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

The pK(a) values of acidic and basic residues buried at the same internal location in a protein are governed by different factors

The pK(a) values of internal ionizable groups are usually very different from the normal pK(a) values of ionizable groups in water. To examine the molecular determinants of pK(a) values of internal groups, we compared the properties of Lys, Asp, and Glu at internal position 38 in staphylococcal nuclease. Lys38 titrates with a normal or elevated pK(a), whereas Asp38 and Glu38 titrate with elevated pK(a) values of 7.0 and 7.2, respectively. In the structure of the L38K variant, the buried amino group of the Lys38 side chain makes an ion pair with Glu122, whereas in the structure of the L38E variant, the buried carboxyl group of Glu38 interacts with two backbone amides and has several nearby carboxyl oxygen atoms. Previously, we showed that the pK(a) of Lys38 is normal owing to structural reorganization and water penetration concomitant with ionization of the Lys side chain. In contrast, the pK(a) values of Asp38 and Glu38 are perturbed significantly owing to an imbalance between favorable polar interactions and unfavorable contributions from dehydration and from Coulomb interactions with surface carboxylic groups. Their ionization is also coupled to subtle structural reorganization. These results illustrate the complex interplay between local polarity, Coulomb interactions, and structural reorganization as determinants of pK(a) values of internal groups in proteins. This study suggests that improvements to computational methods for pK(a) calculations will require explicit treatment of the conformational reorganization that can occur when internal groups ionize.

Protein ionizable groups: pK values and their contribution to protein stability and solubility

The structure, stability, solubility, and function of proteins depend on their net charge and on the ionization state of the individual residues. Consequently, biochemists are interested in the pK values of the ionizable groups in proteins and how these pK values depend on their environment. We review what has been learned about pK values of ionizable groups in proteins from experimental studies and discuss the important contributions they make to protein stability and solubility.

Quantum cluster size and solvent polarity effects on the geometries and Mössbauer properties of the active site model for ribonucleotide reductase intermediate X: a density functional theory study

In studying the properties of metalloproteins using ab initio quantum mechanical methods, one has to focus on the calculations on the active site. The bulk protein and solvent environment is often neglected, or is treated as a continuum dielectric medium with a certain dielectric constant. The size of the quantum cluster of the active site chosen for calculations can vary by including only the first-shell ligands which are directly bound to the metal centers, or including also the second-shell residues which are adjacent to and normally have H-bonding interactions with the first-shell ligands, or by including also further hydrogen bonding residues. It is not well understood how the size of the quantum cluster and the value of the dielectric constant chosen for the calculations will influence the calculated properties. In this paper, we have studied three models (A, B, and C) of different sizes for the active site of the ribonucleotide reductase intermediate X, using density functional theory (DFT) OPBE functional with broken-symmetry methodology. Each model is studied in gas-phase and in the conductor-like screening (COSMO) solvation model with different dielectric constants ε = 4, 10, 20, and 80, respectively. All the calculated Fe-ligand geometries, Heisenberg J coupling constants, and the Mössbauer isomer shifts, quadrupole splittings, and the (57)Fe, (1)H, and (17)O hyperfine tensors are compared. We find that the calculated isomer shifts are very stable. They are virtually unchanged with respect to the size of the cluster and the dielectric constant of the environment. On the other hand, certain Fe-ligand distances are sensitive to both the size of the cluster and the value of ε. ε = 4, which is normally used for the protein environment, appears too small when studying the diiron active site geometry with only the first-shell ligands as seen by comparisons with larger models.

Progress in ab initio QM/MM free-energy simulations of electrostatic energies in proteins: accelerated QM/MM studies of pKa, redox reactions and solvation free energies

Hybrid quantum mechanical/molecular mechanical (QM/MM) approaches have been used to provide a general scheme for chemical reactions in proteins. However, such approaches still present a major challenge to computational chemists, not only because of the need for very large computer time in order to evaluate the QM energy but also because of the need for proper computational sampling. This review focuses on the sampling issue in QM/MM evaluations of electrostatic energies in proteins. We chose this example since electrostatic energies play a major role in controlling the function of proteins and are key to the structure-function correlation of biological molecules. Thus, the correct treatment of electrostatics is essential for the accurate simulation of biological systems. Although we will be presenting different types of QM/MM calculations of electrostatic energies (and related properties) here, our focus will be on pKa calculations. This reflects the fact that pKa's of ionizable groups in proteins provide one of the most direct benchmarks for the accuracy of electrostatic models of macromolecules. While pKa calculations by semimacroscopic models have given reasonable results in many cases, existing attempts to perform pKa calculations using QM/MM-FEP have led to discrepancies between calculated and experimental values. In this work, we accelerate our QM/MM calculations using an updated mean charge distribution and a classical reference potential. We examine both a surface residue (Asp3) of the bovine pancreatic trypsin inhibitor and a residue buried in a hydrophobic pocket (Lys102) of the T4-lysozyme mutant. We demonstrate that, by using this approach, we are able to reproduce the relevant side chain pKa's with an accuracy of 3 kcal/mol. This is well within the 7 kcal/mol energy difference observed in studies of enzymatic catalysis, and is thus sufficient accuracy to determine the main contributions to the catalytic energies of enzymes. We also provide an overall perspective of the potential of QM/MM calculations in general evaluations of electrostatic free energies, pointing out that our approach should provide a very powerful and accurate tool to predict the electrostatics of not only solution but also enzymatic reactions, as well as the solvation free energies of even larger systems, such as nucleic acid bases incorporated into DNA.

Towards a quantitative understanding of protein hydration and volumetric properties

Herein, we probe by pressure perturbation calorimetry (PPC) the coefficient of thermal expansion, the volumetric and the hydration properties of variants of a hyperstable variant of staphylococcal nuclease (SNase), Delta+PHS. The temperature-dependent volumetric properties of the folded and unfolded states of the wild-type protein are calculated with previously published data. The present PPC results are used to interpret the volume diagram and expansivity at a molecular level. We conclude that the expansivity of the unfolded state is, to a first approximation, temperature independent, while that of the folded state decreases with increasing temperature. Our data suggest that at low temperature the defining contribution to DeltaV comes mainly from excluded volume differences and DeltaV for unfolding is negative. In contrast, at high temperatures, differential solvation due to the increased exposed surface area of the unfolded state and, in particular, its larger thermal volume linked to the increased conformational dynamics of the unfolded state ensemble takes over and DeltaV for unfolding eventually becomes positive.

Random walk in orthogonal space to achieve efficient free-energy simulation of complex systems

In the past few decades, many ingenious efforts have been made in the development of free-energy simulation methods. Because complex systems often undergo nontrivial structural transition during state switching, achieving efficient free-energy calculation can be challenging. As identified earlier, the "Hamiltonian" lagging, which reveals the fact that necessary structural relaxation falls behind the order parameter move, has been a primary problem for generally low free-energy simulation efficiency. Here, we propose an algorithm by achieving a random walk in both the order parameter space and its generalized force space; thereby, the order parameter move and the required conformational relaxation can be efficiently synchronized. As demonstrated in both the alchemical transition and the conformational transition, a leapfrog improvement in free-energy simulation efficiency can be obtained; for instance, (i) it allows us to solve a notoriously challenging problem: accurately predicting the pK(a) value of a buried titratable residue, Asp-66, in the interior of the V66E staphylococcal nuclease mutant, and (ii) it allows us to gain superior efficiency over the metadynamics algorithm.

High tolerance for ionizable residues in the hydrophobic interior of proteins

Internal ionizable groups are quite rare in water-soluble globular proteins. Presumably, this reflects the incompatibility between charges and the hydrophobic environment in the protein interior. Here we show that proteins can have an inherently high tolerance for internal ionizable groups. The 25 internal positions in staphylococcal nuclease were substituted one at a time with Lys, Glu, or Asp without abolishing enzymatic activity and without detectable changes in the conformation of the protein. Similar results with substitutions of 6 randomly chosen internal positions in ribonuclease H with Lys and Glu suggest that the ability of proteins to tolerate internal ionizable groups might be a property common to many proteins. Eighty-six of the 87 substitutions made were destabilizing, but in all but one case the proteins remained in the native state at neutral pH. By comparing the stability of each variant protein at two different pH values it was established that the pK(a) values of most of the internal ionizable groups are shifted; many of the internal ionizable groups are probably neutral at physiological pH values. These studies demonstrate that special structural adaptations are not needed for ionizable groups to exist stably in the hydrophobic interior of proteins. The studies suggest that enzymes and other proteins that use internal ionizable groups for functional purposes could have evolved through the random accumulation of mutations that introduced ionizable groups at internal positions, followed by evolutionary adaptation and optimization to modulate stability, dynamics, and other factors necessary for function.

pKa of residue 66 in Staphylococal nuclease. I. Insights from QM/MM simulations with conventional sampling

A combined quantum mechanical/molecular mechanical (QM/MM) potential function is used in a thermodynamic integration approach to calculate the pK(a) of residue 66 in two mutants (V66E, V66D) of Staphylococal nuclease relative to solution. Despite the similarity in chemical nature and experimentally measured pK(a) of the two buried titritable residues, the behaviors of the two mutants and the computed pK(a) values vary greatly in the simulations. For Glu66, the side chain is consistently observed to spontaneously flip out from the protein interior during titration, and the overall protein structure remains stable throughout the simulations. The computed pK(a) shifts using conventional sampling techniques with multiple nanoseconds per lambda window (Set A and B) are generally close to the experimental value, therefore indicating that large-scale conformational rearrangements are not as important for V66E as suggested by the recent study of Warshel and co-worker. For Asp66, by contrast, flipping of the shorter side chain is not sufficient for getting adequate solvent stabilization of the ionized state. As a result, more complex behaviors such as partial unfolding of a nearby beta-sheet region is observed, and the computed pK(a) shift is substantially higher than the experimental value unless Asp66 is biased to adopt the similar configurations as Glu66 in the V66E simulations. Collectively, these studies suggest that the lack of electronic polarization is not expected to be the dominant source of error in microscopic pK(a) shift calculations, while the need of enhanced sampling is more compelling for predicting the pK(a) of buried residues. Furthermore, the comparison between V66E and V66D also highlights that the microscopic interpretation of similar apparent pK(a) values and effective "dielectric constants" of proteins can vary greatly in terms of the residues that make key contributions and the scale of structural/hydration response to titration, the latter of which is difficult to predict a priori. Perturbative analyses of interactions that contribute to the titration free energy point to mutants that can be used to verify the microscopic mechanisms of titration in V66E/D SNase proteins.

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.

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

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

Crystallographic study of hydration of an internal cavity in engineered proteins with buried polar or ionizable groups

Although internal water molecules are essential for the structure and function of many proteins, the structural and physical factors that govern internal hydration are poorly understood. We have examined the molecular determinants of internal hydration systematically, by solving the crystal structures of variants of staphylococcal nuclease with Gln-66, Asn-66, and Tyr-66 at cryo (100 K) and room (298 K) temperatures, and comparing them with existing cryo and room temperature structures of variants with Glu-66, Asp-66, Lys-66, Glu-92 or Lys-92 obtained under conditions of pH where the internal ionizable groups are in the neutral state. At cryogenic temperatures the polar moieties of all these internal side chains are hydrated except in the cases of Lys-66 and Lys-92. At room temperature the internal water molecules were observed only in variants with Glu-66 and Tyr-66; water molecules in the other variants are probably present but they are disordered and therefore undetectable crystallographically. Each internal water molecule establishes between 3 and 5 hydrogen bonds with the protein or with other internal water molecules. The strength of interactions between internal polar side chains and water molecules seems to decrease from carboxylic acids to amides to amines. Low temperature, low cavity volume, and the presence of oxygen atoms in the cavity increase the positional stability of internal water molecules. This set of structures and the physical insight they contribute into internal hydration will be useful for the development and benchmarking of computational methods for artificial hydration of pockets, cavities, and active sites in proteins.

Screening of the active site from water by the incoming ligand triggers catalysis and inhibition in serine proteases

The pKa of the catalytic His57 N(epsilon)H in the tetrahedral complex (TC) of chymotrypsin with trifluoromethyl ketone inhibitors is 4-5 units higher relative to the free enzyme (FE). Such stable TC's, formed with transition state (TS) analog inhibitors, are topologically similar to the catalytic TS. Thus, analysis of this pKa shift may shed light on the role of water solvation in the general base catalysis by histidine. We applied our QM/SCRF(VS) approach to study this shift. The method enables explicit quantum mechanical DFT calculations of large molecular clusters that simulate chemical reactions at the active site (AS) of water solvated enzymes. We derived an analytical expression for the pKa dependence on the degree of water exposure of the ionizable group, and on the total charge in the enzyme AS, Q(A) and Q(B), when the target ionizable functional group (His57 in this study) is in the acidic (A) and basic (B) forms, respectively. Q2(B) > Q2(A) both in the FE and in the TC of chymotrypsin. Therefore, water solvation decreases the relative stability of the protonated histidine in both. Ligand binding reduces the degree of water solvation of the imidazole ring, and consequently elevates the histidine pKa. Thus, the binding of the ligand plays a triggering role that switches on the cascade of catalytic reactions in serine proteases.

Ions in water: Characterizing the forces that control chemical processes and biological structure

The continuum electrostatics model of Debye and Hückel [P. Debye and E. Hückel, On the theory of electrolytes. I. Freezing point depression and related phenomena., Phys. Z. 24 (1923) 185-206.] and its successors utilize a macroscopic dielectric constant and assume that all interactions involving ions are strictly electrostatic, implying that simple ions in water generate electric fields strong enough to orient water dipoles over long distances. However, solution neutron and X-ray diffraction indicate that even di- and tri-valent ions do not significantly alter the density or orientation of water more than two water molecules (5 A) away. Therefore the long range electric fields (generated by simple ions) which can be detected by various resonance techniques such as fluorescence resonance energy transfer over distances of 30 A (about 11 water diameters) or more must be weak relative to the strength of water-water interactions. Two different techniques indicate that the interaction of water with anions is by an approximately linear hydrogen bond, suggesting that the dominant forces on ions in water are short range forces of a chemical nature.

Role of flexibility and polarity as determinants of the hydration of internal cavities and pockets in proteins

Molecular dynamics simulations of Staphylococcal nuclease and of 10 variants with internal polar or ionizable groups were performed to investigate systematically the molecular determinants of hydration of internal cavities and pockets in proteins. In contrast to apolar cavities in rigid carbon structures, such as nanotubes or buckeyballs, internal cavities in proteins that are large enough to house a few water molecules will most likely be dehydrated unless they contain a source of polarity. The water content in the protein interior can be modulated by the flexibility of protein elements that interact with water, which can impart positional disorder to water molecules, or bias the pattern of internal hydration that is stabilized. This might explain differences in the patterns of hydration observed in crystal structures obtained at cryogenic and room temperature conditions. The ability of molecular dynamics simulations to determine the most likely sites of water binding in internal pockets and cavities depends on its efficiency in sampling the hydration of internal sites and alternative protein and water conformations. This can be enhanced significantly by performing multiple molecular dynamics simulations as well as simulations started from different initial hydration states.

Spatial distribution of dielectric shielding in the interior of Pyrococcus furiosus rubredoxin as sampled in the subnanosecond timeframe by hydrogen exchange

Experimental pK values of ionizable sidechains provide the most direct test for models representing dielectric shielding within the interior of a protein. However, only the strongly shifted pK values are particularly useful for discriminating among models. NMR titration studies have usually found only one or two such shifted pK values in each protein, so that the fitting of the experimental data to a uniform internal dielectric (epsilon(int)) model is not well constrained. The observed variation among proteins for such epsilon(int) estimates may reflect nonuniformity of dielectric shielding within each protein interior or qualitative differences between individual proteins. The differential amide kinetic acidities for a series of metal-substituted rubredoxins are shown to be consistent with Poisson-Boltzmann predictions of dielectric shielding that is relatively uniform for all of the amides that are sensitive to the metal charge, a region which corresponds to roughly 1/3 of the internal volume. The effective epsilon(int) values near 6 that are found in this study are significantly lower than many such estimates derived from sidechain pK measurements. The differing timeframes in which dielectric relaxation can respond to the highly transient peptide anion as compared to the longer lived states of the charged sidechains offers an explanation for the lower apparent dielectric constant deduced from these measurements.