Translational Entropy Research Articles

ATP-Induced Conformational Changes of Nucleotide-Binding Domains in an ABC Transporter. Importance of the Water-Mediated Entropic Force

ATP binding cassette (ABC) proteins belong to a superfamily of active transporters. Recent experimental and computational studies have shown that binding of ATP to the nucleotide binding domains (NBDs) of ABC proteins drives the dimerization of NBDs, which, in turn, causes large conformational changes within the transmembrane domains (TMDs). To elucidate the active substrate transport mechanism of ABC proteins, it is first necessary to understand how the NBD dimerization is driven by ATP binding. In this study, we selected MalKs (NBDs of a maltose transporter) as a representative NBD and calculated the free-energy change upon dimerization using molecular mechanics calculations combined with a statistical thermodynamic theory of liquids, as well as a method to calculate the translational, rotational, and vibrational entropy change. This combined method is applied to a large number of snapshot structures obtained from molecular dynamics simulations containing explicit water molecules. The results suggest that the NBD dimerization proceeds with a large gain of water entropy when ATP molecules bind to the NBDs. The energetic gain arising from direct NBD-NBD interactions is canceled by the dehydration penalty and the configurational-entropy loss. ATP hydrolysis induces a loss of the shape complementarity between the NBDs, which leads to the dissociation of the dimer, due to a decrease in the water-entropy gain and an increase in the configurational-entropy loss. This interpretation of the NBD dimerization mechanism in concert with ATP, especially focused on the water-mediated entropy force, is potentially applicable to a wide variety of the ABC transporters.

The Role of Solution Conditions in the Bacteriophage PP7 Capsid Charge Regulation

We investigate and quantify the effects of pH and salt concentration on the charge regulation of the bacteriophage PP7 capsid. These effects are found to be extremely important and substantial, introducing qualitative changes in the charge state of the capsid such as a transition from net-positive to net-negative charge depending on the solution pH. The overall charge of the virus capsid arises as a consequence of a complicated balance with the chemical dissociation equilibrium of the amino acids and the electrostatic interaction between them, and the translational entropy of the mobile solution ions, i.e., counterion release. We show that to properly describe and predict the charging equilibrium of viral capsids in general, one needs to include molecular details as exemplified by the acid-base equilibrium of the detailed distribution of amino acids in the proteinaceous capsid shell.

Nonionic Double and Triple Network Hydrogels of High Mechanical Strength

Among the hydrogels prepared in recent years, double network (DN) hydrogels exhibit the highest compression strength, toughness, and fracture energies. However, synthesis of DN hydrogels with extraordinary mechanical properties is limited to polyelectrolyte networks, which hinders their widespread applications. Herein, we prepared nonionic DN and triple network (TN) hydrogels based on polyacrylamide (PAAm) and poly(N,N-dimethylacrylamide) (PDMA) with a high mechanical strength by sequential polymerization reactions. The TN approach is based on the decrease of the translational entropy of the second monomer upon its polymerization in the first network, so that additional solvent (third monomer) can enter into DN hydrogel to assume its new thermodynamic equilibrium. The first network of TN hydrogels comprises chemically cross-linked PAAm or PDMA while the second and third networks are linear polymers. To increase the degree of inhomogeneity of the first network hydrogel, an oligomeric ethylene glycol dimeth...

Single water entropy: hydrophobic crossover and application to drug binding.

Entropy of water plays an important role in both chemical and biological processes e.g. hydrophobic effect, molecular recognition etc. Here we use a new approach to calculate translational and rotational entropy of the individual water molecules around different hydrophobic and charged solutes. We show that for small hydrophobic solutes, the translational and rotational entropies of each water molecule increase as a function of its distance from the solute reaching finally to a constant bulk value. As the size of the solute increases (0.746 nm), the behavior of the translational entropy is opposite; water molecules closest to the solute have higher entropy that reduces with distance from the solute. This indicates that there is a crossover in translational entropy of water molecules around hydrophobic solutes from negative to positive values as the size of the solute is increased. Rotational entropy of water molecules around hydrophobic solutes for all sizes increases with distance from the solute, indicating the absence of crossover in rotational entropy. This makes the crossover in total entropy (translation + rotation) of water molecule happen at much larger size (>1.5 nm) for hydrophobic solutes. Translational entropy of single water molecule scales logarithmically (Str(QH) = C + kB ln V), with the volume V obtained from the ellipsoid of inertia. We further discuss the origin of higher entropy of water around water and show the possibility of recovering the entropy loss of some hypothetical solutes. The results obtained are helpful to understand water entropy behavior around various hydrophobic and charged environments within biomolecules. Finally, we show how our approach can be used to calculate the entropy of the individual water molecules in a protein cavity that may be replaced during ligand binding.

Logarithmic finite-size effects on interfacial free energies: phenomenological theory and Monte Carlo studies.

The computation of interfacial free energies between coexisting phases (e.g., saturated vapor and liquid) by computer simulation methods is still a challenging problem due to the difficulty of an atomistic identification of an interface and interfacial fluctuations on all length scales. The approach to estimate the interfacial tension from the free-energy excess of a system with interfaces relative to corresponding single-phase systems does not suffer from the first problem but still suffers from the latter. Considering d-dimensional systems with interfacial area L(d-1) and linear dimension L(z) in the direction perpendicular to the interface, it is argued that the interfacial fluctuations cause logarithmic finite-size effects of order ln(L)/L(d-1) and order ln(L(z))/L(d-1), in addition to regular corrections (with leading-order const/L(d-1)). A phenomenological theory predicts that the prefactors of the logarithmic terms are universal (but depend on the applied boundary conditions and the considered statistical ensemble). The physical origin of these corrections are the translational entropy of the interface as a whole, "domain breathing" (coupling of interfacial fluctuations to the bulk order parameter fluctuations of the coexisting domains), and capillary waves. Using a new variant of the ensemble switch method, interfacial tensions are found from Monte Carlo simulations of d = 2 and d = 3 Ising models and a Lennard-Jones fluid. The simulation results are fully consistent with the theoretical predictions.

Estimating Translational and Orientational Entropies Using the k-Nearest Neighbors Algorithm.

Inhomogeneous fluid solvation theory (IFST) and free energy perturbation (FEP) calculations were performed for a set of 20 solutes to compute the hydration free energies. We identify the weakness of histogram methods in computing the IFST hydration entropy by showing that previously employed histogram methods overestimate the translational and orientational entropies and thus underestimate their contribution to the free energy by a significant amount. Conversely, we demonstrate the accuracy of the k-nearest neighbors (KNN) algorithm in computing these translational and orientational entropies. Implementing the KNN algorithm within the IFST framework produces a powerful method that can be used to calculate free-energy changes for large perturbations. We introduce a new KNN approach to compute the total solute-water entropy with six degrees of freedom, as well as the translational and orientational contributions. However, results suggest that both the solute-water and water-water entropy terms are significant and must be included. When they are combined, the IFST and FEP hydration free energies are highly correlated, with an R(2) of 0.999 and a mean unsigned difference of 0.9 kcal/mol. IFST predictions are also highly correlated with experimental hydration free energies, with an R(2) of 0.997 and a mean unsigned error of 1.2 kcal/mol. In summary, the KNN algorithm is shown to yield accurate estimates of the combined translational-orientational entropy and the novel approach of combining distance metrics that is developed here could be extended to provide a powerful method for entropy estimation in numerous contexts.

Measuring Entropy in Molecular Recognition by Proteins.

Molecular recognition by proteins is fundamental to the molecular basis of biology. Dissection of the thermodynamic landscape governing protein-ligand interactions has proven difficult because determination of various entropic contributions is quite challenging. Nuclear magnetic resonance relaxation measurements, theory, and simulations suggest that conformational entropy can be accessed through a dynamical proxy. Here, we review the relationship between measures of fast side-chain motion and the underlying conformational entropy. The dynamical proxy reveals that the contribution of conformational entropy can range from highly favorable to highly unfavorable and demonstrates the potential of this key thermodynamic variable to modulate protein-ligand interactions. The dynamical so-called entropy meter also refines the role of solvent entropy and directly determines the loss in rotational-translational entropy that occurs upon formation of high-affinity complexes. The ability to quantify the roles of entropy through an entropy meter based on measurable dynamical properties promises to highlight its role in protein function.

Soft confinement for polymer solutions

As a model of soft confinement for polymers, we investigated equilibrium shapes of a flexible vesicle that contains a phase-separating polymer solution. To simulate such a system, we combined the phase field theory (PFT) for the vesicle and the self-consistent field theory (SCFT) for the polymer solution. We observed a transition from a symmetric prolate shape of the vesicle to an asymmetric pear shape induced by the domain structure of the enclosed polymer solution. Moreover, when a non-zero spontaneous curvature of the vesicle is introduced, a re-entrant transition between the prolate and the dumbbell shapes of the vesicle is observed. This re-entrant transition is explained by considering the competition between the loss of conformational entropy and that of translational entropy of polymer chains due to the confinement by the deformable vesicle. This finding is in accordance with the recent experimental result reported by Terasawa et al. (Proc. Natl. Acad. Sci. U.S.A., 108 (2011) 5249).

Communication: From rods to helices: Evidence of a screw-like nematic phase

Evidence of a special chiral nematic phase is provided using numerical simulation and Onsager theory for systems of hard helical particles. This phase appears at the high density end of the nematic phase, when helices are well aligned, and is characterized by the C2 symmetry axes of the helices spiraling around the nematic director with periodicity equal to the particle pitch. This coupling between translational and rotational degrees of freedom allows a more efficient packing and hence an increase of translational entropy. Suitable order parameters and correlation functions are introduced to identify this screw-like phase, whose main features are then studied as a function of radius and pitch of the helical particles. Our study highlights the physical mechanism underlying a similar ordering observed in colloidal helical flagella [E. Barry, Z. Hensel, Z. Dogic, M. Shribak, and R. Oldenbourg, Phys. Rev. Lett. 96, 018305 (2006)] and raises the question of whether it could be observed in other helical particle systems, such as DNA, at sufficiently high densities.

Computation of Standard Binding Free Energies of Polar and Charged Ligands to the Glutamate Receptor GluA2

Accurate calculation of the binding affinity of small molecules to proteins has the potential to become an important tool in rational drug design. In this study, we use the free energy perturbation (FEP) method with restraints to calculate the standard binding free energy of five ligands (ACPA, AMPA, CNQX, DNQX, and glutamate) to the glutamate receptor GluA2, which plays an essential role in synaptic transmission. To deal with the convergence problem in FEP calculations with charged ligands, we use a protocol where the ligand is coupled in the binding site while it is decoupled in bulk solution simultaneously. The contributions from the conformational, rotational, and translational entropies to the standard binding free energy are determined by applying/releasing respective restraints to the ligand in bulk/binding site. We also employ the confine-and-release approach, which helps to resolve convergence problems in FEP calculations. Our results are in good agreement with the experimental values for all five ligands, including the charged ones which are often problematic in FEP calculations. We also analyze the different contributions to the binding free energy of each ligand to GluA2 and discuss the nature of these interactions.

Metal complexation and ion hydration in low density hydrothermal fluids: Ab initio molecular dynamics simulation of Cu(I) and Au(I) in chloride solutions (25–1000 °C, 1–5000 bar)

Low-density supercritical fluids are suspected of being able to transport metals, but it is unclear what the speciation/complexation would be in such conditions. In this work, we used ab initio molecular dynamics simulations to investigate the complexation, ion association and hydration of Cu+ and Au+ in NaCl brines as a function of solution density, from ambient to supercritical conditions (to 1000°C, 5000bar). Cu(I) and Au(I) form distorted linear complexes with two chloride ligands (i.e., CuCl2− and AuCl2−) in subcritical chloride brines. We have discovered that these charged complexes remain in high density supercritical fluids even at high temperature; however, with decreasing density, these complexes become progressively neutralized by ion association with Na+ to form low-charge (NanCuCl2)n−1 and (NanAuCl2)n−1 complexes. In these species, the Na+ ion is very weakly bonded in the outer coordination sphere, resulting in highly disordered structures and fast (few picoseconds) exchange among coordinated and solvent Na+ ions. Thermodynamic models to predict the solubility of metals in low-density magmatic or metamorphic fluids must account for these species. In addition, we found that the number of water molecules (i.e., the hydration number) surrounding the Cu+, Au+, Na+ and Cl− ions decreases linearly when fluid density decreases; this supports empirical thermodynamic models that correlate the stability constants of complexation reactions with solvent density. The traditional Born-model description explains the ion association as resulting from the decreased dielectric constant of the solvent. However at a molecular level, the increased ion association results from the increase in translational entropy associated with ion dehydration.

Mechanistic Insights into Carbonyl-Directed Rhodium-Catalyzed Hydroboration: ab Initio Study of a Cyclic γ,δ-Unsaturated Amide.

A two-point binding mechanism for the cationic rhodium(I)-catalyzed carbonyl-directed catalytic asymmetric hydroboration of a cyclic γ,δ-unsaturated amide is investigated using density functional theory. Geometry optimizations and harmonic frequency calculations for the model reaction are carried out using the basis set 6-31+G** for C, O, P, B, N, and H and LANL2DZ for Rh atoms. The Gibbs free energy of each species in THF solvent is obtained based on the single-point energy computed using the PCM model at the ECP28MWB/6-311+G(d,p) level plus the thermal correction to Gibbs free energy by deducting translational entropy contribution. The Rh-catalyzed reaction cycle involves the following sequence of events: (1) chelation of the cyclic γ,δ-unsaturated amide via alkene and carbonyl complexation in a model active catalytic species, [Rh(L2)2S2]+, (2) oxidative addition of pinacol borane (pinBH), (3) migratory insertion of the alkene double bond into Rh–H (preferred pathway) or Rh–B bond, (4) isomerization of the resulting intermediate, and finally, (5) reductive elimination to form the B–C or H–C bond with regeneration of the catalyst. Free energy profiles for potential pathways leading to the major γ-borylated product are computed and discussed in detail. The potential pathways considered include (1) pathways proceeding via migratory insertion into the Rh–H bond (pathways I, I-1, and I-2), (2) a potential pathway proceeding via migratory insertion into the Rh–B bond (pathway II), and two potential competing routes to a β-borylated byproduct (pathway III). The results find that the Rh–H migratory insertion pathway I-2, followed in sequence by an unanticipated isomerization via amide rotation and reductive elimination, is the most favorable reaction pathway. A secondary consequence of amide rotation is access to a competing β-hydride elimination pathway. The pathways computed in this study are supported by and help explain related experimental results.

Born energy, acid-base equilibrium, structure and interactions of end-grafted weak polyelectrolyte layers

This work addresses the effect of the Born self-energy contribution in the modeling of the structural and thermodynamical properties of weak polyelectrolytes confined to planar and curved surfaces. The theoretical framework is based on a theory that explicitly includes the conformations, size, shape, and charge distribution of all molecular species and considers the acid-base equilibrium of the weak polyelectrolyte. Namely, the degree of charge in the polymers is not imposed but it is a local varying property that results from the minimization of the total free energy. Inclusion of the dielectric properties of the polyelectrolyte is important as the environment of a polymer layer is very different from that in the adjacent aqueous solution. The main effect of the Born energy contribution on the molecular organization of an end-grafted weak polyacid layer is uncharging the weak acid (or basic) groups and consequently decreasing the concentration of mobile ions within the layer. The magnitude of the effect increases with polymer density and, in the case of the average degree of charge, it is qualitatively equivalent to a small shift in the equilibrium constant for the acid-base equilibrium of the weak polyelectrolyte monomers. The degree of charge is established by the competition between electrostatic interactions, the polymer conformational entropy, the excluded volume interactions, the translational entropy of the counterions and the acid-base chemical equilibrium. Consideration of the Born energy introduces an additional energetic penalty to the presence of charged groups in the polyelectrolyte layer, whose effect is mitigated by down-regulating the amount of charge, i.e., by shifting the local-acid base equilibrium towards its uncharged state. Shifting of the local acid-base equilibrium and its effect on the properties of the polyelectrolyte layer, without considering the Born energy, have been theoretically predicted previously. Account of the Born energy leads to systematic, but in general small, corrections to earlier theoretical predictions describing the behavior of weak polyelectrolyte layers. However, polyelectrolyte uncharging results in a decrease in the concentration of counterions and inclusion of the Born Energy can result in a substantial decrease of the counterion concentration. The effect of considering the Born energy contribution is explored for end-grafted weak polyelectrolyte layers by calculating experimental observables which are known to depend on the presence of charges within the polyelectrolyte layer: inclusion of the Born energy contribution leads to a decrease in the capacitance of polyelectrolyte-modified electrodes, a decrease of conductivity of polyelectrolyte-modified nanopores and an increase in the repulsion exerted by a planar polyelectrolyte layer confined by an opposing wall.

DFT calculations on kinetic data for some [4+2] reactions in solution.

The reaction mechanisms of [4+2] cycloaddition reactions between dienes and dienophiles have been investigated with several density functional theory (DFT) methods, such as CAM-B3LYP, BMK, M062x wB97x and wB97xd, and the obtained results show that most of the reactions are synchronous or asynchronous. The stability of the transition state is moderated by the interaction of frontier molecular orbitals (FMOs), in which a diene acts as an electron-donating partner and a dienophile acts as an electron-acceptor from the charge transfer direction in the transition state. The activation free energy barriers have been calculated with both gas-phase translational entropy and solution translational entropy, in which those from gas-phase translational entropy (output of the Gaussian job) are far away from the experimental estimations. It has been found that free-energy barriers generated from solution translational entropies with CAM-B3LYP+IDSCRF/6-31G(d), BMK+IDSCRF/6-31G(d) and wB97x+IDSCRF/6-31G(d) are very close to the experimental measurements, but both M062x and wB97xd methods predict too low free-energy barriers for most of the studied reactions. The substituent and solvent effects on reaction dynamic data have also been addressed.

On the mechanism of cold denaturation.

A theoretical rationalization of the occurrence of cold denaturation for globular proteins was devised, assuming that the effective size of water molecules depends upon temperature [G. Graziano, Phys. Chem. Chem. Phys., 2010, 12, 14245-14252]. In the present work, it is shown that the latter assumption is not necessary. By performing the same type of calculations in water, 40% (by weight) methanol, methanol, and carbon tetrachloride, it emerges that cold denaturation occurs only in water due to the special temperature dependence of its density and the small size of its molecules. These two coupled factors determine the magnitude and the temperature dependence of the stabilizing term that measures the gain in configurational/translational entropy of water molecules upon folding of the protein. This term has to be contrasted with the destabilizing contribution measuring the loss in conformational entropy of the polypeptide chain upon folding.

Cluster-continuum quantum mechanical models to guide the choice of anions for Li+-conducting ionomers

A quantum-mechanical investigation on Li poly(ethylene oxide)-based ionomers was performed in the cluster-continuum solvation model (CCM) that includes specific solvation in the first shell surrounding the cation, all surrounded by a polarizable continuum. A four-state model, including a free Li cation, Li(+)-anion pair, triple ion, and quadrupole was used to represent the states of Li(+) within the ionomer in the CCM. The relative energy of each state was calculated for Li(+) with various anions, with dimethyl ether representing the ether oxygen solvation. The population distribution of Li(+) ions among states was estimated by applying Boltzmann statistics to the CCM energies. Entropy difference estimates are needed for populations to better match the true ionomer system. The total entropy change is considered to consist of four contributions: translational, rotational, electrostatic, and solvent immobilization entropies. The population of ion states is reported as a function of Bjerrum length divided by ion-pair separation with/without entropy considered to investigate the transition between states. Predicted concentrations of Li(+)-conducting states (free Li(+) and positive triple ions) are compared among a series of anions to indicate favorable features for design of an optimal Li(+)-conducting ionomer; the perfluorotetraphenylborate anion maximizes the conducting positive triple ion population among the series of anions considered.

A Piston-Rotaxane with Two Potential Stripes: Force Transitions and Yield Stresses

We examine a rod piston-rotaxane system, where the positions of several mobile rings on the axle are controlled by an external force acting on one of the rings. This allows us to access the translational entropy of the rings. For a simple rotaxane molecule with an axle that has uniform ring-axle interactions along its length, the molecule behaves like a miniature piston filled with a one-dimensional ideal gas. We then examine the effect of two stripes on the axle, having different ring-axle interactions with the mobile rings, so that one section is of high energy (repulsive) for the rings and another section is of lower energy (or attractive). This kind of rotaxane can exhibit rapid changes in displacement or force, and in particular, this molecule can exhibit a yield stress in which the piston suddenly compresses under a small increase in the applied force.

A Piston-Rotaxane with Two Potential Stripes: Force Transitions and Yield Stresses

We examine a rod piston-rotaxane system, where the positions of several mobile rings on the axle are controlled by an external force acting on one of the rings. This allows us to access the translational entropy of the rings. For a simple rotaxane molecule with an axle that has uniform ring-axle interactions along its length, the molecule behaves like a miniature piston filled with a one-dimensional ideal gas. We then examine the effect of two stripes on the axle, having different ring-axle interactions with the mobile rings, so that one section is of high energy (repulsive) for the rings and another section is of lower energy (or attractive). This kind of rotaxane can exhibit rapid changes in displacement or force, and in particular, this molecule can exhibit a yield stress in which the piston suddenly compresses under a small increase in the applied force.

Efficient Computation of Small-Molecule Configurational Binding Entropy and Free Energy Changes by Ensemble Enumeration

Here we present a novel, end-point method using the dead-end-elimination and A* algorithms to efficiently and accurately calculate the change in free energy, enthalpy, and configurational entropy of binding for ligand–receptor association reactions. We apply the new approach to the binding of a series of human immunodeficiency virus (HIV-1) protease inhibitors to examine the effect ensemble reranking has on relative accuracy as well as to evaluate the role of the absolute and relative ligand configurational entropy losses upon binding in affinity differences for structurally related inhibitors. Our results suggest that most thermodynamic parameters can be estimated using only a small fraction of the full configurational space, and we see significant improvement in relative accuracy when using an ensemble versus single-conformer approach to ligand ranking. We also find that using approximate metrics based on the single-conformation enthalpy differences between the global minimum energy configuration in the bound as well as unbound states also correlates well with experiment. Using a novel, additive entropy expansion based on conditional mutual information, we also analyze the source of ligand configurational entropy loss upon binding in terms of both uncoupled per degree of freedom losses as well as changes in coupling between inhibitor degrees of freedom. We estimate entropic free energy losses of approximately +24 kcal/mol, 12 kcal/mol of which stems from loss of translational and rotational entropy. Coupling effects contribute only a small fraction to the overall entropy change (1–2 kcal/mol) but suggest differences in how inhibitor dihedral angles couple to each other in the bound versus unbound states. The importance of accounting for flexibility in drug optimization and design is also discussed.

Effects of attractive colloids on the phase separation behaviors of binary polymer blends

The attractive colloids are added as fillers to control the phase behaviors of binary polymer blends. Because the colloids attract both components in the blends, aggregates are formed by the colloids coated with both kinds of polymer brushes. The aggregation results in two contradictory effects on the phase separation. First, the formation of aggregate decreases the translational entropy, which promotes the phase separation. On the other hand, the phase separation causes the extra free energy penalty due to the stretch of the chains attaching on the colloids, which prevents the phase separation. Furthermore, as the concentration or adsorbability of the colloids increases the local fluctuations within the aggregates become important. This results in a transition from the macro-phase separation to the micro-phase separation and the existence of the Lifshitz point. All of these effects lead to diverse phase behaviors in the polymer nanocomposites system. In present work, these behaviors are studied theoretically by the random phase approximation in a model system.