Translational Entropy Research Articles

Molecular driving forces of the pocket–ligand hydrophobic association

Molecular dynamics simulations have shown that the concave pocket–convex ligand hydrophobic association is enthalpy-driven due to water reorganisation [11]. A different theoretical analysis is provided, grounded on the basic notion that the Gibbs energy gain upon association is mainly due to the decrease in the solvent-excluded volume, that translates in a gain of configurational–translational entropy of water molecules. It is also underscored that the reorganisation of water–water H-bonds is characterised by an almost complete enthalpy–entropy compensation and cannot affect the Gibbs energy change.

Salt-doped block copolymers: ion distribution, domain spacing and effective χ parameter

We develop a self-consistent field theory for salt-doped diblock copolymers, such as polyethylene oxide (PEO)–polystyrene with added lithium salts. We account for the inhomogeneous distribution of Li+ ions bound to the ion-dissolving block, the preferential solvation energy of anions in the different block domains, the translational entropy of anions, the ion-pair equilibrium between polymer-bound Li+ and anion, and changes in the χ parameter due to the bound ions. We show that the preferential solvation energy of anions provides a large driving force for microphase separation. Our theory is able to explain many features observed in experiments, particularly the systematic dependence in the effective χ-parameter on the radius of the anions, the observed linear dependence in the effective χ on salt concentration, and increase in the domain spacing of the lamellar phase due to the addition of lithium salts. We also examine the relationship between two definitions of the effective χ parameter, one based on the domain spacing of the ordered phase and the other based on the structure factor in the disordered phase. We argue that the latter is a more fundamental measure of the effective interaction between the two blocks. We show that the ion distribution and the electrostatic potential profile depend strongly on the dielectric contrast between the two blocks and on the ability of the Li+ to redistribute along the backbone of the ion-dissolving block.

A Novel Rotation Mechanism of F1-ATPase Based on the Water-Entropy Effect

We propose a novel picture of the rotation mechanism of F1-ATPase [1]. In the proposal, the key factor is the translational entropy of water, which has been shown to drive a variety of self-assembly processes in biological systems [2]. We calculate the hydration entropies of three different sub-complexes comprising the gamma subunit, one of the beta subunits, and two alpha subunits adjacent to them. The calculation is made using the angle-dependent integral equation theory combined with the multipolar water model [3] and morphometric approach [4]. The major finding is that the packing in F1-ATPase is highly asymmetrical and this asymmetry is ascribed to the water-entropy effect. We discuss how the rotation of the gamma subunit is induced by such chemical processes as ATP binding, ATP hydrolysis, and release of the products. In our picture, the asymmetrical packing plays crucially important roles and the rotation is driven by the water-entropy effect. As part of the demonstration of the validity of our rotation mechanism, we also analyze the water-entropy change in yeast F1-ATPase during 16-degree rotation of the gamma subunit [5]. The result demonstrates the validity of the water-entropy mechanism proposed in Ref 1.[1] T. Yoshidome, Y. Ito, M. Ikeguchi, and M. Kinoshita, J. Am. Chem. Soc. 133, 4030 (2011).[2] M. Kinoshita, Front. Biosci. 14, 3419 (2009).[3] M. Kinoshita, J. Chem. Phys. 128, 024507 (2008).[4] R. Roth, Y. Harano, and M. Kinoshita, Phys. Rev. Lett. 97, 078101 (2006).[5] T. Yoshidome, Y. Ito, N. Matubayasi, M. Ikeguchi, and M. Kinoshita, Soft Matter (to be submitted).

Simulation Analysis of the Temperature Dependence of Lignin Structure and Dynamics

Lignins are hydrophobic, branched polymers that regulate water conduction and provide protection against chemical and biological degradation in plant cell walls. Lignins also form a residual barrier to effective hydrolysis of plant biomass pretreated at elevated temperatures in cellulosic ethanol production. Here, the temperature-dependent structure and dynamics of individual softwood lignin polymers in aqueous solution are examined using extensive (17 μs) molecular dynamics simulations. With decreasing temperature the lignins are found to transition from mobile, extended to glassy, compact states. The polymers are composed of blobs, inside which the radius of gyration of a polymer segment is a power-law function of the number of monomers comprising it. In the low temperature states the blobs are interpermeable, the polymer does not conform to Zimm/Stockmayer theory, and branching does not lead to reduction of the polymer size, the radius of gyration being instead determined by shape anisotropy. At high temperatures the blobs become spatially separated leading to a fractal crumpled globule form. The low-temperature collapse is thermodynamically driven by the increase of the translational entropy and density fluctuations of water molecules removed from the hydration shell, thus distinguishing lignin collapse from enthalpically driven coil-globule polymer transitions and providing a thermodynamic role of hydration water density fluctuations in driving hydrophobic polymer collapse. Although hydrophobic, lignin is wetted, leading to locally enhanced chain dynamics of solvent-exposed monomers. The detailed characterization obtained here provides insight at atomic detail into processes relevant to biomass pretreatment for cellulosic ethanol production and general polymer coil-globule transition phenomena.

Thermodynamics of Ion-Containing Polymer Blends and Block Copolymers

We develop a theory for the thermodynamics of ion-containing polymer blends and diblock copolymers, taking polyethylene oxide (PEO), polystyrene and lithium salts as an example. We account for the tight binding of Li^{+} ions to the PEO, the preferential solvation energy of anions in the PEO domain, the translational entropy of anions, and the ion-pair equilibrium between EO-complexed Li^{+} and anion. Our theory is able to predict many features observed in experiments, particularly the systematic dependence in the effective χ parameter on the size of the anions. Furthermore, comparison with the observed linear dependence in the effective χ on salt concentration yields an upper limit for the binding constant of the ion pair.

Multiscale Modeling of Functionalized Nanocarriers in Targeted Drug Delivery.

Targeted drug delivery using functionalized nanocarriers (NCs) is a strategy in therapeutic and diagnostic applications. In this paper we review the recent development of models at multiple length and time scales and their applications to targeting of antibody functionalized nanocarriers to antigens (receptors) on the endothelial cell (EC) surface. Our mesoscale (100 nm-1 μm) model is based on phenomenological interaction potentials for receptor-ligand interactions, receptor-flexure and resistance offered by glycocalyx. All free parameters are either directly determined from independent biophysical and cell biology experiments or estimated using molecular dynamics simulations. We employ a Metropolis Monte Carlo (MC) strategy in conjunction with the weighted histogram analysis method (WHAM) to compute the free energy landscape (potential of mean force or PMF) associated with the multivalent antigen-antibody interactions mediating the NC binding to EC. The binding affinities (association constants) are then derived from the PMF by computing absolute binding free energy of binding of NC to EC, taking into account the relevant translational and rotational entropy losses of NC and the receptors. We validate our model predictions by comparing the computed binding affinities and PMF to a wide range of experimental measurements, including in vitro cell culture, in vivo endothelial targeting, atomic force microscopy (AFM), and flow chamber experiments. The model predictions agree closely and quantitatively with all types experimental measurements. On this basis, we conclude that our computational protocol represents a quantitative and predictive approach for model driven design and optimization of functionalized NCs in targeted vascular drug delivery.

Elucidating the Energetics of Entropically Driven Protein–Ligand Association: Calculations of Absolute Binding Free Energy and Entropy

The binding of proteins and ligands is generally associated with the loss of translational, rotational, and conformational entropy. In many cases, however, the net entropy change due to binding is positive. To develop a deeper understanding of the energetics of entropically driven protein-ligand binding, we calculated the absolute binding free energies and binding entropies for two HIV-1 protease inhibitors Nelfinavir and Amprenavir using the double-decoupling method with molecular dynamics simulations in explicit solvent. For both ligands, the calculated absolute binding free energies are in general agreement with experiments. The statistical error in the computed ΔG(bind) due to convergence problem is estimated to be ≥2 kcal/mol. The decomposition of free energies indicates that, although the binding of Nelfinavir is driven by nonpolar interaction, Amprenavir binding benefits from both nonpolar and electrostatic interactions. The calculated absolute binding entropies show that (1) Nelfinavir binding is driven by large entropy change and (2) the entropy of Amprenavir binding is much less favorable compared with that of Nelfinavir. Both results are consistent with experiments. To obtain qualitative insights into the entropic effects, we decomposed the absolute binding entropy into different contributions based on the temperature dependence of free energies along different legs of the thermodynamic pathway. The results suggest that the favorable entropic contribution to binding is dominated by the ligand desolvation entropy. The entropy gain due to solvent release from binding site appears to be more than offset by the reduction of rotational and vibrational entropies upon binding.

Calculation of Configurational Entropy with a Boltzmann–Quasiharmonic Model: The Origin of High-Affinity Protein–Ligand Binding

Accurate assessment of configurational entropy remains a large challenge in biology. While many methods exist to calculate configurational entropy, there is a balance between accuracy and computational demands. Here we calculate ligand and protein conformational entropies using the Boltzmann-quasiharmonic (BQH) method, which treats the first-order entropy term by the Boltzmann expression for entropy while determining correlations using the quasiharmonic model. This method is tested by comparison with the exact Clausius expression for entropy on a range of test molecules ranging from small ligands to a protein. Using the BQH method, we then analyze the rotational and translational (R/T) entropy change upon ligand binding for five protein complexes to explore the origins of extremely tight affinity. The results suggest that in these systems such affinity is achieved by a combination of simultaneously maintaining good protein-ligand contacts while allowing significant residual R/T motion of the ligand through suitable protein motions.

Effects of ion solvation on phase equilibrium and interfacial tension of liquid mixtures

We study the bulk thermodynamics and interfacial properties of electrolyte solution mixtures by accounting for electrostatic interaction, ion solvation, and inhomogeneity in the dielectric medium in the mean-field framework. Difference in the solvation energy between the cations and anions is shown to give rise to local charge separation near the interface, and a finite Galvani potential between two coexisting solutions. The ion solvation affects the phase equilibrium of the solvent mixture, depending on the dielectric constants of the solvents, reflecting the competition between the solvation energy and translation entropy of the ions. Miscibility is decreased if both solvents have low dielectric constants and is enhanced if both solvents have high dielectric constant. At the mean-field level, the ion distribution near the interface is determined by two competing effects: accumulation in the electrostatic double layer and depletion in a diffuse interface. The interfacial tension shows a nonmonotonic dependence on the salt concentration: it increases linearly with the salt concentration at higher concentrations and decreases approximately as the square root of the salt concentration for dilute solutions, reaching a minimum near 1 mM. We also find that, for a fixed cation type, the interfacial tension decreases as the size of anion increases. These results offer qualitative explanations within one unified framework for the long-known concentration and ion size effects on the interfacial tension of electrolyte solutions.

Entropy and the driving force for the filling of carbon nanotubes with water

The spontaneous filling of hydrophobic carbon nanotubes (CNTs) by water observed both experimentally and from simulations is counterintuitive because confinement is generally expected to decrease both entropy and bonding, and remains largely unexplained. Here we report the entropy, enthalpy, and free energy extracted from molecular dynamics simulations of water confined in CNTs from 0.8 to 2.7-nm diameters. We find for all sizes that water inside the CNTs is more stable than in the bulk, but the nature of the favorable confinement of water changes dramatically with CNT diameter. Thus we find (i) an entropy (both rotational and translational) stabilized, vapor-like phase of water for small CNTs (0.8-1.0 nm), (ii) an enthalpy stabilized, ice-like phase for medium-sized CNTs (1.1-1.2 nm), and (iii) a bulk-like liquid phase for tubes larger than 1.4 nm, stabilized by the increased translational entropy as the waters sample a larger configurational space. Simulations with structureless coarse-grained water models further reveal that the observed free energies and sequence of transitions arise from the tetrahedral structure of liquid water. These results offer a broad theoretical basis for understanding water transport through CNTs and other nanostructures important in nanofluidics, nanofiltrations, and desalination.

Phase Behavior of Binary Blends of Block Copolymers Having Hydrogen Bonding

Phase behavior of binary blends consisting of high molecular weight polystyrene-block-poly(2-vinylpyridine) copolymer (PS-b-P2VP) and low molecular weight polystyrene-block-poly(4-hydroxystyrene) copolymer (PS-b-PHS) was investigated by using small-angle X-ray scattering and transmission electron microscopy. Both PS-b-P2VP and PS-b-PHS exhibited lamellar microdomain over the entire experimental temperatures up to 300 °C. When the weight fraction of PS-b-PHS in the blend was less than 0.1, the lamellar microdomains were maintained. However, with increasing amount of PS-b-PHS, the microdomains in the blends were transformed to hexagonally packed (HEX) cylindrical microdomains and body-centered cubic (BCC) spherical microdomains. On the other hand, when a relatively high molecular weight of PS-b-PHS was used, the BCC spherical microdomains were not observed even at a large weight fraction of PS-b-PHS in the blend, but HEX cylindrical microdomains were formed. The phase behaviors observed experimentally were rationalized by the results of the self-consistent mean-field theory. Based on the theoretical results, some of the PS-b-PHS dissolves into P2VP microdomains and thereby increases the effective volume of P2VP/PHS phase because of hydrogen bonding between P2VP and PHS blocks. In the case of smaller molecular weight of PS-b-PHS, the dissolved amount is significant because the favorable interactions of P2VP/PHS chains becomes dominant over the relatively small enthalpic penalty between PS/P2VP in the P2VP microdomains as well as the gain of the translational entropy of the PS-b-PHS chains. The increase in the effective volume of the PHS/P2VP phase leads to a transformation of the microdomains in the blends from lamellar to HEX cylindrical microdomains and BCC spherical microdomains. On the other hand, when a relatively higher molecular weight of PS-b-PHS was used, the effective volume of the P2VP/PHS phase was not increased greatly because of a large enthalpic loss arising from the contact between PS/P2VP as well as a small gain in the translational entropy. Consequently, only lamellar and HEX cylindrical microdomains are observed.

Entropy-Based Mechanism of Ribosome-Nucleoid Segregation in E. coli Cells

In Escherichia coli, ribosomes concentrate near the cylindrical wall and at the endcaps, whereas the chromosomal DNA segregates in the more centrally located nucleoid. A simple statistical model recovers the observed ribosome-nucleoid segregation remarkably well. Plectonemic DNA is represented as a hyperbranched hard-sphere polymer, and multiple ribosomes that simultaneously translate the same mRNA strand (polysomes) are represented as freely jointed chains of hard spheres. There are no attractive interactions between particles, only excluded-volume effects. At realistic DNA and ribosome concentrations, segregation arises primarily from two effects: the DNA polymer avoids walls to maximize conformational entropy, and the polysomes occupy the empty space near the walls to maximize translational entropy. In this complex system, maximizing total entropy results in spatial organization of the components. Due to coupling of mRNA to DNA through RNA polymerase, the same entropic effects should favor the placement of highly expressed genes at the interface between the nucleoid and the ribosome-rich periphery. Such a placement would enable efficient cotranscriptional translation and facile transertion of membrane proteins into the cytoplasmic membrane. Finally, in the model, monofunctional DNA polymer beads representing the tips of plectonemes preferentially locate near the cylindrical wall. This suggests that initiation of transcription may occur preferentially near the ribosome-rich periphery.

Lattice model of oligonucleotide hybridization in solution. I. Model and thermodynamics

A coarse-grained lattice model of DNA oligonucleotides is proposed to investigate the general mechanisms by which single-stranded oligonucleotides hybridize to their complementary strands in solution. The model, based on a high-coordination cubic lattice, is simple enough to allow the direct simulation of DNA solutions, yet capturing how the fundamental thermodynamic processes are microscopically encoded in the nucleobase sequences. Physically relevant interactions are considered explicitly, such as interchain excluded volume, anisotropic base-pairing and base-stacking, and single-stranded bending rigidity. The model is studied in detail by a specially adapted Monte Carlo simulation method, based on parallel tempering and biased trials, which is designed to overcome the entropic and enthalpic barriers associated with the sampling of hybridization events of multiple single-stranded chains in solution. This methodology addresses both the configurational complexity of bringing together two complementary strands in a favorable orientation (entropic barrier) and the energetic penalty of breaking apart multiple associated bases in a double-stranded state (enthalpic barrier). For strands with sequences restricted to nonstaggering association and homogeneous pairing and stacking energies, base-pairing is found to dominate the hybridization over the translational and conformational entropy. For strands with sequence-dependent pairing corresponding to that of DNA, the complex dependence of the model's thermal stability on concentration, sequence, and degree of complementarity is shown to be qualitatively and quantitatively consistent both with experiment and with the predictions of statistical mechanical models.

Quantum mechanical binding free energy calculation for phosphopeptide inhibitors of the Lck SH2 domain

The accurate and efficient calculation of binding free energies is essential in computational biophysics. We present a linear-scaling quantum mechanical (QM)-based end-point method termed MM/QM-COSMO to calculate binding free energies in biomolecular systems, with an improved description of entropic changes. Molecular dynamics trajectories are re-evaluated using a semiempirical Hamiltonian and a continuum solvent model; translational and rotational entropies are calculated using configurational integrals, and internal entropy is calculated using the harmonic oscillator approximation. As an application, we studied the binding of a series of phosphotyrosine tetrapeptides to the human Lck SH2 domain, a key component in intracellular signal transduction, modulation of which can have therapeutic relevance in the treatment of cancer, osteoporosis, and autoimmune diseases. Calculations with molecular mechanics Poisson-Boltzmann, and generalized Born surface area methods showed large discrepancies with experimental data stemming from the enthalpic component, in agreement with an earlier report. The empirical force field-based solvent interaction energy scoring function yielded improved results, with average unsigned error of 3.6 kcal/mol, and a better ligand ranking. The MM/QM-COSMO method exhibited the best agreement both for absolute (average unsigned error = 0.7 kcal/mol) and relative binding free energy calculations. These results show the feasibility and promise of a full QM-based end-point method with an adequate balance of accuracy and computational efficiency.

Thermodynamics of water entry in hydrophobic channels of carbon nanotubes

Experiments and computer simulations demonstrate that water spontaneously fills the hydrophobic cavity of a carbon nanotube. To gain a quantitative thermodynamic understanding of this phenomenon, we use the recently developed two phase thermodynamics method to compute translational and rotational entropies of confined water molecules inside single-walled carbon nanotubes and show that the increase in energy of a water molecule inside the nanotube is compensated by the gain in its rotational entropy. The confined water is in equilibrium with the bulk water and the Helmholtz free energy per water molecule of confined water is the same as that in the bulk within the accuracy of the simulation results. A comparison of translational and rotational spectra of water molecules confined in carbon nanotubes with that of bulk water shows significant shifts in the positions of the spectral peaks that are directly related to the tube radius.

Entropic crystal–crystal transitions of Brownian squares

When a monolayer of hard microscale square platelets, produced lithographically, is osmotically concentrated in a flat plane to raise the particle area fraction Φ(A), an order-order transition occurs between a hexagonal rotator crystal and a rhombic crystal. Strikingly, phases having fourfold symmetry are not observed at any Φ(A). The rhombic lattice angle α increases continuously with Φ(A), as the system maximizes its total rotational and translational entropy. A cage model, based on packing rotationally swept squares, or "squaroids," reasonably predicts the measured α(Φ(A)), indicating that rotational entropy and the square particle shape combine to produce the rhombic unit cell.

Study of volume phase transitions in polymeric nanogels by theoretically informed coarse-grained simulations

Polymeric nanogels undergo large reversible volume changes with moderate changes in environmental conditions. We report the results of a theoretically informed coarse-grained Monte Carlo simulation of a polymeric nanogel placed in a solvent bath. The nanogel is modelled as a regular network of Gaussian strands, each discretized with coarse-grained polymer beads (“particles”). The non-bonded interactions are evaluated from the local densities (“fields”), and include the Flory-like contribution for polymer-solvent interactions, the translational entropy of mobile ions, and the Coulomb energy obtained by solution of Poisson's equation. While an artificial grid is employed for convenience of implementation, the results are independent of contour and grid discretization within a resolution specified by the level of coarse-graining. We analyze the effects of degree of crosslinking, solvent quality, and charge fraction of polymer backbone, on the volume phase transition behavior of nanogels.

Thermodynamic Stability of the Nematic Phase and Configurational Entropy of Mesogens: A Molecular Simulation Approach

The thermodynamic stability of the nematic phase and the configurational entropy of mesogens–p-n-pentylbenzoic acid (5BAC) and p-n-hexylbenzoic acid (6BAC)–have been studied. The atomic net charge and dipole moment components at each atomic center have been evaluated using the complete neglect differential overlap (CNDO/2) method. The modified Rayleigh–Schrodinger perturbation theory along with multicentered-multipole expansion method has been employed to evaluate the long-range intermolecular interactions, while a “6-exp” potential function has been assumed for short-range interactions. The interaction energy values obtained with respect to translational and orientational motions during the different modes of molecular interactions have been taken as input to estimate the configurational entropy and Helmholtz free energy at room temperature, nematic–isotropic transition temperature, and above transition temperature. The observed difference in free energy of the molecules between the mesophase and the crystal phase during in-plane interactions suggests the thermodynamic stability of the nematic phase. Further, the comparable values of translational entropy during stacking and in-plane interactions are helpful to understand the relative flexibility/stability of one configuration over the other.

Crucial Importance of Translational Entropy of Water in Rotation Mechanism of F1-ATPase

Crucial Importance of Translational Entropy of Water in Rotation Mechanism of F1-ATPase

Predicting kinetic constants of protein–protein interactions based on structural properties

Elucidating kinetic processes of protein-protein interactions (PPI) helps to understand how basic building blocks affect overall behavior of living systems. In this study, we used structure-based properties to build predictive models for kinetic constants of PPI. A highly diverse PPI dataset, protein-protein kinetic interaction data and structures (PPKIDS), was built. PPKIDS contains 62 PPI with complex structures and kinetic constants measured experimentally. The influence of structural properties on kinetics of PPI was studied using 35 structure-based features, describing different aspects of complex structures. Linear models for the prediction of kinetic constants were built by fitting with selected subsets of structure-based features. The models gave correlation coefficients of 0.801, 0.732, and 0.770 for k(off), k(on), and K(d), respectively, in leave-one-out cross validations. The predictive models reported here use only protein complex structures as input and can be generally applied in PPI studies as well as systems biology modeling. Our study confirmed that different properties play different roles in the kinetic process of PPI. For example, k(on) was affected by overall structural features of complexes, such as the composition of secondary structures, the change of translational and rotational entropy, and the electrostatic interaction; while k(off) was determined by interfacial properties, such as number of contacted atom pairs per 100 Ų. This information provides useful hints for PPI design.