Communication: Free-energy analysis of hydration effect on protein with explicit solvent: equilibrium fluctuation of cytochrome c

The relationship between the protein conformation and the hydration effect is investigated for the equilibrium fluctuation of cytochrome c. To elucidate the hydration effect with explicit solvent, the solvation free energy of the protein immersed in water was calculated using the molecular dynamics simulation coupled with the method of energy representation. The variations of the protein intramolecular energy and the solvation free energy are found to compensate each other in the course of equilibrium structural fluctuation. The roles of the attractive and repulsive components in the protein-water interaction are further examined for the solvation free energy. The attractive component represented as the average sum of protein-water interaction energy is dominated by the electrostatic effect and is correlated to the solvation free energy through the linear-response-type relationship. No correlation with the (total) solvation free energy is seen, on the other hand, for the repulsive component expressed as the excluded-volume effect.     

Numerical analysis of the radial-axial structure transition with an applied field in a nematic droplet

In this paper the director configurations and the free energies of a nematic droplet with a surface normal anchoring condition are calculated numerically. For this surface anchoring, a transition occurs between the radial and axial structures with respect to an applied field. In the calculation of the director configurations, the position of a disclination has been fixed. Comparing the free energies for different disclinations, the stable position which gives the minimum free energy is found. In calculating the free energy of a droplet, it is assumed that the free energy density of the nematic phase does not exceed the isotropic free energy density, so that the large distortion in the vicinity of the disclination causes a nematic-isotropic transition and the free energy density of the disclination core becomes equal to the isotropic free energy density. The director configuration in a droplet is calculated as a function of an applied field for different isotropic free energy densities, elastic constant ratios and droplet shapes. The relation between the radial-axial structure transition and these factors are clarified.     

Modeling the catalysis of anti-cocaine catalytic antibody: competing reaction pathways and free energy barriers

The competing reaction pathways and the corresponding free energy barriers for cocaine hydrolysis catalyzed by an anti-cocaine catalytic antibody, mAb15A10, were studied by using a novel computational strategy based on the binding free energy calculations on the antibody binding with cocaine and transition states. The calculated binding free energies were used to evaluate the free energy barrier shift from the cocaine hydrolysis in water to the antibody-catalyzed cocaine hydrolysis for each reaction pathway. The free energy barriers for the antibody-catalyzed cocaine hydrolysis were predicted to be the corresponding free energy barriers for the cocaine hydrolysis in water plus the calculated free energy barrier shifts. The calculated free energy barrier shift of -6.87 kcal/mol from the dominant reaction pathway of the cocaine benzoyl ester hydrolysis in water to the dominant reaction pathway of the antibody-catalyzed cocaine hydrolysis is in good agreement with the experimentally derived free energy barrier shift of -5.93 kcal/mol. The calculated mutation-caused shifts of the free energy barrier are also reasonably close to the available experimental activity data. The good agreement suggests that the protocol for calculating the free energy barrier shift from the cocaine hydrolysis in water to the antibody-catalyzed cocaine hydrolysis may be used in future rational design of possible high-activity mutants of the antibody as anti-cocaine therapeutics. The general strategy of the free energy barrier shift calculation may also be valuable in studying a variety of chemical reactions catalyzed by other antibodies or proteins through noncovalent bonding interactions with the substrates.     

Synergistic approach to improve "alchemical" free energy calculation in rugged energy surface

The authors present an integrated approach to "alchemical" free energy simulation, which permits efficient calculation of the free energy difference on rugged energy surface. The method is designed to obtain efficient canonical sampling for rapid free energy convergence. The proposal is motivated by the insight that both the exchange efficiency in the presently designed dual-topology alchemical Hamiltonian replica exchange method (HREM), and the confidence of the free energy determination using the overlap histogramming method, depend on the same criterion, viz., the overlaps of the energy difference histograms between all pairs of neighboring states. Hence, integrating these two techniques can produce a joint solution to the problems of the free energy convergence and conformational sampling in the free energy simulations, in which lambda parameter plays two roles to simultaneously facilitate the conformational sampling and improve the phase space overlap for the free energy determination. Specifically, in contrast with other alchemical HREM based free energy simulation methods, the dual-topology approach can ensure robust conformational sampling. Due to these features (a synergistic solution to the free energy convergence and canonical sampling, and the improvement of the sampling efficiency with the dual-topology treatment), the present approach, as demonstrated in the model studies of the authors, is highly efficient in obtaining accurate free energy differences, especially for the systems with rough energy landscapes.     

Melting temperature of Pb nanostructural materials from free energy calculation

The thermodynamic properties of lead, including the entropy, heat capacity, Gibbs free energy, and surface free energy have been studied. Based on bulk thermodynamic properties of lead, Gibbs free energy for nanostructural materials is obtained and used to calculate the size-dependent melting point depression for lead nanostructural materials. The studies indicate that the surface free energy difference between solid phase and liquid phase is a decisive factor for the size-dependent melting of nanostructural materials. The calculated results are in agreement with recent experimental values and the available molecular dynamics simulation data.     

Free energy landscapes of electron transfer system in dipolar environment below and above the rotational freezing temperature

Electron transfer reaction in a polar solvent is modeled by a solute dipole surrounded by dipolar molecules with simple rotational dynamics posted on the three-dimensional distorted lattice sites. The interaction energy between the solute and solvent dipoles as a reaction coordinate is adopted and free energy landscapes are calculated by generating all possible states for a 26 dipolar system and by employing Wang-Landau sampling algorithm for a 92 dipolar system. For temperatures higher than the energy scale of dipole-dipole interactions, the free energy landscapes for the small reaction coordinate region have quadratic shape as predicted by Marcus [Rev. Mod. Phys. 65, 599 (1993)] whereas for the large reaction coordinate region, the landscapes exhibit a nonquadratic shape. When the temperature drops, small notched structures appear on the free energy profiles because of the frustrated interactions among dipoles. The formation of notched structure is analyzed with statistical approach and it is shown that the amplitude of notched structure depend upon the segment size of the reaction coordinate and is characterized by the interaction energy among the dipoles. Using simulated free energy landscapes, the authors calculate the reaction rates as a function of the energy gap for various temperatures. At high temperature, the reactions rates follow a bell shaped (inverted parabolic) energy gap law in the small energy gap regions, while it becomes steeper than the parabolic shape in a large energy gap regions due to the nonquadratic shape of the free energy landscape. The peak position of parabola also changes as the function of temperature. At low temperature, the profile of the reaction rates is no longer smooth because of the many local minima of the free energy landscape.     

Free energy partitioning analysis of the driving forces that determine ion density profiles near the water liquid-vapor interface

Free energy partitioning analysis is employed to explore the driving forces for ions interacting with the water liquid-vapor interface using recently optimized point charge models for the ions and SPC/E water. The Na(+) and I(-) ions are examined as an example kosmotrope/chaotrope pair. The absolute hydration free energy is partitioned into cavity formation, attractive van der Waals, local electrostatic, and far-field electrostatic contributions. We first compute the bulk hydration free energy of the ions, followed by the free energy to insert the ions at the center of a water slab. Shifts of the ion free energies occur in the slab geometry consistent with the SPC/E surface potential of the water liquid-vapor interface. Then the free energy profiles are examined for ion passage from the slab center to the dividing surface. The profiles show that, for the large chaotropic I(-) ion, the relatively flat total free energy profile results from the near cancellation of several large contributions. The far-field electrostatic part of the free energy, largely due to the water liquid-vapor interface potential, has an important effect on ion distributions near the surface in the classical model. We conclude, however, that the individual forms of the local and far-field electrostatic contributions are expected to be model dependent when comparing classical and quantum results. The substantial attractive cavity free energy contribution for the larger I(-) ion suggests that there is a hydrophobic component important for chaotropic ion interactions with the interface.     

Free energy of an inhomogeneous polymer-polymer-solvent system

The free energy of an inhomogeneous polymer-polymer-solvent system has been obtained by extending Debye's approach for a polymer-solvent system. Our ternary result reduces to Debye's result for a binary polymer-solvent system and to McMaster's result for a binary polymer-polymer system at the appropriate limits. Like Debye's work, we neglect the entropic gradient contribution to free energy. Based on the ternary result we suggest a generalized expression for the free energy of multiple polymers dissolved in a common solvent. This expression is used to find the free energy of an inhomogeneous polydispersed polymer-solvent system.     

Heating and flooding: a unified approach for rapid generation of free energy surfaces

We propose a general framework for the efficient sampling of conformational equilibria in complex systems and the generation of associated free energy hypersurfaces in terms of a set of collective variables. The method is a strategic synthesis of the adiabatic free energy dynamics approach, previously introduced by us and others, and existing schemes using Gaussian-based adaptive bias potentials to disfavor previously visited regions. In addition, we suggest sampling the thermodynamic force instead of the probability density to reconstruct the free energy hypersurface. All these elements are combined into a robust extended phase-space formalism that can be easily incorporated into existing molecular dynamics packages. The unified scheme is shown to outperform both metadynamics and adiabatic free energy dynamics in generating two-dimensional free energy surfaces for several example cases including the alanine dipeptide in the gas and aqueous phases and the met-enkephalin oligopeptide. In addition, the method can efficiently generate higher dimensional free energy landscapes, which we demonstrate by calculating a four-dimensional surface in the Ramachandran angles of the gas-phase alanine tripeptide.     

Calculation of thermodynamics of gas-liquid solutions for 15 homologous series of solutes in n-hexadecane via a cavity-interaction-based method

A thermodynamic model is presented for the calculation of ΔG _s ^o (298.15 K) (gas → liquid) and the sum of free energy of cavity formation and interaction. The process of solution is divided into (a) the free energy of immersion solvent molecular volume for pure compounds, (b) the free energy of vaporization, (c) a correction term, and (d) the excess free energy. Likewise, it is shown that the free energy of vaporization and correction term play the main role in a gas-liquid solution. This model calculates the total excess free energy of solution, which is divided into two major parts: a combinatorial portion, which accounts for differences in the sizes of molecules, and a residual portion, which takes into account the different solvent-solvent and solvent-solute interactions. The model was applied to a solution of 15 homologous series of polar and nonpolar solutes in n-hexadecane. The text was submitted by the authors in English.     

The influence of aging on surface free energy of corona treated packaging films

In packaging, plastic films are very often applied as overprinting materials. The printing properties of plastic films depend on the value of the surface free energy. Usually, during storage but before printing, the surface free energy is decreasing as a result of ageing. The aim of this study was to analyse the influence of elevated temperature and UV radiation on ageing properties and variation of the free surface energy for three commercially available plastic films: polyethylene, polypropylene and polyethylene terephthalate. The investigation was done experimentally, and the surface free energy was calculated using two approaches, Owens-Wendt and van Oss-Chaudhury-Good. The time change of polar fractions was also analysed. The calculation results were compared and it was concluded that UV radiation causes more changes in surface free energy than elevated temperature. In some cases, surface free energy values calculated with the applied methods show similar trends.     

FEW: A workflow tool for free energy calculations of ligand binding

In the later stages of drug design projects, accurately predicting relative binding affinities of chemically similar compounds to a biomolecular target is of utmost importance for making decisions based on the ranking of such compounds. So far, the extensive application of binding free energy approaches has been hampered by the complex and time‐consuming setup of such calculations. We introduce the free energy workflow (FEW) tool that facilitates setup and execution of binding free energy calculations with the AMBER suite for multiple ligands. FEW allows performing free energy calculations according to the implicit solvent molecular mechanics (MM‐PB(GB)SA), the linear interaction energy, and the thermodynamic integration approaches. We describe the tool's architecture and functionality and demonstrate in a show case study on Factor Xa inhibitors that the time needed for the preparation and analysis of free energy calculations is considerably reduced with FEW compared to a fully manual procedure. © 2013 Wiley Periodicals, Inc.     

Efficiency of monte carlo minimization procedures and their use in analysis of NMR data obtained from flexible peptides

The Monte Carlo minimization (MCM) method of Li and Scheraga is an efficient tool for generating low energy minimized structures of peptides, in particular the global energy minimum (GEM). In a recent article we proposed an enhancement to MCM, called the free energy Monte Carlo minimization (FMCM) procedure. With FMCM the conformational search is carried out with respect to the harmonic free energy, which approximates the free energy of the potential energy wells around the energy minimized structures (these wells are called localized microstates). In this work we apply both methods to the pentapeptide Leu-enkephalin described by the potential energy function ECEPP, and study their efficiency in identifying the GEM structure as well as the global harmonic free energy (GFM) structure. We also investigate the efficiency of these methods to generate localized microstates, which pertain to different energy and harmonic free energy intervals above the GEM and GFM, respectively. Such microstates constitute an important ingredient of our statistical mechanical methodology for analyzing nuclear magnetic resonance data of flexible peptides. Aspects of this methodology related to the stability properties of the localized microstates are examined. © 1997 by John Wiley & Sons, Inc.     

环形嵌段共聚物的胶束化行为模拟简

利用Monte Carlo模拟,对比了相同组成下环形二嵌段共聚物AB和线形三嵌段共聚物ABA在选择性溶剂中的胶束化行为. 结果发现,相同链组成的环形和线形嵌段共聚物的临界胶束浓度(cmc)的差别与A嵌段的比例(fA)及B嵌段间的吸引强度(ε)密切相关. 在fA较小、ε较大的情况下,相应环形嵌段共聚物的cmc值更小;而在fA较大、ε较小的情况下,线形嵌段共聚物的cmc值更小. 为了进一步理解胶束化行为同fA及ε的关系,计算了胶束化过程中熵和势能部分对自由能的贡献. 结果表明,在所研究的fA和ε范围内,环形嵌段共聚物形成胶束时的熵损失更小,因而从熵贡献角度来看,环形嵌段共聚物更易发生胶束化. 而从势能贡献角度来看,当fA较小、ε较大时,环形嵌段共聚物形成胶束时势能有较大程度的降低,对自由能的贡献更大,因而此时环形嵌段共聚物更易发生胶束化. 而当fA较大、ε较小时,线形嵌段共聚物形成胶束时势能有较大程度的降低,对自由能的贡献更大,因而此时线形嵌段共聚物更易发生胶束化. 由此可见,对体系的胶束化自由能进行系统分析,有助于更好地理解环形和线形嵌段共聚物的胶束化行为.     

Activation free energy for proton transfer in solution

Path integral molecular dynamics methods are employed to compute the free energy for proton transfer reactions for strongly hydrogen bonded systems in a polar solvent. The free energy profile is calculated using several different techniques, including: integration of the mean force acting on the proton path with its centroid constrained at different values, the integral form of the free energy calculation in the constrained-reaction-coordinate-dynamics ensemble and direct simulation of the unconstrained dynamics. The results show that estimates of the free energy barrier obtained by harmonic extrapolation are likely to be in error. Both quantum and classical results for the free energy are obtained and compared with simulations using adiabatic quantum dynamics. Comparison of the quantum and classical results show that there are quantum corrections to the solvent contributions to the free energy.     

Calculation of the hydration free energy difference between pyridine and its methyl-substituted derivatives by computer simulation methods

The difference of the hydration free energy of pyridine and its methyl- and symmetrically dimethyl-substituted derivatives has been calculated by the method of free energy perturbation. To check the precision of the results obtained, we have repeated the calculations using thermodynamic integration over different paths. Besides the hydration free energy, the difference in the energy and entropy of hydration between pyridine and monomethyl- and dimethylpyridines has also been determined. The obtained results clearly show that the hydration free energy of the pyridine derivatives becomes more negative with each additional methyl group. However, the accuracy of the calculation does not allow us to draw any conclusion about the dependence of the hydration free energy on the location of the methyl group. The analysis of the Coulomb and Lennard-Jones contributions to the hydration free energy differences has shown the dominance of the latter term. The comparison of the hydration energy and free energy values has shown that there is a strong compensation effect between the energetic and entropic terms of the free energy. The hydration energy of the solute becomes considerably more negative with each additional methyl group due to the dispersion attraction between the methyl group and the surrounding water molecules. The introduction of a methyl group results in an approximately 30 J/(mol K) decrease of the entropy of hydration, and hence, at 300 K, the entropic contribution to the hydration free energy increases by about 9 kJ/mol. Due to their opposite signs, the entropic and energetic contributions largely cancel each other, resulting in approximately an order of magnitude smaller value for the free energy.     

Identifying low variance pathways for free energy calculations of molecular transformations in solution phase

Improving the efficiency of free energy calculations is important for many biological and materials design applications, such as protein-ligand binding affinities in drug design, partitioning between immiscible liquids, and determining molecular association in soft materials. We show that for any pair potential, moderately accurate estimation of the radial distribution function for a solute molecule is sufficient to accurately estimate the statistical variance of a sampling along a free energy pathway. This allows inexpensive analytical identification of low statistical error free energy pathways. We employ a variety of methods to estimate the radial distribution function (RDF) and find that the computationally cheap two-body "dilute gas" limit performs as well or better than 3D-RISM theory and other approximations for identifying low variance free energy pathways. With a RDF estimate in hand, we can search for pairwise interaction potentials that produce low variance. We give an example of a search minimizing statistical variance of solvation free energy over the entire parameter space of a generalized "soft core" potential. The free energy pathway arising from this optimization procedure has lower curvature in the variance and reduces the total variance by at least 50% compared to the traditional soft core solvation pathway. We also demonstrate that this optimized pathway allows free energies to be estimated with fewer intermediate states due to its low curvature. This free energy variance optimization technique is generalizable to solvation in any homogeneous fluid and for any type of pairwise potential and can be performed in minutes to hours, depending on the method used to estimate g(r).     

On‐the‐fly reconstruction of free‐energy profiles using logarithmic mean‐force dynamics

Mean‐force dynamics (MFD), which is a fictitious dynamics for a set of collective variables on a potential of mean‐force, is a powerful algorithm to efficiently explore free‐energy landscapes. Recently, we have introduced logarithmic MFD (LogMFD) (Morishita et al., Phys. Rev. E 2012, 85, 066702) which overcomes difficulties encounterd in free‐energy calculations using standard approaches such as thermodynamic integration. Here, we present a guide to implementing LogMFD calculations paying attention to the practical issues in choosing the parameters in LogMFD. A primary focus is given to the effect of the parameters on the accuracy of the reconstructed free‐energy profiles. A recipe for reducing the errors due to energy dissipation is presented. We also demonstrate that multidimensional free‐energy landscapes can be reconstructed on‐the‐fly using LogMFD, which cannot be accomplished using any other free‐energy calculation techniques. © 2013 Wiley Periodicals, Inc.     

Modeling size effects on the surface free energy of metallic nanoparticles and nanocavities

Based on the rigorous consideration of the bond broken rule and surface relaxation, a model for the size-dependent surface free energy of face-centered-cubic nanoparticles and nanocavities is presented, where the surface relaxation is calculated by the BOLS relationship. It is found that the surface free energy of nanoparticles and nanocavities represents a reverse size effect-the surface free energy of nanoparticles decreases with the decrease of particle size while it rises with the shrinkage of cavities. The size effect on the surface free energy of nanoparticles and nanocavities is not evident in large size ranges, while it becomes more and more distinct with decreasing size, especially for sizes smaller than 10 nm. The present predictions are in good agreement with the available literature data.     

Improved precision and efficiency of free energy calculations for small systems using lambda-scaled atomic masses and separating conformational and transformational sampling

We present results showing the importance of appropriate treatment of atomic masses in molecular dynamics (MD)-based single topology free-energy perturbations (FEPs) on small molecule systems. The reversibility of gas phase simulations is significantly improved by scaling the atomic mass of mutated atoms with the lambda variable normally used for the scaling of energy terms. Because this effect is less pronounced for solvated systems, it will not cancel in estimates of the relative hydration free energy difference. The advantage of mass scaling is demonstrated by a null mutation of ethane to ethane and the calculation of the relative hydration free energy difference between ethane and n-propane. Furthermore, it is found that the simulation time necessary for converged MD/FEPs is prohibitively large for relative hydration free energy calculations on cyclic alkanes. Therefore, we explore an alternative free energy pathway including strongly constrained conformations to improve convergence in FEP simulations of flexible molecules.