Multibody local approximation: application to conformational entropy calculations on biomolecules

Toward Reliable and Insightful Entropy Calculations on Flexible Molecules

The absolute entropy of a flexible molecule can be approximated by the sum of a rigid-rotor-harmonic-oscillator (RRHO) entropy and a Gibbs-Shannon entropy associated to the Boltzmann distribution for the occupation of the conformational energy levels. Herein, we show that such partitioning, which has received renewed interest, leads to accurate entropies of single molecules of increasing size provided that the conformational part is estimated by means of a set of discretization and expansion techniques that are able to capture the significant correlation effects among the torsional motions. To ensure a reliable entropy estimation, we rely on extensive sampling as that produced by classical molecular dynamics simulations on the microsecond time scale, which is currently affordable for small- and medium-sized molecules. According to test calculations, the gas-phase entropy of simple organic molecules is predicted with a mean unsigned error of 0.9 cal/(mol K) when the RRHO entropies are computed at the B3LYP-D3/cc-pVTZ level. Remarkably, the same protocol gives small errors [<1 cal/(mol K)] for the extremely flexible linear alkane molecules (C_nH_2n+2, n = 14, 16, and 18). Similarly, we obtain well-converged entropies for a more challenging test of drug molecules, which exhibit more pronounced correlation effects. We also perform equivalent entropy calculations on a 76 amino acid protein, ubiquitin, by taking advantage of the cutoff-dependent formulation of an expansion technique (correlation-consistent multibody local approximation, CC-MLA), which incorporates genuine correlation effects among the neighboring dihedral angles. Moreover, we show that insightful descriptors of the coupled torsional motions can be obtained with the CC-MLA approach.

Understanding the Conformational Properties of Fluorinated Polypeptides: Molecular Modelling of Unguisin A

In this work, we investigate the conformational properties of unguisin A, a natural macrocyclic heptapeptide that incorporates a γ-aminobutyric acid (Gaba), and four of its difluorinated stereoisomers at the Gaba residue. According to nuclear magnetic resonance (NMR) experiments, their secondary structure depends dramatically on the stereochemistry of the fluorinated carbon atoms. However, many molecular details of the structure and flexibility of these systems remain unknown, so that a rationale of the conformational changes induced by the fluorine atoms in the macrocycle is still missing. To fill this gap, we apply enhanced molecular dynamics (MD) techniques to explore the peptide conformational space in dimethyl sulfoxide solution followed by 4-8 μs of conventional MD simulations that provide extensive equilibrium sampling. The simulations, which compare reasonably well with the NMR-based observations, show that the secondary structure of the macrocycle is altered substantially upon fluorination, except for the (S,S) diastereomer. It also turns out that the conformations of the fluorinated peptides are visited during the enhanced MD simulation of natural unguisin A, suggesting thus that conformations accessible to the unsubstituted macrocyclic peptide may be selected by fluorination. Therefore, computational characterization of the macrocyclic peptides could be helpful in the rational design of stereoselective fluorinated peptides with fine-tuned conformation and activity.

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.

Entropy of Simulated Liquids Using Multiscale Cell Correlation

Accurately calculating the entropy of liquids is an important goal, given that many processes take place in the liquid phase. Of almost equal importance is understanding the values obtained. However, there are few methods that can calculate the entropy of such systems, and fewer still to make sense of the values obtained. We present our multiscale cell correlation (MCC) method to calculate the entropy of liquids from molecular dynamics simulations. The method uses forces and torques at the molecule and united-atom levels and probability distributions of molecular coordinations and conformations. The main differences with previous work are the consistent treatment of the mean-field cell approximation to the approriate degrees of freedom, the separation of the force and torque covariance matrices, and the inclusion of conformation correlation for molecules with multiple dihedrals. MCC is applied to a broader set of 56 important industrial liquids modeled using the Generalized AMBER Force Field (GAFF) and Optimized Potentials for Liquid Simulations (OPLS) force fields with 1.14*CM1A charges. Unsigned errors versus experimental entropies are 8.7 J K - 1 mol - 1 for GAFF and 9.8 J K - 1 mol - 1 for OPLS. This is significantly better than the 2-Phase Thermodynamics method for the subset of molecules in common, which is the only other method that has been applied to such systems. MCC makes clear why the entropy has the value it does by providing a decomposition in terms of translational and rotational vibrational entropy and topographical entropy at the molecular and united-atom levels.

Affinity Calculations of Cyclodextrin Host-Guest Complexes: Assessment of Strengths and Weaknesses of End-Point Free Energy Methods

The end-point methods like MM/PBSA or MM/GBSA estimate the free energy of a biomolecule by combining its molecular mechanics energy with solvation free energy and entropy terms. On the one hand, their performance largely depends on the particular system of interest, and despite numerous attempts to improve their reliability that have resulted in many variants, there is still no clear alternative to improve their accuracy. On the other hand, the relatively small cyclodextrin host-guest complexes, for which high-quality binding calorimetric data are usually available, are becoming reference models for testing the accuracy of free energy methods. In this work, we further assess the performance of various MM/PBSA-like approaches as applied to cyclodextrin complexes. To this end, we select a set of complexes between β-cyclodextrin and 57 small organic molecules that has been previously studied with the binding energy distribution analysis method in combination with an implicit solvent model ( Wickstrom, L.; He, P.; Gallicchio, E.; Levy, R. M. J. Chem. Theory Comput. 2013 , 9 , 3136 - 3150 ). For each complex, a conventional 1.0 μs molecular dynamics simulation in explicit solvent is performed. Then we employ semiempirical quantum chemical calculations, several variants of the MM-PB(GB)SA methods, entropy estimations, etc., to assess the reliability of the end-point affinity calculations. The best end-point protocol in this study, which combines DFTB3 energies with entropy corrections, yields estimations of the binding free energies that still have substantial errors (RMSE = 2.2 kcal/mol), but it exhibits a good prediction capacity in terms of ligand ranking ( R² = 0.66) that is close to or even better than that of rigorous free energy methodologies. Our results can be helpful to discriminate between the intrinsic limitations of the end-point methods and other sources of error, such as the underlying energy and continuum solvation methods.

Probing the functional conformations of an atypical proline-rich fusion peptide

Simulations confirm a propensity for extended and solvent exposed conformations of the p15 fusion peptide capable of membrane targeting.

Ligand Strain and Entropic Effects on the Binding of Macrocyclic and Linear Inhibitors: Molecular Modeling of Penicillopepsin Complexes

Using extensive molecular dynamics simulations, we investigate the structure and dynamics of the complexes formed between penicillopepsin and two peptidomimetic inhibitors: a linear compound, isovaleryl(P₄)-valine(P₃)-asparagine(P₂)-leucine(P₁)-phosphonate-phenylalanine(P₁'), and its macrocylic analog that includes a methylene bridge between the Asn(P₂) and Phe(P₁') side chains. The macrocyclic inhibitor, which has a 420-fold stronger affinity than that of the acyclic one, has been considered to lower the entropic penalty for binding. To better understand this binding preference, the solution structure of the inhibitors is studied by molecular dynamics simulations. Subsequently, we assess the influence of the enzyme/inhibitor contacts, the enzyme-induced inhibitor strain, the variation of the ligand configurational entropy and the enzyme reorganization by combining molecular-mechanics Poisson-Boltzmann surface area and normal mode calculations with conformational entropy calculations. We find that there is no relevant entropic stabilization on the binding of the cyclic inhibitor with respect to the acyclic analog because the methylene bridge does not reduce appreciably the conformational flexibility of the free inhibitor. The most important factors explaining the stronger affinity of the macrocyclic inhibitor are the conformational filtering and the lower ligand strain induced by the methylene bridge.

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.

Conformational and entropy analyses of extended molecular dynamics simulations of α-, β- and γ-cyclodextrins and of the β-cyclodextrin/nabumetone complex

The conformational entropies of cyclodextrins and of the β-CD/nabumetone complex are assessed by means of extensive MD simulations.

Macromolecular Entropy Can Be Accurately Computed from Force

A method is presented to evaluate a molecule's entropy from the atomic forces calculated in a molecular dynamics simulation. Specifically, diagonalization of the mass-weighted force covariance matrix produces eigenvalues which in the harmonic approximation can be related to vibrational frequencies. The harmonic oscillator entropies of each vibrational mode may be summed to give the total entropy. The results for a series of hydrocarbons, dialanine and a β hairpin are found to agree much better with values derived from thermodynamic integration than results calculated using quasiharmonic analysis. Forces are found to follow a harmonic distribution more closely than coordinate displacements and better capture the underlying potential energy surface. The method's accuracy, simplicity, and computational similarity to quasiharmonic analysis, requiring as input force trajectories instead of coordinate trajectories, makes it readily applicable to a wide range of problems.

Entropy in bimolecular simulations: A comprehensive review of atomic fluctuations-based methods

Entropy of binding constitutes a major, and in many cases a detrimental, component of the binding affinity in biomolecular interactions. While the enthalpic part of the binding free energy is easier to calculate, estimating the entropy of binding is further more complicated. A precise evaluation of entropy requires a comprehensive exploration of the complete phase space of the interacting entities. As this task is extremely hard to accomplish in the context of conventional molecular simulations, calculating entropy has involved many approximations. Most of these golden standard methods focused on developing a reliable estimation of the conformational part of the entropy. Here, we review these methods with a particular emphasis on the different techniques that extract entropy from atomic fluctuations. The theoretical formalisms behind each method is explained highlighting its strengths as well as its limitations, followed by a description of a number of case studies for each method. We hope that this brief, yet comprehensive, review provides a useful tool to understand these methods and realize the practical issues that may arise in such calculations.

Dihedral angle entropy measures for intrinsically disordered proteins

Protein stability is based on a delicate balance between energetic and entropic factors. Intrinsically disordered proteins (IDPs) interacting with a folded partner protein in the act of binding can order the IDP to form the correct functional interface by decrease in the overall free energy. In this work, we evaluate the part of the entropic cost of ordering an IDP arising from their dihedral states. The IDP studied is a leucine zipper dimer that we simulate with molecular dynamics and find that it does show disorder in six phi and psi dihedral angles of the N terminal sequence of one monomer. Essential to ascertain is the degree of disorder in the IDP, and we do so by considering the entire, discretized probability distribution function of N dihedrals with M conformers per dihedral. A compositional clustering method is introduced, whereby the NS = N(M) states are formed from the Cartesian product of each dihedral's conformational space. Clustering is carried out with a version of a k-means algorithm that accounts for the circular nature of dihedral angles. For the 12 dihedrals each found to have three conformers, among the resulting 531441 states, their populations show that the first 100 (500) most populated states account for ∼65% (∼90%) of the entire population, indicating that there are strong dependencies among the dihedrals' conformations. These state populations are used to evaluate a Kullback-Leibler divergence entropy measure and obtain the dihedral configurational entropy S. At 300 K, TS ∼ 3 kcal/mol, showing that IDP entropy, while roughly half that would be expected from independently distributed dihedrals, can be a decisive contributor to the free energy of this IDP binding and ordering.

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.

CENCALC: a computational tool for conformational entropy calculations from molecular simulations

We present the CENCALC software that has been designed to estimate the conformational entropy of single molecules from extended Molecular Dynamics (MD) simulations in the gas-phase or in solution. CENCALC uses both trajectory coordinates and topology information in order to characterize the conformational states of the molecule of interest by discretizing the time evolution of internal rotations. The implemented entropy methods are based on the mutual information expansion, which is built upon the converged probability density functions of the individual torsion angles, pairs of torsions, triads, and so on. Particularly, the correlation-corrected multibody local approximation selects an optimum cutoff in order to retrieve the maximum amount of genuine correlation from a given MD trajectory. We illustrate these capabilities by carrying out conformational entropy calculations for a decapeptide molecule either in its unbound form or in complex with a metalloprotease enzyme. CENCALC is distributed under the GNU public license at http://sourceforge.net/projects/cencalc/.