Role of attractive methane-water interactions in the potential of mean force between methane molecules in water

On the basis of a Gaussian quasichemical model of hydration, a model of non-van der Waals character, we explore the role of attractive methane-water interactions in the hydration of methane and in the potential of mean force between two methane molecules in water. We find that the hydration of methane is dominated by packing and a mean-field energetic contribution. Contributions beyond the mean-field term are unimportant in the hydration phenomena for a hydrophobic solute such as methane. Attractive solute-water interactions make a net repulsive contribution to these pair potentials of mean force. With no conditioning, the observed distributions of binding energies are super-Gaussian and can be effectively modeled by a Gumbel (extreme value) distribution. This further supports the view that the characteristic form of the unconditioned distribution in the high-epsilon tail is due to energetic interactions with a small number of molecules. Generalized extreme value distributions also effectively model the results with minimal conditioning, but in those cases the distributions are sufficiently narrow that the details of their shape are not significant.     

Hydrophobicity within the three-dimensional Mercedes-Benz model: potential of mean force

We use the three-dimensional Mercedes-Benz model for water and Monte Carlo simulations to study the structure and thermodynamics of the hydrophobic interaction. Radial distribution functions are used to classify different cases of the interaction, namely, contact configurations, solvent separated configurations, and desolvation configurations. The temperature dependence of these cases is shown to be in qualitative agreement with atomistic models of water. In particular, while the energy for the formation of contact configurations is favored by entropy, its strengthening with increasing temperature is accounted for by enthalpy. This is consistent with our simulated heat capacity. An important feature of the model is that it can be used to account for well-converged thermodynamics quantities, e.g., the heat capacity of transfer. Microscopic mechanisms for the temperature dependence of the hydrophobic interaction are discussed at the molecular level based on the conceptual simplicity of the model.     

On relationships between surfactant type and globular proteins interactions in solution

The binding of sodium perfluorooctanoate (C8FONa), sodium octanoate (C8HONa), lithium perfluorooctanoate (C8FOLi), and sodium dodecanoate (C12HONa) onto myoglobin, ovalbumin, and catalase in water has been characterized using electrophoretic mobility. The tendency of the protein-surfactant complexes to change their charge in the order catalase < ovalbumin < myoglobin was observed which was related to the contents of alpha-helices in the proteins. alpha-Helices are more hydrophobic than beta-sheets. The effect of surfactant on the zeta potentials follows C8HONa < C8FONa < C8FOLi < C12HONa for catalase and ovalbumin; and C8HONa < C8FOLi < C8FONa < C12HONa for myoglobin. The numbers of binding sites on the proteins were determined from the observed increases of the zeta-potential as a function of surfactant concentration in the regions where the binding was a consequence of the hydrophobic effect. The Gibbs energies of binding of the surfactants onto the proteins were evaluated. For all systems, Gibbs energies are negative and large at low concentrations (where binding to the high energy sites takes place) and become less negative at higher ones. This fact suggests a saturation process. Changes in Gibbs energies with the different proteins and surfactants under study have been found to follow same sequence than that found for the charge. The role of hydrophobic interactions in these systems has been demonstrated to be the predominant.     

Intermolecular interactions of globular proteins in the crystal state

The present work is devoted to investigation of thermal transitions in the crystals of seven proteins to compare the protein globule stability in crystal and solution. Calorimetry methods, electron and optical microscopy, as well as x-ray diffraction studies are used. It is found that protein crystals do not melt and that the destruction of the crystal lattice is a result of protein globule denaturation within the crystal. It is demonstrated that during the heating of pepsin and DF-trypsin crystals it is possible to observe phase transition of the first order. Equilibrium temperatures of protein denaturation in crystals and in solution coincide. The peculiarities of the crystal state are revealed in the increasing thermal transition cooperativity and the system relaxation period.     

BLEEP—potential of mean force describing protein–ligand interactions: I. Generating potential

We have developed BLEEP (biomolecular ligand energy evaluation protocol), an atomic level potential of mean force (PMF) describing protein–ligand interactions. The pair potentials for BLEEP have been derived from high-resolution X-ray structures of protein–ligand complexes in the Brookhaven Protein Data Bank (PDB), with a careful treatment of homology. The use of a broad variety of protein–ligand structures in the derivation phase gives BLEEP more general applicability than previous potentials, which have been based on limited classes of complexes, and thus represents a significant step forward. We calculate the distance distributions in protein–ligand interactions for all 820 possible pairs that can be chosen from our set of 40 different atom types, including polar hydrogen. We then use a reverse Boltzmann methodology to convert these into energy-like pair potential functions. Two versions of BLEEP are calculated, one including and one excluding interactions between protein and water. The pair potentials are found to have the expected forms; polar and hydrogen bonding interactions show minima at short range, around 3.0 Å, whereas a typical hydrophobic interaction is repulsive at this distance, with values above 4.0 Å being preferred. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 1165–1176, 1999     

Toward making the mean spherical approximation of primitive model electrolytes analytic: an analytic approximation of the MSA screening parameter

The mean spherical approximation (MSA) for the primitive model of electrolytes provides reasonable estimates of thermodynamic quantities such as the excess chemical potential and screening length. It is especially widely used because of its explicit formulas so that numerically solving equations is minimized. As originally formulated, the MSA screening parameter Γ (akin to the reciprocal of the Debye screening length) does not have an explicit analytic formula; an equation for Γ must be solved numerically. Here, an analytic approximation for Γ is presented whose relative error is generally ?10(-5). If more accuracy is desired, one step of an iterative procedure (which also produces an explicit formula for Γ) is shown to give relative errors within machine precision in many cases. Even when ion diameter ratios are ～10 and ion valences are ～10, the relative error for the analytic approximation is still ?10(-3) and for the single iterative substitution it is ?10(-9).     

Hydrophobic interactions between methane and a nanoscopic pocket: three dimensional distribution of potential of mean force revealed by computer simulations

We consider a model system of methane molecule and a hemispherical, hydrophobic pocket of an 8 A radius, remaining together in aqueous environment. A spatial map of potential of mean force acting on methane molecule due to presence of pocket is constructed, based on a series of explicit solvent molecular dynamics simulations. A relation between free energy variations associated with methane translocations and accompanying changes in solvent density distribution is analyzed. A funnel-like area where free energy is diminished with respect to bulk is present over the pocket entrance and extends up to 9 A toward the bulk solvent. In order to get into the pocket, however, methane has to cross a free energy barrier, which is more prominent around the circumferential part of pocket entrance, while achieving bulklike free energy values at the very center. As a methane molecule crosses this barrier, the pocket gets completely dehydrated, which leads to "hydrophobic collapse," manifested by a sharp decrease in free energy. We find that the observed free energy changes are closely related to interactions between the methane hydration shell and the surrounding solvent. Results presented here are a continuation of our previous studies of methane-pocket systems.     

Patch controlled attractive electrostatic interactions between similarly charged proteins and adsorbents

A non-uniform distribution of ionic groups on the surface of a protein can lead to attractive electrostatic interactions between itself and an adsorbent, even when the net charge of the protein is of the same type as that of the adsorbent. Chromatographic retention mapping as a function of pH and ionic strength is employed to identify electrostatic repulsion or attraction. Thus, it can be determined whether the global characteristics or an oppositely charged patch controls the adsorptive behavior of a protein on a charged adsorbent.     

Binless estimation of the potential of mean force

For a system in thermal equilibrium, described by classical statistical mechanics, we derive an unbiased estimator for the marginal probability distribution of a coordinate of interest, rho( x). This result provides a "binless" method for estimating the potential of mean force, Phi = -beta (-1) ln rho, eliminating the need to construct histograms or perform numerical thermodynamic integration. In our method, the distribution that we seek to compute is expressed as the sum of a reference distribution, rho 0(x)essentially an initial guess or estimate of rho( x)and a correction term. While the method is valid for arbitrary rho 0, we speculate that an accurate choice of the reference distribution improves the convergence of the method. Using a model molecule, simulated both in vacuum and in solvent, we validate our proposed approach and compare its performance with the histogram and thermodynamic integration methods. We also discuss and validate an extension in which our approach is used in combination with a biasing force, meant to improve uniform sampling of the coordinate of interest.     

Potential of mean force between hydrophobic solutes in the Jagla model of water and implications for cold denaturation of proteins

Using the Jagla model potential we calculate the potential of mean force (PMF) between hard sphere solutes immersed in a liquid displaying water-like properties. Consistent estimates of the PMF are obtained by (a) umbrella sampling, (b) calculating the work done by the mean force acting on the hard spheres as a function of their separation, and (c) determining the position dependent chemical potential after calculating the void space in the liquid. We calculate the PMF for an isobar along which cold denaturation of a model protein has previously been reported. We find that the PMF at contact varies non-monotonically, which is consistent with the observed cold denaturation. The Henry constant also varies non-monotonically with temperature. We find, on the other hand, that a second (solvent separated) minimum of the PMF becomes deeper as temperature decreases. We calculate the solvent-solvent pair correlation functions for solvents near the solute and in the bulk, and show that, as temperature decreases, the two pair correlation functions become indistinguishable, suggesting that the perturbation of solvent structure by the solute diminishes as temperature decreases. The solvent-solute pair correlation function at contact grows as the temperature decreases. We calculate the cavity correlation function and show the development of a solvent-separated peak upon decrease of temperature. These observations together suggest that cold denaturation occurs when the solvent penetrates between hydrophobic solutes in configurations with favorable free energy. Our results thus suggest that cold denatured proteins are structured and that cold denaturation arises from strong solvent-solute interactions, rather than from entropic considerations as in heat denaturation.     

High-resolution mapping of the electrostatic potential in organic thin-film transistors by phase electrostatic force microscopy

We investigate by a scanning probe technique termed phase-electrostatic force microscopy the local electrostatic potential and its correlation to the morphology of the organic semiconductor layer in operating ultra-thin film pentacene field effect transistors. This technique yields a lateral resolution of about 60 nm, allowing us to visualize that the voltage drop across the transistor channel is step-wise. Spatially localized voltage drops, adding up to about 75% of the potential difference between source and drain, are clearly correlated to the morphological domain boundaries in the pentacene film. This strongly supports and gives a direct evidence that in pentacene ultra-thin film transistors charge transport inside the channel is ultimately governed by domain boundaries.     

A sphere-based model for the electrostatics of globular proteins

We propose a model for the electrostatics of globular proteins in which the low dielectric region is replaced by concentric spheres of the appropriate size. The method uses analytical formulas for the dielectric sphere and allows an efficient and accurate treatment of bulk charges. For surface charges, we propose a numerical determination of the sphere radius based on the solvent exposure of the individual atoms. The present implementation of the sphere model yields a good approximation of finite-difference Poisson solvation and interaction energies for a test set of 12 proteins.     

Orientation-dependent potential of mean force for protein folding

We present a solvent-implicit minimalistic model potential among the amino acid residues of proteins, obtained by using the known native structures [deposited in the Protein Data Bank (PDB)]. In this model, the amino acid side chains are represented by a single ellipsoidal site, defined by the group of atoms about the center of mass of the side chain. These ellipsoidal sites interact with other sites through an orientation-dependent interaction potential which we construct in the following fashion. First, the site-site potential of mean force (PMF) between heavy atoms is calculated [following F. Melo and E. Feytsman, J. Mol. Biol. 267, 207 (1997)] from statistics of their distance separation obtained from crystal structures. These site-site potentials are then used to calculate the distance and the orientation-dependent potential between side chains of all the amino acid residues (AAR). The distance and orientation dependencies show several interesting results. For example, we find that the PMF between two hydrophobic AARs, such as phenylalanine, is strongly attractive at short distances (after the obvious repulsive region at very short separation) and is characterized by a deep minimum, for specific orientations. For the interaction between two hydrophilic AARs, such a deep minimum is absent and in addition, the potential interestingly reveals the combined effect of polar (charge) and hydrophobic interactions among some of these AARs. The effectiveness of our potential has been tested by calculating the Z-scores for a large set of proteins. The calculated Z-scores show high negative values for most of them, signifying the success of the potential to identify the native structure from among a large number of its decoy states.     

Crystal nucleation for a model of globular proteins

A continuum model of globular proteins proposed by Talanquer and Oxtoby [J. Chem. Phys. 109, 223 (1998)] is investigated numerically, with particular emphasis on the region near the metastable fluid-fluid coexistence curve. Classical nucleation theory is shown to be invalid not only in the vicinity of the metastable critical point but also close to the liquidus line. An approximate analytic solution is also presented for the shape and properties of the nucleating crystal droplet.     

Effect of electrostatic interaction on the adsorption of globular proteins on octacalcium phosphate crystal film

The electrostatic effect on the adsorption of globular proteins, such as bovine serum albumin (BSA), hen egg white lysozyme (LZM), and beta-lactoglobulin (beta-Lg), on octacalcium phosphate (OCP)-like crystal thin films was investigated. A poorly crystalline thin film was synthesized on a tissue culture polystyrene (TCP) surface and used as a model surface in this study. The solution pH clearly affected the electrostatic properties of both proteins and surface. The adsorbed amounts obtained at quasi-steady state were readily related to the solution pH for each protein. The adsorption rate is fast during the initial period and levels off gradually. The maximum adsorbed mass occurred at pH 7 for BSA and at pH 9 for LZM. beta-Lg adsorbed similar amounts at pHs lower than 9, but the adsorbed mass decreased at pHs higher than 9 where electrostatic repulsion exists. The pH values where the maximum adsorbed mass occurred may be considered as the conditions where electrostatic attraction is most favorable. The adsorbed mass of beta-Lg was the greatest among the proteins of interest while BSA adsorbed the least despite its greater molecular mass. LZM falls into the intermediate region. According to these observations, BSA has undergone conformational changes that prevent further adsorption to a greater extent than the others. A simple relationship between the adsorption rate and the electrostatic properties was not established. However, the order of magnitude of the adsorption rate at the initial period tends to be the same as that of maximum adsorbed mass for each protein.     

Toward the correction of effective electrostatic forces in explicit-solvent molecular dynamics simulations: restraints on solvent-generated electrostatic potential and solvent polarization

Despite considerable advances in computing power, atomistic simulations under nonperiodic boundary conditions, with Coulombic electrostatic interactions and in systems large enough to reduce finite-size associated errors in thermodynamic quantities to within the thermal energy, are still not affordable. As a result, periodic boundary conditions, systems of microscopic size and effective electrostatic interaction functions are frequently resorted to. Ensuing artifacts in thermodynamic quantities are nowadays routinely corrected a posteriori, but the underlying configurational sampling still descends from spurious forces. The present study addresses this problem through the introduction of on-the-fly corrections to the physical forces during an atomistic molecular dynamics simulation. Two different approaches are suggested, where the force corrections are derived from special potential energy terms. In the first approach, the solvent-generated electrostatic potential sampled at a given atom site is restrained to a target value involving corrections for electrostatic artifacts. In the second approach, the long-range regime of the solvent polarization around a given atom site is restrained to the Born polarization, i.e., the solvent polarization corresponding to the ideal situation of a macroscopic system under nonperiodic boundary conditions and governed by Coulombic electrostatic interactions. The restraints are applied to the explicit-water simulation of a hydrated sodium ion, and the effect of the restraints on the structural and energetic properties of the solvent is illustrated. Furthermore, by means of the calculation of the charging free energy of a hydrated sodium ion, it is shown how the electrostatic potential restraint translates into the on-the-fly consideration of the corresponding free-energy correction terms. It is discussed how the restraints can be generalized to situations involving several solute particles. Although the present study considers a very simple system only, it is an important step toward the on-the-fly elimination of finite-size and approximate-electrostatic artifacts during atomistic molecular dynamics simulations.     

Generalizations for the potential of mean force between two isolated colloidal particles from Monte Carlo simulations

A substantial amount of experimental and numerical evidence has shown that the Derjaguin-Landau-Verwey-Overbeek theory is not suitable for describing those colloidal solutions that contain multivalent counterions. Toward improved understanding of such solutions, the authors report Monte Carlo calculations wherein, following Rouzina and Bloomfield, they postulate that, in the absence of van der Waals forces, the overall force between two isolated charged colloidal particles in electrolyte solutions is determined by a dimensionless parameter Gamma=z(2)l(B)/a, which measures the electrostatic repulsion between counterions adsorbed on the macroion surface, where z = counterion valence, l(B)=Bjerrum length, and a = average separation between counterions on the macroion surface calculated as if the macroion were fully neutralized. The authors find, first, that the maximum repulsion between like-charged macroions occurs at Gamma approximately 0.5 and, second, that onset of attraction occurs at Gamma approximately 1.8, essentially independent of the valence and concentration of the surrounding electrolyte. These observations might provide new understanding of interactions between electrostatic double layers and perhaps offer explanations for some electrostatic phenomena related to interactions between DNA molecules or proteins.     

Accuracy assessment of semiempirical molecular electrostatic potential of proteins

 Accurate electrostatic maps of proteins are of great importance in research of protein interaction with ligands, solvent media, drugs, and other biomolecules. The large size of real-life proteins imposes severe limitations on computational methods one can use for obtaining the electrostatic map. Well-known accurate second-order M?ller–Plesset and density functional theory methods are not routinely applicable to systems larger than several hundred atoms. Conventional semiempirical tools, as less resource demanding ones, could be an attractive solution but they do not yield sufficiently accurate calculation results with reference to protein systems, as our analysis demonstrates. The present work performs a thorough analysis of the accuracy issues of the modified neglect of differential overlap type semiempirical Hamiltonians AM1 and PM3 on example of the calculation of the molecular electrostatic potential and the dipole moment of natural amino acids. Real capabilities and limitations of these methods with application to protein modeling are discussed. Received: 26 April 2002 / Accepted: 19 September 2002 / Published online: 14 February 2003     

Ion-specific colloidal aggregation: population balance equations and potential of mean force

Recently reported colloidal aggregation data obtained for different monovalent salts (NaCl, NaNO(3), and NaSCN) and at high electrolyte concentrations are matched with the stochastic solutions of the master equation to obtain bond average lifetimes and bond formation probabilities. This was done for a cationic and an anionic system of similar particle size and absolute charge. Following the series Cl(-), NO(3)(-), SCN(-), the parameters obtained from the fitting procedure to the kinetic data suggest: (i) The existence of a potential of mean force (PMF) barrier and an increasing trend for it for both lattices. (ii) An increasing trend for the PMF at contact, for the cationic system, and a practically constant value for the anionic system. (iii) A decreasing trend for the depth of the secondary minimum. This complex behavior is in general supported by Monte Carlo simulations, which are implemented to obtain the PMF of a pair of colloidal particles immersed in the corresponding electrolyte solution. All these findings contrast the Derjaguin, Landau, Verwey, and Overbeek theory predictions.