Loss Of Configurational Entropy Research Articles

Flocculated Laponite–PEG/PEO dispersions with monovalent salt, a SAXS and simulation study

It is well-known that clay can form lamellar structures i.e. tactoids, and recently it has been shown that the tactoid formation is dependent on the platelet diameter. To the authors knowledge, no tactoid formation has been observed for montmorillonite platelets with a diameter less than 60nm. In this study, small angle X-ray scattering in combination with coarse-grained modeling and molecular dynamics simulations have been utilized to study the sediment of Laponite–polyethylene glycol/polyethylene oxide (PEG/PEO) at elevated salt concentrations (150mM–1M). Laponite consists of platelets with a diameter of 25nm and it is known to have a relatively monodisperse size-distribution. At pH 10, the face of the platelets has a strong negative charge, whereas the rim is slightly positive. Here we show that it is possible to induce tactoids for Laponite if two constraints are fulfilled: (1) addition of high amount of salt such as NaCl, and (2) addition of a neutral polymer such as PEG. The role of the salt is to screen the repulsive interactions between the platelets and the role of the polymer is to bridge the platelets together: hence the loss in configurational entropy of the polymer is counteracted by the gain in attractive polymer–platelet interaction. As the concentration of NaCl and/or PEG increases, the Bragg peak becomes sharper, which is an indication of that larger tactoids are formed.Comparison between Laponite and montmorillonite shows that the interlayer distance between the platelets increases linearly with an increased Debye screening length for both type of clays, whereas the structure peaks of Laponite are broader compared to the montmorillonite. We argue that the main reason to the latter is due to the size of the platelets: (i) smaller platelets are less rotationally restricted and (ii) the effect of positive edge charges is larger when the platelets are smaller, which results in more irregular aggregates. In absence of the polymer, montmorillonite form tactoids above ∼0.3MNaCl whereas Laponite does not.Even though the model used is simple, we find qualitative agreement between experiments and simulations, which verifies that the underlying physics for tactoid formation is captured.

Area per Lipid and Cholesterol Interactions in Membranes from Separated Local-Field 13C NMR Spectroscopy

Investigations of lipid membranes using NMR spectroscopy generally require isotopic labeling, often precluding structural studies of complex lipid systems. Solid-state 13C magic-angle spinning NMR spectroscopy at natural isotopic abundance gives site-specific structural information that can aid in the characterization of complex biomembranes. Using the separated local-field experiment DROSS, we resolved 13C-1H residual dipolar couplings that were interpreted with a statistical mean-torque model. Liquid-disordered and liquid-ordered phases were characterized according to membrane thickness and average cross-sectional area per lipid. Knowledge of such structural parameters is vital for molecular dynamics simulations, and provides information about the balance of forces in membrane lipid bilayers. Experiments were conducted with both phosphatidylcholine (dimyristoylphosphatidylcholine (DMPC) and palmitoyloleoylphosphatidylcholine (POPC)) and egg-yolk sphingomyelin (EYSM) lipids, and allowed us to extract segmental order parameters from the 13C-1H residual dipolar couplings. Order parameters were used to calculate membrane structural quantities, including the area per lipid and bilayer thickness. Relative to POPC, EYSM is more ordered in the ld phase and experiences less structural perturbation upon adding 50% cholesterol to form the lo phase. The loss of configurational entropy is smaller for EYSM than for POPC, thus favoring its interaction with cholesterol in raftlike lipid systems. Our studies show that solid-state 13C NMR spectroscopy is applicable to investigations of complex lipids and makes it possible to obtain structural parameters for biomembrane systems where isotope labeling may be prohibitive.

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.

Perspective: Thermodynamics of structural glasses

According to the conventional wisdom, a (liquid to) glass transition takes place when a liquid, upon cooling, falls out of equilibrium because the observation time becomes less than the structural relaxation time of the liquid. The external time constraint prevents the system from exploring its entire configurational space. The freezing of the slow configurational changes that occur on a time scale longer than the observation time partitions the configurational space into mutually inaccessible components and the system becomes trapped in each component of the partition with a probability that is equal to the probability with which it was exploring that component before partitioning. A glass is such a broken-ergodic system. The volume of the configurational space accessible to a glass is reduced resulting in a loss of configurational entropy during a glass transition. This view of a real glass transition, called the disordered solid (or the broken-ergodic) view, is consistent with all observed features of glass transition such as absence of latent heat and a drop in the heat capacity. Thermodynamics of glassy state based on this and the traditional view are compared. It is concluded that the broken-ergodic view is fundamentally sound but not the traditional view.

Macromolecular depletion as a determinant of RBC adhesive interactions: why blood is thicker than water.

If a surface is in contact with a solution containing macromolecules or proteins, and the loss of configurational entropy of these molecules at the surface is not balanced by adsorption energy, a polymer-poor layer will develop near the surface. If two such layers overlap, an attractive force develops due to the osmotic pressure difference between these depletion zones and the bulk phase. Recent studies have shown that depletion interaction plays a major role in red blood cell (RBC) aggregation and hence it is a major determinate of blood flow stability; depletion interaction also markedly affects RBC adhesion to vascular endothelial cells. Understanding and quantitating factors that regulate depletion in vivo are thus of importance, yet made difficult since only very small changes of the cell surface (e.g., glycocalyx thickness) such as seen during RBC aging can lead to massive changes of depletion interaction and hence cell-cell adhesion. It is suggested that insight into the in vivo relevance of depletion mechanisms may lead to an improved understanding of how and why blood flow is altered in many diseases, and may also provide new biomarkers (e.g., surface properties) that will aid in the development of novel or improved diagnostic and therapeutic tools.

Thermodynamic Analysis of the Binding of 2F5 (Fab and Immunoglobulin G Forms) to Its gp41 Epitope Reveals a Strong Influence of the Immunoglobulin Fc Region on Affinity

Immunotherapies and vaccines based on the induction of broadly neutralizing monoclonal antibodies (bNAbs) have become outstanding strategies against HIV-1. Diverse bNAbs recognizing different regions of the HIV-1 envelope have been identified and extensively studied. However, there is little information about the thermodynamics of binding of these bNAbs and their epitopes. We used isothermal titration calorimetry to characterize thermodynamically the interactions between bNAb2F5 (in both the IgG and Fab forms) and its functional and core epitope peptides. We found that these interactions are enthalpically driven and opposed by a negative entropy change. The highest affinity was found for 2F5 IgG for its functional epitope, indicating that additional interactions involving residues flanking the core epitope contribute strongly to higher affinity. In addition, the strong influence of the Fc region on the binding affinity suggests long-range allosteric effects within IgG. Our results provide useful information for developing new therapeutics against HIV-1 and, in a broader scope, contribute to a better understanding of antigen-antibody recognition.

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.

Effect of undulations on spontaneous braid formation

This paper is an extension of a recent study where it was shown that forces dependent on molecular helical structure may cause two DNA molecules to spontaneously braid [R. Cortini et al., Biophys. J. 101, 875 (2011)]. Here, bending fluctuations of molecular center lines about the braid axis are incorporated into the braiding theory, which may be generalized to other helix-dependent interactions and other helical molecules. The free energy of the pair of molecules is recalculated and compared to its value without incorporating undulations. We find that the loss of configurational entropy due to confinement of the molecules in the braid is quite high. This contribution to the free energy increases the amount of attraction needed for spontaneous braiding due to helix-dependent forces. The theory will be further developed for plectonemes and braids under mechanical forces in later work.

Molecular Dynamics Study of β-Cyclodextrin–Phenylalanine (1:1) Inclusion Complex in Aqueous Medium

Atomistic molecular dynamics (MD) simulations of host-guest inclusion complexes of β-cyclodextrin (BCD) and zwitterionic phenylalanine (zPHE) following two possible orientations of zPHE in aqueous solutions have been carried out. The guest induced structural changes of BCD and the microscopic properties of surrounding water have been explored. The results obtained for the complex molecules were compared with those obtained for free BCD and free zPHE molecules. It is found that irrespective of the orientation, the complexation of BCD and zPHE (1:1) is associated with loss of configurational entropy. Besides, the net configurational entropy is found to differ depending upon its orientation inside the BCD cavity. Interestingly, within the simulation length scale it is found that the relatively hydrophobic aromatic moiety of zPHE prefers to stay within the hydrophobic cavity of BCD, irrespective of its orientation. Further, nonuniform distribution and altered tetrahedral ordering of hydration water molecules around the complex molecules as compared to that around the free forms are correlated well with their conformational flexibilities.

A “configurational entropy-loss” law for non-Arrhenius relaxation in disordered systems

From Nowick's self-induced ordering theory, we develop a new configurational entropy relation to describe the non-Arrhenius temperature (T)-dependent relaxation in disordered systems. Both the loss of configurational entropy and the coupling interaction among relaxing units (RUs) are explicitly introduced in this relation; thus, this relation offers a novel connection, which differs from the Adam-Gibbs (A-G) entropy relation, between kinetics and thermodynamics, and this relation generalizes several well-known currently used relations. The present relation can provide direct and more accurate estimates of (i) the intrinsic activation enthalpy, (ii) the T-evolution of the configurational entropy loss of the system, and (iii) the self-induced ordering temperature Tc, which characterizes the coupling interaction among RUs. The application of the theory to experimental data for the relaxation time in typical organic liquids demonstrates the validity of the theory.

Assembling Wormlike Micelles in Tubular Nanopores by Tuning Surfactant–Wall Interactions

Threadlike molecular assemblies are excluded from narrow pores unless attractive interactions with the confining pore walls compensate for the loss of configurational entropy. Here we show that wormlike surfactant micelles can be assembled in the 8 nm tubular nanopores of SBA-15 silica by adjusting the surfactant-pore-wall interactions. The modulation of the interactions was achieved by coadsorption of a surface modifier that also provides control over the partitioning of wormlike aggregates between the bulk solution and the pore space. We anticipate that the concept of tuning the interactions with the pore wall will be applicable to a wide variety of self-assembling molecules and pores.

Insights from Free-Energy Calculations: Protein Conformational Equilibrium, Driving Forces, and Ligand-Binding Modes

Accurate free-energy calculations provide mechanistic insights into molecular recognition and conformational equilibrium. In this work, we performed free-energy calculations to study the thermodynamic properties of different states of molecular systems in their equilibrium basin, and obtained accurate absolute binding free-energy calculations for protein-ligand binding using a newly developed M2 algorithm. We used a range of Asp-Phe-Gly (DFG)-in/out p38α mitogen-activated protein kinase inhibitors as our test cases. We also focused on the flexible DFG motif, which is closely connected to kinase activation and inhibitor binding. Our calculations explain the coexistence of DFG-in and DFG-out states of the loop and reveal different components (e.g., configurational entropy and enthalpy) that stabilize the apo p38α conformations. To study novel ligand-binding modes and the key driving forces behind them, we computed the absolute binding free energies of 30 p38α inhibitors, including analogs with unavailable experimental structures. The calculations revealed multiple stable, complex conformations and changes in p38α and inhibitor conformations, as well as balance in several energetic terms and configurational entropy loss. The results provide relevant physics that can aid in designing inhibitors and understanding protein conformational equilibrium. Our approach is fast for use with proteins that contain flexible regions for structure-based drug design.

How, when and why proteins collapse: the relation to folding

Unfolded proteins under strongly denaturing conditions are highly expanded. However, when the conditions are more close to native, an unfolded protein may collapse to a compact globular structure distinct from the folded state. This transition is akin to the coil-globule transition of homopolymers. Single-molecule FRET experiments have been particularly conducive in revealing the collapsed state under conditions of coexistence with the folded state. The collapse can be even more readily observed in natively unfolded proteins. Time-resolved studies, using FRET and small-angle scattering, have shown that the collapse transition is a very fast event, probably occurring on the submicrosecond time scale. The forces driving collapse are likely to involve both hydrophobic and backbone interactions. The loss of configurational entropy during collapse makes the unfolded state less stable compared to the folded state, thus facilitating folding.

Free energy calculations offer insights into the influence of receptor flexibility on ligand–receptor binding affinities

Docking algorithms for computer-aided drug discovery and design often ignore or restrain the flexibility of the receptor, which may lead to a loss of accuracy of the relative free enthalpies of binding. In order to evaluate the contribution of receptor flexibility to relative binding free enthalpies, two host-guest systems have been examined: inclusion complexes of α-cyclodextrin (αCD) with 1-chlorobenzene (ClBn), 1-bromobenzene (BrBn) and toluene (MeBn), and complexes of DNA with the minor-groove binding ligands netropsin (Net) and distamycin (Dist). Molecular dynamics simulations and free energy calculations reveal that restraining of the flexibility of the receptor can have a significant influence on the estimated relative ligand-receptor binding affinities as well as on the predicted structures of the biomolecular complexes. The influence is particularly pronounced in the case of flexible receptors such as DNA, where a 50% contribution of DNA flexibility towards the relative ligand-DNA binding affinities is observed. The differences in the free enthalpy of binding do not arise only from the changes in ligand-DNA interactions but also from changes in ligand-solvent interactions as well as from the loss of DNA configurational entropy upon restraining.

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.

Synergistic Regulation and Ligand-Induced Conformational Changes of Tryptophan Synthase

Conformational changes of enzyme complexes are often related to regulating and creating an optimal environment for efficient chemistry. We investigated the synergistic regulation of the tryptophan synthase (TRPS) complex, studied for decades as a model of allosteric regulation and substrate channeling within protein complexes. TRPS is a bifunctional tetrameric alphabetabetaalpha enzyme complex that exhibits cooperative motions of the alpha- and beta-subunits by tightly controlled allosteric interactions. We have delineated the atomically detailed dynamics and conformational changes of TRPS in the absence and presence of substrates using molecular dynamics simulations. The computed energy and entropy associated with the protein motions also offer mechanistic insights into the conformational fluctuations and the ligand binding mechanism. The flexible alpha-L6 loop samples both open and partially closed conformations in the ligand-free state but shifts to fully closed conformations when its substrates are present. The fully closed conformations are induced by favorable protein-ligand interactions but are partly compensated by configurational entropy loss. Considerable local rearrangements exist during ligand binding processes when the system is searching for energy minima. The motion of the region that closes the beta-subunit during catalysis, the COMM domain, couples with the motion of the alpha-subunit, although the fluctuations are smaller than in the flexible loop regions. Because of multiple conformations of ligand-free TRPS in the open and partially closed states, the alpha-L6 loop fluctuations have preferential directionality, which may facilitate the fully closed conformations induced by alpha- and beta-substrates binding to both subunits. Such cooperative and directional motion may be a general feature that contributes to catalysis in many enzyme complexes.

Simple Quantitative Model for the Reversible Association of DNA Coated Colloids

We investigate the reversible association of micrometer-sized colloids coated with complementary single-stranded DNA "sticky ends" as a function of the temperature and the sticky end coverage. We find that even a qualitative description of the dissociation transition curves requires the inclusion of an entropic cost. We develop a simple general model for this cost in terms of the configurational entropy loss due to binding and confinement of the tethered DNA between neighboring particles. With this easy-to-use model, we demonstrate for different kinds of DNA constructs quantitative control over the dissociation temperature and the sharpness of the dissociation curve, both essential properties for complex self-assembly processes.

Polyelectrolyte Adsorption on a Charged Surface. Free Energy Calculation from Monte Carlo Simulations Using Jarzynski Equality

In the present work we develop a Monte Carlo study of polyelectrolyte adsorption on a charged surface by means of a primitive model for the PE and the ions; the small ions are simulated in the Grand Canonical ensemble. Free energy profiles for the adsorption process were obtained using a generalization of Jarzynski equality, where a minimum was found close to surface. This minimum stems from a balance between the loss of configurational entropy by the PE chain upon adsorption, and the gain of translation entropy caused by the release of small ions. The results of the free energy of adsorption are compared with those from a previous work using mean field equations and simulated density profiles.

Entropic contributions and the influence of the hydrophobic environment in promiscuous protein–protein association

The mechanisms by which a promiscuous protein can strongly interact with several different proteins using the same binding interface are not completely understood. An example is protein kinase A (PKA), which uses a single face on its docking/dimerization domain to interact with multiple A-kinase anchoring proteins (AKAP) that localize it to different parts of the cell. In the current study, the configurational entropy contributions to the binding between the AKAP protein HT31 with the D/D domain of RII alpha-regulatory subunit of PKA were examined. The results show that the majority of configurational entropy loss for the interaction was due to decreased fluctuations within rotamer states of the side chains. The result is in contrast to the widely held approximation that the decrease in the number of rotamer states available to the side chains forms the major component. Further analysis showed that there was a direct linear relationship between total configurational entropy and the number of favorable, alternative contacts available within hydrophobic environments. The hydrophobic binding pocket of the D/D domain provides alternative contact points for the side chains of AKAP peptides that allow them to adopt different binding conformations. The increase in binding conformations provides an increase in binding entropy and hence binding affinity. We infer that a general strategy for a promiscuous protein is to provide alternative contact points at its interface to increase binding affinity while the plasticity required for binding to multiple partners is retained. Implications are discussed for understanding and treating diseases in which promiscuous protein interactions are used.

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.