Configurational entropies of lipids in pure and mixed bilayers from atomic-level and coarse-grained molecular dynamics simulations

Single-chain and single-fragment configurational entropies of lipid tails in hydrated lipid bilayers are evaluated from molecular dynamics simulations using the quasi-harmonic approximation. The entropy distribution along individual acyl tails is obtained and compared to that of corresponding hydrocarbon chains in the liquid phase. We consider pure dipalmitoylphosphatidylcholine and mixed dioleoylphosphatidylcholine/dioleoylphosphatidylethanolamine bilayers. The systems are modeled at different levels of spatial resolution: In an atomic-level (AL) model all (heavy) atoms are explicitly simulated; in a coarse-grained (CG) model particles (beads) representing groups of covalently bound atoms are used, which map approximately four non-hydrogen atoms to one interaction site. Single-chain and single-fragment entropies and correlations between the motions of (single) acyl chains are compared. A good correspondence is found between the flexibility of the AL and CG models. The loss in configurational entropy due to the reduction in the number of degrees of freedom upon coarse-graining of the model is estimated. The CG model shows about 4 times faster convergence of the chain entropies than the more detailed AL model. Corrections to the quasi-harmonic entropy estimates were found to be small for the CG model. For the AL model, the correction due to mode anharmonicities is small, but the correction due to pairwise (supralinear) mode correlations is sizable.     

Calculations of solute and solvent entropies from molecular dynamics simulations

The translational, rotational and conformational (vibrational) entropy contributions to ligand-receptor binding free energies are analyzed within the standard formulation of statistical thermodynamics. It is shown that the partitioning of the binding entropy into different components is to some extent arbitrary, but an appropriate method to calculate both translational and rotational entropy contributions to noncovalent association is by estimating the configurational volumes of the ligand in the bound and free states. Different approaches to calculating solute entropies using free energy perturbation calculations, configurational volumes based on root-mean-square fluctuations and covariance matrix based quasiharmonic analysis are illustrated for some simple molecular systems. Numerical examples for the different contributions demonstrate that theoretically derived results are well reproduced by the approximations. Calculation of solvent entropies, either using total potential energy averages or van't Hoff plots, are carried out for the case of ion solvation in water. Although convergence problems will persist for large and complex simulation systems, good agreement with experiment is obtained here for relative and absolute ion hydration entropies. We also outline how solvent and solute entropic contributions are taken into account in empirical binding free energy calculations using the linear interaction energy method. In particular it is shown that empirical scaling of the nonpolar intermolecular ligand interaction energy effectively takes into account size dependent contributions to the binding free energy.     

Nearest-neighbor nonparametric method for estimating the configurational entropy of complex molecules

A method for estimating the configurational (i.e., non-kinetic) part of the entropy of internal motion in complex molecules is introduced that does not assume any particular parametric form for the underlying probability density function. It is based on the nearest-neighbor (NN) distances of the points of a sample of internal molecular coordinates obtained by a computer simulation of a given molecule. As the method does not make any assumptions about the underlying potential energy function, it accounts fully for any anharmonicity of internal molecular motion. It provides an asymptotically unbiased and consistent estimate of the configurational part of the entropy of the internal degrees of freedom of the molecule. The NN method is illustrated by estimating the configurational entropy of internal rotation of capsaicin and two stereoisomers of tartaric acid, and by providing a much closer upper bound on the configurational entropy of internal rotation of a pentapeptide molecule than that obtained by the standard quasi-harmonic method. As a measure of dependence between any two internal molecular coordinates, a general coefficient of association based on the information-theoretic quantity of mutual information is proposed. Using NN estimates of this measure, statistical clustering procedures can be employed to group the coordinates into clusters of manageable dimensions and characterized by minimal dependence between coordinates belonging to different clusters.     

Influence of the solvent representation on vibrational entropy calculations: Generalized born versus distance‐dependent dielectric model

The harmonic model is the most popular approximation for estimating the “configurational” entropy of a solute in molecular mechanics/Poisson‐Boltzmann solvent accessible surface area (MM/PBSA)‐type binding free energy calculations. Here, we investigate the influence of the solvent representation in the harmonic model by comparing estimates of changes in the vibrational entropies for 30 trypsin/ligand complexes on ligand binding. Second derivatives of Amber generalized Born (GB) solvation models are available in the nucleic acid builder code. They allow one to use these models for the calculation of vibrational entropies instead of using a simpler solvation model based on a distance‐dependent dielectric (DDD) constant. Estimates of changes in the vibrational entropies obtained with a DDD model are systematically and significantly larger, by on average, 6 kcal mol^?1 (at T = 300 K), than estimates obtained with a GB model and so are more favorable for complex formation. The difference becomes larger the more the vibrational entropy contribution disfavors complex formation, that is, the larger the ligand is (for the complexes considered here). A structural decomposition of the estimates into per‐residue contributions reveals polar interactions between the ligand and the surrounding protein, in particular involving charged nitrogens, as a main source of the differences. Snapshots minimized with the DDD model showed a structural deviation from snapshots minimized in explicit water that is larger by, on average, 0.5 Å RMSD compared to snapshots that were minimized with GB^HCT. As experimental vibrational entropies of biomacromolecules are elusive, there is no direct way to establish a solvent model's superiority. Thus, we can only recommend using the GB harmonic model for vibrational entropy calculations based on the reasoning that smaller structural deviations should point to the implicit solvent model that closer approximates the energy landscape of the solute in explicit solvent. © 2012 Wiley Periodicals, Inc.     

Predicting hydration free energies with a hybrid QM/MM approach: an evaluation of implicit and explicit solvation models in SAMPL4

The correct representation of solute-water interactions is essential for the accurate simulation of most biological phenomena. Several highly accurate quantum methods are available to deal with solvation by using both implicit and explicit solvents. So far, however, most evaluations of those methods were based on a single conformation, which neglects solute entropy. Here, we present the first test of a novel approach to determine hydration free energies that uses molecular mechanics (MM) to sample phase space and quantum mechanics (QM) to evaluate the potential energies. Free energies are determined by using re-weighting with the Non-Boltzmann Bennett (NBB) method. In this context, the method is referred to as QM-NBB. Based on snapshots from MM sampling and accounting for their correct Boltzmann weight, it is possible to obtain hydration free energies that incorporate the effect of solute entropy. We evaluate the performance of several QM implicit solvent models, as well as explicit solvent QM/MM for the blind subset of the SAMPL4 hydration free energy challenge. While classical free energy simulations with molecular dynamics give root mean square deviations (RMSD) of 2.8 and 2.3 kcal/mol, the hybrid approach yields an improved RMSD of 1.6 kcal/mol. By selecting an appropriate functional and basis set, the RMSD can be reduced to 1 kcal/mol for calculations based on a single conformation. Results for a selected set of challenging molecules imply that this RMSD can be further reduced by using NBB to reweight MM trajectories with the SMD implicit solvent model.     

A Theoretical Study of Ene Reactions in Solution: A Solution‐Phase Translational Entropy Model

Several density functional theory (DFT) methods, such as CAM‐B3LYP, M06, ωB97x, and ωB97xD, are used to characterize a range of ene reactions. The Gibbs free energy, activation enthalpy, and entropy are calculated with both the gas‐ and solution‐phase translational entropy; the results obtained from the solution‐phase translational entropies are quite close to the experimental measurements, whereas the gas‐phase translational entropies do not perform well. For ene reactions between the enophile propanedioic acid (2‐oxo‐1,3‐dimethyl ester) and π donors, the two‐solvent‐involved explicit+implicit model can be employed to obtain accurate activation entropies and free‐energy barriers, because the interaction between the carbonyl oxygen atom and the solvent in the transition state is strengthened with the formation of C?C and O?H bonds. In contrast, an implicit solvent model is adequate to calculate activation entropies and free‐energy barriers for the corresponding reactions of the enophile 4‐phenyl‐1,2,4‐triazoline‐3,5‐dione.     

Free energy, entropy, and induced fit in host-guest recognition: calculations with the second-generation mining minima algorithm

This study applies a novel computational method to study molecular recognition for three sets of synthetic hosts: molecular clips, molecular tweezers, and a synthetic barbiturate receptor. The computed standard free energies of binding for the 12 binding reactions agree closely with experiment and provide insight into the roles of configurational entropy, preorganization, and induced fit in the systems studied. The computed changes in configurational entropy are comparable in magnitude to the changes in mean potential plus solvation energy, and they result primarily from changes in the average width of the energy wells upon binding. A strong correlation is observed between the changes in configurational energy and configurational entropy upon binding, resulting in near-linear compensation analogous to classical entropy-enthalpy compensation.     

Representation of molecular configurations by CAST coding method

A configurational CAST (CAnonical representation of STereochemistry) coding method, which represents relative and absolute configuration, is described. The configurational CAST codes are constructed by canonical rotation of the dihedral angles of the input structure before the CAST codes are assigned. Using the configurational CAST, configurational differences can be distinguished independently of conformational differences. Representation of enantiomers is also achieved by a mirror image conversion method. The CAST representation shows the distinctive characteristics of several diastereomers and conformers that were examined. The method clearly represents the differences in configurations. Applications to organic molecules having complex stereochemistry are also demonstrated.     

Grcarma: A fully automated task‐oriented interface for the analysis of molecular dynamics trajectories

We report the availability of grcarma, a program encoding for a fully automated set of tasks aiming to simplify the analysis of molecular dynamics trajectories of biological macromolecules. It is a cross‐platform, Perl/Tk‐based front‐end to the program carma and is designed to facilitate the needs of the novice as well as those of the expert user, while at the same time maintaining a user‐friendly and intuitive design. Particular emphasis was given to the automation of several tedious tasks, such as extraction of clusters of structures based on dihedral and Cartesian principal component analysis, secondary structure analysis, calculation and display of root‐mean‐square deviation (RMSD) matrices, calculation of entropy, calculation and analysis of variance–covariance matrices, calculation of the fraction of native contacts, etc. The program is free‐open source software available immediately for download. © 2013 Wiley Periodicals, Inc.     

Comparing landscape calculations with calorimetric data on ortho-terphenyl, and the question of the configurational fraction of the excess entropy

Mossa et al. [Phys. Rev. E 65, 041205 (2002)] have calculated the total and configurational entropies of supercooled ortho-terphenyl liquid using the potential-energy landscape formalism and a simplified model of the intermolecular potential. I show here that the agreement of their calculated configurational entropy with the experimental data depends on what is assumed about the configurational fraction of the excess entropy and its temperature dependence. In particular, if the configurational fraction is taken as 0.70 and independent of temperature the agreement is excellent; if a marked temperature dependence of that fraction inferred from calorimetric data is assumed the agreement is only fair at best. This marked temperature dependence of the configurational fraction also implies some implausible behavior of contributions to the excess entropy at the Kauzmann temperature, but no obvious reason for disregarding it presents itself.     

Improved mapping of protein binding sites

Computational mapping methods place molecular probes – small molecules or functional groups – on a protein surface in order to identify the most favorable binding positions by calculating an interaction potential. Mapping is an important step in a number of flexible docking and drug design algorithms. We have developed improved algorithms for mapping protein surfaces using small organic molecules as molecular probes. The calculations reproduce the binding of eight organic solvents to lysozyme as observed by NMR, as well as the binding of four solvents to thermolysin, in good agreement with x-ray data. Application to protein tyrosine phosphatase 1B shows that the information provided by the mapping can be very useful for drug design. We also studied why the organic solvents bind in the active site of proteins, in spite of the availability of alternative pockets that can very tightly accommodate some of the probes. A possible explanation is that the binding in the relatively large active site retains a number of rotational states, and hence leads to smaller entropy loss than the binding elsewhere else. Indeed, the mapping reveals that the clusters of the ligand molecules in the protein's active site contain different rotational-translational conformers, which represent different local minima of the free energy surface. In order to study the transitions between different conformers, reaction path and molecular dynamics calculations were performed. Results show that most of the rotational states are separated by low free energy barriers at the experimental temperature, and hence the entropy of binding in the active site is expected to be high.     

New and fast statistical-thermodynamic method for computation of protein-ligand binding entropy substantially improves docking accuracy

We present a novel method to estimate the contributions of translational and rotational entropy to protein-ligand binding affinity. The method is based on estimates of the configurational integral through the sizes of clusters obtained from multiple docking positions. Cluster sizes are defined as the intervals of variation of center of ligand mass and Euler angles in the cluster. Then we suggest a method to consider the entropy of torsional motions. We validate the suggested methods on a set of 135 PDB protein-ligand complexes by comparing the averaged root-mean square deviations (RMSD) of the top-scored ligand docked positions, accounting and not accounting for entropy contributions, relative to the experimentally determined positions. We demonstrate that the method increases docking accuracy by 10-21% when used in conjunction with the AutoDock docking program, thus reducing the percent of incorrectly docked ligands by 1.4-fold to four-fold, so that in some cases the percent of ligands correctly docked to within an RMSD of 2 A is above 90%. We show that the suggested method to account for entropy of relative motions is identical to the method based on the Monte Carlo integration over intervals of variation of center of ligand mass and Euler angles in the cluster.     

Spatially heterogeneous dynamics and the Adam-Gibbs relation in the Dzugutov liquid

We perform molecular dynamics simulations of a one-component glass-forming liquid and use the inherent structure formalism to test the predictions of the Adam-Gibbs (AG) theory and to explore the possible connection between these predictions and spatially heterogeneous dynamics. We calculate the temperature dependence of the average potential energy of the equilibrium liquid and show that it obeys the Rosenfeld-Tarazona T(3/5) law for low temperature T, while the average inherent structure energy is found to be inversely proportional to temperature at low T, consistent with a Gaussian distribution of potential energy minima. We investigate the shape of the basins around the local minima in configuration space via the average basin vibrational frequency and show that the basins become slightly broader upon cooling. We evaluate the configurational entropy S(conf), a measure of the multiplicity of potential energy minima sampled by the system, and test the validity of the AG relation between S(conf) and the bulk dynamics. We quantify the dynamically heterogeneous motion by analyzing the motion of particles that are mobile on short and intermediate time scales relative to the characteristic bulk relaxation time. These mobile particles move in one-dimensional "strings", and these strings form clusters with a well-defined average cluster size. The AG approach predicts that the minimum size of cooperatively rearranging regions (CRR) of molecules is inversely proportional to S(conf), and recently (Phys. Rev. Lett. 2003, 90, 085506) it has been shown that the mobile-particle clusters are consistent with the CRR envisaged by Adam and Gibbs. We test the possibility that the mobile-particle strings, rather than clusters, may describe the CRR of the Adam-Gibbs approach. We find that the strings also follow a nearly inverse relation with S(conf). We further show that the logarithm of the time when the strings and clusters are maximum, which occurs in the late-beta-relaxation regime of the intermediate scattering function, follows a linear relationship with 1/TS(conf), in agreement with the AG prediction for the relationship between the configurational entropy and the characteristic time for the liquid to undergo a transition to a new configuration. Since strings are the basic elements of the clusters, we propose that strings are a more appropriate measure of the minimum size of a CRR that leads to configurational transitions. That the cluster size also has an inverse relationship with S(conf) may be a consequence of the fact that the clusters are composed of strings.     

Environmental swap energy and role of configurational entropy in transfer of small molecules from water into alkanes

We studied the effect of segmented solvent molecules on the free energy of transfer of small molecules from water into alkanes (hexane, heptane, octane, decane, dodecane, tetradecane, and hexadecane). For these alkanes we measured partition coefficients of benzene, 3-methylindole (3MI), 2,3,4,6-tetrachlorophenol (TeCP), and 2,4,6-tribromophenol (TriBP) at 3, 11, 20, 33 [corrected], and 47 degrees C. For 3MI, TeCP, and TriBP the dependence of free energy of transfer on length of alkane chains was found to be very different from that for benzene. In contrast to benzene, the energy of transfer for 3MI, TeCP, and TriBP was independent of the number of carbons in alkanes. To interpret data, we used the classic Flory-Huggins (FH) theory of concentrated polymer solutions for the alkane phase. For benzene, the measured dependence of energy of transfer on the number of carbons in alkanes agreed well with predictions based on FH model in which the size of alkane segments was obtained from the ratio of molar volumes of alkanes and the solute. We show that for benzene, the energy of transfer can be divided into two components, one called environmental swap energy (ESE), and one representing the contribution of configurational entropy of alkane chains. For 3MI, TeCP, and TriBP the contribution of configurational entropy was not measurable even though the magnitude of the effect predicted from the FH model for short chain alkanes was as much as 20 times greater than experimental uncertainties. From the temperature dependence of ESE we obtained enthalpy and entropy of transfer for benzene, 3MI, TeCP, and TriBP. Experimental results are discussed in terms of a thermodynamic cycle considering creation of cavity, insertion of solute, and activation of solute-medium attractive interactions. Our results suggest that correcting experimental free energy of transfer by Flory-Huggins configurational entropy term is not generally appropriate and cannot be applied indiscriminately.     

(Thermo)dynamic Role of Receptor Flexibility,Entropy, and Motional Correlation in Protein–Ligand Binding

The binding of 2‐amino‐5‐methylthiazole to the W191G cavity mutant of cytochrome c peroxidase is an ideal test case to investigate the entropic contribution to the binding free energy due to changes in receptor flexibility. The dynamic and thermodynamic role of receptor flexibility are studied by 50 ns‐long explicit‐solvent molecular dynamics simulations of three separate receptor ensembles: W191G binding a K⁺ ion, W191G–2a5mt complex with a closed 190–195 gating loop, and apo with an open loop. We employ a method recently proposed to estimate accurate absolute single‐molecule configurational entropies and their differences for systems undergoing conformational transitions. We find that receptor flexibility plays a generally underestimated role in protein–ligand binding (thermo)dynamics and that changes of receptor motional correlation determine such large entropy contributions.     

Importance of entropy in the conformational equilibrium of phenylalanine: a matrix-isolation infrared spectroscopy and density functional theory study

The conformational behavior and infrared spectrum of l-phenylalanine were studied by matrix-isolation infrared spectroscopy and DFT [B3LYP/6-311++G(d,p)] calculations. The fourteen most stable structures were predicted to differ in energy by less than 10 kJ mol(-1), eight of them with abundances higher than 5% at the temperature of evaporation of the compound (423 K). Experimental results suggest that six conformers contribute to the spectrum of the isolated compound, whereas two conformers (IIb(3) and IIIb(3)) relax in matrix to a more stable form (IIb(2)) due to low energy barriers for conformational isomerization (conformational cooling). The two lowest-energy conformers (Ib(1), Ia) differ only in the arrangement of the amino acid group relative to the phenyl ring; they exhibit a relatively strong stabilizing intramolecular hydrogen bond of the O-H...N type and the carboxylic group in the trans configuration (O=C-O-H dihedral angle ca. 180 degrees ). Type II conformers have a weaker H-bond of the N-H...O=C type, but they bear the more favorable cis arrangement of the carboxylic group. Being considerably more flexible, type II conformers are stabilized by entropy and the relative abundances of two conformers of this type (IIb(2) and IIc(1)) are shown to significantly increase with temperature due to entropic stabilization. At 423 K, these conformers are found to be the first and third most abundant species present in the conformational equilibrium, with relative populations of ca. 15% each, whereas their populations could be expected to be only ca. 5% if entropy effects were not taken into consideration. Indeed, phenylalanine can be considered a notable example of a molecule where entropy plays an essential role in determining the relative abundance of the possible low-energy conformational states and then, the thermodynamics of the compound, even at moderate temperatures. Upon UV irradiation (lambda > 235 nm) of the matrix-isolated compound, unimolecular photodecomposition of phenylalanine is observed with production of CO(2) and phenethylamine.     

Associative Versus Dissociative Mechanisms of Phosphate Monoester Hydrolysis: On the Interpretation of Activation Entropies

Phosphate monoester and anhydride hydrolysis is ubiquitous in biology, being involved in, amongst other things, signal transduction, energy production, and the regulation of protein function. Therefore, this reaction has understandably been the focus of intensive research. Nevertheless, the precise mechanism by which phosphate monoester hydrolysis proceeds remains controversial. Traditionally, it has been assumed and frequently implied that a near‐zero activation entropy is indicative of a dissociative pathway. Herein, we examine free‐energy surfaces for the hydrolysis of the methyl phosphate dianion and the methyl pyrophosphate trianion in aqueous solution. In both cases, the reaction can proceed through either compact or expansive concerted (A_ND_N) transition states, with fairly similar barriers. We have evaluated the activation entropies for each transition state and demonstrate that both associative and dissociative transition states have near‐zero entropies of activation that are in good agreement with experimental values. Therefore, we believe that the activation entropy alone is not a useful diagnostic tool, as it depends not only on bond orders at the transition state, but also on other issues that include (but are not limited to) steric factors determining the configurational volumes available to reactants during the reaction, solvation and desolvation effects that may be associated with charge redistribution upon approaching the transition state and entropy changes associated with intramolecular degrees of freedom as the transition state is approached.     

Entropy of water in the hydration layer of major and minor grooves of DNA

Transport properties (translational and rotational) of water in the two grooves of the B-DNA duplex are known to be different from those in the bulk. Here, we use a recently developed theoretical scheme to compute the entropies of water molecules in both of the grooves of DNA and compare them with that in the bulk. The scheme requires as input both translational and rotational velocity autocorrelation function (C(V)(t) and C(omega)(t), respectively) data. These velocity autocorrelation functions were computed from an atomistic MD simulation of a B-DNA duplex (36 base pairs long) in explicit water (TIP3P). The average values of the entropy of water at 300 K in both of the grooves of DNA (the TS value in the major groove is 6.71 kcal/mol and that in the minor groove is 6.41 kcal/mol) are found to be significantly lower than that in bulk water (the TS value is 7.27 kcal/mol). Thus, the entropic contribution to the free energy change (TDeltaS) of transferring a minor groove water molecule to the bulk is 0.86 kcal/mol and of transferring a major groove water to the bulk is 0.56 kcal/mol at 300 K, which is to be compared with 1.44 kcal/mol for melting of ice at 273 K. We also calculate the energy of interaction of each water molecule with the rest of the atoms in the system and hence calculate the chemical potential (Helmholtz free energy per water molecule, A = E - TS) in the different domains. The identical free energy value of water molecules in the different domains proves the robustness of the scheme. We propose that the configurational entropy of water in the grooves can be used as a measure of the mobility (or microviscosity) of water molecules in a given domain.     

A determination of antioxidant efficiencies using ESR and computational methods

Using Transition-State Theory, experimental rate constants, determined over a range of temperatures, for reactions of Vitamin E type antioxidants are analysed in terms of their enthalpies and entropies of activation. It is further shown that computational methods may be employed to calculate enthalpies and entropies, and hence Gibbs free energies, for the overall reactions. Within the linear free energy relationship (LFER) assumption, that the Gibbs free energy of activation is proportional to the overall Gibbs free energy change for the reaction, it is possible to rationalise, and even to predict, the relative contributions of enthalpy and entropy for reactions of interest, involving potential antioxidants. A method is devised, involving a competitive reaction between *CH3 radicals and both the spin-trap PBN and the antioxidant, which enables the relatively rapid determination of a relative ordering of activities for a series of potential antioxidant compounds, and also of their rate constants for scavenging *CH3 radicals (relative to the rate constant for addition of *CH3 to PBN).     

Free energy landscape of protein-like chains with discontinuous potentials

In this article the configurational space of two simple protein models consisting of polymers composed of a periodic sequence of four different kinds of monomers is studied as a function of temperature. In the protein models, hydrogen bond interactions, electrostatic repulsion, and covalent bond vibrations are modeled by discontinuous step, shoulder, and square-well potentials, respectively. The protein-like chains exhibit a secondary alpha helix structure in their folded states at low temperatures, and allow a natural definition of a configuration by considering which beads are bonded. Free energies and entropies of configurations are computed using the parallel tempering method in combination with hybrid Monte Carlo sampling of the canonical ensemble of the discontinuous potential system. The probability of observing the most common configuration is used to analyze the nature of the free energy landscape, and it is found that the model with the least number of possible bonds exhibits a funnel-like free energy landscape at low enough temperature for chains with fewer than 30 beads. For longer proteins, the free landscape consists of several minima, where the configuration with the lowest free energy changes significantly by lowering the temperature and the probability of observing the most common configuration never approaches one due to the degeneracy of the lowest accessible potential energy.