Methodology for the calculation of the potential of mean force for a cation-pi complex in water.

We report potential of mean force (PMF) calculations on the interaction between the p-sulfonatocalix[4]arene and a monovalent cation (Cs(+)). It has been recently shown from microcalorimetry and (133)Cs NMR experiments that the association with Cs(+) is governed by favourable cation-pi interactions and is characterized by the insertion of the cation into the cavity of the macrocycle. We show that the PMF calculation based upon a classical model is not able to reproduce both the thermodynamic properties of association and the insertion of the cation. In order to take into account the different contributions of the cation-pi interactions, we develop a new methodology consisting of changing the standard PMF by an additional contribution resulting from quantum calculations. The calculated thermodynamic properties of association are thus in line with the microcalorimetry and (133)Cs NMR experiments and the structure of the complex at the Gibbs free-energy minimum shows the insertion of the cation into the cavity of the calixarene.     

Effects of the cutoff center on the mean potential and pair distribution functions in liquid water

Molecular dynamics simulations of extended simple point charge (SPC/E) water have been performed to study the effects of the truncation of long-range interactions on some calculated bulk properties of the liquid. The mean potential calculated in liquid water is sensitive to the choice of the cutoff center in the water molecule. The pair distribution function is also dependent on this choice, although not as strongly as the mean potential. An analysis is carried out to understand the origin of these effects. A common cutoff center is at the oxygen atom in the water molecule, but our study shows that this choice does not yield a mean potential value consistent with a more accurate estimate when no cutoff is applied. © 1995 John Wiley & Sons, Inc.     

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.     

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.     

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.     

Implementation of an adaptive umbrella sampling method for the calculation of multidimensional potential of mean force of chemical reactions in solution

We describe the implementation of an adaptive umbrella sampling method, making use of the weighted histogram analysis method, for computing multidimensional potential of mean force for chemical reaction in solution. The approach is illustrated by investigating the effect of aqueous solution on the free energy surface for the proton transfer reaction of [H(3)N-H-NH(3)](+) using a combined quantum mechanical and molecular mechanical AM1/TIP3P potential.     

Structural transitions in interionic force models of liquid AlCl3

We present molecular dynamics simulation of an interionic force model of liquid AlCl₃ along a few paths in the temperature-density plane. These paths include (1) an isobar and an isochore starting from the experimental standard freezing point (SFP), and (2) high-temperature isotherms starting from the isochore passing through the SFP. Our calculations show: the dissociation of dimers and higher molecular clusters into monomers with increasing temperature both along the experimental atmospheric pressure isobar and along the SFP isochore; and the pressure-induced molecular-to-ionic (MI) transition accompanied by, or followed by, solidification on increasing density along the two isotherms. The high-pressure solid structure is of the same layer type, with 6-fold coordinated metal ions, met at standard pressure. Crossing of the mean square displacements of the two ionic species provides a clear signal of the MI transition in the liquid. We discuss the consistency of our results with recent X-ray diffraction experiments on AlCl₃ under pressure.     

Activation energies and potentials of mean force for water cluster evaporation

Activation energies for water cluster evaporation are of interest in many areas of chemical physics. We present the first computation of activation energies for monomer evaporation of small water clusters using the formalism of dynamical nucleation theory (DNT). To this end, individual evaporation rate constants are computed for water clusters (H(2)O)(i), where i=2-10 for temperatures ranging from 243 to 333 K. These calculations employ a parallel sampling technique utilizing a Global Arrays toolkit. The resulting evaporation rate constants for each cluster are then fitted to Arrhenius equations to obtain activation energies. We discuss DNT evaporation rate constants and their relation to potentials of mean force, activation energies, and how to account for nonseparability of the reaction coordinate in the reactant state partition function.     

Defect structures and three-body potential of the mean force for nanoparticles in a nematic host

We have investigated the defect structures and potential of mean force (PMF) of three colloidal spheres immersed in a nematic liquid crystal (for linear and equilateral-triangular configurations), using a coarse-grained theory for the tensor order parameter. At large separations, each sphere is surrounded by an equatorial Saturn ring defect; at short separations, the theory predicts a drastic reorganization of the disclination lines: additional disclination rings are observed in planes perpendicular to the original ones. For both types of configurations, the PMF is attractive and always greater than a pairwise sum of binary PMFs. Comparing the PMFs of linear and equilateral-triangular configurations, we have found the triangular configuration to be more stable than the linear configuration. © 2005 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 43: 1033-1040, 2005     

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.     

Enthalpy-entropy contributions to the potential of mean force of nanoscopic hydrophobic solutes

Entropic and enthalpic contributions to the hydrophobic interaction between nanoscopic hydrophobic solutes, modeled as graphene plates in water, have been calculated using molecular dynamics simulations in the isothermal-isobaric (NPT) ensemble with free energy perturbation methodology. We find the stabilizing contribution to the free energy of association (contact pair formation) to be the favorable entropic part, the enthalpic contribution being highly unfavorable. The desolvation barrier is dominated by the unfavorable enthalpic contribution, despite a fairly large favorable entropic compensation. The enthalpic contributions, incorporating the Lennard-Jones solute-solvent terms, largely determine the stability of the solvent separated configuration. We decompose the enthalpy into a direct solute-solute term, the solute-solvent interactions, and the remainder that contains pressure-volume work as well as contributions due to solvent reorganization. The enthalpic contribution due to changes in water-water interactions arising from solvent reorganization around the solute molecules is shown to have major contribution in the solvent induced enthalpy change.     

Iso-energy contour maps for the interaction of water with DNA double helix in the B conformation

The interaction energy between water with B-DNA double helix is computed for few cylindrical surfaces (enclosing the helix) using analytical pair potentials. The iso-energy contour maps indicate a strong attraction for water extending to three water layers surrounding DNA and very stable bridging structure of water molecules connecting two successive phosphate groups along a single helix in the innermost layer.     

A comparison of methods to compute the potential of mean force.

Most processes occurring in a system are determined by the relative free energy between two or more states because the free energy is a measure of the probability of finding the system in a given state. When the two states of interest are connected by a pathway, usually called reaction coordinate, along which the free-energy profile is determined, this profile or potential of mean force (PMF) will also yield the relative free energy of the two states. Twelve different methods to compute a PMF are reviewed and compared, with regard to their precision, for a system consisting of a pair of methane molecules in aqueous solution. We analyze all combinations of the type of sampling (unbiased, umbrella-biased or constraint-biased), how to compute free energies (from density of states or force averaging) and the type of coordinate system (internal or Cartesian) used for the PMF degree of freedom. The method of choice is constraint-bias simulation combined with force averaging for either an internal or a Cartesian PMF degree of freedom.     

Potential of mean force and pKa profile calculation for a lipid membrane-exposed arginine side chain

The issue of ionizable protein side chains interacting with lipid membranes has been the focus of much attention since the proposal of the paddle model of voltage-gated ion channels, which suggested multiple arginine (Arg) side chains may move through the hydrocarbon core of a lipid membrane. Recent cell biology experiments have also been interpreted to suggest that these side chains would face only small free energy penalties to cross membranes, challenging a long-standing view in membrane biophysics. Here, we employ side chain analog and transmembrane helix models to determine the free energy of an Arg side chain, as a function of protonation state, across a membrane. We observe high free energy barriers for both the charged and neutral states that would prohibit lipid-exposed movement. The mechanisms for charged and neutral Arg transport are, however, very different, with the neutral state experiencing simple dehydration, whereas the charged state experiences a complex mechanism involving connections to the bilayer interfaces that deform the local membrane structure. We employ special methods to ensure sampling of these interfacial connections and decompose the free energy to shed light on the mechanisms. These deformations are found to preferentially stabilize the protonated form, such that the Arg side chain remains almost exclusively charged inside the membrane, with a pKa shift of 相似文献   

On the use of local diffusion models for path ensemble averaging in potential of mean force computations

We use a constant velocity steered molecular dynamics (SMD) simulation of the stretching of deca-alanine in vacuum to demonstrate a technique that can be used to create a surrogate processes approximation (SPA) using the time series that come out of SMD simulations. In this article, the surrogate processes are constructed by first estimating a sequence of local parametric diffusion models along a SMD trajectory and then a single global model is constructed by piecing the local models together through smoothing splines (estimation is made computationally feasible by likelihood function approximations). The SPAs are then "bootstrapped" in order to obtain a plausible range of work values associated with a particular SMD realization. This information is then used to assist in estimating a potential of mean force constructed by appealing to the Jarzynski equality. When this procedure is repeated for a small number of SMD paths, it is shown that the global models appear to come from a single family of closely related diffusion processes. Possible techniques for exploiting this observation are also briefly discussed. The findings of this paper have potential relevance to computationally expensive computer simulations and experimental works involving optical tweezers where it is difficult to collect a large number of samples, but possible to sample accurately and frequently in time.     

The effect of the water/methane interface on methane hydrate cages: the potential of mean force and cage lifetimes

Molecular dynamics simulations were used to determine the influence of a methane-water interface on the position and stability of methane hydrate cages. A potential of mean force was calculated as a function of the separation of a methane hydrate cage and a methane-water interface. The hydrate cages are found to be strongly repelled from the methane gas into the water phase. At low enough temperatures, however, the most favorable location for the hydrate cage is at the interface on the water side. Cage lifetime simulations were performed in bulk water and near a methane-water interface. The methane-water interface increases the cage lifetime by almost a factor of 2 compared to cage lifetimes of cages in bulk water. The potential of mean force and the cage lifetime results give additional explanations for the proposed nucleation of gas hydrates at gas-water interfaces.     

The dynamics of water exchange in gadolinium DOTA complexes studied by transition path sampling and potential of mean force methods

The mechanism of water exchange at the Gd centre of the two isomers of [Gd(iii)DOTA](-) (gadolinate(1-), [1,4,7,10-tetraazacyclododecane-1,4,7,10-tetraacetato(4-)-N1,N4,N7,N10,O1,O4,O7,O10]) has been explored using transition path sampling and potential of mean force methods to sample those regions of phase space inaccessible to standard molecular dynamics simulation. We find that there are definite differences in the details of the solvent rearrangement accompanying the exchange of the capping water molecule for the two isomers. We conclude that these solvent effects, rather than any differences in the binding energy of the capping water, are central in determining the exchange rate. We find that the potential of mean force studies yield absolute and relative rates of water exchange for the two isomers that are in good agreement with experiment.