Structural and ITC Characterization of Peptide-Protein Binding: Thermodynamic Consequences of Cyclization Constraints,a Case Study on Vascular Endothelial Growth Factor Ligands

Macrocyclization constraints are widely used in the design of protein ligands to stabilize their bioactive conformation and increase their affinities. However, the resulting changes in binding entropy can be puzzling and uncorrelated to affinity gains. Here, the thermodynamic (Isothermal Titration Calorimetry) and structural (X-ray, NMR and CD) analysis of a complete series of lactam-bridged peptide ligands of the vascular endothelial growth factor, and their unconstrained analogs are reported. It is shown that differences in thermodynamics arise mainly from the folding energy of the peptide upon binding. The systematic reduction in conformational entropy penalty due to helix pre-organization can be counterbalanced by an unfavorable vibrational entropy change if the constraints are too rigid. The gain in configurational entropy partially escapes the enthalpy/entropy compensation and leads to an improvement in affinity. The precision of the analytical ITC method makes this study a possible benchmark for constrained peptides optimization.     

Concepts in receptor optimization: targeting the RGD peptide

Synthetic receptors have a wide range of potential applications, but it has been difficult to design low molecular weight receptors that bind ligands with high, "proteinlike" affinities. This study uses novel computational methods to understand why it is hard to design a high-affinity receptor and to explore the limits of affinity, with the bioactive peptide RGD as a model ligand. The M2 modeling method is found to yield excellent agreement with experiment for a known RGD receptor and then is used to analyze a series of receptors generated in silico with a de novo design algorithm. Forces driving binding are found to be systematically opposed by proportionate repulsions due to desolvation and entropy. In particular, strong correlations are found between Coulombic attractions and the electrostatic desolvation penalty and between the mean energy change on binding and the cost in configurational entropy. These correlations help explain why it is hard to achieve high affinity. The change in surface area upon binding is found to correlate poorly with affinity within this series. Measures of receptor efficiency are formulated that summarize how effectively a receptor uses surface area, total energy, and Coulombic energy to achieve affinity. Analysis of the computed efficiencies suggests that a low molecular weight receptor can achieve proteinlike affinity. It is also found that macrocyclization of a receptor can, unexpectedly, increase the entropy cost of binding because the macrocyclic structure further restricts ligand motion.     

Extraction of configurational entropy from molecular simulations via an expansion approximation

A method is presented for extracting the configurational entropy of solute molecules from molecular dynamics simulations, in which the entropy is computed as an expansion of multidimensional mutual information terms, which account for correlated motions among the various internal degrees of freedom of the molecule. The mutual information expansion is demonstrated to be equivalent to estimating the full-dimensional configurational probability density function (PDF) using the generalized Kirkwood superposition approximation (GKSA). While the mutual information expansion is derived to the full dimensionality of the molecule, the current application uses a truncated form of the expansion in which all fourth- and higher-order mutual information terms are neglected. Truncation of the mutual information expansion at the nth order is shown to be equivalent to approximating the full-dimensional PDF using joint PDFs with dimensionality of n or smaller by successive application of the GKSA. The expansion method is used to compute the absolute (classical) configurational entropy in a basis of bond-angle-torsion internal coordinates for several small molecules as well as the change in entropy upon binding for a small host-guest system. Convergence properties of the computed entropy values as a function of simulation time are investigated and comparisons are made with entropy values from the second generation Mining Minima software. These comparisons demonstrate a deviation in -TS of no more than about 2 kcal/mol for all cases in which convergence has been obtained.     

Modeling Protein-Ligand Binding by Mining Minima

We present the first application of the mining minima algorithm to protein-small molecule binding. This end-point approach use an empirical force field and implicit solvent models, treats the protein binding-site as fully flexible and estimates free energies as sums over local energy wells. The calculations are found to yield encouraging agreement with experiment for three sets of HIV-1protease inhibitors and a set of phosphodiesterase 10a inhibitors. The contributions of various aspects of the model to its accuracy are examined, and the Poisson-Boltzmann correction is found to be the most critical. Interestingly, the computed changes in configurational entropy upon binding fall roughly along the same entropy-energy correlation previously observed for smaller host-guest systems. Strengths and weaknesses of the method are discussed, as are the prospects for enhancing accuracy and speed.     

(Thermo)dynamic Role of Receptor Flexibility,Entropy, and Motional Correlation in Protein–Ligand Binding

The binding of 2‐amino‐5‐methylthiazole to the W191G cavity mutant of cytochrome c peroxidase is an ideal test case to investigate the entropic contribution to the binding free energy due to changes in receptor flexibility. The dynamic and thermodynamic role of receptor flexibility are studied by 50 ns‐long explicit‐solvent molecular dynamics simulations of three separate receptor ensembles: W191G binding a K⁺ ion, W191G–2a5mt complex with a closed 190–195 gating loop, and apo with an open loop. We employ a method recently proposed to estimate accurate absolute single‐molecule configurational entropies and their differences for systems undergoing conformational transitions. We find that receptor flexibility plays a generally underestimated role in protein–ligand binding (thermo)dynamics and that changes of receptor motional correlation determine such large entropy contributions.     

Host-rotaxanes model proteins that promote ligand association through a favorable change in configurational entropy

Proteins can reduce the entropic penalty for ligand association through a favorable change in configurational entropy. To investigate this process, the Delta G(o), Delta H(o), and DeltaS(o) of complexes formed between host-rotaxanes and guests were determined and compared to discover the relationship between rotaxane-structure and the energies involved in guest-association in water and DMSO. Fluorescence quenching assays provided the association constants. Van't Hoff analysis of variable temperature assays gave the enthalpies of binding. The driving force for the association of a guest and a host-rotaxane can switch from being enthalpically to entropically driven with a change in the solvent or guest. This study shows that a dramatic increase in the entropy of binding can be obtained through the addition of a rotaxane-wheel to a synthetic host. An increased motion of the wheel appears to be the source of the positive binding entropy, which would be an example of favorable configurational entropy promoting complex formation.     

Calculations of solute and solvent entropies from molecular dynamics simulations

The translational, rotational and conformational (vibrational) entropy contributions to ligand-receptor binding free energies are analyzed within the standard formulation of statistical thermodynamics. It is shown that the partitioning of the binding entropy into different components is to some extent arbitrary, but an appropriate method to calculate both translational and rotational entropy contributions to noncovalent association is by estimating the configurational volumes of the ligand in the bound and free states. Different approaches to calculating solute entropies using free energy perturbation calculations, configurational volumes based on root-mean-square fluctuations and covariance matrix based quasiharmonic analysis are illustrated for some simple molecular systems. Numerical examples for the different contributions demonstrate that theoretically derived results are well reproduced by the approximations. Calculation of solvent entropies, either using total potential energy averages or van't Hoff plots, are carried out for the case of ion solvation in water. Although convergence problems will persist for large and complex simulation systems, good agreement with experiment is obtained here for relative and absolute ion hydration entropies. We also outline how solvent and solute entropic contributions are taken into account in empirical binding free energy calculations using the linear interaction energy method. In particular it is shown that empirical scaling of the nonpolar intermolecular ligand interaction energy effectively takes into account size dependent contributions to the binding free energy.     

Prediction of SAMPL3 host-guest affinities with the binding energy distribution analysis method (BEDAM)

BEDAM calculations are described to predict the free energies of binding of a series of anaesthetic drugs to a recently characterized acyclic cucurbituril host. The modeling predictions, conducted as part of the SAMPL3 host-guest affinity blind challenge, are generally in good quantitative agreement with the experimental measurements. The correlation coefficient between computed and measured binding free energies is 70% with high statistical significance. Multiple conformational stereoisomers and protonation states of the guests have been considered. Better agreement is obtained with high statistical confidence under acidic modeling conditions. It is shown that this level of quantitative agreement could have not been reached without taking into account reorganization energy and configurational entropy effects. Extensive conformational variability of the host, the guests and their complexes is observed in the simulations, affecting binding free energy estimates and structural predictions. A conformational reservoir technique is introduced as part of the parallel Hamiltonian replica exchange molecular dynamics BEDAM protocol to fully capture conformational variability. It is shown that these advanced computational strategies lead to converged free energy estimates for these systems, offering the prospect of utilizing host-guest binding free energy data for force field validation and development.     

Designing generalized statistical ensembles for numerical simulations of biopolymers

Conformational properties of polymers, such as average dihedral angles or molecular alpha-helicity, display a rather weak dependence on the detailed arrangement of the elementary constituents (atoms). We propose a computer simulation method to explore the polymer phase space using a variant of the standard multicanonical method, in which the density of states associated to suitably chosen configurational variables is considered in place of the standard energy density of states. This configurational density of states is used in the Metropolis acceptance/rejection test when configurations are generated with the help of a hybrid Monte Carlo algorithm. The resulting configurational probability distribution is then modulated by exponential factors derived from the general principle of the maximal constrained entropy by requiring that certain average configurational quantities take preassigned (possibly temperature dependent) values. Thermal averages of other configurational quantities can be computed by using the probability distributions obtained in this way. Moments of the energy distribution require an extra canonical sampling of the system phase space at the desired temperature, in order to locally thermalize the configurational degrees of freedom. As an application of these ideas we present the study of the structural properties of two simple models: a bead-and-spring model of polyethylene with independent hindered torsions and an all-atom model of alanine and glycine oligomers with 12 amino acids in vacuum.     

Treating entropy and conformational changes in implicit solvent simulations of small molecules

Implicit solvent models are increasingly popular for estimating aqueous solvation (hydration) free energies in molecular simulations and other applications. In many cases, parameters for these models are derived to reproduce experimental values for small molecule hydration free energies. Often, these hydration free energies are computed for a single solute conformation, neglecting solute conformational changes upon solvation. Here, we incorporate these effects using alchemical free energy methods. We find significant errors when hydration free energies are estimated using only a single solute conformation, even for relatively small, simple, rigid solutes. For example, we find conformational entropy (TDeltaS) changes of up to 2.3 kcal/mol upon hydration. Interestingly, these changes in conformational entropy correlate poorly (R2 = 0.03) with the number of rotatable bonds. The present study illustrates that implicit solvent modeling can be improved by eliminating the approximation that solutes are rigid.     

The paradoxical thermodynamic basis for the interaction of ethylene glycol, glycine, and sarcosine chains with bovine carbonic anhydrase II: an unexpected manifestation of enthalpy/entropy compensation

This paper describes a systematic study of the thermodynamics of association of bovine carbonic anhydrase II (BCA) and para-substituted benzenesulfonamides with chains of oligoglycine, oligosarcosine, and oligoethylene glycol of lengths of one to five residues. For all three of these series of ligands, the enthalpy of binding became less favorable, and the entropy less unfavorable, as the chain length of the ligands increased. The dependence on chain length of the enthalpy was almost perfectly compensated by that of the entropy; this compensation resulted in dissociation constants that were independent of chain length for the three series of ligands. Changes in heat capacity were independent of chain length for the three series and revealed that the amount of molecular surface area buried upon protein-ligand complexation did not increase with increasing chain length. Taken together, these data refute a model in which the chains of the ligands interact hydrophobically with the surface of BCA. To explain the data, a model is proposed based on decreasing "tightness" of the protein-ligand interface as the chain length of the ligand increases. This decreasing tightness, as the chain length increases, is reflected in a less favorable enthalpy (due to fewer van der Waals contacts) and a less unfavorable entropy (due to greater mobility of the chain) of binding for ligands with long chains than for those with short chains. Thus, this study demonstrates a surprising example of enthalpy/entropy compensation in a well-defined system. Understanding this compensation is integral to the rational design of high-affinity ligands for proteins.     

Calculation of Host-Guest Binding Affinities Using a Quantum-Mechanical Energy Model

The prediction of protein-ligand binding affinities is of central interest in computer-aided drug discovery, but it is still difficult to achieve a high degree of accuracy. Recent studies suggesting that available force fields may be a key source of error motivate the present study, which reports the first mining minima (M2) binding affinity calculations based on a quantum mechanical energy model, rather than an empirical force field. We apply a semi-empirical quantum-mechanical energy function, PM6-DH+, coupled with the COSMO solvation model, to 29 host-guest systems with a wide range of measured binding affinities. After correction for a systematic error, which appears to derive from the treatment of polar solvation, the computed absolute binding affinities agree well with experimental measurements, with a mean error 1.6 kcal/mol and a correlation coefficient of 0.91. These calculations also delineate the contributions of various energy components, including solute energy, configurational entropy, and solvation free energy, to the binding free energies of these host-guest complexes. Comparison with our previous calculations, which used empirical force fields, point to significant differences in both the energetic and entropic components of the binding free energy. The present study demonstrates successful combination of a quantum mechanical Hamiltonian with the M2 affinity method.     

A novel synthesis of chiral guanidinium receptors and their use in unfolding the energetics of enantiorecognition of chiral carboxylates

A novel synthetic route to the versatile chiral bicyclic guanidinium building block is described making use of l-methionine as a starting material from the natural chiral pool. Furthermore, the synthetic elaboration of this building block is shown in the construction of macrocyclic and open chain hosts, respectively. The host design employs urea functions as the connecting units and supplementary anchor groups for the complexation of anions. The binding studies of these hosts with various chiral and achiral oxoanions are performed by isothermal titration calorimetry. A trend analysis of the binding energetics in an ensemble of structurally similar guests discloses the importance of geometrical confinement of the guest. Association entropy rather than free energy (affinity) is identified as an indicator of structural uniqueness needed to distinguish configurational isomers in the recognition of enantiomeric carboxylates by the chiral guanidinium hosts.     

Calculations of protein-ligand binding entropy of relative and overall molecular motions

In the context of virtual database screening, calculations of protein-ligand binding entropy of relative and overall molecular motions are challenging, owing to the inherent structural complexity of the ligand binding well in the energy landscape of protein-ligand interactions and computing time limitations. We describe a fast statistical thermodynamic method for estimation the binding entropy to address the challenges. The method is based on the integration of the configurational integral over clusters obtained from multiple docked positions. We apply the method in conjunction with 11 popular scoring functions (AutoDock, ChemScore, DrugScore, D-Score, F-Score, G-Score, LigScore, LUDI, PLP, PMF, X-Score) to evaluate the binding entropy of 100 protein-ligand complexes. The averaged values of binding entropy contribution vary from 6.2 to 9.1 kcal/mol, showing good agreement with literature. We calculate positional sizes and the angular volume of the native ligand wells. The averaged geometric mean of positional sizes in principal directions varies from 0.8 to 1.4 A. The calculated range of angular volumes is 3.3-11.8 rad(2). Then we demonstrate that the averaged six-dimensional volume of the native well is larger than the volume of the most populated non-native well in energy landscapes described by all of 11 scoring functions.     

Nearest-neighbor nonparametric method for estimating the configurational entropy of complex molecules

A method for estimating the configurational (i.e., non-kinetic) part of the entropy of internal motion in complex molecules is introduced that does not assume any particular parametric form for the underlying probability density function. It is based on the nearest-neighbor (NN) distances of the points of a sample of internal molecular coordinates obtained by a computer simulation of a given molecule. As the method does not make any assumptions about the underlying potential energy function, it accounts fully for any anharmonicity of internal molecular motion. It provides an asymptotically unbiased and consistent estimate of the configurational part of the entropy of the internal degrees of freedom of the molecule. The NN method is illustrated by estimating the configurational entropy of internal rotation of capsaicin and two stereoisomers of tartaric acid, and by providing a much closer upper bound on the configurational entropy of internal rotation of a pentapeptide molecule than that obtained by the standard quasi-harmonic method. As a measure of dependence between any two internal molecular coordinates, a general coefficient of association based on the information-theoretic quantity of mutual information is proposed. Using NN estimates of this measure, statistical clustering procedures can be employed to group the coordinates into clusters of manageable dimensions and characterized by minimal dependence between coordinates belonging to different clusters.     

The natural DNA bending angle in the lac repressor headpiece-O1 operator complex is determined by protein-DNA contacts and water release

We performed molecular dynamics simulations of the lac repressor headpiece-O1 operator complex for natural, over and underbent DNA to assess the factors that determine the natural DNA bending angle. At the natural angle, the specific and nonspecific contacts between the protein and DNA are optimized. Protein-DNA contacts show different angle dependences in the right and left sites, with the left site generally getting weaker and the right site getting stronger as the bending angle increases. Two entropic factors were identified as well: at the natural bending angle, water release and the quasiharmonic protein configurational entropy are maximized. The gain in protein configurational entropy might stem from an entropy-entropy compensation mechanism, in which a reduction in protein fluctuations is offset by a loss in correlations between the right and left sites.     

Exhaustive search of the configurational space of heat-shock protein 90 with its inhibitor by multicanonical molecular dynamics based dynamic docking

Multicanonical molecular dynamics based dynamic docking was used to exhaustively search the configurational space of an inhibitor binding to the N-terminal domain of heat-shock protein 90 (Hsp90). The obtained structures at 300 K cover a wide structural ensemble, with the top two clusters ranked by their free energy coinciding with the native binding site. The representative structure of the most stable cluster reproduced the experimental binding configuration, but an interesting conformational change in Hsp90 could be observed. The combined effects of solvation and ligand binding shift the equilibrium from a preferred loop-in conformation in the unbound state to an α-helical one in the bound state for the flexible lid region of Hsp90. Thus, our dynamic docking method is effective at predicting the native binding site while exhaustively sampling a wide configurational space, modulating the protein structure upon binding.     

Free energy calculations offer insights into the influence of receptor flexibility on ligand–receptor binding affinities

Docking algorithms for computer-aided drug discovery and design often ignore or restrain the flexibility of the receptor, which may lead to a loss of accuracy of the relative free enthalpies of binding. In order to evaluate the contribution of receptor flexibility to relative binding free enthalpies, two host–guest systems have been examined: inclusion complexes of α-cyclodextrin (αCD) with 1-chlorobenzene (ClBn), 1-bromobenzene (BrBn) and toluene (MeBn), and complexes of DNA with the minor-groove binding ligands netropsin (Net) and distamycin (Dist). Molecular dynamics simulations and free energy calculations reveal that restraining of the flexibility of the receptor can have a significant influence on the estimated relative ligand–receptor binding affinities as well as on the predicted structures of the biomolecular complexes. The influence is particularly pronounced in the case of flexible receptors such as DNA, where a 50% contribution of DNA flexibility towards the relative ligand–DNA binding affinities is observed. The differences in the free enthalpy of binding do not arise only from the changes in ligand–DNA interactions but also from changes in ligand–solvent interactions as well as from the loss of DNA configurational entropy upon restraining.     

Discovery of Elusive K4O6, a Compound Stabilized by Configurational Entropy of Polarons

Synthesis of elusive K₄O₆ has disclosed implications of crucial relevance for new solid materials discovery. K₄O₆ forms in equilibrium from K₂O₂ and KO₂, in an all‐solid state, endothermic reaction at elevated temperature, undergoing back reaction upon cooling to ambient conditions. This tells that the compound is stabilized by entropy alone. Analyzing possible entropic contributions reveals that the configurational entropy of “localized” electrons, i.e., of polaronic quasi‐particles, provides the essential contribution to the stabilization. We corroborate this assumption by measuring the relevant heats of transformation and tracking the origin of entropy of formation computationally. These findings challenge current experimental and computational approaches towards exploring chemical systems for new materials by searching the potential energy landscape: one would fail in detecting candidates that are crucially stabilized by the configurational entropy of localized polarons.