for the prediction of deprotonation free energies, yet the accurate theoretical/computational calculation of pK_a values is recognized as a current challenge. A methodology based on a hybrid quantum/classical approach is here proposed for the computation of deprotonation free energies. The method is applied to calculate the pK_a of formic acid, methylammonium, and methanethiol, providing results in good agreement with the corresponding experimental estimates. The pK_a is also calculated for aspartic acid and lysine as single residues in solution and for three aspartic/glutamic acids inside a well-characterized protein: hen egg white lysozyme. While for small molecules the method is able to deal with multiple protonation states of all titratable groups, this becomes computationally very expensive for proteins. The calculated pK_a values for the single amino acids (except for the zwitterionic aspartic acid) and inside the protein display a systematic shift with respect to the experimental values that suggests that the fine balance between hydrophobic and polar interactions might be not accurately reproduced by the usual classical force-fields, thus affecting the computation of deprotonation free energies. The calculated pK_a shifts inside the protein are in good agreement with the corresponding experimental ones (within 1 pK_a unit), well reproducing the pK_a changes due to the protein environment even in the case of large pK_a shifts.

Development of constant-pH simulation methods in implicit solvent and applications in biomolecular systems

pH is a critical parameter for biological and technological systems directly related with electrical charges. It can give rise to peculiar electrostatic phenomena, which also makes them more challenging. Due to the quantum nature of the process, involving the forming and breaking of chemical bonds, quantum methods should ideally by employed. Nevertheless, due to the very large number of ionizable sites, different macromolecular conformations, salt conditions, and all other charged species, the CPU time cost simply becomes prohibitive for computer simulations, making this a quite complex problem. Simplified methods based on Monte Carlo sampling have been devised and will be reviewed here, highlighting the updated state-of-the-art of this field, advantages, and limitations of different theoretical protocols for biomolecular systems (proteins and nucleic acids). Following a historical perspective, the discussion will be associated with the applications to protein interactions with other proteins, polyelectrolytes, and nanoparticles.

Using polarizable POSSIM force field and fuzzy-border continuum solvent model to calculate pK(a) shifts of protein residues

Our Fuzzy-Border (FB) continuum solvent model has been extended and modified to produce hydration parameters for small molecules using POlarizable Simulations Second-order Interaction Model (POSSIM) framework with an average error of 0.136 kcal/mol. It was then used to compute pKa shifts for carboxylic and basic residues of the turkey ovomucoid third domain (OMTKY3) protein. The average unsigned errors in the acid and base pKa values were 0.37 and 0.4 pH units, respectively, versus 0.58 and 0.7 pH units as calculated with a previous version of polarizable protein force field and Poisson Boltzmann continuum solvent. This POSSIM/FB result is produced with explicit refitting of the hydration parameters to the pKa values of the carboxylic and basic residues of the OMTKY3 protein; thus, the values of the acidity constants can be viewed as additional fitting target data. In addition to calculating pKa shifts for the OMTKY3 residues, we have studied aspartic acid residues of Rnase Sa. This was done without any further refitting of the parameters and agreement with the experimental pKa values is within an average unsigned error of 0.65 pH units. This result included the Asp79 residue that is buried and thus has a high experimental pKa value of 7.37 units. Thus, the presented model is capable or reproducing pKa results for residues in an environment that is significantly different from the solvated protein surface used in the fitting. Therefore, the POSSIM force field and the FB continuum solvent parameters have been demonstrated to be sufficiently robust and transferable. © 2016 Wiley Periodicals, Inc.

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.

Molecular view of ligands specificity for CAG repeats in anti-Huntington therapy

Huntington's disease is a fatal and devastating neurodegenerative genetic disorder for which there is currently no cure. It is characterized by Huntingtin protein's mRNA transcripts with 36 or more CAG repeats. Inhibiting the formation of pathological complexes between these expanded transcripts and target proteins may be a valuable strategy against the disease. Yet, the rational design of molecules specifically targeting the expanded CAG repeats is limited by the lack of structural information. Here, we use well-tempered metadynamics-based free energy calculations to investigate pose and affinity of two ligands targeting CAG repeats for which affinities have been previously measured. The first consists of two 4-guanidinophenyl rings linked by an ester group. It is the most potent ligand identified so far, with Kd = 60(30) nM. The second consists of a 4-phenyl dihydroimidazole and 4-1H-indole dihydroimidazole connected by a C-C bond (Kd = 700(80) nM). Our calculations reproduce the experimental affinities and uncover the recognition pattern between ligands' and their RNA target. They also provide a molecular basis for the markedly different affinity of the two ligands for CAG repeats as observed experimentally. These findings may pave the way for a structure-based hit-to-lead optimization to further improve ligand selectivity toward CAG repeat-containing mRNAs.

First Principles Calculations of Aqueous pKa Values for Organic and Inorganic Acids Using COSMO-RS Reveal an Inconsistency in the Slope of the pKa Scale

The COSMO-RS method, a combination of the quantum chemical dielectric continuum solvation model COSMO with a statistical thermodynamics treatment for more realistic solvation (RS) simulations, has been used for the direct prediction of pKa constants of a large variety of 64 organic and inorganic acids. A highly significant correlation of r(2) = 0.984 with a standard deviation of only 0.49 between the calculated values of the free energies of dissociation and the experimental pKa values was found, without any special adjustment of the method. Thus, we have a theoretical a priori prediction method for pKa, which has the regression constant and the slope as only adjusted parameters. Such a method can be of great value in many areas of physical chemistry, especially in pharmaceutical and agrochemical industry. To our surprise, the slope of pKa vs ΔGdiss is only 58% of the theoretically expected value of 1/RTln(10). A careful analysis with respect to different contributions as well as a comparison with the work of other authors excludes the possibility that the discrepancy is due to weaknesses of the calculation method. Hence, we must conclude that the experimental pKa scale depends differently on the free energy of dissociation than generally assumed.

Predicting pKa in Implicit Solvents: Current Status and Future Directions

Computational prediction of condensed phase acidity is a topic of much interest in the field today. We introduce the methods available for predicting gas phase acidity and pKas in aqueous and non-aqueous solvents including high-level electronic structure methods, empirical linear free energy relationships (LFERs), implicit solvent methods, explicit solvent statistical free energy methods, and hybrid implicit–explicit approaches. The focus of this paper is on implicit solvent methods, and we review recent developments including new electronic structure methods, cluster-continuum schemes for calculating ionic solvation free energies, as well as address issues relating to the choice of proton solvation free energy to use with implicit solvation models, and whether thermodynamic cycles are necessary for the computation of pKas. A comparison of the scope and accuracy of implicit solvent methods with ab initio molecular dynamics free energy methods is also presented. The present status of the theory and future directions are outlined.

Biomolecular electrostatics and solvation: a computational perspective

An understanding of molecular interactions is essential for insight into biological systems at the molecular scale. Among the various components of molecular interactions, electrostatics are of special importance because of their long-range nature and their influence on polar or charged molecules, including water, aqueous ions, proteins, nucleic acids, carbohydrates, and membrane lipids. In particular, robust models of electrostatic interactions are essential for understanding the solvation properties of biomolecules and the effects of solvation upon biomolecular folding, binding, enzyme catalysis, and dynamics. Electrostatics, therefore, are of central importance to understanding biomolecular structure and modeling interactions within and among biological molecules. This review discusses the solvation of biomolecules with a computational biophysics view toward describing the phenomenon. While our main focus lies on the computational aspect of the models, we provide an overview of the basic elements of biomolecular solvation (e.g. solvent structure, polarization, ion binding, and non-polar behavior) in order to provide a background to understand the different types of solvation models.

Reduction potentials and acidity constants of Mn superoxide dismutase calculated by QM/MM free-energy methods

We used two theoretical methods to estimate reduction potentials and acidity constants in Mn superoxide dismutase (MnSOD), namely combined quantum mechanical and molecular mechanics (QM/MM) thermodynamic cycle perturbation (QTCP) and the QM/MM-PBSA approach. In the latter, QM/MM energies are combined with continuum solvation energies calculated by solving the Poisson-Boltzmann equation (PB) or by the generalised Born approach (GB) and non-polar solvation energies calculated from the solvent-exposed surface area. We show that using the QTCP method, we can obtain accurate and precise estimates of the proton-coupled reduction potential for MnSOD, 0.30±0.01 V, which compares favourably with experimental estimates of 0.26-0.40 V. However, the calculated potentials depend strongly on the DFT functional used: The B3LYP functional gives 0.6 V more positive potentials than the PBE functional. The QM/MM-PBSA approach leads to somewhat too high reduction potentials for the coupled reaction and the results depend on the solvation model used. For reactions involving a change in the net charge of the metal site, the corresponding results differ by up to 1.3 V or 24 pK(a) units, rendering the QM/MM-PBSA method useless to determine absolute potentials. However, it may still be useful to estimate relative shifts, although the QTCP method is expected to be more accurate.

Absolute pKa Values and Solvation Structure of Amino Acids from Density Functional Based Molecular Dynamics Simulation

Absolute pKa values of the amino acid side chains of arginine, aspartate, cysteine, histidine, and tyrosine; the C- and N-terminal group of tyrosine; and the tryptophan radical cation are calculated using a revised density functional based molecular dynamics simulation technique introduced previously [ Cheng , J. ; Sulpizi , M. ; Sprik , M. J. Chem. Phys. 2009 , 131 , 154504 ]. In the revised scheme, acid deprotonation is considered as a dissociation rather than a proton transfer reaction, and a correction term for treating the proton as a hydronium ion is suggested. The acidity constants of the amino acids are obtained from the vertical energy gaps for removal or insertion of the acidic proton and the computed solvation free energy of the proton. The unsigned mean error relative to experimental results is 2.1 pKa units with a maximum error of 4.0 pKa units. The estimated mean statistical uncertainty due to the finite length of the trajectories is ±1.1 pKa units. The solvation structures of the protonated and deprotonated amino acids are analyzed in terms of radial distribution functions, which can serve as reference data for future force field developments.

Noncovalent interactions in extended systems described by the effective fragment potential method: theory and application to nucleobase oligomers

The implementation of the effective fragment potential (EFP) method within the Q-CHEM electronic structure package is presented. The EFP method is used to study noncovalent π-π and hydrogen-bonding interactions in DNA strands. Since EFP is a computationally inexpensive alternative to high-level ab initio calculations, it is possible to go beyond the dimers of nucleic acid bases and to investigate the asymptotic behavior of different components of the total interaction energy. The calculations demonstrated that the dispersion energy is a leading component in π-stacked oligomers of all sizes. Exchange-repulsion energy also plays an important role. The contribution of polarization is small in these systems, whereas the magnitude of electrostatics varies. Pairwise fragment interactions (i.e., the sum of dimer binding energies) were found to be a good approximation for the oligomer energy.

Microscopic pKa analysis of Glu286 in cytochrome c oxidase (Rhodobacter sphaeroides): toward a calibrated molecular model

As stringent tests for the molecular model and computational protocol, microscopic pK(a) calculations are performed for the key residue, Glu286, in cytochrome c oxidase (CcO) using a combined quantum mechanical/molecular mechanical (QM/MM) potential and a thermodynamic integration protocol. The impact of the number of water molecules in the hydrophobic cavity and protonation state of several key residues (e.g., His334, Cu(B)-bound water, and PRD(a3)) on the computed microscopic pK(a) values of Glu286 has been systematically examined. To help evaluate the systematic errors in the QM/MM-based protocol, microscopic pK(a) calculations have also been carried out for sites in a soluble protein (Asp70 in T4 lysozyme) and a better-characterized membrane protein (Asp85 in bacteriorhodopsin). Overall, the results show a significant degree of internal consistency and reproducibility that support the effectiveness of the computational framework. Although the number of water molecules in the hydrophobic cavity does not greatly influence the computed pK(a) of Glu286, the protonation states of several residues, some of which are rather far away, have more significant impacts. Adopting the standard protonation state for all titratable residues leaves a large net charge on the system and a significantly elevated pK(a) for Glu286, highlighting that any attempt to address the energetics of proton transfers in CcO at a microscopic level should carefully select the protonation state of residues, even those not in the immediate neighborhood of the active site. The calculations indirectly argue against the deprotonation of His334 for the proton pumping process, although further studies that explicitly compute its pK(a) are required for a more conclusive statement. Finally, the deprotonated Glu286 is found to be in a stable water-mediated connection with PRD(a3) for at least several nanoseconds when this presumed pumping site is protonated. This does not support the proposed role of Glu286 as a robust gating valve that prevents proton leakage, although a conclusive statement awaits a more elaborate characterization of the Glu286-PRD(a3) connectivity with free energy simulations and a protonated PRD(a3). The large sets of microscopic simulations performed here have provided useful guidance to the establishment of a meaningful molecular model and effective computational protocol for explicitly analyzing the proton transfer kinetics in CcO, which is required for answering key questions regarding the pumping function of this fascinating and complex system.

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.

Reproducing basic pKa values for turkey ovomucoid third domain using a polarizable force field

We have extended our previous studies of calculating acidity constants for the acidic residues found in the turkey ovomucoid third domain protein (OMTKY3) by determining the relative pKa values for the basic residues (Lys13, Arg21, Lys29, Lys34, His52, and Lys55). A polarizable force field (PFF) was employed. The values of the pKa were found by direct comparison of energies of solvated protonated and deprotonated forms of the protein. Poisson-Boltzmann (PBF) and surface generalized Born (SGB) continuum solvation models represent the hydration, and a nonpolarizable fixed-charge OPLS-AA force field was used for comparison. Our results indicate that (i) the pKa values of the basic residues can be found in close agreement with the experimental values when a PFF is used in conjunction with the PBF solvation model, (ii) it is sufficient to take into the account only the residues which are in close proximity (hydrogen bonded) to the residue in question, and (iii) the PBF solvation model is superior to the SGB solvation model for these pKa calculations. The average error with the PBF/PFF model is only 0.7 pH unit, compared with 2.2 and 6.1 units for the PBF/OPLS and SGB/OPLS, respectively. The maximum deviation of the PBF/PFF results from the experimental values is 1.7 pH units compared with 6.0 pH units for the PBF/OPLS. Moreover, the best results were obtained while using an advanced nonpolar energy calculation scheme. The overall conclusion is that this methodology and force field are suitable for the accurate assessment of pKa shifts for both acidic and basic protein residues.

Incorporation of a QM/MM buffer zone in the variational double self-consistent field method

The explicit polarization (X-Pol) potential is an electronic-structure-based polarization force field, designed for molecular dynamics simulations and modeling of biopolymers. In this approach, molecular polarization and charge transfer effects are explicitly treated by a combined quantum mechanical and molecular mechanical (QM/MM) scheme, and the wave function of the entire system is variationally optimized by a double self-consistent field (DSCF) method. In the present article, we introduce a QM buffer zone for a smooth transition from a QM region to an MM region. Instead of using the Mulliken charge approximation for all QM/MM interactions, the Coulombic interactions between the adjacent fragments are determined directly by electronic structure theory. The present method is designed to accelerate the speed of convergence of the total energy and charge density of the system.

Protein-protein binding is often associated with changes in protonation state

pK(a) values of ionizable residues have been calculated using the PROPKA method and structures of 75 protein-protein complexes and their corresponding free forms. These pK(a) values were used to compute changes in protonation state of individual residues, net changes in protonation state of the complex relative to the uncomplexed proteins, and the correction to a binding energy calculated assuming standard protonation states at pH 7. For each complex, two different structures for the uncomplexed form of the proteins were used: the X-ray structures determined for the proteins in the absence of the other protein and the individual protein structures taken from the structure of the complex (referred to as unbound and bound structures, respectively). In 28 and 77% of the cases considered here, protein-protein binding is accompanied by a complete (>95%) or significant (>50%) change in protonation state of at least one residue using unbound structures. Furthermore, in 36 and 61% of the cases, protein-protein binding is accompanied by a complete or significant net change in protonation state of the complex relative to the separated monomers. Using bound structures, the corresponding values are 12, 51, 20, and 48%. Comparison to experimental data suggest that using unbound and bound structures lead to over- and underestimation of binding-induced protonation state changes, respectively. Thus, we conclude that protein-protein binding is often associated with changes in protonation state of amino acid residues and with changes in the net protonation state of the proteins. The pH-dependent correction to the binding energy contributes at least one order of magnitude to the binding constant in 45 and 23%, using unbound and bound structures, respectively.

Computational methods for biomolecular electrostatics

An understanding of intermolecular interactions is essential for insight into how cells develop, operate, communicate, and control their activities. Such interactions include several components: contributions from linear, angular, and torsional forces in covalent bonds, van der waals forces, as well as electrostatics. Among the various components of molecular interactions, electrostatics are of special importance because of their long range and their influence on polar or charged molecules, including water, aqueous ions, and amino or nucleic acids, which are some of the primary components of living systems. Electrostatics, therefore, play important roles in determining the structure, motion, and function of a wide range of biological molecules. This chapter presents a brief overview of electrostatic interactions in cellular systems, with a particular focus on how computational tools can be used to investigate these types of interactions.

Extending the Power of Quantum Chemistry to Large Systems with the Fragment Molecular Orbital Method

Following the brief review of the modern fragment-based methods and other approaches to perform quantum-mechanical calculations of large systems, the theoretical development of the fragment molecular orbital method (FMO) is covered in detail, with the emphasis on the physical properties, which can be computed with FMO. The FMO-based polarizable continuum model (PCM) for treating the solvent effects in large systems and the pair interaction energy decomposition analysis (PIEDA) are described in some detail, and a range of applications of FMO to biological studies is introduced. The factors determining the relative stability of polypeptide conformers (alpha-helix, beta-turn, and extended form) are elucidated using FMO/PCM and PIEDA, and the interactions in the Trp-cage miniprotein construct (PDB: 1L2Y) are analyzed using PIEDA.

Prediction and rationalization of protein pKa values using QM and QM/MM methods

We describe the development and application of a computational method for the prediction and rationalization of pKa values of ionizable residues in proteins, based on ab initio quantum mechanics (QM) and the effective fragment potential (EFPs) method (a hybrid QM/MM method). The theoretical developments include (1) a covalent boundary method based on frozen localized orbitals, (2) divide-and-conquer methods for the ab initio computation of protein EFPs consisting of multipoles up to octupoles and dipole polarizability tensors, (3) a method for computing vibrational free energies for a localized molecular region, and (4) solutions of the polarized continuum model of bulk solvation equations for protein-sized systems. The QM-based pKa prediction method is one of the most accurate methods currently available and can be used in cases where other pKa prediction methods fail. Preliminary analysis of the computed results indicate that many pKa values (1) are primarily determined by hydrogen bonds rather than long-range charge-charge interactions and (2) are relatively insensitive to large-scale dynamical fluctuations of the protein structure.

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.

Influence of the Membrane Potential on the Protonation of Bacteriorhodopsin: Insights from Electrostatic Calculations into the Regulation of Proton Pumping

Proton binding and release are elementary steps for the transfer of protons within proteins, which is a process that is crucial in biochemical catalysis and biological energy transduction. Local electric fields in proteins affect the proton binding energy compared to aqueous solution. In membrane proteins, also the membrane potential affects the local electrostatics and can thus be crucial for protein function. In this paper, we introduce a procedure to calculate the protonation probability of titratable sites of a membrane protein in the presence of a membrane potential. In the framework of continuum electrostatics, we use a modified Poisson-Boltzmann equation to include the influence of the membrane potential. Our method considers that in a transmembrane protein each titratable site is accessible for protons from only one side of the membrane depending on the hydrogen bond pattern of the protein. We show that the protonation of sites receiving their protons from different sides of the membrane is differently influenced by the membrane potential. In addition, the effect of the membrane potential is combined with the effect of the pH gradient resulting from proton pumping. Our method is applied to bacteriorhodopsin, a light-activated proton pump. We find that the proton pumping of this protein might be regulated by Asp115, a conserved residue for which no function has been identified yet. According to our calculations, the interaction of Asp115 with Asp85 leads to the protonation of the latter if the pH gradient or the membrane potential becomes too large. Since Asp85 is the primary proton acceptor in the photocycle, bacteriorhodopsin molecules in which Asp85 is protonated cannot pump protons. Furthermore, we estimate how the membrane potential affects the energetics of the individual proton-transfer reactions of the photocycle. Most reactions, except the initial proton transfer from the Schiff base to Asp85, are influenced. Our calculations give new insights into the mechanism with which bacteriorhodopsin senses the membrane potential and the pH gradient and how the proton pumping is regulated by these parameters.

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.

A polarizable continuum approach for the study of heterogeneous dielectric environments

We present a computational method, exploiting some features of the polarizable continuum model (PCM) to describe heterogeneous media; it belongs to the family of electrostatic embedding mixed methods, such as the more common quantum-mechanical (QM)/molecular mechanics approaches, with the electrostatic long range effects accounted for by a polarized continuum instead of atomic point charges. Provided effective dielectric constants are determined for the various parts of the system, the method is much faster than its atomistic counterpart, and allows for high-level QM calculations on the fragment of interest, using all the highly efficient computational tools developed for homogeneous PCM. Two case studies (the calculation of the pKa of solvent exposed acidic residues in a model protein, and the calculation of the electron spin resonance spectrum of a typical spin probe partially embedded in a membrane) are analyzed in some detail, to illustrate the application of the method to complex systems.

Adding explicit solvent molecules to continuum solvent calculations for the calculation of aqueous acid dissociation constants

Aqueous acid dissociation free energies for a diverse set of 57 monoprotic acids have been calculated using a combination of experimental and calculated gas and liquid-phase free energies. For ionic species, aqueous solvation free energies were calculated using the recently developed SM6 continuum solvation model. This model combines a dielectric continuum with atomic surface tensions to account for bulk solvent effects. For some of the acids studied, a combined approach that involves attaching a single explicit water molecule to the conjugate base (anion), and then surrounding the resulting anion-water cluster by a dielectric continuum, significantly improves the agreement between the calculated pK(a) value and experiment. This suggests that for some anions, particularly those concentrating charge on a single exposed heteroatom, augmenting implicit solvent calculations with a single explicit water molecule is required, and adequate, to account for strong short-range hydrogen bonding interactions between the anion and the solvent. We also demonstrate the effect of adding several explicit waters by calculating the pK(a) of bicarbonate (HCO(3)(-)) using as the conjugate base carbonate (CO(3)(2-)) bound by up to three explicit water molecules.

Accurate prediction of basicity in aqueous solution with COSMO-RS

The COSMO-RS method, a combination of the quantum chemical dielectric continuum solvation model COSMO with a statistical thermodynamics treatment for realistic solvation simulations, has been used for the prediction of base pK(a) constants. For a variety of 43 organic bases the directly calculated values of the free energies of dissociation in water showed a very good correlation with experimental base pK(a) values (r2 = 0.98), corresponding to a standard deviation of 0.56 pK(a) units. Thus, we have an a priori prediction method for base pK(a) with the regression constant and the slope as only adjusted parameters. In accord with recent findings for pK(a) acidity predictions, the slope of pK(a) vs. DeltaG(diss) was significantly smaller than the theoretically expected value of 1/RTln(10). The predictivity of the presented method is general and not restricted to certain compound classes, but systematic corrections of 1 and 2 pKa units for secondary and tertiary aliphatic amines are required, respectively. The pK(a) prediction method was validated on a set of 58 complex multifunctional drug-like compounds, yielding an RMS accuracy of 0.66 pK(a) units.

Long dynamics simulations of proteins using atomistic force fields and a continuum representation of solvent effects: calculation of structural and dynamic properties

Long dynamics simulations were carried out on the B1 immunoglobulin-binding domain of streptococcal protein G (ProtG) and bovine pancreatic trypsin inhibitor (BPTI) using atomistic descriptions of the proteins and a continuum representation of solvent effects. To mimic frictional and random collision effects, Langevin dynamics (LD) were used. The main goal of the calculations was to explore the stability of tens-of-nanosecond trajectories as generated by this molecular mechanics approximation and to analyze in detail structural and dynamical properties. Conformational fluctuations, order parameters, cross correlation matrices, residue solvent accessibilities, pKa values of titratable groups, and hydrogen-bonding (HB) patterns were calculated from all of the trajectories and compared with available experimental data. The simulations comprised over 40 ns per trajectory for ProtG and over 30 ns per trajectory for BPTI. For comparison, explicit water molecular dynamics simulations (EW/MD) of 3 ns and 4 ns, respectively, were also carried out. Two continuum simulations were performed on each protein using the CHARMM program, one with the all-atom PAR22 representation of the protein force field (here referred to as PAR22/LD simulations) and the other with the modifications introduced by the recently developed CMAP potential (CMAP/LD simulations). The explicit solvent simulations were performed with PAR22 only. Solvent effects are described by a continuum model based on screened Coulomb potentials (SCP) reported earlier, i.e., the SCP-based implicit solvent model (SCP-ISM). For ProtG, both the PAR22/LD and the CMAP/LD 40-ns trajectories were stable, yielding C(alpha) root mean square deviations (RMSD) of about 1.0 and 0.8 A respectively along the entire simulation time, compared to 0.8 A for the EW/MD simulation. For BPTI, only the CMAP/LD trajectory was stable for the entire 30-ns simulation, with a C(alpha) RMSD of approximately 1.4 A, while the PAR22/LD trajectory became unstable early in the simulation, reaching a C(alpha) RMSD of about 2.7 A and remaining at this value until the end of the simulation; the C(alpha) RMSD of the EW/MD simulation was about 1.5 A. The source of the instabilities of the BPTI trajectories in the PAR22/LD simulations was explored by an analysis of the backbone torsion angles. To further validate the findings from this analysis of BPTI, a 35-ns SCP-ISM simulation of Ubiquitin (Ubq) was carried out. For this protein, the CMAP/LD simulation was stable for the entire simulation time (C(alpha) RMSD of approximately 1.0 A), while the PAR22/LD trajectory showed a trend similar to that in BPTI, reaching a C(alpha) RMSD of approximately 1.5 A at 7 ns. All the calculated properties were found to be in agreement with the corresponding experimental values, although local deviations were also observed. HB patterns were also well reproduced by all the continuum solvent simulations with the exception of solvent-exposed side chain-side chain (sc-sc) HB in ProtG, where several of the HB interactions observed in the crystal structure and in the EW/MD simulation were lost. The overall analysis reported in this work suggests that the combination of an atomistic representation of a protein with a CMAP/CHARMM force field and a continuum representation of solvent effects such as the SCP-ISM provides a good description of structural and dynamic properties obtained from long computer simulations. Although the SCP-ISM simulations (CMAP/LD) reported here were shown to be stable and the properties well reproduced, further refinement is needed to attain a level of accuracy suitable for more challenging biological applications, particularly the study of protein-protein interactions.

AB INITIO QUANTUM CHEMICAL AND MIXED QUANTUM MECHANICS/MOLECULAR MECHANICS (QM/MM) METHODS FOR STUDYING ENZYMATIC CATALYSIS

We describe large scale ab initio quantum chemical and mixed quantum mechanics/molecular mechanics (QM/MM) methods for studying enzymatic reactions. First, technical aspects of the methodology are reviewed, including the hybrid density functional theory (DFT) methods that are typically employed for the QM aspect of the calculations, and various approaches to defining the interface between the QM and MM regions in QM/MM approaches. The modeling of the enzymatic catalytic cycle for three examples--methane monooxygenase, cytochrome P450, and triose phosphate isomerase--are discussed in some depth, followed by a brief summary of other systems that have been investigated by ab initio methods over the past several years. Finally, a discussion of the qualitative and quantitative conclusions concerning enzymatic catalysis that are available from modern ab initio approaches is presented, followed by a conclusion briefly summarizing future prospects.

Solvent Effects on the SN2 Reaction: Application of the Density Functional Theory-Based Effective Fragment Potential Method

The performance of the density functional theory (DFT)-based effective fragment potential (EFP) method is assessed using the S(N)2 reaction: Cl- + nH2O + CH3Br = CH3Cl + Br- + nH2O. The effect of the systematic addition of water molecules on the structures and relative energies of all species involved in the reaction has been studied. The EFP1 method is compared with second-order perturbation theory (MP2) and DFT results for n = 1, 2, and 3, and EFP1 results are also presented for four water molecules. The incremental hydration effects on the barrier height are the same for all methods. However, only full MP2 or MP2 with EFP1 solvent molecules are able to provide an accurate treatment of the transition state (TS) and hence the central barriers. Full DFT and DFT with EFP1 solvent molecules both predict central barriers that are too small. The results illustrate that the EFP1-based DFT method gives reliable results when combined with an accurate quantum mechanical (QM) method, so it may be used as an efficient alternative to fully QM methods in the treatment of larger microsolvated systems.

The triplex-hairpin transition in cytosine-rich DNA

We present a theoretical study of the self-complementary single-stranded 30-mer d(TC*TTC*C*TTTTCCTTCTC*CCGAGAAGGTTTT) (PDB ID: 1b4y) that was designed to form an intramolecular triplex by folding back twice on itself. At neutral pH the molecule exists in a duplex hairpin conformation, whereas at acidic pH the cytosines labeled by an asterisk (*) are protonated, forming Hoogsteen hydrogen bonds with guanine of a GC Watson-Crick basepair to generate a triplex. As a first step in an investigation of the energetics of the triplex-hairpin transition, we applied the Bashford-Karplus multiple site model of protonation to calculate the titration curves for the two conformations. Based on these data, a two-state model is used to study the equilibrium properties of transition. Although this model properly describes the thermodynamics of the protonation-deprotonation steps that drive the folding-unfolding of the oligomer, it cannot provide insight into the time-dependent mechanism of the process. A series of molecular dynamics simulations using the ff94 force field of the AMBER 6.0 package was therefore run to explore the dynamics of the folding/unfolding pathway. The molecular dynamics method was combined with Poisson-Boltzmann calculations to determine when a change in protonation state was warranted during a trajectory. This revealed a sequence of elementary protonation steps during the folding/unfolding transition and suggests a strong coupling between ionization and folding in cytosine-rich triple-helical triplexes.