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

Molecular Dynamics as a Tool for Virtual Ligand Screening

Rational drug design is essential for new drugs to emerge, especially when the structure of a target protein or nucleic acid is known. To that purpose, high-throughput virtual ligand screening campaigns aim at discovering computationally new binding molecules or fragments to modulate particular biomolecular interactions or biological activities, related to a disease process. The structure-based virtual ligand screening process primarily relies on docking methods which allow predicting the binding of a molecule to a biological target structure with a correct conformation and the best possible affinity. The docking method itself is not sufficient as it suffers from several and crucial limitations (lack of full protein flexibility information, no solvation and ion effects, poor scoring functions, and unreliable molecular affinity estimation).At the interface of computer techniques and drug discovery, molecular dynamics (MD) allows introducing protein flexibility before or after a docking protocol, refining the structure of protein-drug complexes in the presence of water, ions, and even in membrane-like environments, describing more precisely the temporal evolution of the biological complex and ranking these complexes with more accurate binding energy calculations. In this chapter, we describe the up-to-date MD, which plays the role of supporting tools in the virtual ligand screening (VS) process.Without a doubt, using docking in combination with MD is an attractive approach in structure-based drug discovery protocols nowadays. It has proved its efficiency through many examples in the literature and is a powerful method to significantly reduce the amount of required wet experimentations (Tarcsay et al, J Chem Inf Model 53:2990-2999, 2013; Barakat et al, PLoS One 7:e51329, 2012; De Vivo et al, J Med Chem 59:4035-4061, 2016; Durrant, McCammon, BMC Biol 9:71-79, 2011; Galeazzi, Curr Comput Aided Drug Des 5:225-240, 2009; Hospital et al, Adv Appl Bioinforma Chem 8:37-47, 2015; Jiang et al, Molecules 20:12769-12786, 2015; Kundu et al, J Mol Graph Model 61:160-174, 2015; Mirza et al, J Mol Graph Model 66:99-107, 2016; Moroy et al, Future Med Chem 7:2317-2331, 2015; Naresh et al, J Mol Graph Model 61:272-280, 2015; Nichols et al, J Chem Inf Model 51:1439-1446, 2011; Nichols et al, Methods Mol Biol 819:93-103, 2012; Okimoto et al, PLoS Comput Biol 5:e1000528, 2009; Rodriguez-Bussey et al, Biopolymers 105:35-42, 2016; Sliwoski et al, Pharmacol Rev 66:334-395, 2014).

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.

Non-bonding energy directed designing of HDAC2 inhibitors through molecular dynamics simulation

Designing an inhibitor having strong affinity in the active site pocket is the cherished goal of structure based drug designing. To achieve this, it is considerably important to predict which structural scaffold is better suited for change to increase affinity. We have explored five HDAC2 co-crystals having PDB ligand code-SHH (vorinostat), LLX, 20Y, IWX (BRD4884) and 6EZ (BRD7232). For analyzing protein-ligand interaction at an atomistic level, we have employed the NAMD molecular dynamics (MD) package. The obtained 100 ns long MD trajectories were subjected to quantitative estimations of non-bonding energies (NBEs) for inferring their interactions with the whole protein or its composite active site (CAS). In addition, relative ΔG_bind was calculated to rank the inhibitors. These inhibitors' NBEs reveal that the phenyl moieties are the major structural scaffold where modifications should be attempted. We designed new compounds (NCs) via introducing hydroxyl groups at 4,5 position of the phenyl moiety of 6EZ, called NC1. Improvement in NC1 further encouraged us for CAP modification by isochromane and isoindoline moieties in place of oxabicyclooctane in NC1, resulting in NC2 and NC3. We also explored trifluoromethyl oxadiazole in 6EZ (NC4 and NC5) and SHH (NC6 and NC7). This moiety acts as a ZBG in NC4 while acting as a part of the foot-pocket in the rest. NC2 and NC6 have highest favorable NBEs among all studied ligands due increased favorable electrostatic contribution. We expect these NBEs data will provide atomistic level insights and benefit in designing new and improved HDAC2 inhibitors. Communicated by Ramaswamy H. Sarma.

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.

Entropies Derived from the Packing Geometries within a Single Protein Structure

A fast, simple, yet robust method to calculate protein entropy from a single protein structure is presented here. The focus is on the atomic packing details, which are calculated by combining Voronoi diagrams and Delaunay tessellations. Even though the method is simple, the entropies computed exhibit an extremely high correlation with the entropies previously derived by other methods based on quasi-harmonic motions, quantum mechanics, and molecular dynamics simulations. These packing-based entropies account directly for the local freedom and provide entropy for any individual protein structure that could be used to compute free energies directly during simulations for the generation of more reliable trajectories and also for better evaluations of modeled protein structures. Physico-chemical properties of amino acids are compared with these packing entropies to uncover the relationships with the entropies of different residue types. A public packing entropy web server is provided at packing-entropy.bb.iastate.edu, and the application programing interface is available within the PACKMAN (https://github.com/Pranavkhade/PACKMAN) package.

Interactions of the Aβ(1-42) Peptide with Boron Nitride Nanoparticles of Varying Curvature in an Aqueous Medium: Different Pathways to Inhibit β-Sheet Formation

The aggregation of amyloid β (Aβ) peptide triggered by its conformational changes leads to the commonly known neurodegenerative disease of Alzheimer's. It is believed that the formation of β sheets of the peptide plays a key role in its aggregation and subsequent fibrillization. In the current study, we have investigated the interactions of the Aβ(1-42) peptide with boron nitride nanoparticles and the effects of the latter on conformational transitions of the peptide through a series of molecular dynamics simulations. In particular, the effects of curvature of the nanoparticle surface are studied by considering boron nitride nanotubes (BNNTs) of varying diameter and also a planar boron nitride nanosheet (BNNS). Altogether, the current study involves the generation and analysis of 9.5 μs of dynamical trajectories of peptide-BNNT/BNNS pairs in an aqueous medium. It is found that BN nanoparticles of different curvatures that are studied in the present work inhibit the conformational transition of the peptide to its β-sheet form. However, such an inhibition effect follows different pathways for BN nanoparticles of different curvatures. For the BNNT with the highest surface curvature, i.e., (3,3) BNNT, the nanoparticle is found to inhibit β-sheet formation by stabilizing the helical structure of the peptide, whereas for planar BNNS, the β-sheet formation is prevented by making more favorable pathways available for transitions of the peptide to conformations of random coils and turns. The BNNTs with intermediate curvatures are found to exhibit diverse pathways of their interactions with the peptide, but in all cases, essentially no formation of the β sheet is found whereas substantial β-sheet formation is observed for Aβ(1-42) in water in the absence of any nanoparticle. The current study shows that BN nanoparticles have the potential to act as effective tools to prevent amyloid formation from Aβ peptides.

Recent Developments in Free Energy Calculations for Drug Discovery

The grand challenge in structure-based drug design is achieving accurate prediction of binding free energies. Molecular dynamics (MD) simulations enable modeling of conformational changes critical to the binding process, leading to calculation of thermodynamic quantities involved in estimation of binding affinities. With recent advancements in computing capability and predictive accuracy, MD based virtual screening has progressed from the domain of theoretical attempts to real application in drug development. Approaches including the Molecular Mechanics Poisson Boltzmann Surface Area (MM-PBSA), Linear Interaction Energy (LIE), and alchemical methods have been broadly applied to model molecular recognition for drug discovery and lead optimization. Here we review the varied methodology of these approaches, developments enhancing simulation efficiency and reliability, remaining challenges hindering predictive performance, and applications to problems in the fields of medicine and biochemistry.

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.

Computer-aided drug design of small molecule inhibitors of the ERCC1-XPF protein-protein interaction

The heterodimer of DNA excision repair protein ERCC-1 and DNA repair endonuclease XPF (ERCC1-XPF) is a 5'-3' structure-specific endonuclease essential for the nucleotide excision repair (NER) pathway, and it is also involved in other DNA repair pathways. In cancer cells, ERCC1-XPF plays a central role in repairing DNA damage induced by chemotherapeutics including platinum-based and cross-linking agents; thus, its inhibition is a promising strategy to enhance the effect of these therapies. In this study, we rationally modified the structure of F06, a small molecule inhibitor of the ERCC1-XPF interaction (Molecular Pharmacology, 84, 2013 and 12), to improve its binding to the target. We followed a multi-step computational approach to investigate potential modification sites of F06, rationally design and rank a library of analogues, and identify candidates for chemical synthesis and in vitro testing. Our top compound, B5, showed an improved half-maximum inhibitory concentration (IC₅₀ ) value of 0.49 µM for the inhibition of ERCC1-XPF endonuclease activit, and lays the foundation for further testing and optimization. Also, the computational approach reported here can be used to develop DNA repair inhibitors targeting the ERCC1-XPF complex.

Identification of potent L,D-transpeptidase 5 inhibitors for Mycobacterium tuberculosis as potential anti-TB leads: virtual screening and molecular dynamics simulations

Virtual screening is a useful in silico approach to identify potential leads against various targets. It is known that carbapenems (doripenem and faropenem) do not show any reasonable inhibitory activities against L,D-transpeptidase 5 (Ldt_Mt5) and also an adduct of meropenem exhibited slow acylation. Since these drugs are active against L,D-transpeptidase 2 (Ldt_Mt2), understanding the differences between these two enzymes is essential. In this study, a ligand-based virtual screening of 12,766 compounds followed by molecular dynamics (MD) simulations was applied to identify potential leads against Ldt_Mt5. To further validate the obtained virtual screening ranking for Ldt_Mt5, we screened the same libraries of compounds against Ldt_Mt2 which had more experimetal and calculated binding energies reported. The observed consistency between the binding affinities of Ldt_Mt2 validates the obtained virtual screening binding scores for Ldt_Mt5. We subjected 37 compounds with docking scores ranging from - 7.2 to - 9.9 kcal mol^-1 obtained from virtual screening for further MD analysis. A set of compounds (n = 12) from four antibiotic classes with ≤ - 30 kcal mol^-1 molecular mechanics/generalized born surface area (MM-GBSA) binding free energies (ΔG_bind) was characterized. A final set of that, all β-lactams (n = 4), was considered. The outcome of this study provides insight into the design of potential novel leads for Ldt_Mt5. Graphical abstract.

Thermodynamics of Conformational Transitions in a Disordered Protein Backbone Model

Conformational entropy is expected to contribute significantly to the thermodynamics of structural transitions in intrinsically disordered proteins or regions in response to protein/ligand binding, posttranslational modifications, and environmental changes. We calculated the backbone (dihedral) conformational entropy of oligoglycine (Gly_N), a protein backbone mimic and model intrinsically disordered region, as a function of chain length (N=3, 4, 5, 10, and 15) from simulations using three different approaches. The backbone conformational entropy scales linearly with chain length with a slope consistent with the entropy of folding of well-structured proteins. The entropic contributions of second-order dihedral correlations are predominantly through intraresidue ϕ-ψ pairs, suggesting that oligoglycine may be thermodynamically modeled as a system of independent glycine residues. We find the backbone conformational entropy to be largely independent of global structural parameters, like the end-to-end distance and radius of gyration. We introduce a framework referred to herein as "ensemble confinement" to estimate the loss (gain) of conformational free energy and its entropic component when individual residues are constrained to (released from) particular regions of the ϕ-ψ map. Quantitatively, we show that our protein backbone model resists ordering/folding with a significant, unfavorable ensemble confinement free energy because of the loss of a substantial portion of the absolute backbone entropy. Proteins can couple this free-energy reservoir to distal binding events as a regulatory mechanism to promote or suppress binding.

Targeting DNA Repair in Tumor Cells via Inhibition of ERCC1-XPF

The ERCC1-XPF heterodimer is a 5'-3' structure-specific endonuclease, which plays an essential role in several DNA repair pathways in mammalian cells. ERCC1-XPF is primarily involved in the repair of chemically induced helix-distorting and bulky DNA lesions, such as cyclobutane pyrimidine dimers (CPDs), and DNA interstrand cross-links. Inhibition of ERCC1-XPF has been shown to potentiate cytotoxicity of platinum-based drugs and cyclophosphamide in cancer cells. In this study, the previously described ERCC1-XPF inhibitor 4-((6-chloro-2-methoxyacridin-9-yl)amino)-2-((4-methylpiperazin-1-yl)methyl)phenol (compound 1) was used as a reference compound. Following the outcome of docking-based virtual screening (VS), we synthesized seven novel derivatives of 1 that were identified in silico as being likely to have high binding affinity for the ERCC1-XPF heterodimerization interface by interacting with the XPF double helix-hairpin-helix (HhH2) domain. Two of the new compounds, 4-((6-chloro-2-methoxyacridin-9-yl)amino)-2-((4-cyclohexylpiperazin-1-yl)methyl)phenol (compound 3) and 4-((6-chloro-2-methoxyacridin-9-yl)amino)-2-((4-(2-(dimethylamino)ethyl) piperazin-1-yl) methyl) phenol (compound 4), were shown to be potent inhibitors of ERCC1-XPF activity in vitro. Compound 4 showed significant inhibition of the removal of CPDs in UV-irradiated cells and the capacity to sensitize colorectal cancer cells to UV radiation and cyclophosphamide.

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.

Binding free energy analysis of protein-protein docking model structures by evERdock

To aid the evaluation of protein-protein complex model structures generated by protein docking prediction (decoys), we previously developed a method to calculate the binding free energies for complexes. The method combines a short (2 ns) all-atom molecular dynamics simulation with explicit solvent and solution theory in the energy representation (ER). We showed that this method successfully selected structures similar to the native complex structure (near-native decoys) as the lowest binding free energy structures. In our current work, we applied this method (evERdock) to 100 or 300 model structures of four protein-protein complexes. The crystal structures and the near-native decoys showed the lowest binding free energy of all the examined structures, indicating that evERdock can successfully evaluate decoys. Several decoys that show low interface root-mean-square distance but relatively high binding free energy were also identified. Analysis of the fraction of native contacts, hydrogen bonds, and salt bridges at the protein-protein interface indicated that these decoys were insufficiently optimized at the interface. After optimizing the interactions around the interface by including interfacial water molecules, the binding free energies of these decoys were improved. We also investigated the effect of solute entropy on binding free energy and found that consideration of the entropy term does not necessarily improve the evaluations of decoys using the normal model analysis for entropy calculation.

Recent Developments and Applications of the MMPBSA Method

The Molecular Mechanics Poisson-Boltzmann Surface Area (MMPBSA) approach has been widely applied as an efficient and reliable free energy simulation method to model molecular recognition, such as for protein-ligand binding interactions. In this review, we focus on recent developments and applications of the MMPBSA method. The methodology review covers solvation terms, the entropy term, extensions to membrane proteins and high-speed screening, and new automation toolkits. Recent applications in various important biomedical and chemical fields are also reviewed. We conclude with a few future directions aimed at making MMPBSA a more robust and efficient method.

Molecular Dynamics as a Tool for Virtual Ligand Screening

Rational drug design is essential for new drugs to emerge, especially when the structure of a target protein or catalytic enzyme is known experimentally. To that purpose, high-throughput virtual ligand screening campaigns aim at discovering computationally new binding molecules or fragments to inhibit a particular protein interaction or biological activity. The virtual ligand screening process often relies on docking methods which allow predicting the binding of a molecule into a biological target structure with a correct conformation and the best possible affinity. The docking method itself is not sufficient as it suffers from several and crucial limitations (lack of protein flexibility information, no solvation effects, poor scoring functions, and unreliable molecular affinity estimation).At the interface of computer techniques and drug discovery, molecular dynamics (MD) allows introducing protein flexibility before or after a docking protocol, refining the structure of protein-drug complexes in the presence of water, ions and even in membrane-like environments, and ranking complexes with more accurate binding energy calculations. In this chapter we describe the up-to-date MD protocols that are mandatory supporting tools in the virtual ligand screening (VS) process. Using docking in combination with MD is one of the best computer-aided drug design protocols nowadays. It has proved its efficiency through many examples, described below.

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.