Modeling implicit reorganization in continuum descriptions of protein-protein interactions

The determination of free energies that govern protein-protein recognition is essential for a detailed molecular understanding of biological specificity. Continuum models of macromolecular interactions, in which the solvent is treated by an implicit representation and the proteins are treated semi-microscopically, are computationally tractable for estimating free energies, yet many questions remain concerning their accuracy. This article reports a continuum analysis of the free-energy changes underlying the binding of 31 interfacial alanine substitutions of two complexes of the antihen egg white lysozyme (HEL) antibody D1.3 bound with HEL or the antibody E5.2. Two implicit schemes for modeling the effects of protein and solvent relaxation were examined, in which the protein environment was treated as either homogeneous with a "protein dielectric constant" of epsilon(p) = 4 or inhomogeneous, with epsilon(p) = 4 for neutral residues and epsilon(p) = 25 for ionized residues. The results showed that the nonuniform dielectric model reproduced the experimental differences better, with an average absolute error of +/-1.1 kcal/mol, compared with +/-1.4 kcal/mol for the uniform model. More importantly, the error for charged residues in the nonuniform model is +/-0.8 kcal/mol and is nearly half of that corresponding to the uniform model. Several substitutions were clearly problematic in determining qualitative trends and probably required explicit structural reorganization at the protein-protein interface.     

A semi-microscopic Monte Carlo study of permeation energetics in a gramicidin-like channel: the origin of cation selectivity.

The influence of a gramicidin-like channel former on ion free energy barriers is studied using Monte Carlo simulation. The model explicitly describes the ion, the water dipoles, and the peptide carbonyls; the remaining degrees of freedom, bulk electrolyte, non-polar lipid and peptide regions, and electronic (high frequency) permittivity, are treated in continuum terms. Contributions of the channel waters and peptide COs are studied both separately and collectively. We found that if constrained to their original orientations, the COs substantially increase the cationic permeation free energy; with or without water present, CO reorientation is crucial for ion-CO interaction to lower cation free energy barriers; the translocation free energy profiles for potassium-, rubidium-, and cesium-like cations exhibit no broad barriers; the lipid-bound peptide interacts more effectively with anions than cations; anionic translocation free energy profiles exhibit well defined maxima. Using experimental data to estimate transfer free energies of ions and water from bulk electrolyte to a non-polar dielectric (continuum lipid), we found reasonable ion permeation profiles; cations bind and permeate, whereas anions cannot enter the channel. Cation selectivity arises because, for ions of the same size and charge, anions bind hydration water more strongly.     

Dielectric behavior of DNA in water–organic co-solvent mixtures

The radiowave dielectric dispersions of DNA in different water–organic co-solvent mixtures have been measured in the frequency range from 100 kHz to 100 MHz, where the polarization mechanism is generally attributed to the confinement of counterions within some specific lengths, either along tangential or perpendicular to the polyion chain. The dielectric dispersions have been analyzed on the basis of two partially different dielectric models, a continuum counterion fluctuation model proposed by Mandel and a discrete charged site model, proposed by Minakata. The influence of the quality of the solvent on the dielectric parameters has been investigated in water–methanol and water–glycerol mixtures at different composition, by varying the permittivity ?_m and the viscosity η of the solvent phase. The analysis of the dielectric spectra in solvents where electrostatic and hydrodynamic interactions vary with the solvent composition suggests that both the two models are able, in principle, to account for the observed high-frequency dielectric behavior. However, while some certain assumptions are necessary about the polyion structure within the Mandel model, no structural prerequisite is needed within the Minakata model, where the polarization mechanism invoked considers a radial counterion exchange with the outer medium, which is largely independent of the local polyion conformation.     

Dielectric properties of aqueous solutions of sonicated DNA above 40 MHz

The dielectric properties of sonicated calf-thymus DNA sodium salt in aqueous solutions have been studied in the frequency range from 40 MHz to 2 GHz by time domain spectroscopy (TDS). A dielectric dispersion not previously reported was found, which has a characteristic frequency of about 150 MHz. All of the dielectric parameters are insensitive to the size of DNA fragments and to helix-to-coil transitions. The study of this dispersion as a function of DNA concentration and temperature allows us to conclude that it may be due to counterion fluctuation on short sections, probably in a direction transverse to the macromolecular axis.     

A new theoretical approach to biological self-assembly

Upon biological self-assembly, the number of accessible translational configurations of water in the system increases considerably, leading to a large gain in water entropy. It is important to calculate the solvation entropy of a biomolecule with a prescribed structure by accounting for the change in water–water correlations caused by solute insertion. Modeling water as a dielectric continuum is not capable of capturing the physical essence of the water entropy effect. As a reliable tool, we propose a hybrid of the angle-dependent integral equation theory combined with a multipolar water model and a morphometric approach. Using our methods wherein the water entropy effect is treated as the key factor, we can elucidate a variety of processes such as protein folding, cold, pressure, and heat denaturating of a protein, molecular recognition, ordered association of proteins such as amyloid fibril formation, and functioning of ATP-driven proteins.     

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.     

Temperature-dependent dielectric properties of slightly hydrated horn keratin

With an aim to reveal the mechanism of protein-water interaction in a predominantly two phase model protein system this study investigates the frequency and temperature dependence of dielectric constant epsilon' and loss factor epsilon' in cow horn keratin in the frequency range 30 Hz to 3 MHz and temperature range 30-200 degrees C at two levels of hydration. These two levels of hydration were achieved by exposing the sample to air at 50% relative humidity (RH) at ambient temperature and by evacuating the sample for 72 h at 105 degrees C. A low frequency dispersion (LFD) and an intermediate frequency alpha-dispersion were the two main dielectric responses observed in the air-dried sample. The LFD and the high frequency arm of the alpha-dispersion followed the same fractional power law of frequency. Within the framework of percolation cluster model these dispersions, respectively have been attributed to percolation of protons between and within the clusters of hydrogen-bonded water molecules bound to polar or ionizable protein components. The alpha-dispersion peak, which results from intra-cluster charge percolation conformed to Cole-Cole modified Debye equation. Temperature dependence of the dielectric constant in the air-dried sample exhibited peaks at 120 and 155 degrees C which have been identified as temperatures of onset of release of water bound to polar protein components in the amorphous and crystalline regions, respectively. An overall rise in the permittivity was observed above 175 degrees C, which has been identified as the onset of chain melting in the crystalline region of the protein.     

Estimation of Free Energy Systematic Errors in Molecular Simulations of Globular Proteins Surrounded by Finite Water Clusters. One Center Multipole Expansion of Reaction Field Differences

Free energy calculated in simulations on the atomic level (Monte Carlo or Molecular Dynamics) has a systematic error, if the water shell surrounding a globular protein is finite. The error (“cluster error”) is equal to a difference of free energies obtained in simulations with an infinite and finite water shell. In this work a continuum dielectric model was used to estimate the “cluster error”. A multipole expansion of the estimate was performed for a water shell with a spherical outer boundary. The expansion has very simple form. Each term is a product of two functions, one of them depending only on the charge's conformation, and the other one only on dielectric properties of the system. There are two practical uses of the expansion. First, it may be used to estimate the “cluster error” in a simulation already made; second, it may be used to plan a simulation in such a way that the “cluster error” is minimal. Numerical values of the largest terms in the multipole expansion corresponding to a typical system in simulations of globular proteins are given.     

A microscopic electrostatic model for the amphotericin B channel.

A microscopic model of an amphotericin B channel is proposed. The structure of the pores is generated using the atomic coordinates of the molecule in the structure determined experimentally by X-ray diffraction. The net charges of the atoms are determined by Mulliken analysis. With these charges the electrostatic energy profiles are calculated for a monovalent ion passing through the channels formed by different number of antibiotic molecules having different radii. The water inside the channel was considered through a continuum medium using the dielectric constant of the bulk, and the membrane contribution was included using the virtual images of the pore in a dielectric slab of epsilon = 3. The model satisfactorily explains the permeability and selectivity characteristics as well as other observations yet unexplained. The electrostatic profiles obtained reinforce the hypothesis of the existence of channels formed by a variable number of units.     

The dielectric increments of aqueous polyelectrolyte solutions: a scaling approach

This paper deals with dielectric dispersion curves (covering a frequency range from a few Hz to 100 MHz) of Na-poly(styrene-sulfonate) of 65,000 < or = Mw < or = 1,060,000 g mol(-1) in aqueous solutions. The values of the low frequency (dielectric increment1) and high frequency (dielectric increment2) dielectric increments, obtained from the experimental curves matched to a superposition of two Cole-Cole equations, have been analyzed in terms of their concentration and molar mass dependence. The concentrations C (g l(-1)) of the various solutions were mostly situated in the transition regime defined by Odijk [T. Odijk, Macromolecules 12 (1979) 688] between the dilute regime (C < Cg*) and the semi-dilute one (C > C**), and wherein the characteristic concentration C* marks the onset of flexibility effects on the polyion behavior. It has been shown that in the concentration range Cg* < C < C** the increments in both frequency domains satisfy a scaling relation dielectric increment(j) = Bj M(nu j) (C/C*)(mu j) with molar mass independent exponents nu j and mu j changing around C*. Their values are different for dielectric increment1 and dielectric increment2, except for mu above C* where both increments appear to become concentration-independent. Below Cg*, in the dilute regime, the two dispersion domains seem to merge. The increment dielectric increment = relative permittivity (0) - high frequency limit of relative permittivity is molar mass independent if scaled to (C/Cg*). The molar mass dependence of the increments as a function of the macromolecular concentration rhoP, dielectric increment or dielectric increment(j) approximately Mgamma (rhoP)mu, also reveals differences between the different concentration regimes. Extrapolation from above Cg* to zero concentration is thus unjustified.     

A precise analytical method for calculating the electrostatic energy of macromolecules in aqueous solution

A new method for calculating the total electrostatic free energy of a macromolecule in solution is presented. It is applicable to molecules of arbitrary shape and size, including membranes or macromolecular assemblies with substrate molecules and ions. The method is derived from integrating the energy density of the electrostatic field and is termed the field energy method. It is based on the dielectric model, in which the solute and the surrounding water are regarded as different continuous dielectrics. The field energy method yields both the interaction energy between all charge pairs and the self energy of single charges, effectively accounting for the interaction with water. First, the dielectric boundary and mirror charges are determined for all charges of the solute. The energy is then given as a simple function of the interatomic distances, and the standard atomic partial charges and volumes. The interaction and self energy are shown to result from three-body and pairwise interactions. Both energy terms explicitly involve apolar atoms, revealing that apolar groups are also subject to electrostatic forces. We applied the field energy method to a spherical model protein. Comparison with the Kirkwood solution shows that errors are within a small percentage. As a further test, the field energy method was used to calculate the electrostatic potential of the protein superoxide dismutase. We obtained good agreement with the result from a program that implements the numerical finite difference algorithm. The field energy method provides a basis for energy minimization and dynamics programs that account for the solvent and screening effect of water at little computational expense.     

The coordination chemistry of the structural zinc ion in alcohol dehydrogenase studied by ab initio quantum chemical calculations

The coordination chemistry of the structural zinc ion in horse liver alcohol dehydrogenase has been examined by quantum chemical geometry optimisations. It is shown that all four cysteine ligands are deprotonated in the enzyme, not only two of them as has been suggested. The Zn-S bond lengths are very sensitive to the theoretical treatment; in vacuum they are predicted to be 15 pm longer than in the crystal structure. Half of this discrepancy is due to electronic correlation, the rest can be attributed to screening of the negative sulphide charges by the enzyme, in particular by N-H-S hydrogen bonds. The potential surface is rather flat, so the large difference in geometry between the crystal and the vacuum structure corresponds to an energy change of less than 35 kJ/mol. The experimental bond lengths can be reproduced only with methods that account explicitly for the enzyme. A dielectric continuum model gives bond lengths which are too long, indicating that the enzyme solvates the coordination sphere better than water. Thus, the structural zinc ion can be used as a sensitive test of methods which try to model the surrounding medium in quantum chemical computations.     

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     

Electrical Properties of the Membranes of the Pleuropneumonia-Like Organism A 5969

The electrical properties of the pleuropneumonia-like organism A 5969 have been determined over the frequency range from 0.5 to 250 Mcps. The frequency dependence of the dielectric constant and conductivity of PPLO suspensions is completely consistent with the existence of a membrane. The PPLO has an internal conductance which in part reflects its ionic equilibrium with normal nutrient and macromolecular constituents. But it is fairly independent from variation in external ionic strength.