A probabilistic method for calculating the energy of hydrophobic interactions

A theoretical approach is developed for estimating the hydrophobic interaction energy on the molecular level. The underlying idea of the model is the fundamental relationship between the probability of the appearance of a solvent-excluded volume in fluid water as a result of density fluctuations, on one hand, and the free energy of primitive hydrophobic solutes, on the other. This probability is estimated basing on an information theory model in combination with experimental data on the distribution density and radial distribution function. The free energy of hydrophobic interactions is determined for a complex that consists of several hard spheres. The critical number of particles in the complex is estimated.     

Pressure- and temperature-induced unfolding studies: thermodynamics of core hydrophobicity and packing of ribonuclease A

In this work we demonstrate that heat and pressure induce only slightly different energetic changes in the unfolded state of RNase A. Using pressure and temperature as denaturants on a significant number of variants, and by determining the free energy of unfolding at different temperatures, we estimated the stability of variants unable to complete the unfolding transition owing to the experimental conditions required for pressure experiments. The overall set of results allowed us to map the contributions to stability of the hydrophobic core residues of RNase A, with the positions most critical for stability being V54, V57, I106 and V108. We also show that the stability differences can be attributed to both hydrophobic interactions and packing density with an equivalent energetic magnitude. The main hydrophobic core of RNase A is tightly packed, as shown by the small-to-large and isosteric substitutions. In addition, we found that large changes in the number of methylene groups have non-additive positive stability interaction energies that are consistent with exquisite tight core packing and rearrangements of van der Waals' interactions in the protein interior, even after drastic deleterious substitutions.     

Use of a potential of mean force to analyze free energy contributions in protein folding.

A method for calculation of the free energy of residues as a function of residue burial is proposed. The method is based on the potential of mean force, with a reaction coordinate expressed by residue burial. Residue burials are calculated from high-resolution protein structures. The largest individual contributions to the free energy of a residue are found to be due to the hydrophobic interactions of the nonpolar atoms, interactions of the main chain polar atoms, and interactions of the charged groups of residues Arg and Lys. The contribution to the free energy of folding due to the uncharged side chain polar atoms is small. The contribution to the free energy of folding due to the main chain polar atoms is favorable for partially buried residues and less favorable or unfavorable for fully buried residues. Comparison of the accessible surface areas of proteins and model spheres shows that proteins deviate considerably from a spherical shape and that the deviations increase with the size of a protein. The implications of these results for protein folding are also discussed.     

Interactions between CD44 protein and hyaluronan: insights from the computational study

CD44 is a protein, being a major cell surface receptor for hyaluronan (HA). Molecular modeling investigation was carried out on the murine CD44 in complex with a HA heptasaccharide in order to: (i) elucidate the nature and dynamics of interactions between the HA chain and CD44; (ii) find out if the existence of two conformational forms of CD44 discovered in the XRD (X-Ray Diffraction) study can be responsible for its switching between low and high affinity for HA. The results indicate that the contact of CD44 with HA is dominated by hydrogen bonding with small contribution of hydrophobic interactions and salt bridges. In addition, the two ('A' and 'B') conformational forms of the HA-CD44 complex reported experimentally by Banerji et al. cannot be observed during simulations when considering the distance between HA and the sidechain of the R45 residue. There exists, however, a free energy barrier associated with the change of the φ dihedral angle value at Y46. Additionally, some thermodynamic parameters (e.g. the Gibbs free energy change) accompanying the HA binding by CD44 were estimated.     

Calculation of the Affinity of the λ Repressor-Operator Complex Based on Free Energy Component Analysis

A calculation of the binding free energy of the u repressor-operator complex is described based on free energy component analysis. The calculations are based on a thermodynamic cycle of seven steps decomposed into a total of 24 individual components. The values of these terms are estimated using a combination of empirical potential functions from AMBER, generalized Born - solvent accessibility calculations, elementary statistical mechanics and semiempirical physicochemical properties. Two alternative approaches are compared, one based on the crystal structure of the complex and the other based on the molecular dynamics simulation of the u repressor-operator complex. The calculated affinity is m 19.7 kcal/mol from the crystal structure calculation and m 17.9 kcal/mol from the MD method. The corresponding experimental affinity of the complex is about m 12.6 kcal/mol, indicating reasonable agreement between theory and experiment, considering the approximations involved in the computational methodology. The results are analyzed in terms of contributions from electrostatics, van der Waal interactions, the hydrophobic effect and solvent release. The capabilities and limitations of free energy component methodology are assessed and discussed on the basis of these results.     

Free-energy contributions to complex formation between botulinum neurotoxin type B and synaptobrevin fragment

Free-energy terms that contribute to complex formation between the catalytic domain of botulinum neurotoxin type B (BoNT/B-L(C)) and a 36-residue synaptobrevin fragment were estimated by using a combination of microscopic simulations and continuum methods. The complex for a non-hydrolyzed substrate was calculated by optimizing an energy function applied to the X-ray co-crystal structure of BoNT/B-L(C) bound with reaction products from a cleaved synaptobrevin peptide, refined to high crystallographic thermal factors. The estimated absolute binding affinity of the simulation structure is in good qualitative agreement with the experimental free energy of Michaelis complex formation, given the approximations of the model calculations. The simulation structure revealed significant complex stabilization from the hydrophobic effect, while the electrostatic cost of releasing water molecules from the interface determined to be highly unfavorable. By partitioning the total electrostatic and hydrophobic terms into residue free-energy contributions, a binding-affinity 'signature' for synaptobrevin was developed from the optimized conformation. The results demonstrate the effect of substrate length on complex formation and identify a peripheral high-affinity binding site near the N-terminal region that might initiate cooperative activation responsible for the large minimal substrate length requirement. The so-called SNARE motif is observed to contribute negligible free energy of binding.     

New version of Monte Carlo expanded ensemble method for precise calculations of free energy difference

A new version of Monte Carlo (MC) expanded ensemble (EE) method is proposed for the calculations of free energy difference (FED) between two different systems with close values of the free energy. In order to check the method the FED between simple model systems (fluid of hard spheres and freely jointed polymer chain of hard spheres) was calculated. The free energy of the mentioned above systems was also calculated by a standard MC EE method in order to compare the results of two simulations. It was shown that the accuracy of a new algorithm is the same as of a standard one. At the same time new version of EE allows us to obtain FED between two systems having quite different structures, but similar free energies, during one simulation run.     

Attempts to delineate the relative contributions of changes in hydrophobicity and packing to changes in stability of ribonuclease S mutants

While the hydrophobic driving force is thought to be a major contributor to protein stability, it is difficult to experimentally dissect out its contribution to the overall free energy of folding. We have made large to small substitutions of buried hydrophobic residues at positions 8 and 13 in the peptide/protein complex, RNase-S, and have characterized the structures by X-ray crystallography. The thermodynamics of association of these mutant S peptides with S protein was measured in the presence of different concentrations of methanol and ethanol. The reduction in the strength of the hydrophobic driving force in the presence of these organic solvents was estimated from surface-tension data as well as from the dependence of the DeltaC(p) of protein/peptide binding on the alcohol concentration. The data indicated a decrease in the strength of the hydrophobic driving force of about 30-40% over a 0-30% range of the alcohol concentration. We observe that large to small substitutions destabilize the protein. However, the amount of destabilization, relative to the wild type, is independent of the alcohol concentration over the range of alcohol concentrations studied. The data clearly indicate that decreased stability of the mutants is primarily due to the loss of packing interactions rather than a reduced hydrophobic driving force and suggest a value of the hydrophobic driving force of less than 18 cal mol(-)(1) A(2).     

Energy of ligand-RNA complex formation

This work presents an energetic analysis of complex formation of eleven ligands with different structure and charge with RNA aptamers. The most entire set of the components of the total Gibbs energy of complex formation has been first calculated using different physical factors: van der Waals, electrostatic and hydrophobic interactions, hydrogen bonds and specific factors being predominantly of entropic character. The calculated Gibbs energy is found to be in a good agreement with experimental data. Different energy components which stabilize and destabilize complexes are lined up according to the degree of importance. The results obtained provide an understanding of the role of different physical interactions in a ligand-RNA complex formation.     

Folding pathway dependence on energetic frustration and interaction heterogeneity for a three-dimensional hydrophobic protein model

Monte Carlo simulations of a hydrophobic protein model of 40 monomers in the cubic lattice are used to explore the effect of energetic frustration and interaction heterogeneity on its folding pathway. The folding pathway is described by the dependence of relevant conformational averages on an appropriate reaction coordinate, pfold, defined as the probability for a given conformation to reach the native structure before unfolding. We compare the energetically frustrated and heterogeneous hydrophobic potential, according to which individual monomers have a higher or lower tendency to form contacts unspecifically depending on their hydrophobicities, to an unfrustrated homogeneous Go-type potential with uniformly attractive native interactions and neutral non-native interactions (called Go1 in this study), and to an unfrustrated heterogeneous potential with neutral non-native interactions and native interactions having the same energy as the hydrophobic potential (called Go2 in this study). Folding kinetics are slowed down dramatically when energetic frustration increases, as expected and previously observed in a two-dimensional model. Contrary to our previous results in two dimensions, however, it appears that the folding pathway and transition state ensemble can be significantly dependent on the energy function used to stabilize the native structure. The sequence of events along the reaction coordinate, or the order along this coordinate in which different regions of the native conformation become structured, turns out to be similar for the hydrophobic and Go2 potentials, but with analogous events tending to occur at lower pfold values in the first case. In particular, the transition state obtained from the ensemble around pfold = 0.5 is more structured for the hydrophobic potential. For Go1, not only the transition state ensemble but the order of events itself is modified, suggesting that interaction heterogeneity, in addition to energetic frustration, can have significant effects on the folding mechanism, most likely by modifying the probability of different contacts in the unfolded state, the starting point for the folding reaction. Although based on a simple model, these results provide interesting insight into how sequence-dependent switching between folding pathways might occur in real proteins.     

Overall charge and local charge density of pectin determines the enthalpic and entropic contributions to complexation with β-lactoglobulin

The complex formation between β-lactoglobulin and pectins of varying overall charge and local charge density were investigated. Isothermal titration calorimetry experiments were carried out to determine the enthalpic contribution to the complex formation at pH 4.25 and various ionic strengths. Complex formation was found to be an exothermic process for all conditions. Combination with previously published binding constants by Sperber et al. (Sperber, B. L. H. M.; Cohen Stuart, M. A.; Schols, H. A.; Voragen, A. G. J.; Norde, W. Biomacromolecules 2009, 10, 3246-3252) allows for the determination of the changes in the Gibbs energy and the change in entropy of the system upon complex formation between β-lactoglobulin and pectin. The local charge density of pectin is found to determine the balance between enthalpic and entropic contributions. For a high local charge density pectin, the main contribution to the Gibbs energy is of an enthalpic nature, supported by a favorable entropy effect due to the release of small counterions. A pectin with a low local charge density has a more even distribution of the enthalpic and entropic part to the change of the Gibbs energy. The enthalpic part is reduced due to the lower charge density, while the relative increase of the entropic contribution is thought to be caused by a change in the location of the binding place for pectin on the β-lactoglobulin molecule. The association of the hydrophobic methyl esters on pectin with an exposed hydrophobic region on β-lg results in the release of water molecules from the hydrophobic region and surrounding the methyl esters of the pectin molecule. An increase in the ionic strength decreases the enthalpic contribution due to the shielding of electrostatic attraction in favor of the entropic contribution, supporting the idea that the release of water molecules from hydrophobic areas plays a part in the complex formation.     

Fine-tuned broad binding capability of human lipocalin-type prostaglandin D synthase for various small lipophilic ligands

The hydrophobic cavity of lipocalin-type prostaglandin D synthase (L-PGDS) has been suggested to accommodate various lipophilic ligands through hydrophobic effects, but its energetic origin remains unknown. We characterized 18 buffer-independent binding systems between human L-PGDS and lipophilic ligands using isothermal titration calorimetry. Although the classical hydrophobic effect was mostly detected, all complex formations were driven by favorable enthalpic gains. Gibbs energy changes strongly correlated with the number of hydrogen bond acceptors of ligand. Thus, the broad binding capability of L-PGDS for ligands should be viewed as hydrophilic interactions delicately tuned by enthalpy–entropy compensation using combined effects of hydrophilic and hydrophobic interactions.     

Study of thermodynamic properties of symmetric and asymmetric electrolyte systems in mixture with neutral components: Monte Carlo simulation results and integral equation predictions

Monte Carlo simulation is conducted to obtain the structure, excess internal energy and Helmholtz energy for systems containing charged and neutral hard spheres of comparable concentrations. The results are compared with the thermodynamic properties predicted by solving Ornstein–Zernike equation with hypernetted chain (HNC) and mean spherical approximation (MSA) closures. The HNC approximation is found to well represent the simulation results for both structure and excess energy, while the excess energy of MSA deviates from the simulation results in the intermediate- and high-density range. A simple modification of MSA, referred to as KMSA, is proposed to accurately predict the excess internal and Helmholtz energy in the studied density range. KMSA is proved to capture the effects of neutral component, size and charge asymmetry, system temperature and dielectric constant of the background solvent, on the excess energy of electrolyte systems.     

Molecular Modeling-Based Energy Analysis of Dimeric Binding of Ligands to the Minor DNA Groove

Molecular-modeling methods have been used to perform energy analysis of dimeric complex formation between lexitropsins and double-stranded DNA. Stabilization of dimeric complexes by hydrophobic and van der Waals interactions has been demonstrated. Electrostatic interactions and the contributions of hydrogen bonds and changes in the number of degrees of freedom had a destabilizing influence. The energy of monomeric and dimeric binding has been compared.     

SNARE Complex Zipping as a Driving Force in the Dilation of Proteinaceous Fusion Pores

The assembly of SNARE proteins into a tight complex has been hypothesized to drive membrane fusion. A model of the initial fusion pore as a proteinaceous channel formed by SNARE proteins places their membrane anchors in separate membranes. This leaves the possibility of a final assembly step that brings the membrane anchors together and drives fusion pore expansion. The present study develops a model for expansion in which the final SNARE complex zipping step drives a transition from a proteinaceous fusion pore to a lipidic fusion pore. An estimate of the energy released upon merger of the helical segments of the SNARE motifs with the helical segments of the membrane anchors indicates that completing the assembly of a few SNARE complexes can overcome the elastic energy that opposes lipid bilayer deformation into a narrow fusion pore. The angle between the helical axes of the membrane anchor and SNARE motif serves as a useful reaction coordinate for this transition. Energy was calculated as a function of this angle, incorporating contributions from membrane bending, SNARE complex assembly, membrane anchor flexing and hydrophobic interactions. The rate of this transition was evaluated as a process of diffusion over the barrier imposed by these combined energies, and the rates estimated were consistent with experimental measurements.     

Free energetics of rigid body association of ubiquitin binding domains: A biochemical model for binding mediated by hydrophobic interaction

Weak intermolecular interactions, such as hydrophobic associations, underlie numerous biomolecular recognition processes. Ubiquitin is a small protein that represents a biochemical model for exploring thermodynamic signatures of hydrophobic association as it is widely held that a major component of ubiquitin's binding to numerous partners is mediated by hydrophobic regions on both partners. Here, we use atomistic molecular dynamics simulations in conjunction with the Adaptive Biasing Force sampling method to compute potentials of mean force (the reversible work, or free energy, associated with the binding process) to investigate the thermodynamic signature of complexation in this well‐studied biochemical model of hydrophobic association. We observe that much like in the case of a purely hydrophobic solute (i.e., graphene, carbon nanotubes), association is favored by entropic contributions from release of water from the interprotein regions. Moreover, association is disfavored by loss of enthalpic interactions, but unlike in the case of purely hydrophobic solutes, in this case protein‐water interactions are lost and not compensated for by additional water‐water interactions generated upon release of interprotein and moreso, hydration, water. We further find that relative orientations of the proteins that mutually present hydrophobic regions of each protein to its partner are favored over those that do not. In fact, the free energy minimum as predicted by a force field based method recapitulates the experimental NMR solution structure of the complex. Proteins 2014; 82:1453–1468. © 2014 Wiley Periodicals, Inc.     

Temperature‐dependent study reveals that dynamics of hydrophobic residues plays an important functional role in the mitochondrial Tim9–Tim10 complex

Protein–protein interaction is a fundamental process in all major biological processes. The hexameric Tim9–Tim10 (translocase of inner membrane) complex of the mitochondrial intermembrane space plays an essential chaperone‐like role during import of mitochondrial membrane proteins. However, little is known about the functional mechanism of the complex because the interaction is weak and transient. This study investigates how electrostatic and hydrophobic interactions affect the conformation and function of the complex at physiological temperatures, using both experimental and computational methods. The results suggest that, first, different complex conformational states exist at equilibrium, and the major difference between these states is the degree of hydrophobic interactions. Second, the conformational change mimics the biological activity of the complex as measured by substrate binding at the same temperatures. Finally, molecular dynamics simulation and detailed energy decomposition analysis provided supporting evidence at the atomic level for the presence of an excited state of the complex, the formation of which is largely driven by the disruption of hydrophobic interactions. Taken together, this study indicates that the dynamics of the hydrophobic residues plays an important role in regulating the function of the Tim9–Tim10 complex. Proteins 2012. © 2011 Wiley Periodicals, Inc.     

Minimal model for studying prion-like folding pathways

The Monte Carlo technique is used to simulate the energy landscape and the folding kinetics of a minimal prion-like protein model. We show that the competition between hydrogen-bonding and hydrophobic interactions yields two energetically favored secondary structures, an alpha-helix and a beta-hairpin. Folding simulations indicate that the probability of reaching the alpha-helix form from a denatured random conformation is much higher than the probability of reaching the beta-sheet form, even though the beta-sheet has a lower energy. The existence of a lower energy beta-sheet state gives the possibility for the normal alpha-helix structure to take a structural transformation into the beta-sheet structure under external influences.     

Solvent effects and polar interactions in the structural stability and dynamics of globular proteins.

Using detailed hydrogen bonding, surface exposure, internal environment, and solvent interaction calculations on several proteins, in conjunction wit data from quantum mechanical hydrogen-bonding studies, various contributions to the free energy of globular estimated and their likely relative significance discussed. A picture emerges of globular proteins as extremely well-fitting jigsaw-puzzles, in which no single driving force dominates the marginal stability of the native conformation. Rather, the folded structure is seen as the result of a complex global maximization of several strongly-interacting driving forces. In particular, the necessity to maintain very efficient internal hydrogen-bonding, and the role of the solvent as a hydrogen-bond sink, are stressed as strong constraints on the (incomplete) maximization of hydrophobic effects. The possible significance of internal dipole-induced dipole interactions is discussed tentatively. Although quantitative estimates of the various contributions remain uncertain, consideration of effective force constants suggests that polar, including solvent, interactions may largely determine the overall curvatures of the native conformation's potential well, and be important in controlling the flexibility of local regions which are important for the exact positioning of groups during enzyme catalysis, as well as the molecule's overall dynamics. In contrast, hydrophobic interactions change less for small geometrical perturbations, and seem more relevant to directing the folding protein. along a path to a region in configurational space where the polar interactions can switch on for the final "docking".