Can Absolute Free Energies of Association Be Estimated from Molecular Mechanical Simulations? The Biotin−Streptavidin System Revisited

Impact of the K24N mutation on the transactivation domain of p53 and its binding to murine double-minute clone 2

The level of the p53 transcription factor is negatively regulated by the E3 ubiquitin ligase murine double-minute clone 2 (MDM2). The interaction between p53 and MDM2 is essential for the maintenance of genomic integrity for most eukaryotes. Previous structural studies revealed that MDM2 binds to p53 transactivation domain (p53TAD) from residues 17 to 29. The K24N mutation of p53TAD changes a lysine at position 24 to an asparagine. This mutation occurs naturally in the bovine family and is also found in a rare form of human gestational cancer called choriocarcinoma. In this study, we have investigated how the K24N mutation affects the affinity, structure, and dynamics of p53TAD binding to MDM2. Nuclear magnetic resonance studies of p53TAD show that the K24N mutant is more flexible and has less transient helical secondary structure than the wild type. Isothermal titration calorimetry measurements demonstrate that these changes in structure and dynamics do not significantly change the binding affinity for p53TAD-MDM2. Finally, free-energy perturbation and standard molecular dynamic simulations suggest the negligible affinity change is due to a compensating interaction energy between the K24N mutant and the MDM2 when it is bound. Overall, the data suggest that the K24N-MDM2 complex is able to, at least partly, compensate for an increase in the conformational entropy in unbound K24N with an increase in the bound-state electrostatic interaction energy.

Efficient Computation of Small-Molecule Configurational Binding Entropy and Free Energy Changes by Ensemble Enumeration

Here we present a novel, end-point method using the dead-end-elimination and A* algorithms to efficiently and accurately calculate the change in free energy, enthalpy, and configurational entropy of binding for ligand–receptor association reactions. We apply the new approach to the binding of a series of human immunodeficiency virus (HIV-1) protease inhibitors to examine the effect ensemble reranking has on relative accuracy as well as to evaluate the role of the absolute and relative ligand configurational entropy losses upon binding in affinity differences for structurally related inhibitors. Our results suggest that most thermodynamic parameters can be estimated using only a small fraction of the full configurational space, and we see significant improvement in relative accuracy when using an ensemble versus single-conformer approach to ligand ranking. We also find that using approximate metrics based on the single-conformation enthalpy differences between the global minimum energy configuration in the bound as well as unbound states also correlates well with experiment. Using a novel, additive entropy expansion based on conditional mutual information, we also analyze the source of ligand configurational entropy loss upon binding in terms of both uncoupled per degree of freedom losses as well as changes in coupling between inhibitor degrees of freedom. We estimate entropic free energy losses of approximately +24 kcal/mol, 12 kcal/mol of which stems from loss of translational and rotational entropy. Coupling effects contribute only a small fraction to the overall entropy change (1–2 kcal/mol) but suggest differences in how inhibitor dihedral angles couple to each other in the bound versus unbound states. The importance of accounting for flexibility in drug optimization and design is also discussed.

Modelling and analysis of early aggregation events of BMHP1-derived self-assembling peptides

Despite the increasing use and development of peptide-based scaffolds in different fields including that of regenerative medicine, the understanding of the factors governing the self-assembly process and the relationship between sequence and properties have not yet been fully understood. BMHP1-derived self-assembling peptides (SAPs) have been developed and characterized showing that biotinylation at the N-terminal cap corresponds to better performing assembly and scaffold biomechanics. In this study, the effects of biotinylation on the self-assembly dynamics of seven BMHP1-derived SAPs have been investigated by molecular dynamics simulations. We confirmed that these SAPs self-assemble into β-structures and that proline acts as a β-breaker of the assembled aggregates. In biotinylated peptides, the formation of ordered β-structured aggregates is triggered by both the establishment of a dense and dynamic H-bonds network and the formation of a 'hydrophobic wall' available to interact with other peptides. Such conditions result from the peculiar chemical composition of the biotinyl-cap, given by the synergic cooperation of the uracil function of the ureido ring with the high hydrophobic portion consisting of the thiophenyl ring and valeryl chain. The inbuilt propensity of biotinylated peptides towards the formation of ordered small aggregates makes them ideal precursors of higher hierarchically organized self-assembled nanostructures as experimentally observed.

Charge-leveling and proper treatment of long-range electrostatics in all-atom molecular dynamics at constant pH

Recent development of constant pH molecular dynamics (CpHMD) methods has offered promise for adding pH-stat in molecular dynamics simulations. However, until now the working pH molecular dynamics (pHMD) implementations are dependent in part or whole on implicit-solvent models. Here we show that proper treatment of long-range electrostatics and maintaining charge neutrality of the system are critical for extending the continuous pHMD framework to the all-atom representation. The former is achieved here by adding forces to titration coordinates due to long-range electrostatics based on the generalized reaction field method, while the latter is made possible by a charge-leveling technique that couples proton titration with simultaneous ionization or neutralization of a co-ion in solution. We test the new method using the pH-replica-exchange CpHMD simulations of a series of aliphatic dicarboxylic acids with varying carbon chain length. The average absolute deviation from the experimental pK(a) values is merely 0.18 units. The results show that accounting for the forces due to extended electrostatics removes the large random noise in propagating titration coordinates, while maintaining charge neutrality of the system improves the accuracy in the calculated electrostatic interaction between ionizable sites. Thus, we believe that the way is paved for realizing pH-controlled all-atom molecular dynamics in the near future.

Toward detection of DNA-bound proteins using solid-state nanopores: insights from computer simulations

Through all-atom molecular dynamics simulations, we explore the use of nanopores in thin synthetic membranes for detection and identification of DNA binding proteins. Reproducing the setup of a typical experiment, we simulate electric field driven transport of DNA-bound proteins through nanopores smaller in diameter than the proteins. As model systems, we use restriction enzymes EcoRI and BamHI specifically and nonspecifically bound to a fragment of dsDNA, and streptavidin and NeutrAvidin proteins bound to dsDNA and ssDNA via a biotin linker. Our simulations elucidate the molecular mechanics of nanopore-induced rupture of a protein-DNA complex, the effective force applied to the DNA-protein bond by the electrophoretic force in a nanopore, and the role of DNA-surface interactions in the rupture process. We evaluate the ability of the nanopore ionic current and the local electrostatic potential measured by an embedded electrode to report capture of DNA, capture of a DNA-bound protein, and rupture of the DNA-protein bond. We find that changes in the strain on dsDNA can reveal the rupture of a protein-DNA complex by altering both the nanopore ionic current and the potential of the embedded electrode. Based on the results of our simulations, we suggest a new method for detection of DNA binding proteins that utilizes peeling of a nicked double strand under the electrophoretic force in a nanopore.

Standard binding free energies from computer simulations: What is the best strategy?

Accurate prediction of standard binding free energies describing protein:ligand association remains a daunting computational endeavor. This challenge is rooted to a large extent in the considerable changes in conformational, translational and rotational entropies underlying the binding process that atomistic simulations cannot easily capture. In spite of significant methodological advances, reflected in a continuously improving agreement with experiment, a characterization of alternate strategies aimed at measuring binding affinities, notably their respective advantages and drawbacks, is somewhat lacking. Here, two distinct avenues to determine the standard binding free energy are compared in the case of a short, proline-rich peptide associating to the Src homology domain 3 of tyrosine kinase Abl. These avenues - one relying upon alchemical transformations and the other on potentials of mean force (PMFs) - invoke a series of geometrical restraints acting on collective variables designed to alleviate sampling limitations inherent to classical molecular dynamics simulations. The experimental binding free energy of ΔG_bind = -7.99 kcal/mol is well reproduced by the two strategies developed herein, with ΔG_bind = -7.7 for the alchemical route and ΔG_bind = -7.8 kcal/mol for the alternate PMF-based route. In detailing the underpinnings of these numerical strategies devised for the accurate determination of standard binding free energies, many practical elements of the proposed rigorous, conceptual framework are clarified, thereby paving way to tackle virtually any recognition and association phenomenon.

Absolute free energy of binding of avidin/biotin, revisited

The binding of biotin to avidin is one of the strongest in nature with absolute free energy of binding, ΔA(0) = -20.4 kcal/mol. Therefore, this complex became a target for a large number of computational studies, which all, however, are based on approximate techniques or simplified models and have led to a wide range of results Therefore, ΔA(0) is calculated here by rigorous statistical mechanical methods and models that consider long-range electrostatics. (1) We apply our method, "hypothetical scanning molecular dynamics with thermodynamic integration" (HSMD-TI) to avidin-biotin modeled by periodic boundary conditions with particle mesh ewald (PME). (2) We apply the double decoupling method (DDM) to this system modeled by the spherical solvent boundary potential (SSBP) and the generalized solvent boundary potential (GSBP). The corresponding results for neutral biotin, ΔA(0) = -29.1 ± 0.8 and -25.2 ± 0.5 kcal/mol are significantly lower than the experimental value; we also provide the result for a charged biotin, ΔA(0) = -33.3 ± 0.8 kcal/mol. It is plausible to suggest that this disagreement with the experiment may stem from ignoring the (positive) contribution of a mobile loop that changes its structure upon ligand binding.

Good practices in free-energy calculations

As access to computational resources continues to increase, free-energy calculations have emerged as a powerful tool that can play a predictive role in a wide range of research areas. Yet, the reliability of these calculations can often be improved significantly if a number of precepts, or good practices, are followed. Although the theory upon which these good practices rely has largely been known for many years, it is often overlooked or simply ignored. In other cases, the theoretical developments are too recent for their potential to be fully grasped and merged into popular platforms for the computation of free-energy differences. In this contribution, the current best practices for carrying out free-energy calculations using free energy perturbation and nonequilibrium work methods are discussed, demonstrating that at little to no additional cost, free-energy estimates could be markedly improved and bounded by meaningful error estimates. Monitoring the probability distributions that underlie the transformation between the states of interest, performing the calculation bidirectionally, stratifying the reaction pathway, and choosing the most appropriate paradigms and algorithms for transforming between states offer significant gains in both accuracy and precision.

Inclusion mechanism of steroid drugs into beta-cyclodextrins. Insights from free energy calculations

The inclusion of hydrocortisone, progesterone, and testosterone into the cavity of beta-cyclodextrin (beta-CD) following two possible orientations was investigated using molecular dynamics simulations and free-energy calculations. The free-energy profiles that delineate the inclusion process were determined using an adaptive biasing force. The present results reveal that although the free-energy surfaces feature two local minima corresponding to a partial and a complete inclusion, the former mode is markedly preferred, irrespective of the orientation. Ranking the propensity of the three steroidal molecules to associate with beta-CD, viz. progesterone>testosterone>hydrocortisone, is shown to be in excellent agreement with experiment. This conclusion is further supported by independent calculations relying on alchemical transformations in conjunction with free energy perturbation, wherein the relative binding free energy for the three steroids was estimated. In addition, decomposition of the potentials of mean force into free-energy contributions and significant decrease in the total hydrophobic surface area suggest that by and large, van der Waals and hydrophobic interactions constitute the main driving forces responsible for the formation of the inclusion complexes. Analysis of their structural features from the molecular dynamics trajectories brings to light different hydrogen-bonding patterns that are characterized by distinct dynamics and stabilities.

Absolute FKBP binding affinities obtained via nonequilibrium unbinding simulations

We compute the absolute binding affinities for two ligands bound to the FKBP protein using nonequilibrium unbinding simulations. The methodology is straightforward requiring little or no modification to many modern molecular simulation packages. The approach makes use of a physical pathway, eliminating the need for complicated alchemical decoupling schemes. We compare our nonequilibrium results to those obtained via a fully equilibrium approach and to experiment. The results of this study suggest that to obtain accurate results using nonequilibrium approaches one should use the stiff-spring approximation with the second cumulant expansion. From this study we conclude that nonequilibrium simulation could provide a simple means to estimate protein-ligand binding affinities.

Mechanism and energetics of charybdotoxin unbinding from a potassium channel from molecular dynamics simulations

Ion channel-toxin complexes are ideal systems for computational studies of protein-ligand interactions, because, in most cases, the channel axis provides a natural reaction coordinate for unbinding of a ligand and a wealth of physiological data is available to check the computational results. We use a recently determined structure of a potassium channel-charybdotoxin complex in molecular dynamics simulations to investigate the mechanism and energetics of unbinding. Pairs of residues on the channel protein and charybdotoxin that are involved in the binding are identified, and their behavior is traced during umbrella-sampling simulations as charybdotoxin is moved away from the binding site. The potential of mean force for the unbinding of charybdotoxin is constructed from the umbrella sampling simulations using the weighted histogram analysis method, and barriers observed are correlated with specific breaking of interactions and influx of water molecules into the binding site. Charybdotoxin is found to undergo conformational changes as a result of the reaction coordinate choice--a nontrivial decision for larger ligands--which we explore in detail, and for which we propose solutions. Agreement between the calculated and the experimental binding energies is obtained once the energetic consequences of these conformational changes are included in the calculations.

Binding free energy prediction in strongly hydrophobic biomolecular systems

We present a comparison of various computational approaches aiming at predicting the binding free energy in ligand-protein systems where the ligand is located within a highly hydrophobic cavity. The relative binding free energy between similar ligands is obtained by means of the thermodynamic integration (TI) method and compared to experimental data obtained through isothermal titration calorimetry measurements. The absolute free energy of binding prediction was obtained on a similar system (a pyrazine derivative bound to a lipocalin) by TI, potential of mean force (PMF) and also by means of the MMPBSA protocols. Although the TI protocol performs poorly either with an explicit or an implicit solvation scheme, the PMF calculation using an implicit solvation scheme leads to encouraging results, with a prediction of the binding affinity being 2 kcal mol(-1) lower than the experimental value. The use of an implicit solvation scheme appears to be well suited for the study of such hydrophobic systems, due to the lack of water molecules within the binding site.

Calculation of absolute ligand binding free energy to a ribosome-targeting protein as a function of solvent model

A comparative analysis is provided of the effect of different solvent models on the calculation of a potential of mean force (PMF) for determining the absolute binding affinity of the small molecule inhibitor pteroic acid bound to ricin toxin A-chain (RTA). Solvent models include the distance-dependent dielectric constant, several different generalized Born (GB) approximations, and a hybrid explicit/GB-based implicit solvent model. We found that the simpler approximation of dielectric screening and a GB model, with Born radii fitted to a switching-window dielectric-boundary surface Poisson solvent model, severely overpredicted the binding affinity as compared to the experimental value, estimated to range from -4.4 to -6.0 kcal/mol. In contrast, GB models that are parametrized to fit the Lee-Richards molecular surface performed much better, predicting binding free energy within 1-3 kcal/mol of experimental estimates. However, the predicted free-energy profiles of these GB models displayed alternative binding modes not observed in the crystal structure. Finally, the most rigorous and computationally costly approach in this work, which used a hybrid explicit/implicit solvent model, correctly determined a binding funnel in the PMF near the crystallographic bound state and predicted an absolute binding affinity that was 2 kcal/mol more favorable than the estimated experimental binding affinity.

Free energy calculations applied to membrane proteins

Selected applications of free energy calculations to the realm of membrane proteins are reviewed. The theoretical underpinnings of these calculations are described, focusing on free energy perturbation and the use of thermodynamic integration to determine free energy changes along well-delineated order parameters. Current strategies for improving the reliability of free energy calculations, while making them somewhat more affordable are outlined. Application of the free energy methodology to understand the structure and function of membrane proteins is illustrated in three concrete examples: The binding of an agonist ligand to a G protein-coupled receptor, the assisted transport of a small permeant through a membrane channel, and the recognition and association of transmembrane alpha-helical domains.

Computation of entropy contribution to protein-ligand binding free energy

The entropy contribution DeltaS to protein-ligand binding free energy is studied for nine protein-lipid complexes. The entropy effect from the loss of the translational/rotational degrees of freedom (DeltaStr) is calculated using the ideal gas approach. The change in the vibrational entropy (DeltaSvib) is calculated using the effective quantum oscillator approach with frequencies derived from the coordinate covariance matrix, so the inharmonic effects are taken into account. The change in the entropy of solvation (DeltaSsolv) is considered using the binomial cell model (developed by the authors) for the hydrophobic effect. The entropy contribution from loss of conformations that are available for the free ligand (DeltaSconf) is also estimated. It is revealed that the negative in view of binding term DeltaStr is only partly compensated by increasing of DeltaSvib, so T(DeltaStr+DeltaSvib+DeltaSconf)<0 for all complexes under investigation, but taking into account DeltaSsolv leads to significantly increased DeltaS. For all complexes except biotin-streptavidin, the results are found to be in reasonable agreement with experimental data.

Calculation of Protein-Ligand Binding Affinities

Accurate methods of computing the affinity of a small molecule with a protein are needed to speed the discovery of new medications and biological probes. This paper reviews physics-based models of binding, beginning with a summary of the changes in potential energy, solvation energy, and configurational entropy that influence affinity, and a theoretical overview to frame the discussion of specific computational approaches. Important advances are reported in modeling protein-ligand energetics, such as the incorporation of electronic polarization and the use of quantum mechanical methods. Recent calculations suggest that changes in configurational entropy strongly oppose binding and must be included if accurate affinities are to be obtained. The linear interaction energy (LIE) and molecular mechanics Poisson-Boltzmann surface area (MM-PBSA) methods are analyzed, as are free energy pathway methods, which show promise and may be ready for more extensive testing. Ultimately, major improvements in modeling accuracy will likely require advances on multiple fronts, as well as continued validation against experiment.

Ligand configurational entropy and protein binding

The restriction of a small molecule's motion on binding to a protein causes a loss of configurational entropy, and thus a penalty in binding affinity. Some energy models used in computer-aided ligand design neglect this entropic penalty, whereas others account for it based on an expected drop in the number of accessible rotamers upon binding. However, the validity of the physical assumptions underlying the various approaches is largely unexamined. The present study addresses this issue by using Mining Minima calculations to analyze the association of amprenavir with HIV protease. The computed loss in ligand configurational entropy is large, contributing approximately 25 kcal/mol (4.184 kJ/kcal) to DeltaG degrees. Most of this loss results from narrower energy wells in the bound state, rather than a drop in the number of accessible rotamers. Coupling among rotation/translation and internal degrees of freedom complicates the decomposition of the entropy change into additive terms. The results highlight the potential to gain affinity by designing conformationally restricted ligands and have implications for the formulation of energy models for ligand scoring.

Unbinding of retinoic acid from the retinoic acid receptor by random expulsion molecular dynamics

Unbinding pathways of retinoic acid (RA) bound to retinoic acid receptor (RAR) have been explored by the random expulsion molecular dynamics (REMD) method. Our results show that RA may exit the binding site of RAR through flexible regions close to the H1-H3 loop and beta-sheets, without displacing H12 from its agonist position. This result may explain kinetic differences between agonist and antagonist ligands observed for other nuclear receptors. The extended and flexible structure of RA initiated a methodological study in a simplified two-dimensional model system. The REMD force should in general be distributed to all atoms of the ligand to obtain the most unbiased results, but for a ligand which is tightly bound in the binding pocket through a strong electrostatic interaction, application of the REMD force on a single atom is preferred.

Grid-based steered thermodynamic integration accelerates the calculation of binding free energies

The calculation of binding free energies is important in many condensed matter problems. Although formally exact computational methods have the potential to complement, add to, and even compete with experimental approaches, they are difficult to use and extremely time consuming. We describe a Grid-based approach for the calculation of relative binding free energies, which we call Steered Thermodynamic Integration calculations using Molecular Dynamics (STIMD), and its application to Src homology 2 (SH2) protein cell signalling domains. We show that the time taken to compute free energy differences using thermodynamic integration can be significantly reduced: potentially from weeks or months to days of wall-clock time. To be able to perform such accelerated calculations requires the ability to both run concurrently and control in realtime several parallel simulations on a computational Grid. We describe how the RealityGrid computational steering system, in conjunction with a scalable classical MD code, can be used to dramatically reduce the time to achieve a result. This is necessary to improve the adoption of this technique and further allows more detailed investigations into the accuracy and precision of thermodynamic integration. Initial results for the Src SH2 system are presented and compared to a reported experimental value. Finally, we discuss the significance of our approach.

Robust and accurate method for free-energy calculation of charged molecular systems

A new approach is presented to eliminate the problem of creation and/or annihilation of atoms in free-energy calculations of charged molecular systems. The method employs a damping potential in the Ewald summation scheme, which is an exact solution of the electrostatics for three-dimensional periodic systems. The proposed method enables entire molecules to be mutated from a noninteracting (ideal) state in an efficient and robust way, thus providing a means by which accurate absolute free energies of structurally complex molecules can be determined. This methodology will enable chemical and phase equilibria to be determined for large molecular species with significant charge distributions, e.g., biomolecules and drugs.

Incorporating the effect of ionic strength in free energy calculations using explicit ions

The incorporation of explicit ions to mimic the effect of ionic strength or to neutralize the overall charge on a system in free energy calculations using molecular dynamics simulations is investigated. The difference in the free energy of hydration between two triosephosphate isomerase inhibitors calculated at five different ion concentrations is used as an example. We show that the free energy difference can be highly sensitive to the presence of explicit ions even in cases where the mutation itself does not involve a change in the overall charge. The effect is most significant if the molecule carries a net charge close to the site mutated. Furthermore, it is shown that the introduction of a small number of ions can lead to very severe sampling problems suggesting that in practical calculations convergence can best be achieved by incorporating either no counterions or by simulating at high ionic strength to ensure sufficient sampling of the ion distribution.

Calculation of affinities of peptides for proteins

Several methodologies were employed to calculate the Gibbs standard free energy of binding for a collection of protein-ligand complexes, where the ligand is a peptide and the protein is representative for various protein families. Almost 40 protein-ligand complexes were employed for a continuum approach, which considers the protein and the peptide at the atomic level, but includes solvent as a polarizable continuum. Five protein-ligand complexes were employed for an all-atom approach that relies on a combination of the double decoupling method with thermodynamic integration and molecular dynamics. These affinities were also computed by means of the linear interaction energy method. Although it generally proved rather difficult to predict the absolute free energies correctly, for some protein families the experimental ranking order was correctly reproduced by the continuum and all-atom approach. Considerable attention has also been given to correctly analyze the affinities of charged peptides, where it is required to judge the effect of one or more ions that are being decoupled in an all-atom approach to preserve electroneutrality. The various methods are further judged upon their merits.

Contributions to the binding free energy of ligands to avidin and streptavidin

The free energy of binding of a ligand to a macromolecule is here formally decomposed into the (effective) energy of interaction, reorganization energy of the ligand and the macromolecule, conformational entropy change of the ligand and the macromolecule, and translational and rotational entropy loss of the ligand. Molecular dynamics simulations with implicit solvation are used to evaluate these contributions in the binding of biotin, biotin analogs, and two peptides to avidin and streptavidin. We find that the largest contribution opposing binding is the protein reorganization energy, which is calculated to be from 10 to 30 kcal/mol for the ligands considered here. The ligand reorganization energy is also significant for flexible ligands. The translational/rotational entropy is 4.5-6 kcal/mol at 1 M standard state and room temperature. The calculated binding free energies are in the correct range, but the large statistical uncertainty in the protein reorganization energy precludes precise predictions. For some complexes, the simulations show multiple binding modes, different from the one observed in the crystal structure. This finding is probably due to deficiencies in the force field but may also reflect considerable ligand flexibility.

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.