Conformational Entropies and Order Parameters: Convergence, Reproducibility, and Transferability

Olanzapine manipulates neuroactive signals and may onset metabolic disturbances

Olanzapine is one of the most prescribed atypical antipsychotics to treat psychiatric illness and is associated with weight gain and metabolic disturbance. The present study investigated the olanzapine-regulated metabolic pathways using functional enrichent analysis including binding affinity with G-protein-coupled receptors (GPCRs). Proteins modulated by olanzapine were retrieved from SwissTargetPrediction, DIGEP-Pred, and BindingDB and then enriched in Search Tool for the Retrieval of Interacting Genes/Proteins (STRING) to assess molecular function, biological process, and cellular components including Kyoto Encyclopedia of Genes and Genomes (KEGG) pathways. We used homology modeling to improve the 3D structure for GPCR synapse proteins including dopamine, serotonin, muscarinic, and histamine receptors which were then optimized using molecular dynamics (MD) simulations. The protein-olanzapine binding mechanisms for different GPCR binders were evaluated using molecular docking; later refined by MD simulations. Binding mechanism of olanzapine with D2, 5HT1A, 5HT2A, 5HT2B, 5HT2C, M1, and M2 receptors were created using homology modeling and optimized using MD simulations. In target identification, it was observed that olanzapine majority targeted G-protein coupled receptors. Further, enrichment analysis identified around 76% of the total genes regulated in molecular function, biological process, and cellular components were common including KEGG pathways. Moreover, it was observed that olanzapine had a major potency over the neurotransmitter synapse including neuroactive signals . Olanzapine-induced weight gain and metabolic alterations could be due to the deregulation of multiple synapses like dopamine, serotonin, muscarinic, and histamine at the feeding center followed by cGMP-PKG, cAMP, and PI3K-Akt signaling pathways. HIGHLIGHTSOlanzapine is used in the management of psychiatric illnesses.Olanzapine causes disturbance in lipids and glucosehomeostasis and manipulates energy expenditure.Olanzapine-induced weight gain may occur due to the deregulation of the multiple synapse and cGMP-PKG, cAMP, and PI3K-Akt signaling pathwayCommunicated by Ramaswamy H. Sarma.

Interplay of halogen bonding and solvation in protein-ligand binding

Halogen bonding is increasingly utilized in efforts to achieve high affinity and selectivity of molecules designed to bind proteins, making it paramount to understand the relationship between structure, dynamics, and thermodynamic driving forces. We present a detailed analysis addressing this problem using a series of protein-ligand complexes involving single halogen substitutions - F, Cl, Br, and I - and nearly identical structures. Isothermal titration calorimetry reveals an increasingly favorable binding enthalpy from F to I that correlates with the halogen size and σ-hole electropositive character, but is partially counteracted by unfavorable entropy, which is constant from F to Cl and Br, but worse for I. Consequently, the binding free energy is roughly equal for Cl, Br, and I. QM and solvation-free-energy calculations reflect an intricate balance between halogen bonding, hydrogen bonds, and solvation. These advances have the potential to aid future drug design initiatives involving halogenated compounds.

How does it really move? Recent progress in the investigation of protein nanosecond dynamics by NMR and simulation

Nuclear magnetic resonance (NMR) spin relaxation experiments currently probe molecular motions on timescales from picoseconds to nanoseconds. The detailed interpretation of these motions in atomic detail benefits from complementarity with the results from molecular dynamics (MD) simulations. In this mini-review, we describe the recent developments in experimental techniques to study the backbone dynamics from ¹⁵N relaxation and side-chain dynamics from ¹³C relaxation, discuss the different analysis approaches from model-free to dynamics detectors, and highlight the many ways that NMR relaxation experiments and MD simulations can be used together to improve the interpretation and gain insights into protein dynamics.

Double Mutant of Chymotrypsin Inhibitor 2 Stabilized through Increased Conformational Entropy

The conformational heterogeneity of a folded protein can affect not only its function but also stability and folding. We recently discovered and characterized a stabilized double mutant (L49I/I57V) of the protein CI2 and showed that state-of-the-art prediction methods could not predict the increased stability relative to the wild-type protein. Here, we have examined whether changed native-state dynamics, and resulting entropy changes, can explain the stability changes in the double mutant protein, as well as the two single mutant forms. We have combined NMR relaxation measurements of the ps-ns dynamics of amide groups in the backbone and the methyl groups in the side chains with molecular dynamics simulations to quantify the native-state dynamics. The NMR experiments reveal that the mutations have different effects on the conformational flexibility of CI2: a reduction in conformational dynamics (and entropy estimated from this) of the native state of the L49I variant correlates with its decreased stability, while increased dynamics of the I57V and L49I/I57V variants correlates with their increased stability. These findings suggest that explicitly accounting for changes in native-state entropy might be needed to improve the predictions of the effect of mutations on protein stability.

How Much Entropy Is Contained in NMR Relaxation Parameters?

Solution-state NMR relaxation experiments are the cornerstone to study internal protein dynamics at an atomic resolution on time scales that are faster than the overall rotational tumbling time τ_R. Since the motions described by NMR relaxation parameters are connected to thermodynamic quantities like conformational entropies, the question arises how much of the total entropy is contained within this tumbling time. Using all-atom molecular dynamics simulations of the T4 lysozyme, we found that entropy buildup is rather fast for the backbone, such that the majority of the entropy is indeed contained in the short-time dynamics. In contrast, the contribution of the slow dynamics of side chains on time scales beyond τ_R on the side-chain conformational entropy is significant and should be taken into account for the extraction of accurate thermodynamic properties.

Dynamics in natural and designed elastins and their relation to elastic fiber structure and recoil

Elastin fibers assemble in the extracellular matrix from the precursor protein tropoelastin and provide the flexibility and spontaneous recoil required for arterial function. Unlike many proteins, a structure-function mechanism for elastin has been elusive. We have performed detailed NMR relaxation studies of the dynamics of the minielastins 24x' and 20x' using solution NMR, and of purified bovine elastin fibers in the presence and absence of mechanical stress using solid state NMR. The low sequence complexity of the minielastins enables us to determine average dynamical timescales and degrees of local ordering in the cross-link and hydrophobic modules separately using NMR relaxation by taking advantage of their residue-specific resolution. We find an extremely high degree of disorder, with order parameters for the entirety of the hydrophobic domains near zero, resembling that of simple chemical polymers and less than the order parameters that have been observed in other intrinsically disordered proteins. We find that average backbone order parameters in natural, purified elastin fibers are comparable to those found in 24x' and 20x' in solution. The difference in dynamics, compared with the minielastins, is that backbone correlation times are significantly slowed in purified elastin. Moreover, when elastin is mechanically stretched, the high chain disorder in purified elastin is retained, showing that any change in local ordering is below that detectable in our experiment. Combined with our previous finding of a 10-fold increase in the ordering of water when fully hydrated elastin fibers are stretched by 50%, these results support the hypothesis that stretch induced solvent ordering, i.e., the hydrophobic effect, is a key player in the elastic recoil of elastin as opposed to configurational entropy loss.

On the Use of Interaction Entropy and Related Methods to Estimate Binding Entropies

Molecular mechanics combined with Poisson-Boltzmann or generalized Born and solvent-accessible area solvation energies (MM/PBSA and MM/GBSA) are popular methods to estimate the free energy for the binding of small molecules to biomacromolecules. However, the estimation of the entropy has been problematic and time-consuming. Traditionally, normal-mode analysis has been used to estimate the entropy, but more recently, alternative approaches have been suggested. In particular, it has been suggested that exponential averaging of the electrostatic and Lennard-Jones interaction energies may provide much faster and more accurate entropies, the interaction entropy (IE) approach. In this study, we show that this exponential averaging is extremely poorly conditioned. Using stochastic simulations, assuming that the interaction energies follow a Gaussian distribution, we show that if the standard deviation of the interaction energies (σIE) is larger than 15 kJ/mol, it becomes practically impossible to converge the interaction entropies (more than 10 million energies are needed, and the number increases exponentially). A cumulant approximation to the second order of the exponential average shows a better convergence, but for σIE > 25 kJ/mol, it gives entropies that are unrealistically large. Moreover, in practical applications, both methods show a steady increase in the entropy with the number of energies considered.

Exploring ligand dynamics in protein crystal structures with ensemble refinement

Understanding the dynamics of ligands bound to proteins is an important task in medicinal chemistry and drug design. However, the dominant technique for determining protein-ligand structures, X-ray crystallography, does not fully account for dynamics and cannot accurately describe the movements of ligands in protein binding sites. In this article, an alternative method, ensemble refinement, is used on six protein-ligand complexes with the aim of understanding the conformational diversity of ligands in protein crystal structures. The results show that ensemble refinement sometimes indicates that the flexibility of parts of the ligand and some protein side chains is larger than that which can be described by a single conformation and atomic displacement parameters. However, since the electron-density maps are comparable and Rfree values are slightly increased, the original crystal structure is still a better model from a statistical point of view. On the other hand, it is shown that molecular-dynamics simulations and automatic generation of alternative conformations in crystallographic refinement confirm that the flexibility of these groups is larger than is observed in standard refinement. Moreover, the flexible groups in ensemble refinement coincide with groups that give high atomic displacement parameters or non-unity occupancy if optimized in standard refinement. Therefore, the conformational diversity indicated by ensemble refinement seems to be qualitatively correct, indicating that ensemble refinement can be an important complement to standard crystallographic refinement as a tool to discover which parts of crystal structures may show extensive flexibility and therefore are poorly described by a single conformation. However, the diversity of the ensembles is often exaggerated (probably partly owing to the rather poor force field employed) and the ensembles should not be trusted in detail.

Entropy-Entropy Compensation between the Protein, Ligand, and Solvent Degrees of Freedom Fine-Tunes Affinity in Ligand Binding to Galectin-3C

Molecular recognition is fundamental to biological signaling. A central question is how individual interactions between molecular moieties affect the thermodynamics of ligand binding to proteins and how these effects might propagate beyond the immediate neighborhood of the binding site. Here, we investigate this question by introducing minor changes in ligand structure and characterizing the effects of these on ligand affinity to the carbohydrate recognition domain of galectin-3, using a combination of isothermal titration calorimetry, X-ray crystallography, NMR relaxation, and computational approaches including molecular dynamics (MD) simulations and grid inhomogeneous solvation theory (GIST). We studied a congeneric series of ligands with a fluorophenyl-triazole moiety, where the fluorine substituent varies between the ortho, meta, and para positions (denoted O, M, and P). The M and P ligands have similar affinities, whereas the O ligand has 3-fold lower affinity, reflecting differences in binding enthalpy and entropy. The results reveal surprising differences in conformational and solvation entropy among the three complexes. NMR backbone order parameters show that the O-bound protein has reduced conformational entropy compared to the M and P complexes. By contrast, the bound ligand is more flexible in the O complex, as determined by ¹⁹F NMR relaxation, ensemble-refined X-ray diffraction data, and MD simulations. Furthermore, GIST calculations indicate that the O-bound complex has less unfavorable solvation entropy compared to the other two complexes. Thus, the results indicate compensatory effects from ligand conformational entropy and water entropy, on the one hand, and protein conformational entropy, on the other hand. Taken together, these different contributions amount to entropy-entropy compensation among the system components involved in ligand binding to a target protein.

Structure-Based Identification of Natural Products as SARS-CoV-2 Mpro Antagonist from Echinacea angustifolia Using Computational Approaches

Coronavirus disease-19 (COVID-19) pandemic, caused by the novel SARS-CoV-2 virus, continues to be a global threat. The number of cases and deaths will remain escalating due to the lack of effective therapeutic agents. Several studies have established the importance of the viral main protease (Mpro) in the replication of SARS-CoV-2 which makes it an attractive target for antiviral drug development, including pharmaceutical repurposing and other medicinal chemistry approaches. Identification of natural products with considerable inhibitory potential against SARS-CoV-2 could be beneficial as a rapid and potent alternative with drug-likeness by comparison to de novo antiviral drug discovery approaches. Thereof, we carried out the structure-based screening of natural products from Echinacea-angustifolia, commonly used to prevent cold and other microbial respiratory infections, targeting SARS-CoV-2 Mpro. Four natural products namely, Echinacoside, Quercetagetin 7-glucoside, Levan N, Inulin from chicory, and 1,3-Dicaffeoylquinic acid, revealed significant docking energy (>-10 kcal/mol) in the SARS-CoV-2 Mpro catalytic pocket via substantial intermolecular contacts formation against co-crystallized ligand (<-4 kcal/mol). Furthermore, the docked poses of SARS-CoV-2 Mpro with selected natural products showed conformational stability through molecular dynamics. Exploring the end-point net binding energy exhibited substantial contribution of Coulomb and van der Waals interactions to the stability of respective docked conformations. These results advocated the natural products from Echinacea angustifolia for further experimental studies with an elevated probability to discover the potent SARS-CoV-2 Mpro antagonist with higher affinity and drug-likeness.

Computational and In Vitro Investigation of (-)-Epicatechin and Proanthocyanidin B2 as Inhibitors of Human Matrix Metalloproteinase 1

Matrix metalloproteinases 1 (MMP-1) energetically triggers the enzymatic proteolysis of extracellular matrix collagenase (ECM), resulting in progressive skin aging. Natural flavonoids are well known for their antioxidant properties and have been evaluated for inhibition of matrix metalloproteins in human. Recently, (-)-epicatechin and proanthocyanidin B2 were reported as essential flavanols from various natural reservoirs as potential anti-inflammatory and free radical scavengers. However, their molecular interactions and inhibitory potential against MMP-1 are not yet well studied. In this study, sequential absorption, distribution, metabolism, and excretion (ADME) profiling, quantum mechanics calculations, and molecular docking simulations by extra precision Glide protocol predicted the drug-likeness of (-)-epicatechin (−7.862 kcal/mol) and proanthocyanidin B2 (−8.145 kcal/mol) with the least reactivity and substantial binding affinity in the catalytic pocket of human MMP-1 by comparison to reference bioactive compound epigallocatechin gallate (−6.488 kcal/mol). These flavanols in docked complexes with MMP-1 were further studied by 500 ns molecular dynamics simulations that revealed substantial stability and intermolecular interactions, viz. hydrogen and ionic interactions, with essential residues, i.e., His218, Glu219, His222, and His228, in the active pocket of MMP-1. In addition, binding free energy calculations using the Molecular Mechanics Generalized Born Surface Area (MM/GBSA) method suggested the significant role of Coulomb interactions and van der Waals forces in the stability of respective docked MMP-1-flavonol complexes by comparison to MMP-1-epigallocatechin gallate; these observations were further supported by MMP-1 inhibition assay using zymography. Altogether with computational and MMP-1–zymography results, our findings support (-)-epicatechin as a comparatively strong inhibitor of human MMP-1 with considerable drug-likeness against proanthocyanidin B2 in reference to epigallocatechin gallate.

Predicting the Most Stable Aptamer/Target Molecule Complex Configuration Using a Stochastic-Tunnelling Basin-Hopping Discrete Molecular Dynamics Method: A Novel Global Minimum Search Method for a Biomolecule Complex

This study proposed a novel global minimum search method for predicting the most stable biomolecule complex, which combines the strengths of three global minimum search methods (stochastic tunnelling, basin hopping, and discrete molecular dynamics) to efficiently improve the spatial domain search ability of the stochastic tunnelling-basin hopping (STUN-BH) method from our previous study. The epithelial cell adhesion molecule (EpCAM, PDB code: 4MZV) was used as a benchmark target molecule for the EpCAM aptamer EpA (Apt_EpA). For the most stable Apt_EpA/EpCAM complex predicted by our new method, the Apt_EpA was attached to the entangling loop fragments of the two EpCAM molecules with the most Apt_EpA residues. After the Apt_EpA/EpCAM complex had equilibrated with the water environment through a molecular dynamics simulation at 300 K for 10 ns, stable hydrogen bonds formed between the bases of Apt_EpA and EpCAM residues of the secondary structures, which included the alpha helix and beta sheet becoming less stable in the water environment. Those hydrogen bonds formed between the bases of Apt_EpA and EpCAM loop fragment residues remained stable in the water environment.

Agonism activities of lyso-phosphatidylcholines (LPC) Ligands binding to peroxisome proliferator-activated receptor gamma (PPARγ)

PPARγ is an isoform of peroxisome proliferator-activated receptor (PPAR) belonging to a super family of nuclear receptors and is a primary target of the effective drug to treat the type II diabetes. The experiments found that Lyso-phosphatidylcholines (LPC) could bind to PPARγ, but the binding modes remain unknown. We used the Molecular Docking and Molecular Dynamic (MD) simulations to study the binding of four LPC ligands (LPC16:0, LPC18:0, LPC18:1-1 and LPC18:1-2) to PPARγ. The two-step MD simulations were employed to determine the final binding modes. The 20 ns MD simulations for four final LPC-PPARγ complexes were performed to analyze their structures, the binding key residues, and agonism activities. The results reveal that three LPC ligands (LPC16:0, LPC18:0 and LPC18:1-1) bind to Arm II and III regions of the Ligand Binding Domain (LBD) pocket, whereas they do not interact with Tyr473 of Helix 12 (H12). In contrast, LPC18:1-2 can form the hydrogen bonds with Tyr473 and bind into Arm I and II regions. Comparing with the paradigm systems of the full agonist (Rosiglitazone-PPARγ) and the partial agonist (MRL24-PPARγ), our results indicate that LPC16:0, LPC18:0 and LPC18:1-1 could be the potential partial agonists and LPC18:1-2 could be a full agonist. The in-depth analysis of the residue fluctuations and structure alignment confirm the present prediction of the LPC agonism activities.Communicated by Ramaswamy H. Sarma.

Interplay between Conformational Entropy and Solvation Entropy in Protein-Ligand Binding

Understanding the driving forces underlying molecular recognition is of fundamental importance in chemistry and biology. The challenge is to unravel the binding thermodynamics into separate contributions and to interpret these in molecular terms. Entropic contributions to the free energy of binding are particularly difficult to assess in this regard. Here we pinpoint the molecular determinants underlying differences in ligand affinity to the carbohydrate recognition domain of galectin-3, using a combination of isothermal titration calorimetry, X-ray crystallography, NMR relaxation, and molecular dynamics simulations followed by conformational entropy and grid inhomogeneous solvation theory (GIST) analyses. Using a pair of diastereomeric ligands that have essentially identical chemical potential in the unbound state, we reduced the problem of dissecting the thermodynamics to a comparison of the two protein-ligand complexes. While the free energies of binding are nearly equal for the R and S diastereomers, greater differences are observed for the enthalpy and entropy, which consequently exhibit compensatory behavior, ΔΔ H°(R - S) = -5 ± 1 kJ/mol and - TΔΔ S°(R - S) = 3 ± 1 kJ/mol. NMR relaxation experiments and molecular dynamics simulations indicate that the protein in complex with the S-stereoisomer has greater conformational entropy than in the R-complex. GIST calculations reveal additional, but smaller, contributions from solvation entropy, again in favor of the S-complex. Thus, conformational entropy apparently dominates over solvation entropy in dictating the difference in the overall entropy of binding. This case highlights an interplay between conformational entropy and solvation entropy, pointing to both opportunities and challenges in drug design.

Are crystallographic B-factors suitable for calculating protein conformational entropy?

It is not possible to obtain reliable entropy estimates from crystallographic B-factors even with re-refined or room-temperature crystal structures.

Assessing the performance of MM/PBSA and MM/GBSA methods. 7. Entropy effects on the performance of end-point binding free energy calculation approaches

Entropy effects play an important role in drug-target interactions, but the entropic contribution to ligand-binding affinity is often neglected by end-point binding free energy calculation methods, such as MM/GBSA and MM/PBSA, due to the expensive computational cost of normal mode analysis (NMA). Here, we systematically investigated entropy effects on the prediction power of MM/GBSA and MM/PBSA using >1500 protein-ligand systems and six representative AMBER force fields. Two computationally efficient methods, including NMA based on truncated structures and the interaction entropy approach, were used to estimate the entropic contributions to ligand-target binding free energies. In terms of the overall accuracy, we found that, for the minimized structures, in most cases the inclusion of the conformational entropies predicted by truncated NMA (enthalpynmode_min_9Å) compromises the overall accuracy of MM/GBSA and MM/PBSA compared with the enthalpies calculated based on the minimized structures (enthalpymin). However, for the MD trajectories, the binding free energies can be improved by the inclusion of the conformation entropies predicted by either truncated-NMA for a relatively high dielectric constant (εin = 4) or the interaction entropy method for εin = 1-4. In terms of reproducing the absolute binding free energies, the binding free energies estimated by including the truncated-NMA entropies based on the MD trajectories (ΔGnmode_md_9Å) give the lowest average absolute deviations against the experimental data among all the tested strategies for both MM/GBSA and MM/PBSA. Although the inclusion of the truncated NMA based on the MD trajectories (ΔGnmode_md_9Å) for a relatively high dielectric constant gave the overall best result and the lowest average absolute deviations against the experimental data (for the ff03 force field), it needs too much computational time. Alternatively, considering that the interaction entropy method does not incur any additional computational cost and can give comparable (at high dielectric constant, εin = 4) or even better (at low dielectric constant, εin = 1-2) results than the truncated-NMA entropy (ΔGnmode_md_9Å), the interaction entropy approach is recommended to estimate the entropic component for MM/GBSA and MM/PBSA based on MD trajectories, especially for a diverse dataset. Furthermore, we compared the predictions of MM/GBSA with six different AMBER force fields. The results show that the ff03 force field (ff03 for proteins and gaff with AM1-BCC charges for ligands) performs the best, but the predictions given by the tested force fields are comparable, implying that the MM/GBSA predictions are not very sensitive to force fields.

Self-Consistent Framework Connecting Experimental Proxies of Protein Dynamics with Configurational Entropy

The recently developed NMR techniques enable estimation of protein configurational entropy change from the change in the average methyl order parameters. This experimental observable, however, does not directly measure the contribution of intramolecular couplings, protein main-chain motions, or angular dynamics. Here, we carry out a self-consistent computational analysis of the impact of these missing contributions on an extensive set of molecular dynamics simulations of different proteins undergoing binding. Specifically, we compare the configurational entropy change in protein complex formation as obtained by the maximum information spanning tree approximation (MIST), which treats the above entropy contributions directly, and the change in the average NMR methyl and NH order parameters. Our parallel implementation of MIST allows us to treat hard angular degrees of freedom as well as couplings up to full pairwise order explicitly, while still involving a high degree of sampling and tackling molecules of biologically relevant sizes. First, we demonstrate a remarkably strong linear relationship between the total configurational entropy change and the average change in both methyl and backbone-NH order parameters. Second, in contrast to canonical assumptions, we show that the main-chain and angular terms contribute significantly to the overall configurational entropy change and also scale linearly with it. Consequently, linear models starting from the average methyl order parameters are able to capture the contribution of main-chain and angular terms well. After applying the quantum-mechanical harmonic oscillator entropy formalism, we establish a similarly strong linear relationship for X-ray crystallographic B-factors. Finally, we demonstrate that the observed linear relationships remain robust against drastic undersampling and argue that they reflect an intrinsic property of compact proteins. Despite their remarkable strength, however, the above linear relationships yield estimates of configurational entropy change whose accuracy appears to be sufficient for qualitative applications only.

Assessing the stability of free-energy perturbation calculations by performing variations in the method

We have calculated relative binding affinities for eight tetrafluorophenyl-triazole-thiogalactoside inhibitors of galectin-3 with the alchemical free-energy perturbation approach. We obtain a mean absolute deviation from experimental estimates of only 2-3 kJ/mol and a correlation coefficient (R²) of 0.5-0.8 for seven relative affinities spanning a range of up to 11 kJ/mol. We also studied the effect of using different methods to calculate the charges of the inhibitor and different sizes of the perturbed group (the atoms that are described by soft-core potentials and are allowed to have differing coordinates). However, the various approaches gave rather similar results and it is not possible to point out one approach as consistently and significantly better than the others. Instead, we suggest that such small and reasonable variations in the computational method can be used to check how stable the calculated results are and to obtain a more accurate estimate of the uncertainty than if performing only one calculation with a single computational setup.

The Dynameomics Entropy Dictionary: A Large-Scale Assessment of Conformational Entropy across Protein Fold Space

Molecular dynamics (MD) simulations contain considerable information with regard to the motions and fluctuations of a protein, the magnitude of which can be used to estimate conformational entropy. Here we survey conformational entropy across protein fold space using the Dynameomics database, which represents the largest existing data set of protein MD simulations for representatives of essentially all known protein folds. We provide an overview of MD-derived entropies accounting for all possible degrees of dihedral freedom on an unprecedented scale. Although different side chains might be expected to impose varying restrictions on the conformational space that the backbone can sample, we found that the backbone entropy and side chain size are not strictly coupled. An outcome of these analyses is the Dynameomics Entropy Dictionary, the contents of which have been compared with entropies derived by other theoretical approaches and experiment. As might be expected, the conformational entropies scale linearly with the number of residues, demonstrating that conformational entropy is an extensive property of proteins. The calculated conformational entropies of folding agree well with previous estimates. Detailed analysis of specific cases identifies deviations in conformational entropy from the average values that highlight how conformational entropy varies with sequence, secondary structure, and tertiary fold. Notably, α-helices have lower entropy on average than do β-sheets, and both are lower than coil regions.

Conformational Entropy of Intrinsically Disordered Proteins from Amino Acid Triads

This work quantitatively characterizes intrinsic disorder in proteins in terms of sequence composition and backbone conformational entropy. Analysis of the normalized relative composition of the amino acid triads highlights a distinct boundary between globular and disordered proteins. The conformational entropy is calculated from the dihedral angles of the middle amino acid in the amino acid triad for the conformational ensemble of the globular, partially and completely disordered proteins relative to the non-redundant database. Both Monte Carlo (MC) and Molecular Dynamics (MD) simulations are used to characterize the conformational ensemble of the representative proteins of each group. The results show that the globular proteins span approximately half of the allowed conformational states in the Ramachandran space, while the amino acid triads in disordered proteins sample the entire range of the allowed dihedral angle space following Flory’s isolated-pair hypothesis. Therefore, only the sequence information in terms of the relative amino acid triad composition may be sufficient to predict protein disorder and the backbone conformational entropy, even in the absence of well-defined structure. The predicted entropies are found to agree with those calculated using mutual information expansion and the histogram method.

Loss of conformational entropy in protein folding calculated using realistic ensembles and its implications for NMR-based calculations

The loss of conformational entropy is a major contribution in the thermodynamics of protein folding. However, accurate determination of the quantity has proven challenging. We calculate this loss using molecular dynamic simulations of both the native protein and a realistic denatured state ensemble. For ubiquitin, the total change in entropy is TΔSTotal = 1.4 kcal⋅mol(-1) per residue at 300 K with only 20% from the loss of side-chain entropy. Our analysis exhibits mixed agreement with prior studies because of the use of more accurate ensembles and contributions from correlated motions. Buried side chains lose only a factor of 1.4 in the number of conformations available per rotamer upon folding (ΩU/ΩN). The entropy loss for helical and sheet residues differs due to the smaller motions of helical residues (TΔShelix-sheet = 0.5 kcal⋅mol(-1)), a property not fully reflected in the amide N-H and carbonyl C=O bond NMR order parameters. The results have implications for the thermodynamics of folding and binding, including estimates of solvent ordering and microscopic entropies obtained from NMR.

Analysis of factors influencing hydration site prediction based on molecular dynamics simulations

Water contributes significantly to the binding of small molecules to proteins in biochemical systems. Molecular dynamics (MD) simulation based programs such as WaterMap and WATsite have been used to probe the locations and thermodynamic properties of hydration sites at the surface or in the binding site of proteins generating important information for structure-based drug design. However, questions associated with the influence of the simulation protocol on hydration site analysis remain. In this study, we use WATsite to investigate the influence of factors such as simulation length and variations in initial protein conformations on hydration site prediction. We find that 4 ns MD simulation is appropriate to obtain a reliable prediction of the locations and thermodynamic properties of hydration sites. In addition, hydration site prediction can be largely affected by the initial protein conformations used for MD simulations. Here, we provide a first quantification of this effect and further indicate that similar conformations of binding site residues (RMSD < 0.5 Å) are required to obtain consistent hydration site predictions.

Sampling Assessment for Molecular Simulations Using Conformational Entropy Calculations

The extent and significance of conformational sampling is a major factor determining the reliability of long-scale molecular simulations of large and flexible biomolecules. Although several methods have been proposed to quantify the effective sample size of molecular simulations by transforming root mean squared distances between pairs of configurations into statistical/probabilistic quantities, there is still no standard technique for measuring the size of sampling. In this work, we study conformational entropy (Sconform) as a purely informational and probabilistic measure of sampling that does not require the adoption of any clustering protocol or distance metric between configurations. In addition Sconform, which is calculated from the probability mass functions associated with discretized dihedral angles, offers other potential advantages for sampling assessment (e.g., global character, thermodynamic significance, relationship with internal degrees of freedom, etc.). The utility of Sconform in sampling assessment is illustrated by carrying out test calculations on configurations produced by two extended molecular dynamics simulations, namely, a 2.0 μs trajectory of a highly flexible 17-residue peptide and the trajectory data set of the 1.0 ms bovine pancreatic trypsin inhibitor simulation provided by the D. E. Shaw research group.

Correlation as a determinant of configurational entropy in supramolecular and protein systems

For biomolecules in solution, changes in configurational entropy are thought to contribute substantially to the free energies of processes like binding and conformational change. In principle, the configurational entropy can be strongly affected by pairwise and higher-order correlations among conformational degrees of freedom. However, the literature offers mixed perspectives regarding the contributions that changes in correlations make to changes in configurational entropy for such processes. Here we take advantage of powerful techniques for simulation and entropy analysis to carry out rigorous in silico studies of correlation in binding and conformational changes. In particular, we apply information-theoretic expansions of the configurational entropy to well-sampled molecular dynamics simulations of a model host–guest system and the protein bovine pancreatic trypsin inhibitor. The results bear on the interpretation of NMR data, as they indicate that changes in correlation are important determinants of entropy changes for biologically relevant processes and that changes in correlation may either balance or reinforce changes in first-order entropy. The results also highlight the importance of main-chain torsions as contributors to changes in protein configurational entropy. As simulation techniques grow in power, the mathematical techniques used here will offer new opportunities to answer challenging questions about complex molecular systems.