Calculation Of Configurational Entropy Research Articles

Exploring Pyrrolo-Fused Heterocycles as Promising Anticancer Agents: An Integrated Synthetic, Biological, and Computational Approach.

Five new series of pyrrolo-fused heterocycles were designed through a scaffold hybridization strategy as analogs of the well-known microtubule inhibitor phenstatin. Compounds were synthesized using the 1,3-dipolar cycloaddition of cycloimmonium N-ylides to ethyl propiolate as a key step. Selected compounds were then evaluated for anticancer activity and ability to inhibit tubulin polymerization in vitro. Notably, pyrrolo[1,2-a]quinoline 10a was active on most tested cell lines, performing better than control phenstatin in several cases, most notably on renal cancer cell line A498 (GI50 27 nM), while inhibiting tubulin polymerization in vitro. In addition, this compound was predicted to have a promising ADMET profile. The molecular details of the interaction between compound 10a and tubulin were investigated through in silico docking experiments, followed by molecular dynamics simulations and configurational entropy calculations. Of note, we found that some of the initially predicted interactions from docking experiments were not stable during molecular dynamics simulations, but that configurational entropy loss was similar in all three cases. Our results suggest that for compound 10a, docking experiments alone are not sufficient for the adequate description of interaction details in terms of target binding, which makes subsequent scaffold optimization more difficult and ultimately hinders drug design. Taken together, these results could help shape novel potent antiproliferative compounds with pyrrolo-fused heterocyclic cores, especially from an in silico methodological perspective.

A universal configurational entropy metric for high-entropy materials

In recent years, high-entropy materials have become increasingly complex, outgrowing simple solid-solution based calculations of configurational entropy, and with that, the previously established metrics for high-, medium-, and low-entropy. Those previous metrics are derived from metals-specific approximations of the entropy of fusion and the internal energy of a monatomic gas, and thus, do not apply to other materials. A new entropy metric is proposed that is universally applicable for crystalline materials, from simple solid-solution high-entropy alloys to high-entropy materials with complex crystal structures and multiple sublattices. Additionally, the methods for calculating configurational entropy in complex crystal structures are discussed.

Novel rare-earth and transition metal-based entropy stabilized oxides with spinel structure

Entropy stabilization was attempted through the addition of 5 different rare earth cations, 5 different transition metal cations and all these 10 cations in the form of three compounds based on NiFe2O4. Equiatomic ratios of these cations were used to maximize the entropy stabilization effect. The compounds attempted and investigated were NiFe1.9(Dy0.02Er0.02Gd0.02Ho0.02Tb0.02)O4, (Co0.2Cr0.2Fe0.2Mn0.2Ni0.2)Fe2O4 and (Co0.2Cr0.2Fe0.2Mn0.2Ni0.2)Fe1.9(Dy0.02Er0.02Gd0.02Ho0.02Tb0.02)O4. Synthesis of these compounds were carried out by sol–gel technique and all the compounds were found to crystallize in Fd3¯mspinel structure indicating the entropy stabilization. An appropriate calculation of configurational entropy of mixing employing the sublattice model is demonstrated for such entropy stabilized oxides.

Updated method considers system’s structure when calculating entropy

Calculations of configurational entropy traditionally do not take the symmetries of a system into account, which can have nonnegligible effects on other parameters.

Beyond amino acid sequence: disulfide bonds and the origins of the extreme amyloidogenic properties of insulin's H-fragment.

The presence of disulfide bonds affects the protein stability and therefore tendency to misfold and form amyloid-like fibrils. Insulin's three disulfide bridges stabilize the native state and prevent aggregation. Partial proteolysis of insulin releases highly amyloidogenic and inherently disordered two-chain 'H-fragment' retaining insulin's Cys7A-Cys7B and Cys6A-Cys11A disulfide bonds. The abrupt self-association of H-fragment monomers into fibrils is suppressed in the presence of disulfide-reducing agent. These circumstances make the H-fragment an interesting model to study the impact of disulfide bonds on amyloidogenesis beyond the 'stabilization-of-the-native-state' paradigm. Here, we investigate fibrillization of various synthetic peptides derived from the H-fragment through modifications of Cys7A-Cys7B/Cys6A-Cys11A bonds. In comparison to H-fragment, aggregation of a two-chain 'AB' analog lacking Cys6A-Cys11A bond is decelerated, while the alternative removal of Cys7A-Cys7B bond releases a non-aggregating B-chain and a highly amyloidogenic 'ACC' fragment containing the intrachain Cys6A-Cys11A bond. Our analysis, supported by calculations of configurational entropy, suggests that Cys6A-Cys11A bond is a key factor behind the explosive self-association of H-fragment. The bond restricts the conformational space probed by nucleating monomers which is reflected by an approximately 2.4 kJ·mol-1 K-1 decrease in entropy. The fact that the intact Cys6A-Cys11A bond promotes fibrillization of the H-fragment is remarkable in light of the previously established role of the same disulfide bond in preventing formation of insulin fibrils. Our results imply that a single disulfide bond within a folded protein and its fragment may play entirely different roles in aggregation and that this role may evolve with progressing phases of misfolding.

Calculation of Configurational Entropy in Complex Landscapes

Entropy and the second law of thermodynamics are fundamental concepts that underlie all natural processes and patterns. Recent research has shown how the entropy of a landscape mosaic can be calculated using the Boltzmann equation, with the entropy of a lattice mosaic equal to the logarithm of the number of ways a lattice with a given dimensionality and number of classes can be arranged to produce the same total amount of edge between cells of different classes. However, that work seemed to also suggest that the feasibility of applying this method to real landscapes was limited due to intractably large numbers of possible arrangements of raster cells in large landscapes. Here I extend that work by showing that: (1) the proportion of arrangements rather than the number with a given amount of edge length provides a means to calculate unbiased relative configurational entropy, obviating the need to compute all possible configurations of a landscape lattice; (2) the edge lengths of randomized landscape mosaics are normally distributed, following the central limit theorem; and (3) given this normal distribution it is possible to fit parametric probability density functions to estimate the expected proportion of randomized configurations that have any given edge length, enabling the calculation of configurational entropy on any landscape regardless of size or number of classes. I evaluate the boundary limits (4) for this normal approximation for small landscapes with a small proportion of a minority class and show it holds under all realistic landscape conditions. I further (5) demonstrate that this relationship holds for a sample of real landscapes that vary in size, patch richness, and evenness of area in each cover type, and (6) I show that the mean and standard deviation of the normally distributed edge lengths can be predicted nearly perfectly as a function of the size, patch richness and diversity of a landscape. Finally, (7) I show that the configurational entropy of a landscape is highly related to the dimensionality of the landscape, the number of cover classes, the evenness of landscape composition across classes, and landscape heterogeneity. These advances provide a means for researchers to directly estimate the frequency distribution of all possible macrostates of any observed landscape, and then directly calculate the relative configurational entropy of the observed macrostate, and to understand the ecological meaning of different amounts of configurational entropy. These advances enable scientists to take configurational entropy from a concept to an applied tool to measure and compare the disorder of real landscapes with an objective and unbiased measure based on entropy and the second law.

A hierarchy-based solution to calculate the configurational entropy of landscape gradients

The second law of thermodynamics is fundamental in landscape ecology, and Shannon entropy has been employed as an important means of analyzing landscape patterns. However, the thermodynamic basis of Shannon entropy has been recently questioned because such entropy considers only probability and not configurational information. As a result, Boltzmann entropy (also called configurational entropy), which is the basic measure in thermodynamics, has been revisited, and some thoughts on its calculation have been put forward. Nevertheless, a comprehensive calculation method is still lacking. The objective of this study is to propose a feasible solution for the calculation of configurational entropy for landscape gradients. To calculate the configurational entropy, the first step is to define a good macrostate and then to determine the number of microstates. The macrostate of a landscape gradient is defined as its abstract (i.e., upscaled) representation. The number of microstates is calculated by determining all the possible ways of downscaling from the macrostate to the original. Both simulated and real-life landscape patterns were used for experimental validation. The results show that the entropy calculated using the proposed method successfully captures the disorder of landscape gradients in terms of both composition and configuration. Configurational entropy, calculated using the proposed method, can serve as a thermodynamics-based metric to describe gradient-based landscapes and their changes across space and through time. With this metric, it becomes possible to interpret landscape ecological processes based on thermodynamic insights.

Structural properties of amyloid β(1-40) dimer explored by replica exchange molecular dynamics simulations.

Replica exchange molecular dynamics simulations (300 ns) were used to study the dimerization of amyloid β(1-40) (Aβ(1-40)) polypeptide. Configurational entropy calculations revealed that at physiological temperature (310 K, 37°C) dynamic dimers are formed by randomly docked monomers. Free energy of binding of the two chains to each other was -93.56 ± 6.341 kJmol-1 . Prevalence of random coil conformations was found for both chains with the exceptions of increased β-sheet content from residues 16-21 and 29-32 of chain A and residues 15-21 and 30-33 of chain B with β-turn/β-bend conformations in both chains from residues 1-16, 21-29 of chain A, 1-16, and 21-29 of chain B. There is a mixed β-turn/β-sheet region from residues 33-38 of both chains. Analysis of intra- and interchain residue distances shows that, although the individual chains are highly flexible, the dimer system stays in a loosely packed antiparallel β-sheet configuration with contacts between residues 17-21 of chain A with residues 17-21 and 31-36 of chain B as well as residues 31-36 of chain A with residues 17-21 and 31-36 of chain B. Based on dihedral principal component analysis, the antiparallel β-sheet-loop-β-sheet conformational motif is favored for many low energy sampled conformations. Our results show that Aβ(1-40) can form dynamic dimers in aqueous solution that have significant conformational flexibility and are stabilized by collapse of the central and C-terminal hydrophobic cores with the expected β-sheet-loop-β-sheet conformational motif. Proteins 2017; 85:1024-1045. © 2017 Wiley Periodicals, Inc.

PARENT: A Parallel Software Suite for the Calculation of Configurational Entropy in Biomolecular Systems.

Accurate estimation of configurational entropy from the in silico-generated biomolecular ensembles, e.g., from molecular dynamics (MD) trajectories, is dependent strongly on exhaustive sampling for physical reasons. This, however, creates a major computational problem for the subsequent estimation of configurational entropy using the Maximum Information Spanning Tree (MIST) or Mutual Information Expansion (MIE) approaches for internal molecular coordinates. In particular, the available software for such estimation exhibits serious limitations when it comes to molecules with hundreds or thousands of atoms, because of its reliance on a serial program architecture. To overcome this problem, we have developed a parallel, hybrid MPI/openMP C++ implementation of MIST and MIE, called PARENT, which is particularly optimized for high-performance computing and provides efficient estimation of configurational entropy in different biological processes (e.g., protein-protein interactions). In addition, PARENT also allows for a detailed mapping of intramolecular allosteric networks. Here, we benchmark the program on a set of 1-μs-long MD trajectories of 10 different protein complexes and their components, demonstrating robustness and good scalability. A direct comparison between MIST and MIE on the same dataset demonstrates a superior convergence behavior for the former approach, when it comes to total simulation length and configurational-space binning.

Unravelling the specific site preference in doping of calcium hydroxyapatite with strontium from ab initio investigations and Rietveld analyses

Strontium can be substituted into the calcium sublattice of hydroxyapatite without a solubility limit. However, recent ab initio simulations carried out at 0 K report endothermic nature of this process. There is also striking discrepancy between experimentally observed preference of Sr doping at Ca-II sites and the first principles calculations, which indicate that a Ca-I site is preferred energetically for the Sr substitution. In this paper we combine insights from Density Functional Theory simulations and regular configurational entropy calculations to determine the site preference of Sr doping in the range of 0-100 at% at finite temperatures. In addition, samples of Sr-HA are synthesized and refinement of the relevant structural information provides benchmark information on the experimental unit cell parameters of Sr-HA. We find that the contribution of the entropy of mixing can efficiently overcome the endothermic excess energy at a temperature typical of the calcining step in the synthesis route of hydroxyapatite (700-950 °C). We observe that the most preferential substitution pattern is mixed substitution of Sr regardless of the concentration. For a wet chemical method, carried out at a moderate temperature (90 °C), the mixed doping is still slightly favourable at higher Sr-concentrations, except the range at 20% Sr, where Site II substitution is not restricted energetically and equally possible as the mixed doping. We observe a close correspondence between our theoretical results and available experimental data. Hence it should be possible to apply this theory to other divalent dopants in HA, such as Zn(2+), Mg(2+), Pb(2+), Cu(2+), Ba(2+), Cd(2+) etc.

Slow motions in the hydrophobic core of chicken villin headpiece subdomain and their contributions to configurational entropy and heat capacity from solid-state deuteron NMR measurements.

We have investigated microsecond to millisecond time scale dynamics in several key hydrophobic core methyl groups of chicken villin headpiece subdomain protein (HP36) using a combination of single-site labeling, deuteron solid-state NMR line shape analysis, and computational modeling. Deuteron line shapes of hydrated powder samples are dominated by rotameric jumps and show a large variability of rate constants, activation energies, and rotameric populations. Site-specific activation energies vary from 6 to 38 kJ/mol. An additional mode of diffusion on a restricted arc is significant for some sites. In dry samples, the dynamics is quenched. Parameters of the motional models allow for calculations of configurational entropy and heat capacity, which, together with the rate constants, allow for observation of interplay between thermodynamic and kinetic picture of the landscape. Mutations at key phenylalanine residues at both distal (F47L&F51L) and proximal (F58L) locations to a relatively rigid side chain of L69 have a pronounced effect on alleviating the rigidity of this side chain at room temperature and demonstrate the sensitivity of the hydrophobic core environment to such perturbations.

An optimized MM/PBSA virtual screening approach applied to an HIV‐1 gp41 fusion peptide inhibitor

VIRus Inhibitory Peptide (VIRIP), a 20 amino acid peptide, binds to the fusion peptide (FP) of human immunodeficiency virus type 1 (HIV-1) gp41 and blocks viral entry. VIRIP derivatives with improved antiviral activity have been developed, and one of those derivatives has recently proven effective and safe in a phase 1/2 clinical trial. Here, molecular dynamics were executed in combination with molecular mechanics/Poisson-Boltzmann surface area (MM/PBSA) free energy calculations to explore the binding interaction between VIRIP derivatives and gp41 FP. A promising correlation between antiviral activity and simulated binding free energy was established thanks to restriction of the flexibility of the peptides, inclusion of configurational entropy calculations, and the use of multiple internal dielectric constants for the MM/PBSA calculations depending on the amino acid sequence. Based on these results, a virtual screening experiment was carried out to design VIRIP analogs with further improved antiretroviral activity. A selection of peptides was tested for inhibitory activity and several VIRIP derivatives were identified with significantly enhanced activity compared to the reference peptides. The results demonstrate that computational modeling strategies using an adapted MM/PBSA methodology improve the accuracy of binding free energy calculations of peptide complexes compared to the classic MM/PBSA protocol. As such, this virtual screening approach generated HIV-1 gp41 FP inhibitors with improved antiviral activity that could be useful for future clinical applications.

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.

Configurational Entropy Reallocation and Complex Loop Dynamics of the Mosquito-Stage Pvs25 Protein Complexed with the Fab Fragment of the Malaria Transmission Blocking Antibody 2A8.

Pvs25 is a protein of unique 3D structure, and it is characterized by the presence of repeated EGF-like domains and 11 disulfide bonds. It is a very important candidate for the transmission-blocking malaria vaccine, as it plays an important role in mosquito infection by Plasmodium parasites. Recently, the X-ray structure of the protein complexed with the transmission blocking antibody 2A8 has been reported. In this study, we report the loop reorganization of the Pvs25 protein based on configurational entropy calculations and dihedral principal component analysis as revealed from the protein complex and free molecular dynamics simulations. While the total entropy of the protein was estimated to be almost the same in the free and complex trajectories, the partition of the entropy contribution in the loop fragments of the protein revealed interesting entropy reallocation after the 2A8 antibody binding. Interestingly, the 51-71 protein loop experienced a significant reduction in its configurational entropy, while other parts of the protein did not show any difference in it, or even showed an entropy increase. This trend in entropy redistribution was found to be in direct relationship with specific interactions with the antibody's binding site. Results from root-mean-square fluctuations/deviations and dihedral angle principal component analysis further support this finding.

The configurational entropy of mixing of interstitials solid solutions

A simple and general expression for the configurational entropy of mixing of multicomponent interstitials solid solutions is deduced based on a probabilistic approach to compute the configurational entropy. The good agreement obtained with previous experimental and theoretical works based on Monte Carlo and partial configurational entropy calculations in binary systems show that the expression is suitable to be used as a reference state to compute the excess thermodynamic functions in multicomponent interstitials solid solutions. The results show that the configurational entropy can be computed to a high degree of accuracy in some systems without the use of a computer.

CONFIGURATIONAL ENTROPY IN GENERALIZED LATTICE-GAS MODELS

In this review, we present our recent results concerning accurate calculations of configurational entropy in generalized lattice-gas models. The calculations are based on the use of the thermodynamic integration method. Different applications (or systems) have been considered. Namely, systems in presence of (i) anisotropy, (ii) energetic heterogeneity, (iii) geometric heterogeneity, and (iv) multisite-occupancy adsorption. Total energy is calculated by means of the Monte Carlo simulation. Then the entropy is obtained by using thermodynamic integration starting at a known reference state. In case (iv), the method relies upon the definition of an artificial Hamiltonian associated with the system of interest for which the entropy of a reference state can be exactly known. Thermodynamic integration is then applied to obtain the entropy in a given state of the system of interest. A rich variety of behaviors is found and analyzed in the context of the lattice-gas theory.

Overcoming Misconceptions about Configurational Entropy in Condensed Phases

Configurational and thermal entropy yield identical numerical values for ΔS only when the system's "dimensionless" energy gaps (Δε /kT ) between the accessible quantized energy levels are minimized by temperature to nearly infinitesimal values so that the spreading of energy among the system's microstates becomes effectively classical and equiprobable. The molecular partition function provides the numerical value for the effective number of both accessible states and spatial configurations per molecule, for which this condition is valid at a given temperature. Considering the phenomenon of mixing and standard molar entropy values leads to the conclusion that configurational entropy calculations are significant and thermodynamically valid because of their fundamental connection to the process of random energy re-distributions in a system, via the available modes of molecular motion.

Efficient calculation of configurational entropy from molecular simulations by combining the mutual‐information expansion and nearest‐neighbor methods

Changes in the configurational entropies of molecules make important contributions to the free energies of reaction for processes such as protein-folding, noncovalent association, and conformational change. However, obtaining entropy from molecular simulations represents a long-standing computational challenge. Here, two recently introduced approaches, the nearest-neighbor (NN) method and the mutual-information expansion (MIE), are combined to furnish an efficient and accurate method of extracting the configurational entropy from a molecular simulation to a given order of correlations among the internal degrees of freedom. The resulting method takes advantage of the strengths of each approach. The NN method is entirely nonparametric (i.e., it makes no assumptions about the underlying probability distribution), its estimates are asymptotically unbiased and consistent, and it makes optimum use of a limited number of available data samples. The MIE, a systematic expansion of entropy in mutual information terms of increasing order, provides a well-characterized approximation for lowering the dimensionality of the numerical problem of calculating the entropy of a high-dimensional system. The combination of these two methods enables obtaining well-converged estimations of the configurational entropy that capture many-body correlations of higher order than is possible with the simple histogramming that was used in the MIE method originally. The combined method is tested here on two simple systems: an idealized system represented by an analytical distribution of six circular variables, where the full joint entropy and all the MIE terms are exactly known, and the R,S stereoisomer of tartaric acid, a molecule with seven internal-rotation degrees of freedom for which the full entropy of internal rotation has been already estimated by the NN method. For these two systems, all the expansion terms of the full MIE of the entropy are estimated by the NN method and, for comparison, the MIE approximations up to third order are also estimated by simple histogramming. The results indicate that the truncation of the MIE at the two-body level can be an accurate, computationally nondemanding approximation to the configurational entropy of anharmonic internal degrees of freedom. If needed, higher-order correlations can be estimated reliably by the NN method without excessive demands on the molecular-simulation sample size and computing time.

Thermodynamic modeling of the Cu–Ag–Au system using the cluster/site approximation

The cluster/site approximation (CSA), which takes short-range order (SRO) into account in a computationally efficient manner in configurational entropy calculations on alloys, has been used to describe the fcc-type phases in the Cu–Ag–Au system. The CSA-calculated phase diagrams and thermodynamic properties are in good agreement with the available experimental data. Importantly, this CSA description requires no reciprocal interaction parameters as has been the case for descriptions based on the compound energy formulism (CEF). For this reason, the CSA description is expected to be more reliable when used for describing multi-component systems.

Inherent Structure Approach to the Study of Glass-Forming Liquids

The inherent structure approach, wherein thermodynamic and structural changes in glass-forming liquids are analyzed in terms of local potential energy minima that the liquid samples, has recently been applied extensively to the study of thermodynamic aspects of glass-forming liquids. The evaluation of the configurational entropy, which arises from the multiplicity of local energy minima, plays a central role in such analyses. Results are presented here concerning the calculation of configurational entropy based on computer simulations of a model liquid; and the application of the inherent structure formalism to the study of the glass transition locus, and the fragility of glass-forming liquids.