Hydration and conformational equilibria of simple hydrophobic and amphiphilic solutes.

We consider whether the continuum model of hydration optimized to reproduce vacuum-to-water transfer free energies simultaneously describes the hydration free energy contributions to conformational equilibria of the same solutes in water. To this end, transfer and conformational free energies of idealized hydrophobic and amphiphilic solutes in water are calculated from explicit water simulations and compared to continuum model predictions. As benchmark hydrophobic solutes, we examine the hydration of linear alkanes from methane through hexane. Amphiphilic solutes were created by adding a charge of +/-1e to a terminal methyl group of butane. We find that phenomenological continuum parameters fit to transfer free energies are significantly different from those fit to conformational free energies of our model solutes. This difference is attributed to continuum model parameters that depend on solute conformation in water, and leads to effective values for the free energy/surface area coefficient and Born radii that best describe conformational equilibrium. In light of these results, we believe that continuum models of hydration optimized to fit transfer free energies do not accurately capture the balance between hydrophobic and electrostatic contributions that determines the solute conformational state in aqueous solution.     

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.     

Advanced models based on the integral equation theory of molecular liquids for calculating the hydration of ions

An advanced model based on the integral equation theory of molecular liquids has been developed. The model is a modification of the RISM/HNC method in which the solvent electrostatic potential is approximated by a linear dependence on the solute charge, while the solvent response of the solute is calculated by introducing the empirical repulsive bridge function accounting steric constrains. The hydration energies for a series of atomic and molecular ions have been calculated. The results of the calculations deviate only by a few percent from the experimental data.     

The effect of vicinal polar and charged groups on hydrophobic hydration.

The use of a linear relationship between free energy of hydrophobic hydration and solvent-accessible apolar surface area has been helpful in interpreting the thermodynamics of biological macromolecules. However, a recent study (Y.-K. Cheng, P. J. Rossky, Nature 1998, Vol. 392, pp. 696-699) has established a substantial enthalpic dependence on biomolecular surface topography, originating from solvent hydrogen-bonding loss in a restrictive geometry. In this study, we use molecular dynamics simulations of 2-Zn insulin in water solvent to explore the further effect of vicinal polar or charged groups on hydrophobic hydration at a biomolecular surface. In contrast to the case for solvent more isolated from such polar solute influences, the binding energies of the water that is proximal to the hydrophobic dimeric interface of insulin and vicinal to polar and charged groups are comparable to the bulk solvent value, a result of compensating interaction primarily with the solute counterions. The results suggest a special importance for such polar/charged groups in biological processes involving hydrophobic surface regions of restricted geometry and also suggest a general route for tuning the hydrophobicity of interfaces.     

Ab initio self-consistent field and potential-dependent partial equalization of orbital electronegativity calculations of hydration properties of N-acetyl-N'-methyl-alanineamide

Using a recently developed parallel computation algorithm, ab initio self-consistent field (SCF) calculations were carried out to estimate the relative hydration energies for 12 low-energy conformations of N-acetyl-N'-methyl-alanineamide. The requisite SCF calculations were carried out using 6-31G and 6-31G* basis sets, both in the absence and presence of a perturbing potential arising from a model solvent. The alpha R, alpha L, polyproline II (PII), and pi helical conformations were preferentially stabilized by the solvent potential, whereas conformations with intramolecular hydrogen-bonding C5 and C7 were preferred in the gas phase. Average vicinal nmr coupling constants (JNH-C alpha H), calculated using the total energies of the various solvated conformations, were consistent with observed coupling constants for this peptide in aqueous solution. Substantial alteration of the solute charge density occurred upon equilibration with the reaction field, as was exemplified in changes both in the molecular dipole moments and in atom-centered multipoles, when the molecule was transferred from a medium of low dielectric constant to one of high dielectric constant. In order to model these changes in charge density with an empirical scheme, we have implemented a novel monopolar representation of the solute charge density based on a potential-dependent form of partial equalization of orbital electronegativities (PDPEOE). In the atom-centered point charge PDPEOE representation, charge flows from one region of the solute to another in response to external fields. Hydration energies calculated using the PDPEOE representation are similar to those calculated by the SCF procedure. Also, the PDPEOE calculations yielded changes in molecular dipole moments upon solvation that agreed closely with the changes in the calculated ab initio SCF dipole moments.     

Influence of hydration on the conformational stability and formation of bends in terminally blocked dipeptides

The conformations of 23 terminally blocked dipeptide sequences were examined by conformational energy calculations that included the effects of the aqueous solvent. Starting structures were derived from combinations of minimum-energy conformations of hydrated single residues. Their conformational energies were then minimized using the ECEPP potential (Empirical Conformational Energy Program for Peptides) with hydration included. Short-range interactions dominate in stabilizing the conformations of the hydrated dipeptides. Differences between conformational stabilities of hydrated and unhydrated dipeptides in many cases are due to the competition of solute–water and intramolecular hydrogen bonds. In other cases, perturbation of the hydration shell of the solute by close approach of solute atoms alters conformational preferences. Probabilities of formation of bends were calculated and compared to the corresponding quantities for unhydrated dipeptides and to those calculated from x-ray structures. For bends in dipeptides containing two nonpolar amino acids, computations omitting hydration yield better results. However, better agreement with experimental (x-ray) bend probabilities for dipeptides containing glycine or polar amino acids is obtained only in some sequences when hydration is included. The results are rationalized by the observation that, in proteins, bends containing nonpolar sequences occur on the inside, shielded from the solvent. Bends containing glycine or polar amino acids occur frequently on the surface of the protein, but they are not completely hydrated.     

Pseudo-dynamic contact surface areas: estimation of apolar bonding

A new Monte Carlo based algorithm has been written for the computation of pseudo-dynamic contact surface areas. The linear correlation of this contact area with solute transfer free energies (water leads to organic liquid) is established for apolar amino acid side chains. The slope of these linear plots, deltaGosp, is a unitary free energy which has potential use in the estimation of apolar bond free energies in proteins. The magnitude of deltaGosp is dependent upon the nature of the organic solvent involved in the transfer process, varying from 86 to 130 cal/A2. Analogues linear correlations with the same range of deltaGosp values are observed for inhibitors of protein-catalyzed reactions.     

Model for the conformational analysis of hydrated peptides. Effect of hydration on the conformational stability of the terminally blocked residues of the 20 naturally occurring amino acids

The effects of hydration are included in empirical conformational energy computations on oligopeptides by means of a modified hydration-shell model. Free energy terms are introduced to account for “specific hydration” due to water–solute hydrogen bonding and for “nonspecific hydration” describing the interaction of the solute with water molecules in a first-neighbor shell. The dielectric constant has been doubled (over the value used for calculations in the absence of water) to take into account the presence of solvent. Computations were carried out for the N-acetyl-N′-methylamides of the 20 naturally occurring amino acids. Conformational energy maps are compared with similar maps calculated in the absence of hydration. Minimum-energy conformations are located and compared with the corresponding minima for unhydrated peptides in terms of ordering with respect to potential energy, the dihedral angles at the minima, and the presence of intramolecular hydrogen bonds. The Boltzmann factors for various conformational regions are altered significantly on hydration in some cases. These changes can be explained in terms of differences in the hydration free energy terms for various conformations.     

An efficient protocol for obtaining accurate hydration free energies using quantum chemistry and reweighting from molecular dynamics simulations

The non-Boltzmann Bennett (NBB) free energy estimator method is applied to 21 molecules from the blind subset of the SAMPL4 challenge. When NBB is applied with the SMD implicit solvent model, and the OLYP/DZP level of quantum chemistry, highly accurate hydration free energy calculations are obtained with respect to experiment (RMSD = 0.89 kcal·mol⁻¹). Other quantum chemical methods are also tested, and the effects of solvent model, density functional, basis set are explored in this benchmarking study, providing a framework for improvements in calculating hydration free energies. We provide a practical guide for using the best QM-NBB protocols that are consistently more accurate than either pure QM or pure MM alone. In situations where high accuracy hydration free energy predictions are needed, the QM-NBB method with SMD implicit solvent should be the first choice of quantum chemists.     

Ultracentrifugal compression of gels

It has been observed that the amount of water associated with casein micelles is markedly higher when measured by intrinsic viscosity than by water content of the pelleted casein. A possible explanation of this discrepancy is that water is squeezed out of the pellet by the ultracentrifugal field. We have calculated the potential magnitude of this effect by considering a simple model system: an elastic gel swollen by solvent and compressed by the centrifugal field. Equilibrium is reached when the sum of the ultracentrifugal, elastic, and mixing free energies is a minimum. The equilibrium degree of swelling is calculated as a function of rotor speed, thickness of the unsolvated gel material, and enthalpy of mixing of solvent and solute. Sizable compressions can occur for highly swollen gels, if the enthalpy of mixing is moderately positive. Casein micelles from intrinsic viscosity measurements have a “hydration” of about 3.7 g water/g protein, corresponding to a swelling ratio of 6.3 relative to the dry protein. The observed pellet hydration is 1.9 g/g. Under the conditions of the experiment, pelleting at 25,000 rpm is predicted to decrease the swelling ratio to 4.9, and the hydration to 2.7 g/g, about half the observed decrease, if the enthalpy of mixing is 0.5 RT/mol segment. These calculations may be relevent to the deterination by pelleting of the solvent content of other biopolymer gels.     

The calculated free energy effects of 5-methyl cytosine on the B to Z transition in DNA

We have examined the free energy effects of 5-methylation of cytosine on the B in equilibrium Z conformational equilibrium in DNA. Free energy differences were calculated using the free energy perturbation approach, which uses an easily derived equation from classical statistical mechanics to relate the free energy difference between two states to the ensemble average of the potential energy difference between the states. Calculations were carried both in explicit solvent and (for comparison) in vacuo. The free energy values obtained for the explicit solvent systems are total free energies, with contributions from all parts of the system (solvent + solute), and so are relevant to the B in equilibrium Z transitions observed under real (physiological) conditions. We calculate that in solution, methylation makes the B in equilibrium Z transition more favorable by about -0.4 kcal/mole base pair (bp) in free energy. This value compares well with approximate experimentally derived values of about -0.3 kcal/mole-bp. We also discuss a method for determining the free energy difference between conformational states poorly maintained by a potential energy model. Finally, the effects of methylation on the melting temperature of DNA are examined.     

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.     

Solvent-induced free energy landscape and solute-solvent dynamic coupling in a multielement solute

Molecular dynamics simulations using a simple multielement model solute with internal degrees of freedom and accounting for solvent-induced interactions to all orders in explicit water are reported. The potential energy landscape of the solute is flat in vacuo. However, the sole untruncated solvent-induced interactions between apolar (hydrophobic) and charged elements generate a rich landscape of potential of mean force exhibiting typical features of protein landscapes. Despite the simplicity of our solute, the depth of minima in this landscape is not far in size from free energies that stabilize protein conformations. Dynamical coupling between configurational switching of the system and hydration reconfiguration is also elicited. Switching is seen to occur on a time scale two orders of magnitude longer than that of the reconfiguration time of the solute taken alone, or that of the unperturbed solvent. Qualitatively, these results are unaffected by a different choice of the water-water interaction potential. They show that already at an elementary level, solvent-induced interactions alone, when fully accounted for, can be responsible for configurational and dynamical features essential to protein folding and function.     

Does Flory–Huggins theory help in interpreting solute partitioning experiments?

The development of Flory–Huggins (FH) theory is reviewed, particularly with regard to the molecular significance of the interaction parameter that scales the contact interaction of solute and solvent. The chemical potential given by FH theory for an “idealute” solute is then compared with that provided by a more general, statistical thermodynamic approach. It is found that the FH contact term does not directly correspond to the solvation free energy. The significance of this result for the interpretation of free energies of transfer of a solute from one solvent to another is examined. It is shown that neither the earlier recommended standard free energy change for the process (using the infinitely dilute reference state, mole fraction units) nor the recently recommended FH-corrected standard free energy change provides the solvation energy desired. Instead, the standard free energy using the infinitely dilute reference state and molarity units, as long advocated by Ben-Naim, provides the desired solvation free energy. Correction of extant values, based on mole fraction units, is easily made. However, application of such results to problems of protein folding is not straightforward. © 1994 John Wiley & Sons, Inc.     

A computer simulation of functional group contributions to free energy in water and a DPPC lipid bilayer

A series of all-atom molecular dynamics simulations has been performed to evaluate the contributions of various functional groups to the free energy of solvation in water and a dipalmitoylphospatidylcholine lipid bilayer membrane and to the free energies of solute transfer (Delta(DeltaG(o))X) from water into the ordered-chain interior of the bilayer. Free energies for mutations of the alpha-H atom in p-toluic acid to six different substituents (-CH3, -Cl, -OCH3, -CN, -OH, -COOH) were calculated by a combined thermodynamic integration and perturbation method and compared to literature results from vapor pressure measurements, partition coefficients, and membrane transport experiments. Convergence of the calculated free energies was indicated by substantial declines in standard deviations for the calculated free energies with increased simulation length, by the independence of the ensemble-averaged Boltzmann factors to simulation length, and the weak dependence of hysteresis effects on simulation length over two different simulation lengths and starting from different initial configurations. Calculated values of Delta(DeltaG(o))X correlate linearly with corresponding values obtained from lipid bilayer transport experiments with a slope of 1.1 and from measurements of partition coefficients between water and hexadecane or decadiene, with slopes of 1.1 and 0.9, respectively. Van der Waals interactions between the functional group of interest and the acyl chains in the ordered chain region account for more than 95% of the overall potential energy of interaction. These results support the view that the ordered chain region within the bilayer interior is the barrier domain for transport and that solvation interactions within this region resemble those occurring in a nonpolar hydrocarbon.     

Prediction of protein-protein binding free energies

We present an energy function for predicting binding free energies of protein-protein complexes, using the three-dimensional structures of the complex and unbound proteins as input. Our function is a linear combination of nine terms and achieves a correlation coefficient of 0.63 with experimental measurements when tested on a benchmark of 144 complexes using leave-one-out cross validation. Although we systematically tested both atomic and residue-based scoring functions, the selected function is dominated by residue-based terms. Our function is stable for subsets of the benchmark stratified by experimental pH and extent of conformational change upon complex formation, with correlation coefficients ranging from 0.61 to 0.66.     

Function of the F1-motor (F1-ATPase) of ATP synthase by apolar-polar repulsion through internal interfacial water

THESIS: Within the structurally-confined internal aqueous cavity of the F1-motor of ATP synthase, function results from free energy changes that shift the balance between interfacial charge hydration and interfacial hydrophobic hydration. TRANSITION STATE DESCRIPTION: At the beta-P end of ADP x Mg occurs an inorganic phosphate, P(i). This P(i) resides at the base of a water-filled cleft that functions like an aperture to focus, into an aqueous chamber, a competition for hydration (an apolar-polar repulsion) between charged phosphate and hydrophobic surface of the gamma-rotor. Two means available for the phosphate and the hydrophobic surface to improve their hydration free energies are physically to separate by rotation of the gamma-rotor or chemically to combine P(i) with ADP to form less charged ATP. This proposal derives from calculated changes in Gibbs free energy for hydrophobic association of amino acid side chains and chemical modifications thereof and from experimentally demonstrated water-mediated repulsion between hydrophobic and charged sites that resulted from extensive studies on designed elastic-contractile model proteins.     

Effects of base substituents on the hydration of B- and Z-DNA: correlations to the B- to Z-DNA transition.

We present a study of how substituent groups of naturally occurring and modified nucleotide bases affect the degree of hydration of right-handed B-DNA and left-handed Z-DNA. A comparison of poly(dG-dC) and poly(dG-dm5C) titrations with the lipotropic salts of the Hofmeister series infers that the methyl stabilization of cytosines as Z-DNA is primarily a hydrophobic effect. The hydration free energies of various alternating pyrimidine-purine sequences in the two DNA conformations were calculated as solvent free energies from solvent accessible surfaces. Our analysis focused on the N2 amino group of purine bases that sits in the minor groove of the double helix. Removing this amino group from guanine to form inosine (I) destabilizes Z-DNA, while adding this group to adenines to form 2-aminoadenine (A') stabilizes Z-DNA. These predictions were tested by comparing the salt concentrations required to crystallize hexanucleotide sequences that incorporate d(CG), d(CI), d(TA) and d(TA') base pairs as Z-DNA. Combining the current results with our previous analysis of major groove substituents, we derived a thermodynamic cycle that relates the systematic addition, deletion, or substitution of each base substituent to the B- to Z-DNA transition free energy.