Why is it so difficult to simulate entropies, free energies, and their differences?

Validation of Molecular Simulation: An Overview of Issues

Computer simulation of molecular systems enables structure-energy-function relationships of molecular processes to be described at the sub-atomic, atomic, supra-atomic, or supra-molecular level. To interpret results of such simulations appropriately, the quality of the calculated properties must be evaluated. This depends on the way the simulations are performed and on the way they are validated by comparison to values Q^exp of experimentally observable quantities Q. One must consider 1) the accuracy of Q^exp , 2) the accuracy of the function Q(r^N ) used to calculate a Q-value based on a molecular configuration r^N of N particles, 3) the sensitivity of the function Q(r^N ) to the configuration r^N , 4) the relative time scales of the simulation and experiment, 5) the degree to which the calculated and experimental properties are equivalent, and 6) the degree to which the system simulated matches the experimental conditions. Experimental data is limited in scope and generally corresponds to averages over both time and space. A critical analysis of the various factors influencing the apparent degree of (dis)agreement between simulations and experiment is presented and illustrated using examples from the literature. What can be done to enhance the validation of molecular simulation is also discussed.

Thermodynamic characterization of new positive allosteric modulators binding to the glutamate receptor A2 ligand-binding domain: combining experimental and computational methods unravels differences in driving forces

Positive allosteric modulation of the ionotropic glutamate receptor GluA2 presents a potential treatment of cognitive disorders, for example, Alzheimer's disease. In the present study, we describe the synthesis, pharmacology, and thermodynamic studies of a series of monofluoro-substituted 3,4-dihydro-2H-1,2,4-benzothiadiazine 1,1-dioxides. Measurements of ligand binding by isothermal titration calorimetry (ITC) showed similar binding affinities for the modulator series at the GluA2 LBD but differences in the thermodynamic driving forces. Binding of 5c (7-F) and 6 (no-F) is enthalpy driven, and 5a (5-F) and 5b (6-F) are entropy driven. For 5d (8-F), both quantities were equal in size. Thermodynamic integration (TI) and one-step perturbation (OSP) were used to calculate the relative binding affinity of the modulators. The OSP calculations had a higher predictive power than those from TI, and combined with the shorter total simulation time, we found the OSP method to be more effective for this setup. Furthermore, from the molecular dynamics simulations, we extracted the enthalpies and entropies, and along with the ITC data, this suggested that the differences in binding free energies are largely explained by the direct ligand-surrounding enthalpies. Furthermore, we used the OSP setup to predict binding affinities for a series of polysubstituted fluorine compounds and monosubstituted methyl compounds and used these predictions to characterize the modulator binding pocket for this scaffold of positive allosteric modulators.

Biophysical and physicochemical methods differentiate highly ligand-efficient human D-amino acid oxidase inhibitors

Many early drug research efforts are too reductionist thereby not delivering key parameters such as kinetics and thermodynamics of target-ligand binding. A set of human D-Amino Acid Oxidase (DAAO) inhibitors 1-6 was applied to demonstrate the impact of key biophysical techniques and physicochemical methods in the differentiation of chemical entities that cannot be adequately distinguished on the basis of their normalized potency (ligand efficiency) values. The resulting biophysical and physicochemical data were related to relevant pharmacodynamic and pharmacokinetic properties. Surface Plasmon Resonance data indicated prolonged target-ligand residence times for 5 and 6 as compared to 1-4, based on the observed k(off) values. The Isothermal Titration Calorimetry-derived thermodynamic binding profiles of 1-6 to the DAAO enzyme revealed favorable contributions of both ΔH and ΔS to their ΔG values. Surprisingly, the thermodynamic binding profile of 3 elicited a substantially higher favorable contribution of ΔH to ΔG in comparison with the structurally closely related fused bicyclic acid 4. Molecular dynamics simulations and free energy calculations of 1, 3, and 4 led to novel insights into the thermodynamic properties of the binding process at an atomic level and in the different thermodynamic signatures of 3 and 4. The presented holistic approach is anticipated to facilitate the identification of compounds with best-in-class properties at an early research stage.

Current trends in finite-time thermodynamics

The cornerstone of finite-time thermodynamics is all about the price of haste and how to minimize it. Reversible processes may be ultimately efficient, but they are unrealistically slow. In all situations-chemical, mechanical, economical-we pay extra to get the job done quickly. Finite-time thermodynamics can be used to develop methods to limit that extra expenditure, be it in energy, entropy production, money, or something entirely different. Finite-time thermodynamics also includes methods to calculate the optimal path or mode of operation to achieve this minimal expenditure. The concept is to place the system of interest in contact with a time-varying environment which will coax the system along the desired path, much like guiding a horse along by waving a carrot in front of it.

A molecular mechanics model for imatinib and imatinib:kinase binding

Imatinib is an important anticancer drug, which binds specifically to the Abl kinase and blocks its signalling activity. To model imatinib:protein interactions, we have developed a molecular mechanics force field for imatinib and four close analogues, which is consistent with the CHARMM force field for proteins and nucleic acids. Atomic charges and Lennard-Jones parameters were derived from a supermolecule ab initio approach. We considered the ab initio energies and geometries of a probe water molecule interacting with imatinib fragments at 32 different positions. We considered both a neutral and a protonated imatinib. The final RMS deviation between the ab initio and force field energies, averaged over both forms, was 0.2 kcal/mol. The model also reproduces the ab initio geometry and flexibility of imatinib. To apply the force field to imatinib:Abl simulations, it is also necessary to determine the most likely imatinib protonation state when it binds to Abl. This was done using molecular dynamics free energy simulations, where imatinib is reversibly protonated during a series of MD simulations, both in solution and in complex with Abl. The simulations indicate that imatinib binds to Abl in its protonated, positively-charged form. To help test the force field and the protonation prediction, we did MD free energy simulations that compare the Abl binding affinities of two imatinib analogs, obtaining good agreement with experiment. Finally, two new imatinib variants were considered, one of which is predicted to have improved Abl binding. This variant could be of interest as a potential drug.

Emerging methods for ensemble-based virtual screening

Ensemble based virtual screening refers to the use of conformational ensembles from crystal structures, NMR studies or molecular dynamics simulations. It has gained greater acceptance as advances in the theoretical framework, computational algorithms, and software packages enable simulations at longer time scales. Here we focus on the use of computationally generated conformational ensembles and emerging methods that use these ensembles for discovery, such as the Relaxed Complex Scheme or Dynamic Pharmacophore Model. We also discuss the more rigorous physics-based computational techniques such as accelerated molecular dynamics and thermodynamic integration and their applications in improving conformational sampling or the ranking of virtual screening hits. Finally, technological advances that will help make virtual screening tools more accessible to a wider audience in computer aided drug design are discussed.

Tetracycline-tet repressor binding specificity: insights from experiments and simulations

Tetracycline (Tc) antibiotics have been put to new uses in the construction of artificial gene regulation systems, where they bind to the Tet repressor protein (TetR) and modulate its affinity for DNA. Many Tc variants have been produced, both to overcome bacterial resistance and to achieve a broad range of binding strengths. To better understand TetR-Tc binding, we investigate a library of 16 tetracyclines, using fluorescence experiments and molecular dynamics free energy simulations (MDFE). The relative TetR binding free energies are computed by reversibly transforming one Tc variant into another during the simulation, with no adjustable parameters. The chemical variations involve polar and nonpolar substitutions along one entire edge of the elongated Tc structure, which provides many of the protein-ligand contacts. The binding constants span five orders of magnitude. The simulations reproduce the experimental binding free energies, when available, within the uncertainty of either method (+/-0.5 kcal/mol), and reveal many additional details. Contributions of individual Tc substituents are evaluated, along with their additivity and transferability among different positions on the Tc scaffold; differences between D- and B-class repressors are quantified. With increasing computer power, the MDFE approach provides an attractive complement to experiment and should play an increasing role in the understanding and engineering of protein-ligand recognition.

Absolute Single-Molecule Entropies from Quasi-Harmonic Analysis of Microsecond Molecular Dynamics: Correction Terms and Convergence Properties

The convergence properties of the absolute single-molecule configurational entropy and the correction terms used to estimate it are investigated using microsecond molecular dynamics simulation of a peptide test system and an improved methodology. The results are compared with previous applications for systems of diverse chemical nature. It is shown that (i) the effect of anharmonicity is small, (ii) the effect of pairwise correlation is typically large, and (iii) the latter affects to a larger extent the entropy estimate of thermodynamic states characterized by a higher motional correlation. The causes of such deviations from a quasi-harmonic behavior are explained. This improved approach provides entropies also for molecular systems undergoing conformational transitions and characterized by highly frustrated energy surfaces, thus not limited to systems sampling a single quasi-harmonic basin. Overall, this study emphasizes the need for extensive phase-space sampling in order to obtain a reliable estimation of entropic contributions.

Binding mechanisms in supramolecular complexes

Supramolecular chemistry has expanded dramatically in recent years both in terms of potential applications and in its relevance to analogous biological systems. The formation and function of supramolecular complexes occur through a multiplicity of often difficult to differentiate noncovalent forces. The aim of this Review is to describe the crucial interaction mechanisms in context, and thus classify the entire subject. In most cases, organic host-guest complexes have been selected as examples, but biologically relevant problems are also considered. An understanding and quantification of intermolecular interactions is of importance both for the rational planning of new supramolecular systems, including intelligent materials, as well as for developing new biologically active agents.

Independent-Trajectories Thermodynamic-Integration Free-Energy Changes for Biomolecular Systems: Determinants of H5N1 Avian Influenza Virus Neuraminidase Inhibition by Peramivir

Free-energy changes are essential physicochemical quantities for understanding most biochemical processes. Yet, the application of accurate thermodynamic-integration (TI) computation to biological and macromolecular systems is limited by finite-sampling artifacts. In this paper, we employ independent-trajectories thermodynamic-integration (IT-TI) computation to estimate improved free-energy changes and their uncertainties for (bio)molecular systems. IT-TI aids sampling statistics of the thermodynamic macrostates for flexible associating partners by ensemble averaging of multiple, independent simulation trajectories. We study peramivir (PVR) inhibition of the H5N1 avian influenza virus neuraminidase flexible receptor (N1). Binding site loops 150 and 119 are highly mobile, as revealed by N1-PVR 20-ns molecular dynamics. Due to such heterogeneous sampling, standard TI binding free-energy estimates span a rather large free-energy range, from a 19% underestimation to a 29% overestimation of the experimental reference value (−62.2 ± 1.8 kJ mol⁻¹). Remarkably, our IT-TI binding free-energy estimate (−61.1 ± 5.4 kJ mol⁻¹) agrees with a 2% relative difference. In addition, IT-TI runs provide a statistics-based free-energy uncertainty for the process of interest. Using ∼800 ns of overall sampling, we investigate N1-PVR binding determinants by IT-TI alchemical modifications of PVR moieties. These results emphasize the dominant electrostatic contribution, particularly through the N1 E277−PVR guanidinium interaction. Future drug development may be also guided by properly tuning ligand flexibility and hydrophobicity. IT-TI will allow estimation of relative free energies for systems of increasing size, with improved reliability by employing large-scale distributed computing.

Binding of tetracyclines to elongation factor Tu, the Tet repressor, and the ribosome: a molecular dynamics simulation study

Tetracycline (Tc) is a broad-spectrum antibiotic that kills bacteria by interrupting protein biosynthesis. It is thought that the bacteriostatic action of Tc is associated with its binding to the acceptor site (or A site) in the bacterial ribosome, interfering with the attachment of aminoacyl-tRNA. Recently, however, the crystal structure of a complex between Tc and trypsin-modified elongation factor Tu (tm-EF-Tu) was determined, raising the question of whether Tc binding to EF-Tu has a role in its inhibition of protein synthesis. We address this question using computer simulations. As controls, we first compute relative ribosome binding free energies for seven Tc variants for which experimental data are available, obtaining good agreement. We then consider the binding of Tc to both the trypsin-modified and unmodified EF-Tu-GDP complexes. We show that the direct contribution of EF-Tu to the binding free energy is negligible; rather, the binding can be solely attributed to interactions of Tc with a bridging Mg(2+) ion and the GDP phosphate groups. The effects of trypsin modification are modest. Further, our calculations show that EF-Tu does not exhibit any binding preference for Tc over the nonantibiotic, 4-dedimethyl-Tc, and EF-Tu does not bind the Tc analogue tigecycline, which is a potent antibiotic. In contrast, both the ribosome and the Tet Repressor protein (involved in Tc resistance) do show a binding preference for Tc over 4-dedimethyl-Tc, and the ribosome prefers to bind tigecycline over Tc. Overall, our results provide insights into the binding properties of tetracyclines and support the idea that EF-Tu is not their primary target.

An improved relaxed complex scheme for receptor flexibility in computer-aided drug design

The interactions among associating (macro)molecules are dynamic, which adds to the complexity of molecular recognition. While ligand flexibility is well accounted for in computational drug design, the effective inclusion of receptor flexibility remains an important challenge. The relaxed complex scheme (RCS) is a promising computational methodology that combines the advantages of docking algorithms with dynamic structural information provided by molecular dynamics (MD) simulations, therefore explicitly accounting for the flexibility of both the receptor and the docked ligands. Here, we briefly review the RCS and discuss new extensions and improvements of this methodology in the context of ligand binding to two example targets: kinetoplastid RNA editing ligase 1 and the W191G cavity mutant of cytochrome c peroxidase. The RCS improvements include its extension to virtual screening, more rigorous characterization of local and global binding effects, and methods to improve its computational efficiency by reducing the receptor ensemble to a representative set of configurations. The choice of receptor ensemble, its influence on the predictive power of RCS, and the current limitations for an accurate treatment of the solvent contributions are also briefly discussed. Finally, we outline potential methodological improvements that we anticipate will assist future development.

Comparison of atomic-level and coarse-grained models for liquid hydrocarbons from molecular dynamics configurational entropy estimates

Molecular liquids can be modeled at different levels of spatial resolution. In atomic-level (AL) models, all (heavy) atoms can be explicitly simulated. In coarse-grained (CG) models, particles (beads) that represent groups of covalently bound atoms are used as elementary units. Ideally, a CG model should reproduce the thermodynamic and structural properties of the corresponding AL model after mapping to the lower-resolution scale. In the present work, two such models are investigated: (i) the classical GROMOS atomic-level model; (ii) a CG model recently proposed by Marrink et al., which maps approximately four non-hydrogen atoms to one bead [J. Phys. Chem. B 2004, 108, 750]. The study is restricted to n-alkanes whose aliphatic fragments are abundantly found in lipids of biological interest. Additionally, cis-9-octadecene is included, as a template chain of the lipid dioleoylphosphatidylcholine (DOPC). The two representations of molecules in the liquid phase are compared in terms of average molecular structures, extent of configurational space sampled, and single-molecule entropies. An approximate method is used to estimate the rotational contributions to the absolute configurational entropy. Good correspondence between the AL and CG representations is found. The loss in configurational entropy due to the reduction in degrees of freedom upon coarse-graining of the model is estimated.

Phase-space overlap measures. I. Fail-safe bias detection in free energies calculated by molecular simulation

We consider ways to quantify the overlap of the parts of phase space important to two systems, labeled A and B. Of interest is how much of the A-important phase space lies in that important to B, and how much of B lies in A. Two measures are proposed. The first considers four total-energy distributions, formed from all combinations made by tabulating either the A-system or the B-system energy when sampling either the A or B system. Measures for A in B and B in A are given by two overlap integrals defined on pairs of these distributions. The second measure is based on information theory, and defines two relative entropies which are conveniently expressed in terms of the dissipated work for free-energy perturbation (FEP) calculations in the A-->B and B-->A directions, respectively. Phase-space overlap is an important consideration in the performance of free-energy calculations. To demonstrate this connection, we examine bias in FEP calculations applied to a system of independent particles in a harmonic potential. Systems are selected to represent a range of overlap situations, including extreme subset, subset, partial overlap, and nonoverlap. The magnitude and symmetry of the bias (A-->B vs B-->A) are shown to correlate well with the overlap, and consequently with the overlap measures. The relative entropies are used to scale the amount of sampling to obtain a universal bias curve. This result leads to develop a simple heuristic that can be applied to determine whether a work-based free-energy measurement is free of bias. The heuristic is based in part on the measured free energy, but we argue that it is fail-safe inasmuch as any bias in the measurement will not promote a false indication of accuracy.

Comparison of Thermodynamic Properties of Coarse-Grained and Atomic-Level Simulation Models

Thermodynamic data are often used to calibrate or test amomic-level (AL) force fields for molecular dynamics (MD) simulations. In contrast, the majority of coarse-grained (CG) force fields do not rely extensively on thermodynamic quantities. Recently, a CG force field for lipids, hydrocarbons, ions, and water, in which approximately four non-hydrogen atoms are mapped onto one interaction site, has been proposed and applied to study various aspects of lipid systems. To date, no extensive investigation of its capability to describe salvation thermodynamics has been undertaken. In the present study, a detailed picture of vaporization, solvation, and phase-partitioning thermodynamics for liquid hydrocarbons and water was obtained at CG and AL resolutions, in order to compare the two types or models and evaluate their ability to describe thermodynamic properties in the temperature range between 263 and 343 K. Both CG and AL models capture the experimental dependence of the thermodynamic properties on the temperature, albeit a systematically weaker dependence is found for the CG model. Moreover, deviations are found for solvation thermodynamics and for the corresponding enthalpy-entropy compensation for the CG model. Particularly water/oil repulsion seems to be overestimated. However, the results suggest that the thermodynamic properties considered should be reproducible by a CG model provided it is reparametrized on the basis of these liquid-phase properties.

On side-chain conformational entropy of proteins

The role of side-chain entropy (SCE) in protein folding has long been speculated about but is still not fully understood. Utilizing a newly developed Monte Carlo method, we conducted a systematic investigation of how the SCE relates to the size of the protein and how it differs among a protein's X-ray, NMR, and decoy structures. We estimated the SCE for a set of 675 nonhomologous proteins, and observed that there is a significant SCE for both exposed and buried residues for all these proteins-the contribution of buried residues approaches approximately 40% of the overall SCE. Furthermore, the SCE can be quite different for structures with similar compactness or even similar conformations. As a striking example, we found that proteins' X-ray structures appear to pack more "cleverly" than their NMR or decoy counterparts in the sense of retaining higher SCE while achieving comparable compactness, which suggests that the SCE plays an important role in favouring native protein structures. By including a SCE term in a simple free energy function, we can significantly improve the discrimination of native protein structures from decoys.

Monte Carlo sampling of near-native structures of proteins with applications

Since a protein's dynamic fluctuation inside cells affects the protein's biological properties, we present a novel method to study the ensemble of near-native structures (NNS) of proteins, namely, the conformations that are very similar to the experimentally determined native structure. We show that this method enables us to (i) quantify the difficulty of predicting a protein's structure, (ii) choose appropriate simplified representations of protein structures, and (iii) assess the effectiveness of knowledge-based potential functions. We found that well-designed simple representations of protein structures are likely as accurate as those more complex ones for certain potential functions. We also found that the widely used contact potential functions stabilize NNS poorly, whereas potential functions incorporating local structure information significantly increase the stability of NNS.

Configurational entropy change of netropsin and distamycin upon DNA minor-groove binding

Binding of a small molecule to a macromolecular target reduces its conformational freedom, resulting in a negative entropy change that opposes the binding. The goal of this study is to estimate the configurational entropy change of two minor-groove-binding ligands, netropsin and distamycin, upon binding to the DNA duplex d(CGCGAAAAACGCG).d(CGCGTTTTTCGCG). Configurational entropy upper bounds based on 10-ns molecular dynamics simulations of netropsin and distamycin in solution and in complex with DNA in solution were estimated using the covariance matrix of atom-positional fluctuations. The results suggest that netropsin and distamycin lose a significant amount of configurational entropy upon binding to the DNA minor groove. The estimated changes in configurational entropy for netropsin and distamycin are -127 J K(-1) mol(-1) and -104 J K(-1) mol(-1), respectively. Estimates of the configurational entropy contributions of parts of the ligands are presented, showing that the loss of configurational entropy is comparatively more pronounced for the flexible tails than for the relatively rigid central body.

Efficient calculation of many stacking and pairing free energies in DNA from a few molecular dynamics simulations

Through the use of the one-step perturbation approach, 130 free energies of base stacking and 1024 free energies of base pairing in DNA have been calculated from only five simulations of a nonphysical reference state. From analysis of a diverse set of 23 natural and unnatural bases, it appears that stacking free energies and stacking conformations play an important role in pairing of DNA nucleotides. On the one hand, favourable pairing free energies were found for bases that do not have the possibility to form canonical hydrogen bonds, while on the other hand, good hydrogen-bonding possibilities do not guarantee a favourable pairing free energy if the stacking of the bases dictates an unfavourable conformation. In this application, the one-step perturbation approach yields a wealth of both energetic and structural information at minimal computational cost.

Optimizing the driving function for nonequilibrium free-energy calculations in the linear regime: a variational approach

We consider the issue of optimizing linear-regime nonequilibrium simulations to estimate free-energy differences. In particular, we focus on the problem of finding the best-possible driving function lambda(t) that, for a given thermodynamic path, simulation algorithm, and amount of computational effort, minimizes dissipation. From the fluctuation-dissipation theorem it follows that, in the linear-response regime, the dissipation is controlled by the magnitude and characteristic correlation time of the equilibrium fluctuations in the driving force. As a result, the problem of finding the optimal switching scheme involves the solution of a standard problem in variational calculus: the minimization of a functional with respect to the switching function. In practice, the minimization involves solving the associated Euler-Lagrange equation subject to a set of boundary conditions. As a demonstration we apply the approach to the simple, yet illustrative problem of computing the free-energy difference between two classical harmonic oscillators with very different characteristic frequencies.

Protonation States of Methionine Aminopeptidase and Their Relevance for Inhibitor Binding and Catalytic Activity

We have performed a computational study of different protomeric states of the methionine aminopeptidase active site using a combined quantum-mechanical/molecular mechanical simulation approach. The aim of this study was to clarify the native protonation state of the enzyme, which is needed for the development of novel irreversible inhibitors that can possibly be used as antiangiogenic and antibiotic drugs by virtual screening and other drug design methods. The results of the simulations indicated that two protonation states are possible without disturbing the overall geometry of the active site. We then verified experimentally the presence of the two protonation states by studying the substrate hydrolysis and inhibitor binding reactions at different pH values and come to the conclusion that one of the protomeric states is relevant for inhibitor binding, whereas the other is relevant for substrate hydrolysis. This result has implications for the development of other inhibitors of this class of enzymes and adds a new perspective to the pharmacological properties of the antiangiogenic drug fumagillin, which is an irreversible inhibitor of the human methionine aminopeptidase type II.

Free energy simulations come of age: protein-ligand recognition

In recent years, molecular dynamics simulations of biomolecular free energy differences have benefited from significant methodological advances and increased computer power. Applications to molecular recognition provide an understanding of the interactions involved that goes beyond, and is an important complement to, experimental studies. Poisson-Boltzmann electrostatic models provide a faster and simpler free energy method in cases where electrostatic interactions are important. We illustrate both molecular dynamics and Poisson-Boltzmann methods with a detailed study of amino acid recognition by aspartyl-tRNA synthetase, whose specificity is important for maintaining the integrity of the genetic code.

Molecular dynamics simulations

Molecular dynamics simulations have become a standard tool for the investigation of biomolecules. Simulations are performed of ever bigger systems using more realistic boundary conditions and better sampling due to longer sampling times. Recently, realistic simulations of systems as complex as transmembrane channels have become feasible. Simulations aid our understanding of biochemical processes and give a dynamic dimension to structural data; for example, the transformation of harmless prion protein into the disease-causing agent has been modeled.