Specific ion effects in solutions of globular proteins: comparison between analytical models and simulation

Monte Carlo simulations have been performed for ion distributions outside a single globular macroion and for a pair of macroions, in different salt solutions. The model that we use includes both electrostatic and van der Waals interactions between ions and between ions and macroions. Simulation results are compared with the predictions of the Ornstein-Zernike equation with the hypernetted chain closure approximation and the nonlinear Poisson-Boltzmann equation, both augmented by pertinent van der Waals terms. Ion distributions from analytical approximations are generally very close to the simulation results. This demonstrates that properties that are related to ion distributions in the double layer outside a single interface can to a good approximation be obtained from the Poisson-Boltzmann equation. We also present simulation and integral equation results for the mean force between two globular macroions (with properties corresponding to those of hen-egg-white lysozyme protein at pH 4.3) in different salt solutions. The mean force and potential of mean force between the macroions become more attractive upon increasing the polarizability of the counterions (anions), in qualitative agreement with experiments. We finally show that the deduced second virial coefficients agree quite well with experimental results.     

Thermodynamic properties of van der Waals fluids from Monte Carlo simulations and perturbative Monte Carlo theory

Computer simulations have been performed for fluids with van der Waals potential, that is, hard spheres with attractive inverse power tails, to determine the equation of state and the excess energy. On the other hand, the first- and second-order perturbative contributions to the energy and the zero- and first-order perturbative contributions to the compressibility factor have been determined too from Monte Carlo simulations performed on the reference hard-sphere system. The aim was to test the reliability of this "exact" perturbation theory. It has been found that the results obtained from the Monte Carlo perturbation theory for these two thermodynamic properties agree well with the direct Monte Carlo simulations. Moreover, it has been found that results from the Barker-Henderson [J. Chem. Phys. 47, 2856 (1967)] perturbation theory are in good agreement with those from the exact perturbation theory.     

Source of the ice-binding specificity of antifreeze protein type I

Antifreeze proteins (AFPs) are a group of structurally very diverse proteins with the unique capability of binding to the surface of seed ice crystals and inhibiting ice crystal growth. The AFPs bind with high affinity to specific planes of the ice crystal. Previously, this affinity of AFPs has been ascribed to the formation of multiple hydrogen bonds across the protein-ice interface, but more recently van der Waals interactions have been suggested to be the dominant energetic factors for the adsorption. To determine whether van der Waals interactions are also responsible for the binding specificities of AFPs, the protein-ice interaction of the helical AFP Type I from winter flounder (HPLC6) was studied using a Monte Carlo rigid body docking approach. HPLC6 binds in the [1102] direction of the [2021] plane, with the Thr-Ala-Asn surface comprising the protein's binding face. The binding of HPLC6 to this ice plane is highly preferred, but the protein is also found to bind favorably to the [1010] prism plane using a different protein surface comprised of Thr and Ala residues. The results show that van der Waals interactions, despite accounting for most of the intermolecular energy (>80%), are not sufficient to completely explain the AFP binding specificity.     

Monte Carlo studies of isomers, structures, and properties in benzene-cyclohexane clusters: computation strategy and application to the dimer and trimer, (C6H6)(C6H12)n, n = 1-2

Low-temperature isomeric energies, structures, and properties of benzene-cyclohexane clusters are investigated via Monte Carlo simulations. The Monte Carlo strategy is first documented and then applied to (C(6)H(6))(C(6)H(12)) and (C(6)H(6))(C(6)H(12))(2) using four different potential energy surfaces. Results identify a single parallel-displaced dimer isomer. MP2 optimizations and frequency calculations support the Monte Carlo dimer structure and identify the van der Waals mode observed in vibronic spectra. Caloric simulations identify two temperatures where structural transitions occur and imply an experimental temperature below 10 K for dimers in cold supersonic expansions. The (C(6)H(6))(C(6)H(12))(2) studies identify eight independent trimer isomers: three form parallel-stacked (sandwich) arrangements with the two cyclohexane moieties related through a D(6)(h) transformation. The remaining five trimer isomers are trigonal, with no overall symmetry. Caloric studies indicate that the sandwich and trigonal isomeric classes coexist independently below 60 K, consistent with trimer vibronic spectra that contain two independent van der Waals progressions.     

Calculation of solvation free energy from quantum mechanical charge density and continuum dielectric theory

We have combined ultrasoft pseudopotential density functional theory utilizing plane wave basis with a Poisson-Boltzmann/solvent-accessible surface area (PB/SA) model to calculate the solvation free energy of small neutral organic compounds in water. The solute charge density obtained from density functional theory was directly used in solving the Poisson-Boltzmann equation to obtain the reaction field. The polarized electronic wave function of the solute in the solvent was solved by including the reaction field in the density functional Hamiltonian. The quantum mechanical and Poisson-Boltzmann equations were solved self-consistently until the charge density and reaction field converged. Using the solute charge density directly instead of a point-charge representation permitted asymmetric distortion and spreading out of the electron cloud. Because the electron density could leave the van der Waals surface to penetrate into the high-dielectric solvent, the reaction field generated by this density was generally smaller than that obtained by using the point-charge representation. In applying this model to calculate the solvation free energy of 31 small neutral organic molecules spanning a range of 25 kcal/mol, we obtained a root-mean-square error of only 1.3 kcal/mol if we allowed one adjustable parameter to shift the calculated solvation free energy.     

Computational studies on thermodynamic properties, effective diameters, and free volume of argon using an ab initio potential

A quantum mechanical derived ab initio interaction potential for the argon dimer was tested in molecular simulations to reproduce the thermophysical properties of the vapor-liquid phase equilibria using the Gibbs ensemble Monte Carlo simulations as well as the liquid and supercritical equation of state using the NVT Monte Carlo simulations. The ab initio interaction potential was taken from the literature. A recently developed theory [R. Laghaei et al., J. Chem. Phys. 124, 154502 (2006)] was used to compute the effective diameters of argon in fluid phases and the results were subsequently applied in the generic van der Waals theory to compute the free volume of argon. The calculated densities of the coexisting phases, the vapor pressure, and the equation of state show excellent agreement with experimental values. The effective diameters and free volumes of argon are given over a wide range of densities and temperatures. An empirical formula was used to fit the effective diameters as a function of density and temperature. The computed free volume will be used in future investigations to calculate the transport properties of argon.     

Voids, generic van der Waals equation of state, and transport coefficients of liquids

In this Perspective, we discuss the role of voids in transport processes in liquids and the manner in which the concept of voids enters the generic van der Waals equation of state and the modified free volume theory. The density fluctuation theory is then discussed and we show how the density fluctuation theory can be made a molecular theory with the help of the modified free volume theory and the generic van der Waals equation of state. The confluence of the aforementioned three theories makes it possible to calculate the transport coefficients of liquids by using the information on the equilibrium pair correlation function, which can be calculated either by an integral equation theory or Monte Carlo simulations. A number of relations between transport coefficients are also presented, which are derived on the basis of the density fluctuation theory. Since they can be used to obtain one transport coefficient from another they can be very useful in handling experimental and theoretical data. An application of the modified free volume theory to polymer melts is discussed as an example for a theory of transport properties of complex liquids.     

Nanoscale electrostatic actuators in liquid electrolytes

Equilibrium and energy analyses were performed for an electrostatic actuator consisting of two plane parallel electrodes, operated using DC voltages, separated by a liquid electrolyte. One electrode is fixed, and the other electrode is connected to a spring and is free to move. The mechanical equilibrium includes the spring force, the van der Waals force, and the electrochemical force as given by the solution of the linearized Poisson-Boltzmann equation. The electrode separation is determined as a function of the applied potential, the natural (i.e., zeta) potential, the Debeye length, the initial electrode separation, the spring constant, and the Hamaker constant. The actuator exhibits the classical "pull-in" instability. The natural potential increases the critical applied potential but does not significantly affect the critical separation. For zero natural potential, the spring constant does not affect the critical separation. Ratios of the maximum spring energy, the maximum van der Waals energy, and the maximum electrochemical energy were plotted as functions of the Hamaker constant and the initial electrode separation.     

Hybrid Hamiltonian replica exchange molecular dynamics simulation method employing the Poisson-Boltzmann model

A hybrid Hamiltonian replica exchange molecular dynamics simulation scheme based on explicit water model hybrided with Poisson-Boltzmann model is brought out. In this method the motions of atoms are governed by potential energy obtained from explicit water model. However, the exchanges between different replicas under different temperatures are controlled by the solvation energies of the solute calculated using the Poisson-Boltzmann model. In order to get the correct canonical ensembles, the van der Waals radii, which are used to define the dielectric boundary, have to be optimized. The conformational spaces of three distinct pentapeptides, Met-enkephalin, alanine 5, and glycine 5, are explored. We find that with the optimized radii the structural ensembles are nearly identical to those obtained by standard replica exchange simulations while the number of replica needed is reduced greatly.     

New-generation amber united-atom force field

We have developed a new-generation Amber united-atom force field for simulations involving highly demanding conformational sampling such as protein folding and protein-protein binding. In the new united-atom force field, all hydrogens on aliphatic carbons in all amino acids are united with carbons except those on Calpha. Our choice of explicit representation of all protein backbone atoms aims at minimizing perturbation to protein backbone conformational distributions and to simplify development of backbone torsion terms. Tests with dipeptides and solvated proteins show that our goal is achieved quite successfully. The new united-atom force field uses the same new RESP charging scheme based on B3LYP/cc-pVTZ//HF/6-31g** quantum mechanical calculations in the PCM continuum solvent as that in the Duan et al. force field. van der Waals parameters are empirically refitted starting from published values with respect to experimental solvation free energies of amino acid side-chain analogues. The suitability of mixing new point charges and van der Waals parameters with existing Amber covalent terms is tested on alanine dipeptide and is found to be reasonable. Parameters for all new torsion terms are refitted based on the new point charges and the van der Waals parameters. Molecular dynamics simulations of three small globular proteins in the explicit TIP3P solvent are performed to test the overall stability and accuracy of the new united-atom force field. Good agreements between the united-atom force field and the Duan et al. all-atom force field for both backbone and side-chain conformations are observed. In addition, the per-step efficiency of the new united-atom force field is demonstrated for simulations in the implicit generalized Born solvent. A speedup around two is observed over the Duan et al. all-atom force field for the three tested small proteins. Finally, the efficiency gain of the new united-atom force field in conformational sampling is further demonstrated with a well-known toy protein folding system, an 18 residue polyalanine in distance-dependent dielectric. The new united-atom force field is at least a factor of 200 more efficient than the Duan et al. all-atom force field for ab initio folding of the tested peptide.     

A perturbation method for the Ornstein-Zernike equation and the generic van der Waals equation of state for a square well potential model

We calculate the generic van der Waals parameters A and B for a square well model by means of a perturbation theory. To calculate the pair distribution function or the cavity function necessary for the calculation of A and B, we have used the Percus-Yevick integral equation, which is put into an equivalent form by means of the Wiener-Hopf method. This latter method produces a pair of integral equations, which are solved by a perturbation method treating the Mayer function or the well width or the functions in the square well region exterior to the hard core as the perturbation. In the end, the Mayer function times the well width is identified as the perturbation parameter in the present method. In this sense, the present perturbation method is distinct from the existing thermodynamic perturbation theory, which expands the Helmholtz free energy in a perturbation series with the inverse temperature treated as an expansion parameter. The generic van der Waals parameters are explicitly calculated in analytic form as functions of reduced temperature and density. The van der Waals parameters are recovered from them in the limits of vanishing density and high temperature. The equation of state thus obtained is tested against Monte Carlo simulation results and found reliable, provided that the temperature is in the supercritical regime. By scaling the packing fraction with a temperature-dependent hard core, it is suggested to construct an equation of state for fluids with a temperature-dependent hard core that mimicks a soft core repulsive force on the basis of the equation of state derived for the square well model.     

Theoretical and computational investigations on thermodynamic properties, effective site diameters, and molecular free volume of carbon disulfide fluid

A newly proposed theory [R. Laghaei et al., J. Chem. Phys. 124, 154502 (2006)] was extended to polyatomics and applied to compute the density and temperature dependence of the effective site diameters of carbon disulfide fluids. The generic van der Waals (GvdW) theory was also extended to polyatomics in order to calculate the GvdW parameters and the molecular free volume using the effective site diameters as the repulsion-attraction separation distance. A three-site Lennard-Jones potential available in the literature was slightly modified and used in Monte Carlo simulations to obtain the functions appearing in the effective site diameter and GvdW expressions. The interaction potential was examined to reproduce the fluid phase thermodynamic properties using Gibbs ensemble Monte Carlo simulations and also the equation of state in the liquid phase using NVT Monte Carlo (NVT-MC) simulations. Comparison between the simulation results and experimental data shows excellent agreement for the densities of the coexisting phases, the vapor pressure, properties of the predicted critical point, and the equation of state. NVT-MC simulations were performed over a wide range of densities and temperatures in sub- and supercritical regions to compute the effective site diameters, the GvdW parameters, and the molecular free volume. The molecular structure in terms of the site-site pair correlation functions, the density dependence of the effective site diameters, and the density and temperature dependence of the GvdW parameters and molecular free volume were studied and discussed. The GvdW parameters were fitted to empirical expressions as a function of density and temperature. The computed molecular free volume will be used in future investigations to study the transport properties of carbon disulfide.     

Corresponding-states laws for protein solutions

The solvent around protein molecules in solutions is structured and this structuring introduces a repulsion in the intermolecular interaction potential at intermediate separations. We use Monte Carlo simulations with isotropic, pair-additive systems interacting with such potentials. We test if the liquid-liquid and liquid-solid phase lines in model protein solutions can be predicted from universal curves and a pair of experimentally determined parameters, as done for atomic and colloid materials using several laws of corresponding states. As predictors, we test three properties at the critical point for liquid-liquid separation: temperature, as in the original van der Waals law, the second virial coefficient, and a modified second virial coefficient, all paired with the critical volume fraction. We find that the van der Waals law is best obeyed and appears more general than its original formulation: A single universal curve describes all tested nonconformal isotropic pair-additive systems. Published experimental data for the liquid-liquid equilibrium for several proteins at various conditions follow a single van der Waals curve. For the solid-liquid equilibrium, we find that no single system property serves as its predictor. We go beyond corresponding-states correlations and put forth semiempirical laws, which allow prediction of the critical temperature and volume fraction solely based on the range of attraction of the intermolecular interaction potential.     

Molecular theory of thermal conductivity of the Lennard-Jones fluid

In this paper the thermal conductivity of the Lennard-Jones fluid is calculated by applying the combination of the density-fluctuation theory, the modified free volume theory of diffusion, and the generic van der Waals equation of state. A Monte Carlo simulation method is used to compute the equilibrium pair-correlation function necessary for computing the mean free volume and the coefficient in the potential-energy and virial contributions to the thermal conductivity. The theoretical results are compared with our own molecular dynamics simulation results and with those reported in the literature. They agree in good accuracy over wide ranges of density and temperature examined in molecular dynamics simulations. Thus the combined theory represents a molecular theory of thermal conductivity of the Lennard-Jones fluid and by extension simple fluids, which enables us to compute the nonequilibrium quantity by means of the Monte Carlo simulations for the equilibrium pair-correlation function.     

Computer simulations of ligand–protein binding with ensembles of protein conformations: A Monte Carlo study of HIV‐1 protease binding energy landscapes

We present the results of molecular docking simulations with HIV‐1 protease for the sb203386 and skf107457 inhibitors by Monte Carlo simulated annealing. A simplified piecewise linear energy function, the standard AMBER force field, and the AMBER force field with solvation and a soft‐core smoothing component are employed in simulations with a single‐protein conformation to determine the relationship between docking simulations with a simple energy function and more realistic force fields. The temperature‐dependent binding free energy profiles of the inhibitors interacting with a single protein conformation provide a detailed picture of relative thermodynamic stability and a distribution of ligand binding modes in agreement with experimental crystallographic data. Using the simplified piecewise linear energy function, we also performed Monte Carlo docking simulations with an ensemble of protein conformations employing preferential biased sampling of low‐energy protein conformations, and the results are analyzed in connection with the free energy profiles. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 72: 73–84, 1999     

Solvation studies of a zinc finger protein in hydrated ionic liquids

The solvation of the zinc finger protein with the PDB-ID “5ZNF” in hydrated ionic liquids was studied at varying water content. 1-Ethyl-3-methylimidazolium and trifluoromethanesulfonate were the cation and anion, respectively. The protein stability as well as the solvation structure, the shell dynamics and the shell resolved dielectric properties were investigated by means of molecular dynamics simulations. The lengths of the respective trajectories extended up to 200 nanoseconds in order to cover the complete solvent dynamics. Considering the above mentioned properties as a function of the water content they all exhibit a maximum or minimum at the very same mole fraction. While the exact value x(H(2)O) = 0.927 depends on the underlying force field, its origin may be traced back to the competition between the van der Waals and the electrostatic energy of the protein as well as to the transition from aqueous dielectric screening to ionic charge screening with decreasing water content. The parameter-free Voronoi decomposition of space served as a basis for the analysis of most results. In particular, solvation shells were naturally inferred from this concept. In addition to the molecular analysis a mesoscopic view is given in terms of dielectric properties. Thereby, the net dielectric constant is decomposed into contributions from the protein, the first and second solvation shells as well as the bulk. Cross-terms between these components are given, too.     

Coulomb replica‐exchange method: Handling electrostatic attractive and repulsive forces for biomolecules

We propose a new type of the Hamiltonian replica‐exchange method (REM) for molecular dynamics (MD) and Monte Carlo simulations, which we refer to as the Coulomb REM (CREM). In this method, electrostatic charge parameters in the Coulomb interactions are exchanged among replicas while temperatures are exchanged in the usual REM. By varying the atom charges, the CREM overcomes free‐energy barriers and realizes more efficient sampling in the conformational space than the REM. Furthermore, this method requires only a smaller number of replicas because only the atom charges of solute molecules are used as exchanged parameters. We performed Coulomb replica‐exchange MD simulations of an alanine dipeptide in explicit water solvent and compared the results with those of the conventional canonical, replica exchange, and van der Waals REMs. Two force fields of AMBER parm99 and AMBER parm99SB were used. As a result, the CREM sampled all local‐minimum free‐energy states more frequently than the other methods for both force fields. Moreover, the Coulomb, van der Waals, and usual REMs were applied to a fragment of an amyloid‐β peptide (Aβ) in explicit water solvent to compare the sampling efficiency of these methods for a larger system. The CREM sampled structures of the Aβ fragment more efficiently than the other methods. We obtained β‐helix, α‐helix, 3₁₀‐helix, β‐hairpin, and β‐sheet structures as stable structures and deduced pathways of conformational transitions among these structures from a free‐energy landscape. © 2012 Wiley Periodicals, Inc.     

Molecular dynamics simulations for water and ions in protein crystals

The spatial and temporal properties of water and ions in bionanoporous materials-protein crystals-have been investigated using molecular dynamics simulations. Three protein crystals are considered systematically with different morphologies and chemical topologies: tetragonal lysozyme, orthorhombic lysozyme, and tetragonal thermolysin. It is found that the thermal fluctuations of C(alpha) atoms in the secondary structures of protein molecules are relatively weak due to hydrogen bonding. The solvent-accessible surface area per residue is nearly identical in the three protein crystals; the hydrophobic and hydrophilic residues in each crystal possess approximately the same solvent-accessible surface area. Water distributes heterogeneously and has different local structures within the biological nanopores of the three protein crystals. The mobility of water and ions in the crystals is enhanced as the porosity increases and also by the fluctuations of protein atoms particularly in the two lysozyme crystals. Anisotropic diffusion is found preferentially along the pore axis, as experimentally observed. The anisotropy of the three crystals increases in the order: tetragonal thermolysin < tetragonal lysozyme < orthorhombic lysozyme.     

QM/MM-PBSA method to estimate free energies for reactions in proteins

We have developed a method to estimate free energies of reactions in proteins, called QM/MM-PBSA. It estimates the internal energy of the reactive site by quantum mechanical (QM) calculations, whereas bonded, electrostatic, and van der Waals interactions with the surrounding protein are calculated at the molecular mechanics (MM) level. The electrostatic part of the solvation energy of the reactant and the product is estimated by solving the Poisson-Boltzmann (PB) equation, and the nonpolar part of the solvation energy is estimated from the change in solvent-accessible surface area (SA). Finally, the change in entropy is estimated from the vibrational frequencies. We test this method for five proton-transfer reactions in the active sites of [Ni,Fe] hydrogenase and copper nitrite reductase. We show that QM/MM-PBSA reproduces the results of a strict QM/MM free-energy perturbation method with a mean absolute deviation (MAD) of 8-10 kJ/mol if snapshots from molecular dynamics simulations are used and 4-14 kJ/mol if a single QM/MM structure is used. This is appreciably better than the original QM/MM results or if the QM energies are supplemented with a point-charge model, a self-consistent reaction field, or a PB model of the protein and the solvent, which give MADs of 22-36 kJ/mol for the same test set.