The non-polar solvent potential of mean force for the dimerization of alanine dipeptide: the role of solute-solvent van der Waals interactions

The non-polar component of the potential of mean force of dimerization of alanine dipeptide has been calculated in explicit solvent by free energy perturbation. We observe that the calculated PMF is inconsistent with a non-polar hydration free energy model based solely on the solute surface area. The non-linear behavior of the solute-solvent van der Waals energy is primarily responsible for the non-linear dependence of the potential of mean force with respect to the surface area. The calculated potential of mean force is reproduced by an implicit solvent model based on a solvent continuum model for the solute-solvent van der Waals interaction energy and the surface area for the work of forming the solute cavity.     

Description of hydration free energy density as a function of molecular physical properties

A method to calculate the solvation free energy density (SFED) at any point in the cavity surface or solvent volume surrounding a solute is proposed. In the special case in which the solvent is water, the SFED is referred to as the hydration free energy density (HFED). The HFED is described as a function of some physical properties of the molecules. These properties are represented by simple basis functions. The hydration free energy of a solute was obtained by integrating the HFED over the solvent volume surrounding the solute, using a grid model. Of 34 basis functions that were introduced to describe the HFED, only six contribute significantly to the HFED. These functions are representations of the surface area and volume of the solute, of the polarization and dispersion of the solute, and of two types of electrostatic interactions between the solute and its environment. The HFED is described as a linear combination of these basis functions, evaluated by summing the interaction energy between each atom of the solute with a grid point in the solvent, where each grid point is a representation of a finite volume of the solvent. The linear combination coefficients were determined by minimizing the error between the calculated and experimental hydration free energies of 81 neutral organic molecules that have a variety of functional groups. The calculated hydration free energies agree well with the experimental results. The hydration free energy of any other solute molecule can then be calculated by summing the product of the linear combination coefficients and the basis functions for the solute.     

A molecular basis for an irreversible thermodynamic description on non-electrolyte permeation through lipid bilayers

The non-electrolyte permeability of liposomal membranes has been investigated according to the concepts of irreversible thermodynamics. A strong interaction between the permeation of solute and water was observed. This solute-solvent interaction can be fully described by assuming that a number of water molecules will copermeate with each molecule of solute. This number of copermeating water molecules is independent of the nature of the permeant and of temperature, but depends on the osmotic concentration of impermeants inside the liposomes.     

154 Protein folding and binding funnels: a common driving force and a common mechanism

Under the free energy landscape theory, both the protein-folding and protein–ligand binding processes are driven by the decrease in total Gibbs free energy of the protein-solvent or protein–ligand-solvent system, which involves the non-complementary changes between the entropy and enthalpy, ultimately leading to a global free energy minimization of these thermodynamic systems (Ji & Liu, 2011; Liu et al., 2012; Yang, Ji & Liu, 2012). In the case of protein folding, the lowering of the system free energy coupled with the gradual reduction in conformational degree of freedom of the folding intermediates determines that the shape of the free energy landscape for protein folding must be funnel-like (Dill & Chan, 1997), rather than non-funneled shapes (Ben-Naim, 2012). In the funnel-like free energy landscape, protein folding can be viewed as going down the hill via multiple parallel routes from a vast majority of individual non-native states on surface outside the funnel to the native states located around the bottom of the funnel. The first stage of folding, i.e. the rapid hydrophobic collapse process, is driven by the solvent entropy maximization. Concretely, the water molecules squeeze and sequestrate the hydrophobic amino acid side chains within the interior of the folding intermediates while exposing the polar and electrostatically charged side chains on the intermediate surface so as to minimize the solvent-accessible surface area of the solute and thus, the minimal contacts between the folding intermediates and the water molecules. This will maximize the entropy of the solvent, thus contributing substantially to lowering of the system free energy due to an absolute advantage of the solvent in both quantity and mass (Yang, Ji & Liu, 2012). The resulting molten globule states (Ohgushi & Wada, 1983), within which a few transient secondary structural components and tertiary contacts have been formed but many native contacts or close residue–residue interactions has yet to form, need to be further sculptured into the native states. This is a relatively slow “bottleneck” process because the competitive interactions between protein residues within the folding intermediates and between residues and water molecules may repeat many rounds to accumulate a large enough number of stable noncovalent bonds capable of counteracting the conformational entropy loss of the intermediates, thus putting this bottleneck stage under the enthalpy control (i.e. negative enthalpy change), contributing further to the lowering of the system free energy. Although the protein–ligand association occurs around the rugged bottom of the free energy landscape, the exclusion of water from the binding interfaces and the formation of noncovalent bonds between the two partners can still lower the system free energy. In conjunction with the loss of the rotational and translational degrees of freedom of the two partners as well as the loss of the conformational entropy of the protein, these processes could merge, downwards expand, and further narrow the free energy wells within which the protein–ligand binding process takes place, thereby making them look like a funnel, which we term the binding funnel. In this funnel, the free energy downhill process follows a similar paradigm to the protein-folding process. For example, if the initial collisions/contacts occur between the properly complementary interfaces of the protein and ligand, a large amount of water molecules (which usually form a water network around the solute surface) will be displaced to suit the need for maximizing the solvent entropy. This process is similar to that of the hydrophobic collapse during protein folding, resulting in a loosely associated protein–ligand complex that needs also to be further adapted into a tight complex, i.e. the second step which is mainly driven by the negative enthalpy change through intermolecular competitive interactions to gradually accumulate the noncovalent bonds and ultimately, to stabilize the complex at a tightly bound state. Taken together, we conclude that whether in the protein-folding or in the protein–ligand binding process, both the entropy-driven first step and the enthalpy-driven second step contribute to the lowering of the system free energy, resulting in the funnel-like folding or binding free energy landscape.     

Shape and energetics of a cavity in c-Myb probed by natural and non-natural amino acid mutations.

The shape and the energetics of a functional cavity in the R2 subdomain (90-141) of the c-Myb DNA-binding domain were investigated by spectroscopy and thermodynamic analysis. We focused on the valine 103 residue located in front of the cavity. Nine mutants, in which valine 103 was substituted with alanine, 2-aminobutyric acid, norvaline, norleucine, leucine, isoleucine, allo -isoleucine, cyclohexylglycine, and cyclohexylalanine, were chemically synthesized and analyzed. These mutants provided a wide distribution of sizes which ranged from forming additional cavity space to filling and overflowing the cavity space. Temperature-scanning circular dichroism measurements and differential scanning calorimetry revealed a linear relationship between the van't Hoff enthalpy and the thermal transition temperature for the cavity-filling mutations. On the other hand, the mutants with side-chains larger than the side-chain of leucine resulted in a relatively low transition enthalpy and temperature, most likely due to the exposure of the side-chain to solvent and the increase in the entropy of the folded states. Branching at the beta-carbon atom reduced the unfolding free energy due to the steric constraint in the cavity. In particular, the mutational elongation of the side-chain from beta-carbon to the trans -to-CO direction proved to be more hindered than that from beta-carbon to the trans -to-NH. The unfolding free energy versus side-chain volume formed a bell-shaped plot with a maximum free energy for the leucine mutant. The difference in the transition free energy for cavity-filling mutants with beta-unbranched side-chains were two to four times larger than the difference in the transfer energy from organic solvent to water. Therefore, the increase in unfolding free energy would most likely be attributed to van der Waals interactions in the cavity wall, which would be a origin of stabilization by the sliding of tryptophan 95 into the cavity upon DNA binding.     

Thermodynamics of binding water and solute to myosin

From the isopiestic measurements of the extents of adsorption of water vapour by fish myosin at various values of water activities at three different temperatures, the changes in free energy, enthalpy and entropy of dehydration of the protein have been calculated. Extents of excess binding of solvent and solute to myosin have also been determined from isopiestic experiments in the presence of different inorganic salts, sucrose and urea respectively. Mols of water and solute respectively bound in absolute amounts to myosin have been evaluated from these data in limited range of solute concentrations. Free energy changes at different concentrations of these solutes have also been evaluated and their relations with ‘salting-in’ and ‘salting-out’ phenomena have been discussed. The order of the values of the standard free energy change for excess binding calculated with respect to an unified thermodynamic scale are found to be consistent with relative reactivity of binding water to myosin in the presence of inorganic salts, sucrose and urea. Part of this work was presented at the 20th Annual Convention of Chemists of the Indian Chemical Society, Cuttack, 26th-30th December 1983.     

Effect of temperature on the modification of spectral properties of tryptophan aqueous solutions induced by water previously treated with laser radiation

It was shown that preliminary exposure of a solvent (water) to low-intensity laser radiation reduces the tryptophan fluorescence intensity, and this fluorescence quenching effect is retained throughout the temperature range explored (from 8 up to 50 degrees C). The effects found are interpreted as resulting from changes in solvent properties induced by the action of electromagnetic radiation on interaction of water molecules with solute.     

Atomic-scale analysis of the solvation thermodynamics of hydrophobic hydration.

Molecular dynamics simulations are used to model the transfer thermodynamics of krypton from the gas phase into water. Extra long, nanosecond simulations are required to reduce the statistical uncertainty of the calculated "solvation" enthalpy to an acceptable level. Thermodynamic integration is used to calculate the "solvation" free energy, which together with the enthalpy is used to calculate the "solvation" entropy. A comparison series of simulations are conducted using a single Lennard-Jones sphere model of water to identify the contribution of hydrogen bonding to the thermodynamic quantities. In contrast to the classical "iceberg" model of hydrophobic hydration, the favorable enthalpy change for the transfer process at room temperature is found to be due primarily to the strong van der Waals interaction between the solute and solvent. Although some stabilization of hydrogen bonding does occur in the solvation shell, this is overshadowed by a destabilization due to packing constraints. Similarly, whereas some of the unfavorable change in entropy is attributed to the reduced rotational motion of the solvation shell waters, the major component is due to a decrease in the number of positional arrangements associated with the translational motions.     

Conformational statistics of pectin substances in solution by a Metropolis Monte Carlo study

Using a Metropolis Monte Carlo algorithm, various average properties of several pectic polysaccharide models were calculated based on the conformational energies for parent disaccharides. The relaxed potential energy surfaces of disaccharides were calculated using the CHARMm force field. Solvent effects were evaluated by calculating a solvation energy for each conformational state by estimating contributions from a cavity formation, and from the electrostatic and dispersion interactions between solvent and solute molecules. The behavior of the mean characteristic ratio, the squared radius of gyration, and the persistence length versus chain length is discussed for various structural models, temperature, and solvent. It is found that the unrefined model of the alternating co-polymer polygalacto-galacturonic acid in vacuum is consistent with the experimentally measured dimension of pectin in salt excess. This model is used to generate computer images of the characteristic conformation of pectin chain.     

Simple combined explicit/implicit water model

A simple combined water model (SCW model) for the calculation of the hydration free energy is presented. In the frame of the model a solute is placed in the centre of the spherical cavity with explicit water molecules, which are considered at the atomistic level. Rigid wall potential at the boundary of the cavity restricts the moving of the explicit water molecules. Water outside the sphere is considered as the conducting continuum (implicit part of the model). Simulation is performed in the frame of the NVT ensemble (constant number of particles, volume and temperature), density of water is fixed and equal to experimental value 1 g/cm³. The energy of electrostatic interaction of atomic point charges of the explicit water molecules with conducting continuum is calculated analytically by means of the image charges method. It provides high computational efficiency of the SCW model. For the averaging of the calculated thermodynamic and structural values over microstates of the system the thermodynamic integration method is used. The possible using of SCW for the docking problem is discussed.     

Optimized atomic radii for protein continuum electrostatics solvation forces

Recently, we presented a Green's function approach for the calculation of analytic continuum electrostatic solvation forces based on numerical solutions of the finite-difference Poisson-Botzmann (FDPB) equation [Im et al., Comp. Phys. Comm. 111 (1998) 59]. In this treatment the analytic forces were explicitly defined as the first derivative of the FDPB continuum electrostatic free energy with respect to the coordinates of the solute atoms. A smooth intermediate region for the solute-solvent dielectric boundary needed to be introduced to avoid abrupt discontinuous variations in the solvation free energy and forces as a function of the atomic positions. In the present paper we extend the set of optimized radii, which was previously parametrized from molecular dynamics free energy simulations of the 20 standard amino acids with explicit solvent molecules [Nina et al., J. Phys. Chem. 101 (1997) 5239], to yield accurate solvation free energy by taking the influence of the smoothed dielectric region into account.     

A desolvation barrier to hydrophobic cluster formation may contribute to the rate-limiting step in protein folding.

To gain insight into the free energy changes accompanying protein hydrophobic core formation, we have used computer simulations to study the formation of small clusters of nonpolar solutes in water. A barrier to association is observed at the largest solute separation that does not allow substantial solvent penetration. The barrier reflects an effective increase in the size of the cavity occupied by the expanded but water-excluding cluster relative to both the close-packed cluster and the fully solvated separated solutes; a similar effect may contribute to the barrier to protein folding/unfolding. Importantly for the simulation of protein folding without explicit solvent, we find that the interactions between nonpolar solutes of varying size and number can be approximated by a linear function of the molecular surface, but not the solvent-accessible surface of the solutes. Comparison of the free energy of cluster formation to that of dimer formation suggests that the assumption of pair additivity implicit in current protein database derived potentials may be in error.     

On the origins of the hydrophobic effect: observations from simulations of n-dodecane in model solvents.

The importance of the small size of a water molecule as contributing to the hydrophobic effect is examined from simulations of n-dodecane in different solvents. The earlier observations of the origin of hydrophobicity, derived from cavity formations by Pratt and Pohorille (1992, Proc. Natl. Acad. Sci. USA. 89:2995-2999) and Madan and Lee (1994, Biophys. Chem, 51:279-289), are shown to be largely consistent for a hydrocarbon-induced water pocket. In effect, the small size of a water molecule limits the probability (and hence free energy) of finding an appropriate void in the fluid that will accommodate a solute. In this work a simulated collapse of an n-dodecane molecule in H2O, CCl4, and a water-like Lennard-Jones solvent indicates that the induced entropy and enthalpy changes are qualitatively similar for hydrogen-bonded and Lennard-Jones water solvents. These results suggest that a large part of the hydrophobic response of solutes in aqueous solutions is due to the small size of the solvent. Important quantitative differences between the studied water solvents indicate that the hydrogen-bonded properties for water are still needed to determine the overall hydrophobic response.     

Structural parameterization of the binding enthalpy of small ligands

A major goal in ligand and drug design is the optimization of the binding affinity of selected lead molecules. However, the binding affinity is defined by the free energy of binding, which, in turn, is determined by the enthalpy and entropy changes. Because the binding enthalpy is the term that predominantly reflects the strength of the interactions of the ligand with its target relative to those with the solvent, it is desirable to develop ways of predicting enthalpy changes from structural considerations. The application of structure/enthalpy correlations derived from protein stability data has yielded inconsistent results when applied to small ligands of pharmaceutical interest (MW < 800). Here we present a first attempt at an empirical parameterization of the binding enthalpy for small ligands in terms of structural information. We find that at least three terms need to be considered: (1) the intrinsic enthalpy change that reflects the nature of the interactions between ligand, target, and solvent; (2) the enthalpy associated with any possible conformational change in the protein or ligand upon binding; and, (3) the enthalpy associated with protonation/deprotonation events, if present. As in the case of protein stability, the intrinsic binding enthalpy scales with changes in solvent accessible surface areas. However, an accurate estimation of the intrinsic binding enthalpy requires explicit consideration of long-lived water molecules at the binding interface. The best statistical structure/enthalpy correlation is obtained when buried water molecules within 5-7 A of the ligand are included in the calculations. For all seven protein systems considered (HIV-1 protease, dihydrodipicolinate reductase, Rnase T1, streptavidin, pp60c-Src SH2 domain, Hsp90 molecular chaperone, and bovine beta-trypsin) the binding enthalpy of 25 small molecular weight peptide and nonpeptide ligands can be accounted for with a standard error of +/-0.3 kcal x mol(-1).     

Implicit solvent models

Implicit solvent models for biomolecular simulations are reviewed and their underlying statistical mechanical basis is discussed. The fundamental quantity that implicit models seek to approximate is the solute potential of mean force, which determines the statistical weight of solute conformations, and which is obtained by averaging over the solvent degrees of freedom. It is possible to express the total free energy as the reversible work performed in two successive steps. First, the solute is inserted in the solvent with zero atomic partial charges; second, the atomic partial charges of the solute are switched from zero to their full values. Consequently, the total solvation free energy corresponds to a sum of non-polar and electrostatic contributions. These two contributions are often approximated by simple geometrical models (such as solvent exposed area models) and by macroscopic continuum electrostatics, respectively. One powerful route is to approximate the average solvent density distribution around the solute, i.e. the solute-solvent density correlation functions, as in statistical mechanical integral equations. Recent progress with semi-analytical approximations makes continuum electrostatics treatments very efficient. Still more efficient are fully empirical, knowledge-based models, whose relation to explicit solvent treatments is not fully resolved, however. Continuum models that treat both solute and solvent as dielectric continua are also discussed, and the relation between the solute fluctuations and its macroscopic dielectric constant(s) clarified.     

Hydrophobicity of benzene

The present work tries to clarify the molecular origin of the poor solubility of benzene in water. The transfer of benzene from pure liquid phase into water is dissected in two processes: transfer from gas phase to pure liquid benzene; and transfer from gas phase to liquid water. The two solvation processes are analyzed in the temperature range 5-100 degrees C according to Lee's Theory. The solvation Gibbs energy change is determined by the balance between the work of cavity creation in the solvent, and the dispersive interactions of the inserted benzene molecule with the surrounding solvent molecules. The purely structural solvent reorganization upon solute insertion proves to be a compensating process. The analysis shows that the work of cavity creation is larger in water than in benzene, whereas the attractive energetic interactions are stronger in benzene than in water; this scenario is true at any temperature. Therefore, both terms act in the same direction, contrasting the transfer of benzene from pure liquid phase into water and determining its hydrophobicity.     

Quantifying the Entropy of Binding for Water Molecules in Protein Cavities by Computing Correlations

Protein structural analysis demonstrates that water molecules are commonly found in the internal cavities of proteins. Analysis of experimental data on the entropies of inorganic crystals suggests that the entropic cost of transferring such a water molecule to a protein cavity will not typically be greater than 7.0 cal/mol/K per water molecule, corresponding to a contribution of approximately +2.0 kcal/mol to the free energy. In this study, we employ the statistical mechanical method of inhomogeneous fluid solvation theory to quantify the enthalpic and entropic contributions of individual water molecules in 19 protein cavities across five different proteins. We utilize information theory to develop a rigorous estimate of the total two-particle entropy, yielding a complete framework to calculate hydration free energies. We show that predictions from inhomogeneous fluid solvation theory are in excellent agreement with predictions from free energy perturbation (FEP) and that these predictions are consistent with experimental estimates. However, the results suggest that water molecules in protein cavities containing charged residues may be subject to entropy changes that contribute more than +2.0 kcal/mol to the free energy. In all cases, these unfavorable entropy changes are predicted to be dominated by highly favorable enthalpy changes. These findings are relevant to the study of bridging water molecules at protein-protein interfaces as well as in complexes with cognate ligands and small-molecule inhibitors.     

Monte Carlo Study of Structural and Thermodynamic Properties of Liquid Chloroform Using a Five Site Model

A five site potential model combining Lennard–Jones plus Coulomb potential functions has been developed for chloroform molecule. The partial charges needed for Coulombic interactions were derived using the chelpg procedure implemented in the gaussian 92 program. These calculations were performed at the MP2 level with MC-311G* basis set for Cl and 6-311G** for C and H atoms. The parameters for the Lennard–Jones potentials were optimized to reproduce experimental values for the density and enthalpy of vaporization of the pure liquid at 298 K and 1 atm. The statistical mechanics calculations were performed with the Monte Carlo method in the isothermic and isobaric (NpT) ensemble. Besides the values obtained for density, ρ, and molar enthalpy of vaporization at constant pressure, Δ H_V, for liquid chloroform, results for molar volume, V_m, molar heat capacity, C_p, isobaric thermal expansivity, α_p, and isothermal compressibility, κ_T, for this pure liquid are also in very good agreement with experimental observations. Size effects on the values of thermodynamic properties were investigated. The potential model was also tested by computing the free energy for solvating one chloroform molecule into its own liquid at 298 K using a statistical perturbation approach. The result obtained compares well with the experimental value. Site–site pair correlation functions were calculated and are in good accordance with theoretical results available in the literature. Dipole–dipole correlation functions for the present five site model were also calculated at different carbon–carbon distances. These correlations were compared to those obtained using the four site model reported in the literature. An investigation of the solvent dependence of the relative free energy for cis/trans conversion of a hypothetical solute in TIP4P water and chloroform was accomplished. The results show strong interaction of water and chloroform molecules with the gauche conformer. The value obtained for the free energy barrier for cis/trans rotation in TIP4P water is higher than that for chloroform. This result is in agreement with the continuous theory for solvation as the conformer with higher dipole moment is more favoured by the solvent with higher dieletric constant. The results also show an increase in entropy as the solute goes from the cis to the trans geometry and this result is more appreciable in the aqueous solution. This revised version was published online in June 2006 with corrections to the Cover Date.