What are the dielectric “constants” of proteins and how to validate electrostatic models?

Implicit models for evaluation of electrostatic energies in proteins include dielectric constants that represent effect of the protein environment. Unfortunately, the results obtained by such models are very sensitive to the value used for the dielectric constant. Furthermore, the factors that determine the optimal value of these constants are far from being obvious. This review considers the meaning of the protein dielectric constants and the ways to determine their optimal values. It is pointed out that typical benchmarks for validation of electrostatic models cannot discriminate between consistent and inconsistent models. In particular, the observed pK(a) values of surface groups can be reproduced correctly by models with entirely incorrect physical features. Thus, we introduce a discriminative benchmark that only includes residues whose pK(a) values are shifted significantly from their values in water. We also use the semimacroscopic version of the protein dipole Langevin dipole (PDLD/S) formulation to generate a series of models that move gradually from microscopic to fully macroscopic models. These include the linear response version of the PDLD/S models, Poisson Boltzmann (PB)-type models, and Tanford Kirkwwod (TK)-type models. Using our different models and the discriminative benchmark, we show that the protein dielectric constant, epsilon(p), is not a universal constant but simply a parameter that depends on the model used. It is also shown in agreement with our previous works that epsilon(p) represents the factors that are not considered explicitly. The use of a discriminative benchmark appears to help not only in identifying nonphysical models but also in analyzing effects that are not reproduced in an accurate way by consistent models. These include the effect of water penetration and the effect of the protein reorganization. Finally, we show that the optimal dielectric constant for self-energies is not the optimal constant for charge-charge interactions.     

The effect of protein relaxation on charge-charge interactions and dielectric constants of proteins.

The effect of the reorganization of the protein polar groups on charge-charge interaction and the corresponding effective dielectric constant (epsilon(eff)) is examined by the semimicroscopic version of the Protein Dipole Langevin Dipoles (PDLD/S) method within the framework of the Linear Response Approximation (LRA). This is done by evaluating the interactions between ionized residues in the reaction center of Rhodobacter sphaeroides, while taking into account the protein reorganization energy. It is found that an explicit consideration of the protein relaxation leads to a significant increase in epsilon(eff) and that semimicroscopic models that do not take this relaxation into account force one to use a large value for the so-called "protein dielectric constant," epsilon(p), of the Poisson-Boltzmann model or for the corresponding epsilon(in) in the PDLD/S model. An additional increase in epsilon(eff) is expected from the reorganization of ionized residues and from changes in the degree of water penetration. This finding provides further support for the idea that epsilon(in) (or epsilon(p)) represents contributions that are not considered explicitly. The present study also provides a systematic illustration of the nature of epsilon(eff), supporting our previously reported view that charge-charge interactions correspond to a large value of this "dielectric constant," even in protein interiors. It is also pointed out that epsilon(eff) for the interaction between ionizable groups in proteins is very different from the effective dielectric constant, epsilon'(eff), that determines the free energy of ion pairs in proteins (epsilon'(eff) reflects the effect of preoriented protein dipoles). Finally, the problems associated with the search for a general epsilon(in) are discussed. It is clarified that the epsilon(in) that reproduces the effect of protein relaxation on charge-charge interaction is not equal to the epsilon(in) that reproduces the corresponding effect upon formation of individual charges. This reflects fundamental inconsistencies in attempts to cast microscopic concepts in a macroscopic model. Thus one should either use a large epsilon(in) for charge-charge interactions and a small epsilon(in) for charge-dipole interactions or consider the protein relaxation microscopically.     

Role of the dielectric constants of membrane proteins and channel water in ion permeation

Using both analytical solutions obtained from simplified systems and numerical results from more realistic cases, we investigate the role played by the dielectric constant of membrane proteins epsilon(p) and pore water epsilon(w) in permeation of ions across channels. We show that the boundary and its curvature are the crucial factors in determining how an ion's potential energy depends on the dielectric constants near an interface. The potential energy of an ion outside a globular protein has a dominant 1/epsilon(w) dependence, but this becomes 1/epsilon(p) for an ion inside a cavity. For channels, where the boundaries are in between these two extremes, the situation is more complex. In general, we find that variations in epsilon(w) have a much larger impact on the potential energy of an ion compared to those in epsilon(p). Therefore a better understanding of the effective epsilon(w) values employed in channel models is desirable. Although the precise value of epsilon(p) is not a crucial determinant of ion permeation properties, it still needs to be chosen carefully when quantitative comparisons with data are made.     

Dielectric properties of proteins from simulation: the effects of solvent, ligands, pH, and temperature

We have used a standard Fr?hlich-Kirkwood dipole moment fluctuation model to calculate the static dielectric permittivity, epsilon(0), for four different proteins, each of which was simulated under at least two different conditions of pH, temperature, solvation, or ligand binding. For the range of proteins and conditions studied, we calculate values for epsilon(0) between 15 and 40. Our results show, in agreement with prior work, that the behavior of charged residues is the primary determinant of the effective permittivity. Furthermore, only environmental changes that alter the properties of charged residues exert a significant effect on epsilon. In contrast, buried water molecules or ligands have little or no effect on protein dielectric properties.     

A fast and simple method to calculate protonation states in proteins.

A simple model for electrostatic interactions in proteins, based on a distance and position dependent screening of the electrostatic potential, is presented. It is applied in conjunction with a Monte Carlo algorithm to calculate pK(alpha) values of ionizable groups in proteins. The purpose is to furnish a simple, fast, and sufficiently accurate model to be incorporated into molecular dynamic simulations. This will allow for dynamic protonation calculations and for coupling between changes in structure and protonation state during the simulation. The best method of calculating protonation states available today is based on solving the linearized Poisson-Boltzmann equation on a finite difference grid. However, this model consumes far too much computer time to be a practical alternative. Tests are reported for fixed structures on bacteriorhodopsin, lysozyme, myoglobin, and calbindin. The studies include comparisons with Poisson-Boltzmann calculations with dielectric constants 4 and 20 inside the protein, a model with uniform dielectric constant 80 and distance-dependent dielectric models. The accuracy is comparable to that of Poisson-Boltzmann calculations with dielectric constant 20, and it is considerably better than that with epsilon = 4. The time to calculate the protonation at one pH value is at least 100 times less than that of a Poisson-Boltzmann calculation. Proteins 1999;36:474-483.     

The alpha helix dipole: screened out?

Aligned alpha helix peptide dipoles sum to a "macroscopic" dipole parallel to the helix axis that has been implicated in protein folding and function. However, in aqueous solution the dipole is counteracted by an electrostatic reaction field generated by the solvent, and the strength of the helix dipole may reduce drastically from its value in vacuum. Here, using atomic-detail helix models and Poisson-Boltzmann continuum electrostatics calculations, the net effective dipole moment, mu(eff), is calculated. Some initially surprising results are found. Whereas in vacuum mu(eff) increases with helix length, the opposite is found to be the case for transmembrane helices. In soluble proteins, mu(eff) is found to vary strongly with the orientation and position of the helix relative to the aqueous medium. A set of rules is established to estimate of the strength of mu(eff) from graphical inspection of protein structures.     

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.     

Comparison of calculation and experiment implicates significant electrostatic contributions to the binding stability of barnase and barstar

The contributions of electrostatic interactions to the binding stability of barnase and barstar were studied by the Poisson-Boltzmann model with three different protocols: a), the dielectric boundary specified as the van der Waals (vdW) surface of the protein along with a protein dielectric constant (epsilon (p)) of 4; b), the dielectric boundary specified as the molecular (i.e., solvent-exclusion (SE)) surface along with epsilon (p) = 4; and c), "SE + epsilon (p) = 20." The "vdW + epsilon (p) = 4" and "SE + epsilon (p) = 20" protocols predicted an overall electrostatic stabilization whereas the "SE + epsilon (p) = 4" protocol predicted an overall electrostatic destabilization. The "vdW + epsilon (p) = 4" protocol was most consistent with experiment. It quantitatively reproduced the observed effects of 17 mutations neutralizing charged residues lining the binding interface and the measured coupling energies of six charge pairs across the interface and reasonably rationalized the experimental ionic strength and pH dependences of the binding constant. In contrast, the "SE + epsilon (p) = 4" protocol predicted significantly larger coupling energies of charge pairs whereas the "SE + epsilon (p) = 20" protocol did not predict any pH dependence. This study calls for further scrutiny of the different Poisson-Boltzmann protocols and demonstrates potential danger in drawing conclusions on electrostatic contributions based on a particular calculation protocol.     

How to measure and predict the molar absorption coefficient of a protein.

The molar absorption coefficient, epsilon, of a protein is usually based on concentrations measured by dry weight, nitrogen, or amino acid analysis. The studies reported here suggest that the Edelhoch method is the best method for measuring epsilon for a protein. (This method is described by Gill and von Hippel [1989, Anal Biochem 182:319-326] and is based on data from Edelhoch [1967, Biochemistry 6:1948-1954]). The absorbance of a protein at 280 nm depends on the content of Trp, Tyr, and cystine (disulfide bonds). The average epsilon values for these chromophores in a sample of 18 well-characterized proteins have been estimated, and the epsilon values in water, propanol, 6 M guanidine hydrochloride (GdnHCl), and 8 M urea have been measured. For Trp, the average epsilon values for the proteins are less than the epsilon values measured in any of the solvents. For Tyr, the average epsilon values for the proteins are intermediate between those measured in 6 M GdnHCl and those measured in propanol. Based on a sample of 116 measured epsilon values for 80 proteins, the epsilon at 280 nm of a folded protein in water, epsilon (280), can best be predicted with this equation: epsilon (280) (M-1 cm-1) = (#Trp)(5,500) + (#Tyr)(1,490) + (#cystine)(125) These epsilon (280) values are quite reliable for proteins containing Trp residues, and less reliable for proteins that do not. However, the Edelhoch method is convenient and accurate, and the best approach is to measure rather than predict epsilon.     

Folding of DsbB in mixed micelles: a kinetic analysis of the stability of a bacterial membrane protein

Measuring the stability of integrated membrane proteins under equilibrium conditions is hampered by the nature of the proteins' amphiphilic environment. While intrinsic fluorescence is a useful probe for structural changes in water-soluble proteins, the fluorescence of membrane proteins is sensitive to changes in lipid and detergent composition. As an attempt to overcome this problem, I present a kinetic analysis of the folding of a membrane protein, disulfide bond reducing protein B (DsbB), in a mixed micelle system consisting of varying molar ratios of sodium dodecyl sulfate (SDS) and dodecyl maltoside (DM). This analysis incorporates both folding and unfolding rates, making it possible to determine both the stability of the native state and the process by which the protein folds. Refolding and unfolding occur on the second to millisecond timescale and involve only one relaxation phase, when monitored by conventional stopped-flow. The kinetic data indicate that denaturation occurs around 0.3 mole fraction of SDS, in agreement with CD analysis and acrylamide quenching data. The rate constants have been fit to a three-state folding scheme involving the SDS-denatured state, the native state and an unfolding intermediate that accumulates only under unfolding conditions at high mole fractions of SDS. The stability of DsbB is around 4.4 kcal/mol in DM, and this is halved upon reduction of the two periplasmic disulfide bonds, and is sensitive to mutagenesis. With the caveat that kinetic data are always open to alternative interpretations, time-resolved studies in mixed micelles provide a useful approach to measure membrane protein stability over a wide range of concentrations of SDS and DM, as well as a framework for the future characterization of the DsbB folding mechanism.     

Determination of the folding of proteins as a function of denaturants, osmolytes or ligands using circular dichroism

Circular dichroism (CD) is an excellent tool for examining the interactions and stability of proteins. This protocol covers methods to obtain and analyze circular dichroism spectra to measure changes in the folding of proteins as a function of denaturants, osmolytes or ligands. Applications include determination of the free energy of folding of a protein, the effects of mutations on protein stability and the estimation of binding constants for the interactions of proteins with other proteins, DNA or ligands, such as substrates or inhibitors. The experiments require 2-5 h.     

Expanding the pressure technique: insights into protein folding from combined use of pressure and chemical denaturants

The fundamental principles derived from in vitro protein folding experiments have practical application in understanding the pathology of diseases of protein misfolding and for the development of industrial processes to produce proteins as pharmaceuticals and biotechnological reagents. High pressure as a tool to denature or disaggregate proteins offers a number of unique advantages. The emphasis of this review is on how low concentrations of chemical denaturants can be used in combination with high pressure to extend the range and scope of this useful technique. This approach has already been used in a number of studies, which are discussed here in the context of the questions they address. These include: the origin of the volume change observed on protein unfolding, pressure-induced formation of partially structured intermediates, pressure-induced dissociation of oligomeric and aggregated proteins, and the use of volume changes to probe the structure of the transition state. Wider use of hydrostatic pressure as a denaturation tool, facilitated by combination with chemical denaturants, is likely to bring significant advances to our understanding of protein structure, stability and folding, particularly in relation to proteins associated with the amyloid and prion diseases.     

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

The dielectric properties of proteins are poorly understood and difficult to describe quantitatively. This limits the accuracy of methods for structure-based calculation of electrostatic energies and pK(a) values. The pK(a) values of many internal groups report apparent protein dielectric constants of 10 or higher. These values are substantially higher than the dielectric constants of 2-4 measured experimentally with dry proteins. The structural origins of these high apparent dielectric constants are not well understood. Here we report on structural and equilibrium thermodynamic studies of the effects of pH on the V66D variant of staphylococcal nuclease. In a crystal structure of this protein the neutral side chain of Asp-66 is buried in the hydrophobic core of the protein and hydrated by internal water molecules. Asp-66 titrates with a pK(a) value near 9. A decrease in the far UV-CD signal was observed, concomitant with ionization of this aspartic acid, and consistent with the loss of 1.5 turns of alpha-helix. These data suggest that the protein dielectric constant needed to reproduce the pK(a) value of Asp-66 with continuum electrostatics calculations is high because the dielectric constant has to capture, implicitly, the energetic consequences of the structural reorganization that are not treated explicitly in continuum calculations with static structures.     

The structure and dipole moment of globular proteins in solution and crystalline states: use of NMR and X-ray databases for the numerical calculation of dipole moment

The large dipole moment of globular proteins has been well known because of the detailed studies using dielectric relaxation and electro-optical methods. The search for the origin of these dipolemoments, however, must be based on the detailed knowledge on protein structure with atomic resolutions. At present, we have two sources of information on the structure of protein molecules: (1) x-ray databases obtained in crystalline state; (2) NMR databases obtained in solution state. While x-ray databases consist of only one model, NMR databases, because of the fluctuation of the protein folding in solution, consist of a number of models, thus enabling the computation of dipole moment repeated for all these models. The aim of this work, using these databases, is the detailed investigation on the interdependence between the structure and dipole moment of protein molecules. The dipole moment of protein molecules has roughly two components: one dipole moment is due to surface charges and the other, core dipole moment, is due to polar groups such as N--H and C==O bonds. The computation of surface charge dipole moment consists of two steps: (A) calculation of the pK shifts of charged groups for electrostatic interactions and (B) calculation of the dipole moment using the pK corrected for electrostatic shifts. The dipole moments of several proteins were computed using both NMR and x-ray databases. The dipole moments of these two sets of calculations are, with a few exceptions, in good agreement with one another and also with measured dipole moments.     

Protein Folding in the Cell: On the Mechanisms of Its Acceleration

The mechanisms responsible for protein folding in the cell can be divided in two groups. The ones in the first group would be those preventing the aggregation of unfolded polypeptide chains or of incompletely folded proteins, as well as the mechanisms which provide for the energy-consuming unfolding of incorrectly folded structures, giving them a chance to begin a new folding cycle. Mechanisms of this type do not affect the rate of folding (it occurs spontaneously), yet considerably increase the efficiency of the entire process. By contrast, the mechanisms belonging to second group actually accelerate protein folding by exerting a direct influence on the rate-limiting steps of the overall reaction. Although not a conventional one, such a classification helps define the topic of this review. Its main purpose is to discuss the ability of chaperonins (and that of some chaperones) to interact directly with substrate proteins in the course of their folding and thus accelerate the rate-limiting steps of that process. (Mechanisms of protein folding acceleration produced by the action of enzymes, e.g., peptidyl-prolyl cis/trans isomerase and protein disulfide isomerase, are not considered in this review.) Specific cases demonstrating an accelerated folding of some proteins encapsulated in the bacterial chaperonin GroEL cavity are considered, and the conditions favoring such acceleration are examined. Experimental data supporting the notion that the structure and functional properties of GroEL are not optimal for an effective folding of many of its substrate proteins is discussed. The current status of research on the mechanism behind the active participation of different subunits of eucaryotic cytosol chaperonin (CCT) in the final steps of the folding of actin and tubulin is reviewed. Particular attention is devoted to steric chaperones, which dramatically accelerate the formation of the native structure of their substrate proteins by stabilizing certain folding intermediates. The structural foundations underlying the effect of the subtilisin pro-domain on the folding of the mature enzyme are considered. The prospects of future studies into the mechanisms responsible for accelerating protein folding in the cell are commented upon.     

Protein redesign.

It has been demonstrated that, for a number of proteins, it is possible to dramatically alter the connectivities between elements of secondary structure. Remarkably large loop insertions are tolerated and many redesigns have generated proteins that successfully fold to stable, active structures. Some redesigns have been entirely the choice of the investigators, whereas others have incorporated a randomization and selection step to identify optimal sequences. These studies have provided basic guidelines for the rational manipulation of protein structure and stability, they have allowed the dissection of folding pathways and they have generated proteins with the potential for practical therapeutic applications.     

Semi-continuum electrostatic calculations of redox potentials in photosystem I

The midpoint redox potentials (E(m)) of all cofactors in photosystem I from Synechococcus elongatus as well as of the iron-sulfur (Fe(4)S(4)) clusters in two soluble ferredoxins from Azotobacter vinelandii and Clostridium acidiurici were calculated within the framework of a semi-continuum dielectric approach. The widely used treatment of proteins as uniform media with single dielectric permittivity is oversimplified, particularly, because permanent charges are considered both as a source for intraprotein electric field and as a part of dielectric polarizability. Our approach overcomes this inconsistency by using two dielectric constants: optical epsilon(o)=2.5 for permanent charges pre-existing in crystal structure, and static epsilon(s) for newly formed charges. We also take into account a substantial dielectric heterogeneity of photosystem I revealed by photoelectric measurements and a liquid junction potential correction for E(m) values of relevant redox cofactors measured in aprotic solvents. We show that calculations based on a single permittivity have the discrepancy with experimental data larger than 0.7 V, whereas E(m) values calculated within our approach fall in the range of experimental estimates. The electrostatic analysis combined with quantum chemistry calculations shows that (i) the energy decrease upon chlorophyll dimerization is essential for the downhill mode of primary charge separation between the special pair P(700) and the primary acceptor A(0); (ii) the primary donor is apparently P(700) but not a pair of accessory chlorophylls; (iii) the electron transfer from the A branch quinone Q(A) to the iron-sulfur cluster F(X) is most probably downhill, whereas that from the B branch quinone Q(B) to F(X) is essentially downhill.     

Downhill versus Barrier-Limited Folding of BBL 2: Mechanistic Insights from Kinetics of Folding Monitored by Independent Tryptophan Probes

Barrier-free downhill folding has been proposed for the peripheral subunit-binding domain BBL. To date, ultrafast kinetic experiments on BBL, which are crucial for a mechanistic understanding of folding, have been hampered by the lack of good intrinsic spectroscopic probes. Here, we present a detailed kinetic characterization of three single-point tryptophan mutants of BBL that have suitable fluorescence properties for following microsecond and nanosecond folding kinetics using temperature jump fluorescence spectroscopy. Experiments were performed at pH 7, which is optimal for stability and minimizes complications that arise from the presence of an alternative native-state conformation of BBL at lower pH. We examined the dependence of rate and equilibrium constants on concentration of denaturant and found that they follow well-established laws allowing kinetic transients to be related to events in folding and compared with equilibrium data. Logarithms of rate constants versus denaturant concentration yielded plots (chevrons) that are characteristic of barrier-limited folding for all mutants investigated, including a truncated sequence that was previously used in the proposal of downhill folding. The thermodynamic quantities calculated from the rate constants were in excellent agreement with those directly determined from equilibrium denaturation based on empirical two-state equations. We found that sequence truncation of BBL as used in studies proposing downhill folding leads to a large loss in helical content and protein stability, which were exacerbated at the low pH used in those studies. The kinetics and equilibria of folding of BBL fit to conventional barrier-limited kinetics.