Entropy calculations on the molten globule state of a protein: side-chain entropies of alpha-lactalbumin.

We present entropy estimates based on molecular dynamics simulations of models of the molten globule state of the protein alpha-lactalbumin at low pH. The entropy calculations use the covariance matrix of atom-positional fluctuations and yield the complete configurational entropy. The configurational entropy of the entire protein and of each of its side chains is calculated. Exposed side chains show a larger entropy compared to buried side chains. A comparison to data from rotamer counting is made and significant differences are found.     

Entropy calculations on a reversibly folding peptide: changes in solute free energy cannot explain folding behavior

The configurational entropy of a beta-heptapeptide in solution at four different temperatures is calculated. The contributions of the backbone and of the side-chain atoms to the total peptide entropy are analyzed separately and the effective contribution to the entropy arising from correlations between these terms determined. The correlation between the backbone and side-chain atoms amounts to about 17% and is rather insensitive to the temperature. The correlation of motion within the backbone and within side-chains is much larger and decreases with temperature. As the peptide reversibly folds at higher temperatures, its change in entropy and enthalpy upon folding is analyzed. The change in entropy and enthalpy upon folding of the peptide alone cannot account for the observed change in free energy on folding of the peptide in solution. Enthalpic and entropic contributions of the solvent thus also play a key role. Proteins 2001;43:45-56.     

To the necessity of checking the validity of some premises of nonequilibrium thermodynamics

Biophysics - Examples are given where the Prigogine principle—minimum entropy production in an open system at steady state, the basis of many issues of nonequilibrium...     

Non Equilibrium Molecular Dynamics Simulation of Coupled Heat- and Mass Transport Across a Liquid/Vapor Interface

Abstract

The transport properties of bulk liquid, gas and at the gas/liquid interface were studied for two binary Lennard-Jones/spline mixtures by use of nonequilibrium molecular dynamics. One of the mixtures was an ideal isotope mixture, the other a non-ideal mixture. The simulations gave the thermal conductivity, mutual diffusion coefficient, heat flux, mass flux, and the changes in these quantities across the interface. The local entropy production was expressed in terms of fluxes and thermodynamic forces, and numercial estimates are given. It was shown that the largest contribution to the total entropy production occurs in the vapor phase under the chosen conditions. We expect, however that if the mass flux were larger, the major contribution to the entropy production would come from the liquid phase.     

Relationship between thermodynamic driving force and one-way fluxes in reversible processes

Chemical reaction systems operating in nonequilibrium open-system states arise in a great number of contexts, including the study of living organisms, in which chemical reactions, in general, are far from equilibrium. Here we introduce a theorem that relates forward and reverse fluxes and free energy for any chemical process operating in a steady state. This relationship, which is a generalization of equilibrium conditions to the case of a chemical process occurring in a nonequilibrium steady state in dilute solution, provides a novel equivalent definition for chemical reaction free energy. In addition, it is shown that previously unrelated theories introduced by Ussing and Hodgkin and Huxley for transport of ions across membranes, Hill for catalytic cycle fluxes, and Crooks for entropy production in microscopically reversible systems, are united in a common framework based on this relationship.     

Binding of bovine pancreatic trypsin inhibitor to trypsinogen: spectroscopic and volumetric studies

We have investigated the binding of bovine pancreatic trypsin inhibitor (BPTI) to bovine trypsinogen by combining ultrasonic velocimetry, high precision densimetry, and fluorescence spectroscopy. We report the changes in volume, adiabatic compressibility, van't Hoff enthalpy, entropy, and free energy that accompany the association of the two proteins at 25 degrees C and pH 8.0. We have used the measured changes in volume and compressibility in conjunction with available structural data to characterize the binding-induced changes in the hydration properties and intrinsic packing of the two proteins. Our estimate reveals that 110 +/- 40 water molecules become released to the bulk from the hydration shells of BPTI and trypsinogen. Furthermore, we find that the intrinsic coefficient of adiabatic compressibility of the two proteins decreases by 14 +/- 2%, which is suggestive of the binding-induced rigidification of the proteins' interior. BPTI-trypsinogen association is an entropy-driven event which proceeds with an unfavorable change in enthalpy. The favorable change in entropy results from partial compensation between two predominant terms. Namely, a large favorable change in hydrational entropy slightly prevails over a close in magnitude but opposite in sign change in configurational entropy. The reduction in configurational entropy and, consequently, protein dynamics is consistent with the observed decrease in intrinsic compressibility. In general, results of this work emphasize the vital role that water plays in modulating protein recognition events.     

On the entropy of protein folding.

The failure to appreciate that the hydration of polar groups is a major contribution to the entropy of protein unfolding has led to considerable underestimates for the loss of configurational freedom when a protein chain folds.     

Partitioning of tryptophan side-chain analogs between water and cyclohexane.

We have measured the partitioning of the tryptophan side-chain analogs 3-methylindole and N-methylindole between water and cyclohexane over the temperature range 8-55 degrees C to investigate the relative contribution of the imine-NH- to the free energy of transfer. We take advantage of the fact that the indole imine nitrogen is blocked by a methyl group in N-methylindole. Unlike previous studies, we take into account the water present in the cyclohexane phase. Free energies of partitioning were calculated using mole-fraction, volume-fraction, and Flory-Huggins-corrected volume-fraction partition coefficients [De Young, L. R., & Dill, K. A. (1990) J. Phys. Chem. 94, 801-809; Sharp, K. A., Nicholls, A., Friedman, R., & Honig, B. (1991) Biochemistry 30, 9686-9697]. These approaches account for configurational entropy changes in different ways and thus lead to different values for the calculated free energies of transfer. There is a 2-3-fold difference in the free energies calculated from our measurements, using the different units. Independent of units, the partitioning of both compounds involves identical entropy changes. However, 3-methylindole has an additional unfavorable enthalpic contribution to partitioning into cyclohexane of +1.6 kcal/mol (independent of units) which is presumably the cost of removing the indole -NH- group from water and transferring it to cyclohexane. In cyclohexane, 3-methylindole forms hydrogen bonds with water that cause water to copartition into cyclohexane with the solute. A method is described which allows the partitioning process to be examined independent of subsequent interactions with water in the solvent.     

Theoretical model for the equilibrium behavior of DNA superhelices

A statistical-mechanical model for superhelical DNA is presented. The partition function for a DNA superhelix is written by using a combinatorial approach in order to allow for the known relation between the number of superhelical twists and the states of the base pairs in the double helix. While the theory allows any factors which might contribute to the free energy of superhelical twisting to be included in the statistical weights of the superhelical twists, only the reduction in configurational entropy is considered in this paper. Similarities between an imperfectly matched DNA double helix and a DNA superhelix are used in the derivation of expressions for the entropy of superhelical DNA. Although the partition function is presented in a general form, permitting many equilibrium properties of DNA superhelices to be treated, the application considered in this paper is the calculation of helix–coil transition curves. Several experimentally observed features of such transitions are predicted. For example, the curves are bimodal, with an early and a late transition relative to that of a nicked molecule. The results are very sensitive to the volume within which two parts of the double helix must meet when forming a superhelical twist. The free energy of superhelix formation is calculated, and the results are compared with those obtained from the data of Bauer and Vinograd for ethidium bromide intercalation. In the present model, the free energy increases less sharply with an increase in the number of superhelical twists than observed experimentally, indicating that factors other than configurational entropy probably make important contributions to the free energy of superhelix formation.     

Accounting for Ligand Conformational Restriction in Calculations of Protein-Ligand Binding Affinities

The conformation adopted by a ligand on binding to a receptor may differ from its lowest-energy conformation in solution. In addition, the bound ligand is more conformationally restricted, which is associated with a configurational entropy loss. The free energy change due to these effects is often neglected or treated crudely in current models for predicting binding affinity. We present a method for estimating this contribution, based on perturbation theory using the quasi-harmonic model of Karplus and Kushick as a reference system. The consistency of the method is checked for small model systems. Subsequently we use the method, along with an estimate for the enthalpic contribution due to ligand-receptor interactions, to calculate relative binding affinities. The AMBER force field and generalized Born implicit solvent model is used. Binding affinities were estimated for a test set of 233 protein-ligand complexes for which crystal structures and measured binding affinities are available. In most cases, the ligand conformation in the bound state was significantly different from the most favorable conformation in solution. In general, the correlation between measured and calculated ligand binding affinities including the free energy change due to ligand conformational change is comparable to or slightly better than that obtained by using an empirically-trained docking score. Both entropic and enthalpic contributions to this free energy change are significant.     

Cold unfolding of β-hairpins: a molecular-level rationalization

Isolated β-hairpins in water have a temperature dependence of their conformational stability qualitatively resembling that of globular proteins, showing both cold and hot unfolding transitions. It is shown that a molecular-level rationalization of this cold unfolding can be provided extending the approach devised for globular proteins (Graziano G. Phys Chem Chem Phys 2010; 12:14245-14252). The decrease in the solvent-excluded volume upon folding, measured by the decrease in the solvent accessible surface area, produces a gain in configurational/translational entropy of water molecules that is the main stabilizing contribution of the folded conformation. This always stabilizing Gibbs energy contribution has a parabolic-like temperature dependence in water and is exactly counterbalanced at two temperatures (i.e., the cold and hot unfolding temperatures) by the always destabilizing Gibbs energy contribution due to the loss in conformational degrees of freedom of the peptide chain.     

On the interpretation of the observed linear free energy relationship in phosphate hydrolysis: a thorough computational study of phosphate diester hydrolysis in solution

The hydrolysis of phosphate esters is crucially important to biological systems, being involved in, among other things, signaling, energy transduction, biosynthesis, and the regulation of protein function. Despite this, there are many questions that remain unanswered in this important field, particularly with regard to the preferred mechanism of hydrolysis of phosphate esters, which can proceed through any of multiple pathways that are either associative or dissociative in nature. Previous comparisons of calculated and observed linear free energy relationships (LFERs) for phosphate monoester dianions with different leaving groups showed that the TS character gradually changes from associative to dissociative with the increasing acidity of the leaving group, while reproducing the experimental LFER. Here, we have generated ab initio potential energy surfaces for the hydrolysis of phosphate diesters in solution, with a variety of leaving groups. Once again, the reaction changes from a compact concerted pathway to one that is more expansive in character when the acidity of the leaving group increases. When such systems are examined in solution, it is essential to take into consideration the contribution of solute to the overall activation entropy, which remains a major computational challenge. The popular method of calculating the entropy using a quasi-harmonic approximation appears to markedly overestimate the configurational entropy for systems with multiple occupied energy wells. We introduce an improved restraint release approach for evaluating configurational entropies and apply this approach to our systems. We demonstrate that when this factor is taken into account, it is possible to reproduce the experimental LFER for this system with reasonable accuracy.     

The free energy of a liquid when viewed as a population of overlapping clusters

The expression of the free energy of a liquid in terms of an explicit decomposition of the particle configurations into local coordination clusters is examined. We argue that the major contribution to the entropy associated with structural fluctuations arises from the local athermal constraints imposed by the overlap of adjacent coordination shells. In the context of the recently developed Favoured Local Structure model [Soft Matt. 11, 3322 (2015)], we derive explicit expressions for the structural energy and entropy in the high-temperature limit, compare this approximation with simulation data and consider the extension of this free energy to the case of spatial inhomogeneity in the distribution of local structures.     

The thermodynamics of protein-protein recognition as characterized by a combination of volumetric and calorimetric techniques: the binding of turkey ovomucoid third domain to alpha-chymotrypsin

We have used ultrasonic velocimetry, high-precision densimetry, and fluorescence spectroscopy, in conjunction with isothermal titration and differential scanning calorimetry, to characterize the binding of turkey ovomucoid third domain (OMTKY3) to alpha-chymotrypsin. We report the changes in volume and adiabatic compressibility that accompany the association of these proteins at 25 degrees C and pH 4.5. In addition, we report the changes in free energy, enthalpy, entropy, and heat capacity upon the binding of OMTKY3 to alpha-chymotrypsin over a temperature range of 20-40 degrees C. Our volume and compressibility data, in conjunction with X-ray crytsallographic data on the OMTKY3-alpha-chymotrypsin complex, suggest that 454(+/-22) water molecules are released to the bulk state upon the binding of OMTKY3 to alpha-chymotrypsin. Furthermore, these volumetric data suggest that the intrinsic compressibility of the two proteins decreases by 7%. At each temperature studied, OMTKY3 association with alpha-chymotrypsin is entropy driven with a large, unfavorable enthalpy contribution. The observed entropy of the binding reflects interplay between two very large favorable and unfavorable terms. The favorable term reflects an increase in the hydrational entropy resulting from release to the bulk of 454 water molecules. The unfavorable term is related to a decrease in the configurational entropy and, consequently, a decrease in the conformational dynamics of the two proteins. In general, we discuss the relationship between macroscopic and microscopic properties, in particular, identifying and quantifying the role of hydration in determining the thermodynamics of protein recognition as reflected in volumetric and calorimetric parameters.     

Configurational entropy change of netropsin and distamycin upon DNA minor-groove binding

Binding of a small molecule to a macromolecular target reduces its conformational freedom, resulting in a negative entropy change that opposes the binding. The goal of this study is to estimate the configurational entropy change of two minor-groove-binding ligands, netropsin and distamycin, upon binding to the DNA duplex d(CGCGAAAAACGCG).d(CGCGTTTTTCGCG). Configurational entropy upper bounds based on 10-ns molecular dynamics simulations of netropsin and distamycin in solution and in complex with DNA in solution were estimated using the covariance matrix of atom-positional fluctuations. The results suggest that netropsin and distamycin lose a significant amount of configurational entropy upon binding to the DNA minor groove. The estimated changes in configurational entropy for netropsin and distamycin are -127 J K(-1) mol(-1) and -104 J K(-1) mol(-1), respectively. Estimates of the configurational entropy contributions of parts of the ligands are presented, showing that the loss of configurational entropy is comparatively more pronounced for the flexible tails than for the relatively rigid central body.     

Dynamics and relative stabilities of parallel- and antiparallel-stranded DNA duplexes.

The dynamics and stability of four DNA duplexes are studied by means of molecular dynamics simulations. The four molecules studied are combinations of 4, 15 bases long, single-stranded oligomers, F1, F2, F3, and F4. The sequence of these single strand oligomers are chosen such that F1-F2 and F3-F4 form parallel (ps) DNA double helices, whereas F1-F4 and F2-F3 form antiparallel-stranded (aps) DNA double helices. Simulations were done at low (100 K) and room (300 K) temperatures. At low temperatures the dynamics are quasi-harmonic and the analysis of the trajectories gives good estimates of the low frequency vibrational modes and density of states. These are used to estimate the linear (harmonic) contribution of local fluctuations to the configurational entropy of the systems. Estimates of the differences in enthalpy between ps and aps duplexes show that aps double helices are more stable than the corresponding ps duplexes, in agreement with experiments. At higher temperatures, the distribution of the fluctuations around the average structures are multimodal and estimates of the configurational entropy cannot be obtained. The multi-basin, nonlinear character of the dynamics at 300 K is established using a novel method which extracts large amplitude nonlinear motions from the molecular dynamics trajectories. Our analysis shows that both ps DNA exhibit much larger fluctuations than the two aps DNA. The large fluctuations of ps DNA are explained in terms of correlated transitions in the beta, epsilon, and zeta backbone dihedral angles.     

Numerical modeling of the formation of steady-state nonequilibrium distributions of particles interacting through a power-law potential

The formation of a steady-state nonequilibrium distribution function of particles interacting through the repulsive potential U ~ α/r ^β(1≤β≤4), which operates at an infinite range, is studied numerically. The collisional particle dynamics in such a system is investigated using a spatially homogeneous nonlinear collision integral in the Landau-Fokker-Planck form, which is a model Boltzmann collision integral for arbitrary potentials of interaction accompanied by little momentum transfer between particles in collisions. Numerical modeling is based on completely conservative difference schemes. It is shown that the principal condition for the existence of steady-state nonequilibrium distributions is the presence of a particle or an energy flux oriented in the proper manner in momentum space. A steady-state local distribution exists inside the momentum interval between the energy source and sink and has the form of a gradually decreasing function. Since a radical change in the distribution function under nonequilibrium conditions leads to an anomalous enhancement of the conduction of a medium and its emission characteristics, the results obtained can be used, e.g., to predict the behavior of semiconductors with an intrinsic or extrinsic conductivity under the action of particle fluxes or electromagnetic radiation.     

Loss of translational entropy in molecular associations

Molecular associations in solution are opposed by the loss of entropy (DeltaS) that results from the restriction of motion of each component in the complex. Theoretical estimates of DeltaS are essential for rationalizing binding affinities, as well as for calculating entropic contribution to enzyme catalysis. Recently a statistical-mechanical framework has been proposed for estimating efficiently the translational entropy loss (DeltaS(trsl)), while taking explicitly into account the complex intermolecular interactions between the solute and the solvent. This framework relates the translational entropy of a solute in solution to its "free volume," defined as the volume accessible to the center of mass of the solute in the presence of the solvent and calculated by using an extension of the cell model (CM) for condensed phases. The translational entropy of pure water, estimated with the CM algorithm, shows good agreement with the experimental information. The free volume of various solutes in water, calculated within the CM by using molecular dynamics simulations with explicit solvent, displays a strong correlation with the solutes' polar and total surface areas. This correlation is used to propose a parameterization that can be used to calculate routinely the translational entropy of a solute in water. We also applied the CM formalism to calculate the free volume and translational entropy loss (DeltaS(trsl)) on binding of benzene to a cavity in a mutant T4-lysozyme. Our results agree with previously published estimates of the binding of benzene to this mutant T4-lysozyme. These and other considerations suggest that the cell model is a simple yet efficient theoretical framework to evaluate the translational entropy loss on molecular association in solution.     

Configurational entropy modulates the mechanical stability of protein GB1

Configurational entropy plays important roles in defining the thermodynamic stability as well as the folding/unfolding kinetics of proteins. Here we combine single-molecule atomic force microscopy and protein engineering techniques to directly examine the role of configurational entropy in the mechanical unfolding kinetics and mechanical stability of proteins. We used a small protein, GB1, as a model system and constructed four mutants that elongate loop 2 of GB1 by 2, 5, 24 and 46 flexible residues, respectively. These loop elongation mutants fold properly as determined by far-UV circular dichroism spectroscopy, suggesting that loop 2 is well tolerant of loop insertions without affecting GB1′s native structure. Our single-molecule atomic force microscopy results reveal that loop elongation decreases the mechanical stability of GB1 and accelerates the mechanical unfolding kinetics. These results can be explained by the loss of configurational entropy upon closing an unstructured flexible loop using classical polymer theory, highlighting the important role of loop regions in the mechanical unfolding of proteins. This study not only demonstrates a general approach to investigating the structural deformation of the loop regions in mechanical unfolding transition state, but also provides the foundation to use configurational entropy as an effective means to modulate the mechanical stability of proteins, which is of critical importance towards engineering artificial elastomeric proteins with tailored nanomechanical properties.     

Calculation of cyclodextrin binding affinities: energy, entropy, and implications for drug design

The second generation Mining Minima method yields binding affinities accurate to within 0.8 kcal/mol for the associations of alpha-, beta-, and gamma-cyclodextrin with benzene, resorcinol, flurbiprofen, naproxen, and nabumetone. These calculations require hours to a day on a commodity computer. The calculations also indicate that the changes in configurational entropy upon binding oppose association by as much as 24 kcal/mol and result primarily from a narrowing of energy wells in the bound versus the free state, rather than from a drop in the number of distinct low-energy conformations on binding. Also, the configurational entropy is found to vary substantially among the bound conformations of a given cyclodextrin-guest complex. This result suggests that the configurational entropy must be accounted for to reliably rank docked conformations in both host-guest and ligand-protein complexes. In close analogy with the common experimental observation of entropy-enthalpy compensation, the computed entropy changes show a near-linear relationship with the changes in mean potential plus solvation energy.