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.     

Entropy in biological binding processes: Estimation of translational entropy loss

The loss of translational degrees of freedom makes an important, unfavorable contribution to the free energy of binding. Examination of experimental values suggest that calculation of this entropy using the Sackur–Tetrode equation produces largely overestimated values. Better agreement is obtained using the cratic entropy. Theoretical considerations suggest that the volumes available for the movement of a ligand in solution and in a complex are rather similar, suggesting also that the cratic entropy provides the best estimate of the loss of translational entropy. © 1994 John Wiley & Sons, Inc.     

The entropic cost of protein-protein association: a case study on acetylcholinesterase binding to fasciculin-2

Protein-protein association is accompanied by a large reduction in translational and rotational (external) entropy. Based on a 15 ns molecular dynamics simulation of acetylcholinesterase (AChE) in complex with fasciculin 2 (Fas2), we estimate the loss in external entropy using quasiharmonic analysis and histogram-based approximations of the probability distribution function. The external entropy loss of AChE-Fas2 binding, ~30 cal/mol K, is found to be significantly larger than most previously characterized protein-ligand systems. However, it is less than the entropy loss estimated in an earlier study by A. V. Finkelstein and J. Janin, which was based on atomic motions in crystals.     

Entropic benefit of a cross-link in protein association

We introduce a method to estimate the loss of configurational entropy upon insertion of a cross-link to a dimeric system. First, a clear distinction is established between the loss of entropy upon tethering and binding, two quantities that are often considered to be equivalent. By comparing the probability distribution of the center-to-center distances for untethered and cross-linked versions, we are able to calculate the loss of translational entropy upon cross-linking. The distribution function for the untethered helices is calculated from the probability that a given helix is closer to its partner than to all other helices, the "Nearest Neighbor" method. This method requires no assumptions about the nature of the solvent, and hence resolves difficulties normally associated with calculations for systems in liquids. Analysis of the restriction of angular freedom upon tethering indicates that the loss of rotational entropy is negligible. The method is applied in the context of the folding of a ten turn helical coiled coil with the tether modeled as a Gaussian chain or a flexible amino acid chain. After correcting for loop closure entropy in the docked state, we estimate the introduction of a six-residue tether in the coiled coil results in an effective concentration of the chain to be about 4 or 100 mM, depending upon whether the helices are denatured or pre-folded prior to their association. Thus, tethering results in significant stabilization for systems with millimolar or stronger dissociation constants.     

Contributions to the binding free energy of ligands to avidin and streptavidin

The free energy of binding of a ligand to a macromolecule is here formally decomposed into the (effective) energy of interaction, reorganization energy of the ligand and the macromolecule, conformational entropy change of the ligand and the macromolecule, and translational and rotational entropy loss of the ligand. Molecular dynamics simulations with implicit solvation are used to evaluate these contributions in the binding of biotin, biotin analogs, and two peptides to avidin and streptavidin. We find that the largest contribution opposing binding is the protein reorganization energy, which is calculated to be from 10 to 30 kcal/mol for the ligands considered here. The ligand reorganization energy is also significant for flexible ligands. The translational/rotational entropy is 4.5-6 kcal/mol at 1 M standard state and room temperature. The calculated binding free energies are in the correct range, but the large statistical uncertainty in the protein reorganization energy precludes precise predictions. For some complexes, the simulations show multiple binding modes, different from the one observed in the crystal structure. This finding is probably due to deficiencies in the force field but may also reflect considerable ligand flexibility.     

Entropy contributions to rate accelerations of intramolecular reactions in water vs non-structured solvents

The calculations of Page and Jencks showing that translational entropy factors may play a significant role in enzymatic rate accelerations are extended to the solvent water. The picture changes significantly due to the large entropy terms associated with solvation of hydrocarbons. While translational entropy should still contribute significantly to enzymatic rate accelerations, it is argued that this effect will appear in ΔH^≠ rather than ΔS^≠.     

Association entropy in adsorption processes

The association of two species to form a bound complex, e.g., the binding of a ligand to a protein or the adsorption of a peptide on a lipid membrane, involves an entropy loss, reflecting the conversion of free translational and rotational degrees of freedom into bound motions. Previous theoretical estimates of the standard entropy change in bimolecular binding processes, DeltaS(o), have been derived from the root-mean-square fluctuations in protein crystals, suggesting DeltaS(o) approximately -50 e.u., i.e., TDeltaS degrees approximately -25 kT = -15 kcal/mol. In this work we focus on adsorption, rather than binding processes. We first present a simple statistical-thermodynamic scheme for calculating the adsorption entropy, including its resolution into translational and rotational contributions, using the known distance-orientation dependent binding (adsorption) potential. We then utilize this scheme to calculate the free energy of interaction and entropy of pentalysine adsorption onto a lipid membrane, obtaining TDeltaS(o) approximately -1.7 kT approximately -1.3 kcal/mol. Most of this entropy change is due to the conversion of one free translation into a bound motion, the rest arising from the confinement of two rotational degrees of freedom. The smaller entropy loss in adsorption compared to binding processes arises partly because a smaller number of degrees of freedom become restricted, but mainly due to the fact that the binding potential is much "softer."     

Translational-entropy gain of solvent upon protein folding

We show that even in the complete absence of potential energies among the atoms in a protein-aqueous solution system, there is a physical factor that favors the folded state of the protein. It is a gain in the translational entropy (TE) of water originating from the translational movement of water molecules. An elaborate statistical-mechanical theory is employed to analyze the TE of water in which a protein or peptide with a prescribed conformation is immersed. It is shown that if the number of residues is sufficiently large, the TE gain is powerful enough to compete with the conformational-entropy loss upon folding. For protein G we have tested over 100 compact conformations generated by a computer simulation with the all-atom potentials as well as the native structure. A significant finding is that the largest TE is attained in the native structure. The translational movement of water molecules is quite effective in achieving the tight packing in the interior of a natural protein. These results are true only when the solvent is water whose molecular size is the smallest among the ordinary liquids in nature.     

Direct estimation of entropy loss due to reduced translational and rotational motions upon molecular binding

The entropic cost due to the loss of translational and rotational (T-R) degree of freedom upon binding has been well recognized for several decades. Tightly bound ligands have higher entropic costs than loosely bound ligands. Quantifying the ligand's residual T-R motions after binding, however, is not an easy task. We describe an approach that uses a reduced Hessian matrix to estimate the contributions due to translational and rotational degrees of freedom to entropy change upon molecular binding. The calculations use a harmonic model for the bound state but only include the T-R degrees of freedom. This approximation significantly speeds up entropy calculations because only 6 x 6 matrices need to be treated, which makes it easier to be used in computer-aided drug design for studying many ligands. The methodological connection with other methods is discussed as well. We tested this approximation by applying it to study the binding of ATP, peptide inhibitor (PKI), and several bound water molecules to protein kinase A (PKA). These ligands span a wide range in size. The model gave reasonable estimates of the residual T-R entropy of bound ligands or water molecules. The residual T-R entropy demonstrated a wide range of values, e.g., 4 to 16 cal/K.mol for the bound water molecules of PKA.     

On the effect of hydrostatic pressure on the conformational stability of globular proteins

The model developed for cold denaturation (Graziano, PCCP 2010, 12, 14245‐14252) is extended to rationalize the dependence of protein conformational stability upon hydrostatic pressure, at room temperature. A pressure− volume work is associated with the process of cavity creation for the need to enlarge the liquid volume against hydrostatic pressure. This contribution destabilizes the native state that has a molecular volume slightly larger than the denatured state due to voids existing in the protein core. Therefore, there is a hydrostatic pressure value at which the pressure−volume contribution plus the conformational entropy loss of the polypeptide chain are able to overwhelm the stabilizing gain in translational entropy of water molecules, due to the decrease in water accessible surface area upon folding, causing denaturation. © 2015 Wiley Periodicals, Inc. Biopolymers 103: 711–718, 2015.     

The role of entropy in the discrimination between CO and O2 in myoglobin

Using stopped-flow rapid mixing and flash photolysis techniques, the dissociation rate coefficients of horse carbonmonoxy myoglobin (hMbCO) and oxygenated myoglobin (hMbO₂) in aqueous solution have been determined as a function of temperature between 274 and 342 K. From the Arrhenius plot, an activation enthalpy for dissociation of 74 kJ/mol was obtained for both ligands. The pronounced kinetic differences arise from markedly different pre-exponentials. We compare the Arrhenius parameters with those of the association reaction, as measured at cryogenic temperatures. In our analysis we conclude that the entropy loss upon binding of O₂ is twice as large as that for CO. Taking reasonable estimates for the frequency factor, the transition state entropy in hMbO₂ is located roughly half way in between the entropies of the bound and unbound states. By contrast, the entropy of the transition state in hMbCO appears to be identical to that of the bound state. Possible structural reasons for the different behavior are discussed. Received: 13 January 1997 / Accepted: 24 April 1997     

Whole-field integration, not detailed analysis, is used by the crab optokinetic system to separate rotation and translation in optic flow

For optimal visual control of compensatory eye movements during locomotion it is necessary to distinguish the rotational and translational components of the optic flow field. Optokinetic eye movements can reduce the rotational component only, making the information contained in the translational flow readily available to the animal. We investigated optokinetic eye rotation in the marble rock crab, Pachygrapsus marmoratus, during translational movement, either by displacing the animal or its visual surroundings. Any eye movement in response to such stimuli is taken as an indication that the system is unable to separate the translational and the rotational components in the optic flow in a mathematically perfect way. When the crabs are translated within a pseudo-natural environment, eye movements are negligible, especially during sideways translation. When, however, crabs were placed in a gangway between two elongated rectangular sidewalls carrying dotted patterns which were translated back and forth, marked eye movements were elicited, depending on the translational velocity. To resolve this discrepancy, we tested several hypotheses about mechanisms using detailed analysis of the optic flow or whole-field integration. We found that the latter are sufficient to explain the efficient separation of translation and rotation of crabs in quasi-natural situations. Accepted: 6 May 1997     

Contribution of translational and rotational motions to molecular association in aqueous solution

Much uncertainty and controversy exist regarding the estimation of the enthalpy, entropy, and free energy of overall translational and rotational motions of solute molecules in aqueous solutions, quantities that are crucial to the understanding of molecular association/recognition processes and structure-based drug design. A critique of the literature on this topic is given that leads to a classification of the various views. The major stumbling block to experimentally determining the translational/rotational enthalpy and entropy is the elimination of vibrational perturbations from the measured effects. A solution to this problem, based on a combination of energy equi-partition and enthalpy-entropy compensation, is proposed and subjected to verification. This method is then applied to analyze experimental data on the dissociation/unfolding of dimeric proteins. For one translational/rotational unit at 1 M standard state in aqueous solution, the results for enthalpy (H degrees (tr)), entropy (S degrees (tr)), and free energy (G degrees (tr)) are H (degrees) (tr) = 4.5 +/- 1.5RT, S (degrees) (tr) = 5 +/- 4R, and G (degrees) (tr) = 0 +/- 5RT. Therefore, the overall translational and rotational motions make negligible contribution to binding affinity (free energy) in aqueous solutions at 1 M standard state.     

Buried surface area,conformational entropy,and protein stability

In this paper, how to calculate the loss of conformational entropy owing to protein folding from data on protein stability, the disulfide bonding pattern and the buried surface area is suggested. The average values of the initiation constant and entropy loss per residue for the helix-coil transition, calculated according to the suggested approach, are in a good agreement with available data. It is shown that the loss of conformational entropy for proteins and protein fragments, thus calculated, can be quite accurately approximated by three straight lines, each representing the loss of conformational entropy in a given range of molecular weights. The first line corresponds to the formation of α helices, the second to proteins with molecular weights below 10,000, and the third to proteins with molecular weights above this value. The standard error varied from 1.3 kcal/mol, for the first region, to between ～3 and 5 kcal/mol for the third. Use of these approximating linear functions is proposed for the calculation of protein stabilities from x-ray data. Such calculations, performed for more than 20 proteins and their fragments, have led to at least qualitative agreement between the calculated stabilities and available data for more than 90% of the cases. While quantitative predictions are at the limit of the method's accuracy, predictions of the probabilities of proteins or their fragments to be stable are expected to give a correct answer in ～95% of the cases.     

Calculating the free energy of association of transmembrane helices

A large number of experimental studies have been devoted to quantifying the interaction between transmembrane (TM) helices in detergent micelles and, more recently, in bilayers. Theoretical calculation of association free energy of TM helices would be useful for predicting the propensity of given sequences to oligomerize and for understanding the difference between association in micelles and in bilayers. In this article, the theoretical foundation for calculating the standard association free energy of TM helices is laid out and is applied to glycophorin A in both micelles and bilayers. The standard association free energy is decomposed into the effective energy, translational, rotational, and conformational entropy terms. The effective energy of association is obtained by molecular dynamics simulations in an implicit membrane model. The translational and rotational entropy of association is calculated from the probability distribution of the translational and rotational degrees of freedom obtained from the molecular dynamics simulations. The side-chain conformational entropy of association is estimated from the probability distribution obtained by rigid rotation of all side-chain dihedral angles. The calculated standard association free energy of glycophorin A in N-dodecylphosphocholine micelles is in good agreement with the experimental value. The translational entropy cost is larger, whereas the rotational entropy cost is smaller in bilayers than in micelles. The standard association free energy in 1,2-dimyristoyl-sn-glycero-3-phosphocholine bilayers is calculated to be approximately 1.3 kcal/mol more favorable than in N-dodecylphosphocholine micelles, consistent with available experimental data.     

Conformational entropy limits the transition from nucleation to elongation in amyloid aggregation

The formation of β-sheet-rich amyloid fibrils in Alzheimer’s disease and other neurodegenerative disorders is limited by a slow nucleation event. To understand the initial formation of β-sheets from disordered peptides, we used all-atom simulations to parameterize a lattice model that treats each amino acid as a binary variable with β- and non-β-sheet states. We show that translational and conformational entropy give the nascent β-sheet an anisotropic surface tension that can be used to describe the nucleus with 2D classical nucleation theory. Since translational entropy depends on concentration, the aspect ratio of the critical β-sheet changes with protein concentration. Our model explains the transition from the nucleation phase to elongation as the point where the β-sheet core becomes large enough to overcome the conformational entropy cost to straighten the terminal molecule. At this point the β-strands in the nucleus spontaneously elongate, which results in a larger binding surface to capture new molecules. These results suggest that nucleation is relatively insensitive to sequence differences in coaggregation experiments because the nucleus only involves a small portion of the peptide.     

Calculation of the free energy of association for protein complexes.

We have developed a method for calculating the association energy of quaternary complexes starting from their atomic coordinates. The association energy is described as the sum of two solvation terms and an energy term to account for the loss of translational and rotational entropy. The calculated solvation energy, using atomic solvation parameters and the solvent accessible surface areas, has a correlation of 96% with experimentally determined values. We have applied this methodology to examine intermediates in viral assembly and to assess the contribution isomerization makes to the association energy of molecular complexes. In addition, we have shown that the calculated association can be used as a predictive tool for analyzing modeled molecular complexes.     

The different faces of mass action in virus assembly

The spontaneous encapsulation of genomic and non-genomic polyanions by coat proteins of simple icosahedral viruses is driven, in the first instance, by electrostatic interactions with polycationic RNA binding domains on these proteins. The efficiency with which the polyanions can be encapsulated in vitro, and presumably also in vivo, must in addition be governed by the loss of translational and mixing entropy associated with co-assembly, at least if this co-assembly constitutes a reversible process. These forms of entropy counteract the impact of attractive interactions between the constituents and hence they counteract complexation. By invoking mass action-type arguments and a simple model describing electrostatic interactions, we show how these forms of entropy might settle the competition between negatively charged polymers of different molecular weights for co-assembly with the coat proteins. In direct competition, mass action turns out to strongly work against the encapsulation of RNAs that are significantly shorter, which is typically the case for non-viral (host) RNAs. We also find that coat proteins favor forming virus particles over nonspecific binding to other proteins in the cytosol even if these are present in vast excess. Our results rationalize a number of recent in vitro co-assembly experiments showing that short polyanions are less effective at attracting virus coat proteins to form virus-like particles than long ones do, even if both are present at equal weight concentrations in the assembly mixture.     

Visual position stabilization in the hummingbird hawk moth, Macroglossum stellatarum L. I. Behavioural analysis

Optomotor responses of freely flying hawk moths, Macroglossum stellatarum, were characterized while the animals were hovering in front of and feeding on a dummy flower. Compensatory translational and rotational movements of the hawk moth were elicited by vertical grating patterns moving horizontally, mimicking imposed rotational and translational displacements of the animal in the horizontal plane. Oscillatory translational and rotational pattern motion leads to compensatory responses that peak in the frequency range between 2 Hz and 4 Hz. The control systems mediating the translational and rotational components of the optomotor response do not seem to influence each other. The system mediating translational responses is more sensitive in the fronto-lateral part of the visual field than in the lateral part; the opposite is true for the rotational system. The sensitivity of the translational system does not change along the vertical, whereas the rotational system is much more sensitive to motion in the dorsal than in the ventral part of the visual field. These sensitivity gradients may reflect an adaptation to the specific requirements of position stabilization in front of flowers during feeding. Accepted: 13 August 1997