Estimating the dielectric constant of the channel protein and pore

When modelling biological ion channels using Brownian dynamics (BD) or Poisson–Nernst–Planck theory, the force encountered by permeant ions is calculated by solving Poisson’s equation. Two free parameters needed to solve this equation are the dielectric constant of water in the pore and the dielectric constant of the protein forming the channel. Although these values can in theory be deduced by various methods, they do not give a reliable answer when applied to channel-like geometries that contain charged particles. To determine the appropriate values of the dielectric constants, here we solve the inverse problem. Given the structure of the MthK channel, we attempt to determine the values of the protein and pore dielectric constants that minimize the discrepancies between the experimentally-determined current–voltage curve and the curve obtained from BD simulations. Two different methods have been applied to determine these values. First, we use all possible pairs of the pore dielectric constant of water, ranging from 20 to 80 in steps of 10, and the protein dielectric constant of 2–10 in steps of 2, and compare the simulated results with the experimental values. We find that the best agreement is obtained with experiment when a protein dielectric constant of 2 and a pore water dielectric constant of 60 is used. Second, we employ a learning-based stochastic optimization algorithm to pick out the optimum combination of the two dielectric constants. From the algorithm we obtain an optimum value of 2 for the protein dielectric constant and 64 for the pore dielectric constant.     

Molecular dynamics studies of a DNA-binding protein: 2. An evaluation of implicit and explicit solvent models for the molecular dynamics simulation of the Escherichia coli trp repressor.

Although aqueous simulations with periodic boundary conditions more accurately describe protein dynamics than in vacuo simulations, these are computationally intensive for most proteins. Trp repressor dynamic simulations with a small water shell surrounding the starting model yield protein trajectories that are markedly improved over gas phase, yet computationally efficient. Explicit water in molecular dynamics simulations maintains surface exposure of protein hydrophilic atoms and burial of hydrophobic atoms by opposing the otherwise asymmetric protein-protein forces. This properly orients protein surface side chains, reduces protein fluctuations, and lowers the overall root mean square deviation from the crystal structure. For simulations with crystallographic waters only, a linear or sigmoidal distance-dependent dielectric yields a much better trajectory than does a constant dielectric model. As more water is added to the starting model, the differences between using distance-dependent and constant dielectric models becomes smaller, although the linear distance-dependent dielectric yields an average structure closer to the crystal structure than does a constant dielectric model. Multiplicative constants greater than one, for the linear distance-dependent dielectric simulations, produced trajectories that are progressively worse in describing trp repressor dynamics. Simulations of bovine pancreatic trypsin were used to ensure that the trp repressor results were not protein dependent and to explore the effect of the nonbonded cutoff on the distance-dependent and constant dielectric simulation models. The nonbonded cutoff markedly affected the constant but not distance-dependent dielectric bovine pancreatic trypsin inhibitor simulations. As with trp repressor, the distance-dependent dielectric model with a shell of water surrounding the protein produced a trajectory in better agreement with the crystal structure than a constant dielectric model, and the physical properties of the trajectory average structure, both with and without a nonbonded cutoff, were comparable.     

Can a simple function for the dielectric response model electrostatic effects in globular proteins?

The relationship between the effective dielectric constant that models the electrostatic effect from a charged side chain in a protein was evaluated both experimentally and theoretically. Experimental values were obtained from the shifts in pKa that resulted from point mutations of side chains in subtilisin. Theoretical values were obtained from an iterative solution to Poisson's equation that considers the dielectric response of the protein and the solvent together with charge positions. There is no simple relationship between the effective dielectric constant and the distance from the charge responsible for the interactions. For some charge positions a linear but not a direct proportional relationship of the effective dielectric with distance of separation was observed. Thus, simple models such as a linear distance-dependence for the dielectric response are not suitable to evaluate electrostatic effects in proteins.     

Electrostatic contribution to the binding stability of protein-protein complexes

To investigate roles of electrostatic interactions in protein binding stability, electrostatic calculations were carried out on a set of 64 mutations over six protein-protein complexes. These mutations alter polar interactions across the interface and were selected for putative dominance of electrostatic contributions to the binding stability. Three protocols of implementing the Poisson-Boltzmann model were tested. In vdW4 the dielectric boundary between the protein low dielectric and the solvent high dielectric is defined as the protein van der Waals surface and the protein dielectric constant is set to 4. In SE4 and SE20, the dielectric boundary is defined as the surface of the protein interior inaccessible to a 1.4-A solvent probe, and the protein dielectric constant is set to 4 and 20, respectively. In line with earlier studies on the barnase-barstar complex, the vdW4 results on the large set of mutations showed the closest agreement with experimental data. The agreement between vdW4 and experiment supports the contention of dominant electrostatic contributions for the mutations, but their differences also suggest van der Waals and hydrophobic contributions. The results presented here will serve as a guide for future refinement in electrostatic calculation and inclusion of nonelectrostatic effects.     

Control of the redox potential of cytochrome c and microscopic dielectric effects in proteins

X-ray structural information provides the opportunity to explore quantitatively the relation between the microenvironments of heme proteins and their redox potentials. This can be done by considering the protein as a "solvent" for its redox center and calculating the difference between the electrostatic energy of the reduced and oxidized heme. Such calculations are presented here, applying the protein dipoles-Langevin dipoles (PDLD) model to cytochrome c. The calculations focus on an evaluation of the difference between the redox potentials of cytochrome c and the octapeptide-methionine complex formed by hydrolysis of cytochrome c. The corresponding difference (approximately 7 kcal/mol) is accounted for by the PDLD calculations. It is found that the protein provides basically a low dielectric environment for the heme, which destabilizes the oxidized heme (relative to its energy in water). The effect of the charged propionic acids on the heme is examined in a preliminary way. It is found that the negative charges of these groups are in a hydrophilic rather than a hydrophobic environment and that the protein-water system provides an effective high dielectric constant for their interaction with the heme. The dual nature of the dielectric effect of the cytochrome (a low dielectric constant for the self-energy of the heme and a high dielectric constant for charge-charge interactions) is discussed. The findings of this work are consistent with the difference between the folding energies of the reduced and oxidized cytochrome c.     

On the importance of being zwitterionic: enzymatic catalysis of decarboxylation and deprotonation of cationic carbon

Carbanion ylides are strongly stabilized by electrostatic interactions between opposing charges at neighboring atoms and this stabilizing electrostatic interaction increases with decreasing dielectric constant of the medium through which the charges interact. Consequently, there is a large increase in the thermodynamic driving force, with decreasing dielectric constant of the reaction medium, for deprotonation of cationic carbon acids and decarboxylation to form related ylides. This favors catalysis of the formation of unstable ylides at enzyme active sites of low dielectric constant. A brief survey of enzymes that catalyze deprotonation of cationic carbon acids and related decarboxylation reactions shows catalysis generally occurs for substrates that are bound in a deep pocket on the protein, with an apparent dielectric constant that is much lower than for the solvent water. In several cases, proton transfer is to a catalytic residue that is relatively weakly solvated in water. We suggest that there is a strong advantage for evolution of protein catalysts that utilize weakly solvated basic side chains which are relatively easily buried in nonpolar active sites that are favorable for zwitterion formation.     

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.     

Ion-binding properties of calbindin D9k: a Monte Carlo simulation study

Monte Carlo simulations are used to calculate the binding constant of two Ca2+ ions to the protein bovine calbindin D9k. The change in binding constant with respect to mutation of charged amino acids, presence of various electrolytes, protein concentration, solution pH, and competitive binding of monovalent ions is investigated. Each of these factors may have a large influence on the binding constant. The simulations are performed in a dielectric continuum model, the so-called primitive model of electrolyte theory, with a fixed protein structure and a uniform dielectric permittivity. The calculated binding constants are in excellent agreement with experimental data and describe changes in the binding constant over six orders of magnitude.     

Simulation of the diffusion-controlled reaction between superoxide and superoxide dismutase. II. Detailed models

A two-stage Brownian dynamics simulation method is used to study the diffusion-influenced bimolecular reaction between superoxide and superoxide dismutase (SOD). The crystal structure of the dimeric enzyme is used in constructing detailed topographical and electrostatic models. Several electrostatic models are considered. In the most realistic, the excluded volume of the protein, which is impermeable to penetration by mobile ions, is assigned a dielectric constant of 2 and the surrounding “solvent” is assigned a value of 78. A finite difference method is used to solve the linearized Poisson–Boltzmann equation. For native SOD, the simulations reproduce the pronounced salt dependence of the rate constant observed experimentally. This salt dependence is attributed to electrostatic interactions between enzyme and substrate that are inherently attractive and amplified by the low dielectric constant of the protein interior. The simulation method is also applied to a modified enzyme, acylated SOD.     

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.     

Experimental evaluation of the effective dielectric constant of proteins

Chemical modifications that alter the net charge of residues in reduction-oxidation proteins influence the redox potential of the protein by changing the electrostatic potential at the redox center. If the locations of the modified charges are known, the shift in redox potential may be used to determine the effective dielectric constant for the interactions between the redox center and modified residues. From the shift in redox potential upon charge neutralization of specific lysines in the hemoprotein cytochrome c, an effective dielectric constant of approximately 50 is calculated for the electrostatic interaction between the modified lysines and heme iron in the native protein.     

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.     

An electrostatic basis for the stability of thermophilic proteins

Two factors provide key contributions to the stability of thermophilic proteins relative to their mesophilic homologues: electrostatic interactions of charged residues in the folded state and the dielectric response of the folded protein. The dielectric response for proteins in a "thermophilic series" globally modulates the thermal stability of its members, with the calculated dielectric constant for the protein increasing from mesophiles to hyperthermophiles. This variability results from differences in the distribution of charged residues on the surface of the protein, in agreement with structural and genetic observations. Furthermore, the contribution of electrostatic interactions to the stability of the folded state is more favorable for thermophilic proteins than for their mesophilic homologues. This leads to the conclusion that electrostatic interactions play an important role in determining the stability of proteins at high temperatures. The interplay between electrostatic interactions and dielectric response also provides further rationalization for the enhanced stability of thermophilic proteins with respect to cold-denaturation. Taken together, the distribution of charged residues and their fluctuations have been shown to be factors in modulating protein stability over the entire range of biologically relevant temperatures.     

Microscopic and semimacroscopic redox calculations: what can and cannot be learned from continuum models

 The major role of electrostatic effects in the control of redox potentials in proteins is now widely appreciated. However, the evaluation and conceptualization of the actual electrostatic contributions is far from trivial, and some models still overlook the nature of electrostatic effects in proteins. This commentary considers different contributions to redox potentials of proteins and discusses the ability of different models to capture these contributions. It is pointed out that macroscopic models which consider the protein as a medium of uniform low dielectric constant cannot reproduce the proper physics of redox proteins. In particular, it is pointed out that the crucial effects of the protein permanent dipoles must be taken into account explicitly and that these permanent dipoles involve effective dielectric constants that are different from those for ionized residues. It is also argued that the reorganization of the protein upon change of oxidation states or ionization of protein residues should be taken into account in redox calculations. The role of water penetration and the inadequacy of describing electrostatic effects by solvent accessibility is briefly mentioned. The nature and the meaning of the "dielectric constant" that should be used in redox calculations are also discussed. It is pointed out that the "dielectric constant" ε_p used in current discretized continuum (DC) models is simply a representation of the contributions which are treated implicitly and not the proper dielectric constant of the protein. It is then explained that the need to use a large "dielectric constant" in DC models reflects, among other factors, the implicit representation of the reorganization of permanent dipoles, and that even an explicit treatment of the fluctuations of ionized surface residues will lead to incorrect results when one uses ε_p=εˉ in continuum treatments. Finally, it is suggested that although the discussion and classification of different contributions to redox potentials is very useful, only the evaluation of the totality of the protein contributions (rather than an arbitrary subset) can lead to a quantitative understanding of redox proteins. Received, accepted: 26 November 1996     

Explicit solvent models in protein pKa calculations.

Continuum methods for calculation of protein electrostatics treat buried and ordered water molecules by one of two approximations; either the dielectric constant of regions containing ordered water molecules is equal to the bulk solvent dielectric constant, or it is equal to the protein dielectric constant though no fixed atoms are used to represent water molecules. A method for calculating the titration behavior of individual residues in proteins has been tested on models of hen egg white lysozyme containing various numbers of explicit water molecules. Water molecules were included based on hydrogen bonding, solvent accessibility, and/or proximity to titrating groups in the protein. Inclusion of water molecules significantly alters the calculated titration behavior of individual titrating sites, shifting calculated pKa values by up to 0.5 pH unit. Our results suggest that approximately one water molecule within hydrogen-bonding distance of each charged group should be included in protein electrostatics calculations.     

Role of the protein's low dielectric constant in the functioning of the photosynthetic reaction center

The high efficiency of the energy storage in the photosynthetic reaction center (RC) is determined by a successful competition of electron transfer from bacteriopheophytin to quinone, as compared to backward recombination of the primary charge-separated state. This relationship is caused by a fine matching of the reorganization energy and the free energy gap making the forward processes activationless, and hence very fast, and mismatching of these two quantities for the backreaction, therefore retarding it strongly. In this study, we show that this matching is due to a low dielectric constant of the RC's protein core because a low dielectric affects strongly electrostatic polarization components of both the reorganization energy and the equilibrium free energy of reaction. If the protein and membrane were replaced by a homogeneous medium with a high dielectric constant, the effective energy storage would be impractical.     

Electrochromic shift of chlorophyll absorption in photosystem I from Synechocystis sp. PCC 6803: a probe of optical and dielectric properties around the secondary electron acceptor

Nanosecond absorption dynamics at approximately 685 nm after excitation of photosystem I (PS I) from Synechocystis sp. PCC 6803 is consistent with electrochromic shift of absorption bands of the Chl a pigments in the vicinity of the secondary electron acceptor A(1). Based on experimental optical data and structure-based simulations, the effective local dielectric constant has been estimated to be between 3 and 20, which suggests that electron transfer in PS I is accompanied by considerable protein relaxation. Similar effective dielectric constant values have been previously observed for the bacterial photosynthetic reaction center and indicate that protein reorganization leading to effective charge screening may be a necessary structural property of proteins that facilitate the charge transfer function. The data presented here also argue against attributing redmost absorption in PS I to closely spaced antenna chlorophylls (Chls) A38 and A39, and suggest that optical transitions of these Chls, along with that of connecting chlorophyll (A40) lie in the range 680-695 nm.     

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.     

Electrostatics of proteins: Description in terms of two dielectric constants simultaneously

In the semi-continuum treatment of the energetics of charge formation (or transfer) inside a protein, two components of the energy are inevitably present: the energy of interaction of the ion with the pre-existing intraprotein electric field, and the energy due to polarization of the medium by the newly formed charge. The pre-existing field is set up by charges (partial or full) of the protein atoms fixed in a definite structure. The calculation of this field involves only the electronic polarization (the optical dielectric constant ϵ_o) of the protein because the polarization due to shifts of heavy atoms has already been accounted for by their equilibrium coordinates. At the same time, the aqueous surroundings should be described by the static constant ϵ_sw, as the positions of water molecules are not fixed. The formation of a new charge, absent in the equilibrium X-ray structure, results in shifts of electrons and polar atoms, i.e., it involves all kinds of medium polarization described by the static dielectric constant of protein ϵ_s. Thus, in calculations of the total energy, two different dielectric constants of the protein are operative simultaneously. This differs from a widely used algorithm employing one effective dielectric constant for both components of the ion's energy. Proteins: 28:174–182, 1997. © 1997 Wiley-Liss Inc.