Balanced and Bias-Corrected Computation of Conformational Entropy Differences for Molecular Trajectories

Total free energy analysis of fully hydrated proteins

The total free energy of a hydrated biomolecule and its corresponding decomposition of energy and entropy provides detailed information about regions of thermodynamic stability or instability. The free energies of four hydrated globular proteins with different net charges are calculated from a molecular dynamics simulation, with the energy coming from the system Hamiltonian and entropy using multiscale cell correlation. Water is found to be most stable around anionic residues, intermediate around cationic and polar residues, and least stable near hydrophobic residues, especially when more buried, with stability displaying moderate entropy-enthalpy compensation. Conversely, anionic residues in the proteins are energetically destabilized relative to singly solvated amino acids, while trends for other residues are less clear-cut. Almost all residues lose intraresidue entropy when in the protein, enthalpy changes are negative on average but may be positive or negative, and the resulting overall stability is moderate for some proteins and negligible for others. The free energy of water around single amino acids is found to closely match existing hydrophobicity scales. Regarding the effect of secondary structure, water is slightly more stable around loops, of intermediate stability around β strands and turns, and least stable around helices. An interesting asymmetry observed is that cationic residues stabilize a residue when bonded to its N-terminal side but destabilize it when on the C-terminal side, with a weaker reversed trend for anionic residues.

Decoupled coordinates for machine learning-based molecular fragment linking

Recent developments in machine learning-based molecular fragment linking have demonstrated the importance of informing the generation process with structural information specifying the relative orientation of the fragments to be linked. However, such structural information has so far not been provided in the form of a complete relative coordinate system. We present a decoupled coordinate system consisting of bond lengths, bond angles and torsion angles, and show that it is complete. By incorporating this set of coordinates in a linker generation framework, we show that it has a significant impact on the quality of the generated linkers. To elucidate the advantages of such a coordinate system, we investigate the amount of reliable information within the different types of degrees of freedom using both detailed ablation studies and an information-theoretical analysis. The presented benefits suggest the application of a complete and decoupled relative coordinate system as a standard good practice in linker design.

Cyclic peptides: backbone rigidification and capability of mimicking motifs at protein-protein interfaces

Cyclization is commonly employed in efforts to improve the target binding affinity of peptide-based probes and therapeutics. Many structural motifs have been identified at protein-protein interfaces and provide promising targets for inhibitor design using cyclic peptides. Cyclized peptides are generally assumed to be rigidified relative to their linear counterparts. This rigidification potentially pre-organizes the molecules to interact properly with their targets. However, the actual impact of cyclization on, for example, peptide configurational entropy, is currently poorly understood in terms of both its magnitude and molecular-level origins. Moreover, even with thousands of desired structural motifs at hand, it is currently not possible to a priori identify the ones that are most promising to mimic using cyclic peptides nor to select the ideal linker length. Instead, labor-intensive chemical synthesis and experimental characterization of various cyclic peptide designs are required, in hopes of finding one with improved target affinity. Herein, using molecular dynamics simulations of polyglycines, we elucidated how head-to-tail cyclization impacts peptide backbone dihedral entropy and developed a simple strategy to rapidly screen for structures that can be reliably mimicked by preorganized cyclic peptides. As expected, cyclization generally led to a reduction in backbone dihedral entropy; notably, however, this effect was minimal when the length of polyglycines was >9 residues. We also found that the reduction in backbone dihedral entropy upon cyclization of small polyglycine peptides does not result from more restricted distributions of the dihedrals; rather, it was the correlations between specific dihedrals that caused the decrease in configurational entropy in the cyclic peptides. Using our comprehensive cyclo-Gn structural ensembles, we obtained a holistic picture of what conformations are accessible to cyclic peptides. Using "hot loops" recently identified at protein-protein interfaces as an example, we provide clear guidelines for choosing the "easiest" hot loops for cyclic peptides to mimic and for identifying appropriate cyclic peptide lengths. In conclusion, our results provide an understanding of the thermodynamics and structures of this interesting class of molecules. This information should prove particularly useful for designing cyclic peptide inhibitors of protein-protein interactions.

Application and Comprehensive Analysis of Neighbor Approximated Information Theoretic Configurational Entropy Methods to Protein-Ligand Binding Cases

The binding entropy is an important thermodynamic quantity which has numerous applications in studies of the biophysical process, and configurational entropy is often one of the major contributors in it. Therefore, its accurate estimation is important, though it is challenging mostly due to sampling limitations, anharmonicity, and multimodality of atomic fluctuations. The present work reports a Neighbor Approximated Maximum Information Spanning Tree (A-MIST) method for conformational entropy and presents its performance and computational advantage over conventional Mutual Information Expansion (MIE) and Maximum Information Spanning Tree (MIST) for two protein-ligand binding cases: indirubin-5-sulfonate to Plasmodium falciparum Protein Kinase 5 (PfPK5) and P. falciparum RON2-peptide to P. falciparum Apical Membrane Antigen 1 (PfAMA1). Important structural regions considering binding configurational entropy are identified, and physical origins for such are discussed. A thorough performance evaluation is done of a set of four entropy estimators (Maximum Likelihood (ML), Miller-Madow (MM), Chao-Shen (CS), and James and Stein shrinkage (JS)) with known varying degrees of sensitivity of the entropy estimate on the extent of sampling, each with two schemes for discretization of fluctuation data of Degrees of Freedom (DFs) to estimate Probability Density Functions (PDFs). Our comprehensive evaluation of influences of variations of parameters shows Neighbor Approximated MIE (A-MIE) outperforms MIE in terms of convergence and computational efficiency. In the case of A-MIE/MIE, results are sensitive to the choice of root atoms, graph search algorithm used for the Bond-Angle-Torsion (BAT) conversion, and entropy estimator, while A-MIST/MIST are not. A-MIST yields binding entropy within 0.5 kcal/mol of MIST with only 20-30% computation. Moreover, all these methods have been implemented in an OpenMP/MPI hybrid parallel C++11 code, and also a python package for data preprocessing and entropy contribution analysis is developed and made available. A comparative analysis of features of current implementation and existing tools is also presented.

Entropy of Proteins Using Multiscale Cell Correlation

A new multiscale method is presented to calculate the entropy of proteins from molecular dynamics simulations. Termed Multiscale Cell Correlation (MCC), the method decomposes the protein into sets of rigid-body units based on their covalent-bond connectivity at three levels of hierarchy: molecule, residue, and united atom. It evaluates the vibrational and topographical entropy from forces, torques, and dihedrals at each level, taking into account correlations between sets of constituent units that together make up a larger unit at the coarser length scale. MCC gives entropies in close agreement with normal-mode analysis and smaller than those using quasiharmonic analysis as well as providing much faster convergence. Moreover, MCC provides an insightful decomposition of entropy at each length scale and for each type of amino acid according to their solvent exposure and whether they are terminal residues. While the residue entropy depends weakly on solvent exposure, there is greater variation in entropy components for larger, more polar amino acids, which have increased conformational entropy but reduced vibrational entropy with greater solvent exposure.

Configurational Entropy Components and Their Contribution to Biomolecular Complex Formation

Configurational entropy change is a central constituent of the free energy change in noncovalent interactions between biomolecules. Due to both experimental and computational limitations, however, the impact of individual contributions to configurational entropy change remains underexplored. Here, we develop a novel, fully analytical framework to dissect the configurational entropy change of binding into contributions coming from molecular internal and external degrees of freedom. Importantly, this framework accounts for all coupled and uncoupled contributions in the absence of an external field. We employ our parallel implementation of the maximum information spanning tree algorithm to provide a comprehensive numerical analysis of the importance of the individual contributions to configurational entropy change on an extensive set of molecular dynamics simulations of protein binding processes. Contrary to commonly accepted assumptions, we show that different coupling terms contribute significantly to the overall configurational entropy change. Finally, while the magnitude of individual terms may be largely unpredictable a priori, the total configurational entropy change can be well approximated by rescaling the sum of uncoupled contributions from internal degrees of freedom only, providing support for NMR-based approaches for configurational entropy change estimation.

An NMR Confirmation for Increased Folded State Entropy Following Loop Truncation

Previous studies conducted on flexible loop regions in proteins revealed that the energetic consequences of changing loop length predominantly arise from the entropic cost of ordering a loop during folding. However, in an earlier study of human acylphosphatase (hmAcP) using experimental and computational approaches, we showed that thermodynamic stabilization upon loop truncation can be attributed mainly to the increased entropy of the folded state. Here, using 15N NMR spectroscopy, we studied the effect of loop truncation on hmAcP backbone dynamics on the picosecond-nanosecond timescale with the aim of confirming the effect of folded state entropy on protein stability. NMR-relaxation-derived N-H squared generalized order parameters reveal that loop truncation results in a significant increase in protein conformational flexibility. Comparison of these results with previously acquired all-atom molecular dynamics simulation, analyzed here in terms of squared generalized NMR order parameters, demonstrates general agreement between the two methods. The NMR study not only provides direct evidence for the enhanced conformational entropy of the folded state of hmAcP upon loop truncation but also gives a quantitative measure of the observed effects.

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.

Entropy Transfer between Residue Pairs and Allostery in Proteins: Quantifying Allosteric Communication in Ubiquitin

It has recently been proposed by Gunasakaran et al. that allostery may be an intrinsic property of all proteins. Here, we develop a computational method that can determine and quantify allosteric activity in any given protein. Based on Schreiber's transfer entropy formulation, our approach leads to an information transfer landscape for the protein that shows the presence of entropy sinks and sources and explains how pairs of residues communicate with each other using entropy transfer. The model can identify the residues that drive the fluctuations of others. We apply the model to Ubiquitin, whose allosteric activity has not been emphasized until recently, and show that there are indeed systematic pathways of entropy and information transfer between residues that correlate well with the activities of the protein. We use 600 nanosecond molecular dynamics trajectories for Ubiquitin and its complex with human polymerase iota and evaluate entropy transfer between all pairs of residues of Ubiquitin and quantify the binding susceptibility changes upon complex formation. We explain the complex formation propensities of Ubiquitin in terms of entropy transfer. Important residues taking part in allosteric communication in Ubiquitin predicted by our approach are in agreement with results of NMR relaxation dispersion experiments. Finally, we show that time delayed correlation of fluctuations of two interacting residues possesses an intrinsic causality that tells which residue controls the interaction and which one is controlled. Our work shows that time delayed correlations, entropy transfer and causality are the required new concepts for explaining allosteric communication in proteins.

On the coupling between the dynamics of protein and water

The solvation entropy of flexible protein regions is higher than that of rigid regions and contributes differently to the overall thermodynamic stability.

Vibrational entropy of a protein: large differences between distinct conformations

In this article, it is investigated whether vibrational entropy (VE) is an important contribution to the free energy of globular proteins at ambient conditions. VE represents the major configurational-entropy contribution of these proteins. By definition, it is an average of the configurational entropies of the protein within single minima of the energy landscape, weighted by their occupation probabilities. Its large part originates from thermal motion of flexible torsion angles giving rise to the finite peak widths observed in torsion angle distributions. While VE may affect the equilibrium properties of proteins, it is usually neglected in numerical calculations as its consideration is difficult. Moreover, it is sometimes believed that all well-packed conformations of a globular protein have similar VE anyway. Here, we measure explicitly the VE for six different conformations from simulation data of a test protein. Estimates are obtained using the quasi-harmonic approximation for three coordinate sets, Cartesian, bond-angle-torsion (BAT), and a new set termed rotamer-degeneracy lifted BAT coordinates by us. The new set gives improved estimates as it overcomes a known shortcoming of the quasi-harmonic approximation caused by multiply populated rotamer states, and it may serve for VE estimation of macromolecules in a very general context. The obtained VE values depend considerably on the type of coordinates used. However, for all coordinate sets we find large entropy differences between the conformations, of the order of the overall stability of the protein. This result may have important implications on the choice of free energy expressions used in software for protein structure prediction, protein design, and NMR refinement.

Estimating Translational and Orientational Entropies Using the k-Nearest Neighbors Algorithm

Inhomogeneous fluid solvation theory (IFST) and free energy perturbation (FEP) calculations were performed for a set of 20 solutes to compute the hydration free energies. We identify the weakness of histogram methods in computing the IFST hydration entropy by showing that previously employed histogram methods overestimate the translational and orientational entropies and thus underestimate their contribution to the free energy by a significant amount. Conversely, we demonstrate the accuracy of the k-nearest neighbors (KNN) algorithm in computing these translational and orientational entropies. Implementing the KNN algorithm within the IFST framework produces a powerful method that can be used to calculate free-energy changes for large perturbations. We introduce a new KNN approach to compute the total solute-water entropy with six degrees of freedom, as well as the translational and orientational contributions. However, results suggest that both the solute-water and water-water entropy terms are significant and must be included. When they are combined, the IFST and FEP hydration free energies are highly correlated, with an R(2) of 0.999 and a mean unsigned difference of 0.9 kcal/mol. IFST predictions are also highly correlated with experimental hydration free energies, with an R(2) of 0.997 and a mean unsigned error of 1.2 kcal/mol. In summary, the KNN algorithm is shown to yield accurate estimates of the combined translational-orientational entropy and the novel approach of combining distance metrics that is developed here could be extended to provide a powerful method for entropy estimation in numerous contexts.

Spatial Decomposition of Translational Water-Water Correlation Entropy in Binding Pockets

A number of computational tools available today compute the thermodynamic properties of water at surfaces and in binding pockets by using inhomogeneous solvation theory (IST) to analyze explicit-solvent simulations. Such methods enable qualitative spatial mappings of both energy and entropy around a solute of interest and can also be applied quantitatively. However, the entropy estimates of existing methods have, to date, been almost entirely limited to the first-order terms in the IST’s entropy expansion. These first-order terms account for localization and orientation of water molecules in the field of the solute but not for the modification of water–water correlations by the solute. Here, we present an extension of the Grid Inhomogeneous Solvation Theory (GIST) approach which accounts for water–water translational correlations. The method involves rewriting the two-point density of water in terms of a conditional density and utilizes the efficient nearest-neighbor entropy estimation approach. Spatial maps of this second order term, for water in and around the synthetic host cucurbit[7]uril and in the binding pocket of the enzyme Factor Xa, reveal mainly negative contributions, indicating solute-induced water–water correlations relative to bulk water; particularly strong signals are obtained for sites at the entrances of cavities or pockets. This second-order term thus enters with the same, negative, sign as the first order translational and orientational terms. Numerical and convergence properties of the methodology are examined.

Shortening a loop can increase protein native state entropy

Protein loops are essential structural elements that influence not only function but also protein stability and folding rates. It was recently reported that shortening a loop in the AcP protein may increase its native state conformational entropy. This effect on the entropy of the folded state can be much larger than the lower entropic penalty of ordering a shorter loop upon folding, and can therefore result in a more pronounced stabilization than predicted by polymer model for loop closure entropy. In this study, which aims at generalizing the effect of loop length shortening on native state dynamics, we use all-atom molecular dynamics simulations to study how gradual shortening a very long or solvent-exposed loop region in four different proteins can affect their stability. For two proteins, AcP and Ubc7, we show an increase in native state entropy in addition to the known effect of the loop length on the unfolded state entropy. However, for two permutants of SH3 domain, shortening a loop results only with the expected change in the entropy of the unfolded state, which nicely reproduces the observed experimental stabilization. Here, we show that an increase in the native state entropy following loop shortening is not unique to the AcP protein, yet nor is it a general rule that applies to all proteins following the truncation of any loop. This modification of the loop length on the folded state and on the unfolded state may result with a greater effect on protein stability.

Replica-exchange molecular dynamics simulations of cellulose solvated in water and in the ionic liquid 1-butyl-3-methylimidazolium chloride

Ionic liquids have become a popular solvent for cellulose pretreatment in biorefineries due to their efficiency in dissolution and their reusability. Understanding the interactions between cations, anions, and cellulose is key to the development of better solvents and the improvement of pretreatment conditions. While previous studies described the interactions between ionic liquids and cellulose fibers, shedding light on the initial stages of the cellulose dissolution process, we study the end state of that process by exploring the structure and dynamics of a single cellulose decamer solvated in 1-butyl-3-methyl-imidazolium chloride (BmimCl) and in water using replica-exchange molecular dynamics. In both solvents, global structural features of the cellulose chain are similar. However, analyses of local structural properties show that cellulose explores greater conformational variability in the ionic liquid than in water. For instance, in BmimCl the cellulose intramolecular hydrogen bond O3H'···O5 is disrupted more often resulting in greater flexibility of the solute. Our results indicate that the cellulose chain is more dynamic in BmimCl than in water, which may play a role in the favorable dissolution of cellulose in the ionic liquid. Calculation of the configurational entropy of the cellulose decamer confirms its higher conformational flexibility in BmimCl than in water at elevated temperatures.

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.

Effect of subdomain interactions on methyl group dynamics in the hydrophobic core of villin headpiece protein

Thermostable villin headpiece protein (HP67) consists of the N-terminal subdomain (residues 10-41) and the autonomously folding C-terminal subdomain (residues 42-76) which pack against each other to form a structure with a unified hydrophobic core. The X-ray structures of the isolated C-terminal subdomain (HP36) and its counterpart in HP67 are very similar for the hydrophobic core residues. However, fine rearrangements of the free energy landscape are expected to occur because of the interactions between the two subdomains. We detect and characterize these changes by comparing the µs-ms time scale dynamics of the methyl-bearing side chains in isolated HP36 and in HP67. Specifically, we probe three hydrophobic side chains at the interface of the two subdomains (L42, V50, and L75) as well as at two residues far from the interface (L61 and L69). Solid-state deuteron NMR techniques are combined with computational modeling for the detailed characterization of motional modes in terms of their kinetic and thermodynamic parameters. The effect of interdomain interactions on side chain dynamics is seen for all residues but L75. Thus, changes in dynamics because of subdomain interactions are not confined to the site of perturbation. One of the main results is a two- to threefold increase in the value of the activation energies for the rotameric mode of motions in HP67 compared with HP36. Detailed analysis of configurational entropies and heat capacities complement the kinetic view of the degree of the disorder in the folded state.

Entropy-enthalpy transduction caused by conformational shifts can obscure the forces driving protein-ligand binding

Molecular dynamics simulations of unprecedented duration now can provide new insights into biomolecular mechanisms. Analysis of a 1-ms molecular dynamics simulation of the small protein bovine pancreatic trypsin inhibitor reveals that its main conformations have different thermodynamic profiles and that perturbation of a single geometric variable, such as a torsion angle or interresidue distance, can select for occupancy of one or another conformational state. These results establish the basis for a mechanism that we term entropy-enthalpy transduction (EET), in which the thermodynamic character of a local perturbation, such as enthalpic binding of a small molecule, is camouflaged by the thermodynamics of a global conformational change induced by the perturbation, such as a switch into a high-entropy conformational state. It is noted that EET could occur in many systems, making measured entropies and enthalpies of folding and binding unreliable indicators of actual thermodynamic driving forces. The same mechanism might also account for the high experimental variance of measured enthalpies and entropies relative to free energies in some calorimetric studies. Finally, EET may be the physical mechanism underlying many cases of entropy-enthalpy compensation.