Free energy derivatives: A new method for probing the convergence problem in free energy calculations

Assessing the Performance of Screening MM/PBSA in Protein-Ligand Interactions

Accurate calculation of the binding free energies between a protein and a ligand is the primary objective of structure-based drug design, but it still remains a challenging problem. In this work, we apply the screening molecular mechanics/Poisson Boltzmann surface area (MM/PBSA) method to calculate the binding affinity of protein-ligand interactions. Our results show that the performance of the screening MM/PBSA is better than that of the standard MM/PBSA, especially in a charged-ligand system. In addition, we also investigate the effect of the solute dielectric constant on the results, and find that the optimal solute dielectric constants are different between the neutral-ligand system and the charged-ligand system. Moreover, we also evaluate the effect of the atomic-charge methods on the performance of the screening MM/PBSA. The present study demonstrates that the screening MM/PBSA should be a reliable method for calculating binding energy of biosystems.

Quantifying the Binding Interaction between the Hypoxia-Inducible Transcription Factor and the von Hippel-Lindau Suppressor

The hypoxia-inducible transcription factors (HIF) play a central role in the human oxygen sensing signaling pathway. The binding of the von Hippel-Lindau tumor suppressor protein (pVHL)-ElonginC-ElonginB complex (VCB) to HIF-1α is highly selective for the trans-4-hydroxylation form of when Pro564 in the C-terminal oxygen-dependent degradation domain (ODDD) of HIF-1α. The binding of HIFα for VCB is increased by ∼1000-fold upon addition of a single hydroxyl group to either of two conserved proline-residues. Here, we address how this addition governs selective recognition and characterizes the strength of the interaction of this "switch-like" signaling event. A new set of molecular mechanics parameters for 4-hydroxyproline has been developed following the CHARMM force field philosophy. Using the free energy perturbation (FEP) formalism, the difference in the binding free energies between HIF-1α in the nonhydroxylated and hydroxylated forms with the VCB complex was estimated using over 3 μs of MD trajectories. These results can favorably be compared to an experimental value of ∼4 kcal mol(-1). It is observed that the optimized hydrogen bonding network to the buried hydroxyprolyl group confers precise discrimination between hydroxylated and unmodified prolyl residues. These observations provide insight that will aid in developing therapeutic agents that block HIF-α recognition by pVHL.

Designing molecular complexes using free-energy derivatives from liquid-state integral equation theory

Complex formation between molecules in solution is the key process by which molecular interactions are translated into functional systems. These processes are governed by the binding or free energy of association which depends on both direct molecular interactions and the solvation contribution. A design goal frequently addressed in pharmaceutical sciences is the optimization of chemical properties of the complex partners in the sense of minimizing their binding free energy with respect to a change in chemical structure. Here, we demonstrate that liquid-state theory in the form of the solute-solute equation of the reference interaction site model provides all necessary information for such a task with high efficiency. In particular, computing derivatives of the potential of mean force (PMF), which defines the free-energy surface of complex formation, with respect to potential parameters can be viewed as a means to define a direction in chemical space toward better binders. We illustrate the methodology in the benchmark case of alkali ion binding to the crown ether 18-crown-6 in aqueous solution. In order to examine the validity of the underlying solute-solute theory, we first compare PMFs computed by different approaches, including explicit free-energy molecular dynamics simulations as a reference. Predictions of an optimally binding ion radius based on free-energy derivatives are then shown to yield consistent results for different ion parameter sets and to compare well with earlier, orders-of-magnitude more costly explicit simulation results. This proof-of-principle study, therefore, demonstrates the potential of liquid-state theory for molecular design problems.

How does the dehydration change the host-guest association under homogeneous and heterogeneous conditions?

In this study, the thermodynamic properties of association of some inorganic ions (ClO4(-) and SO4(2-)) with β-cyclodextrins (β-CD) in aqueous solution are determined under both free β-CD and surface confined β-CD conditions using atomistic simulations. The potential of mean force (PMF) is calculated as a function of the environment and the thermodynamic properties of association are deduced by integrating the free energy profiles. No inclusion complex between SO4(2-) and β-CD is detected. Nevertheless, the PMF curve obtained for gold-confined CD seems to evidence a small minimum at a larger separation distance that shows specific interactions such as hydrogen bonding outside the cavity. As concerns ClO4(-), our simulations reveal the formation of an inclusion complex with free β-CD in perfect agreement with the available experimental results. Nevertheless, we do not detect any formation of the host-guest inclusion complex under heterogeneous conditions. Finally, the differences observed as a function of the anions are interpreted through an atomistic description. The general trend of weaker complex stabilities with the increasing free energy of hydration of the anions is found in homogeneous systems.

Solvation properties of N-acetyl-β-glucosamine: molecular dynamics study incorporating electrostatic polarization

N-Acetyl-β-glucosamine (NAG) is an important moiety of glycoproteins and is involved in many biological functions. However, conformational and dynamical properties of NAG molecules in aqueous solution, the most common biological environment, remain ambiguous due to limitations of experimental methods. Increasing efforts are made to probe structural properties of NAG and NAG-containing macromolecules, like peptidoglycans and polymeric chitin, at the atomic level using molecular dynamics simulations. In this work, we develop a polarizable carbohydrate force field for NAG and contrast simulation results of various properties using this novel force field and an analogous nonpolarizable (fixed charge) model. Aqueous solutions of NAG and its oligomers are investigated; we explore conformational properties (rotatable bond geometry), electrostatic properties (dipole moment distribution), dynamical properties (self-diffusion coefficient), hydrogen bonding (water bridge structure and dynamics), and free energy of hydration. The fixed-charge carbohydrate force field exhibits deviations from the gas phase relative rotation energy of exocyclic hydroxymethyl side chain and of chair/boat ring distortion. The polarizable force field predicts conformational properties in agreement with corresponding first-principles results. NAG-water hydrogen bonding pattern is studied through radial distribution functions (RDFs) and correlation functions. Intermolecular hydrogen bonding between solute and solvent is found to stabilize NAG solution structures while intramolecular hydrogen bonds define glycosidic linkage geometry of NAG oligomers. The electrostatic component of hydration free energy is highly dependent on force field atomic partial charges, influencing a more favorable free energy of hydration in the fixed-charge model compared to the polarizable model.

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.

Expressions for local contributions to the surface tension from the virial route

The expression of the surface tension using the virial route has been reinvestigated in order to establish a local version of the surface tension and of its long-range corrections. In fact, giving a local surface tension is very important for the simulation from a methodological viewpoint. It is also of basic interest to associate the profile of the intrinsic part of the surface tension with that of the long-range corrections to make the surface tension calculation consistent between the different approaches that can be used. Working expressions for two-phase systems interacting through dispersion-repulsion (Lennard-Jones) and Coulombic (Ewald summation) interactions are proposed. Different operational expressions of the surface tension are compared in the cases of n -pentane, carbon dioxide, and water liquid-vapor equilibria for which the orders of magnitude between the electrostatic and dispersion forces are different.

Polarizable empirical force field for the primary and secondary alcohol series based on the classical Drude model

A polarizable empirical force field based on the classical Drude oscillator has been developed for the aliphatic alcohol series. The model is optimized with emphasis on condensed-phase properties and is validated against a variety of experimental data. Transferability of the developed parameters is emphasized by the use of a single electrostatic model for the hydroxyl group throughout the alcohol series. Aliphatic moiety parameters were transferred from the polarizable alkane parameter set, with only the Lennard-Jones parameters on the carbon in methanol optimized. The developed model yields good agreement with pure solvent properties with the exception of the heats of vaporization of 1-propanol and 1-butanol, which are underestimated by approximately 6%; special LJ parameters for the oxygen in these two molecules that correct for this limitation are presented. Accurate treatment of the free energies of aqueous solvation required the use of atom-type specific O(alcohol)-O(water) LJ interaction terms, with specific terms used for the primary and secondary alcohols. With respect to gas phase properties the polarizable model overestimates experimental dipole moments and quantum mechanical interaction energies with water by approximately 10 and 8 %, respectively, a significant improvement over 44 and 46 % overestimations of the corresponding properties in the CHARMM22 fixed-charge additive model. Comparison of structural properties of the polarizable and additive models for the pure solvents and in aqueous solution shows significant differences indicating atomic details of intermolecular interactions to be sensitive to the applied force field. The polarizable model predicts pure solvent and aqueous phase dipole moment distributions for ethanol centered at 2.4 and 2.7 D, respectively, a significant increase over the gas phase value of 1.8 D, whereas in a solvent of lower polarity, benzene, a value of 1.9 is obtained. The ability of the polarizable model to yield changes in dipole moment as well as the reproduction of a range of condensed phase properties indicates its utility in the study of the properties of alcohols in a variety of condensed phase environments as well as representing an important step in the development of a comprehensive force field for biological molecules.

Calculation of the absolute thermodynamic properties of association of host-guest systems from the intermolecular potential of mean force

The authors report calculations of the intermolecular potential of mean force (PMF) in the case of the host-guest interaction. The host-guest system is defined by a water soluble calixarene and a cation. With an organic cation such as the tetramethylammonium cation, the calixarene forms an insertion complex, whereas with the Lanthane cation, the supramolecular assembly is an outer-sphere complex. The authors apply a modified free energy perturbation method and the force constraint technique to establish the PMF profiles as a function of the separation distance between the host and guest. They use the PMF profile for the calculation of the absolute thermodynamic properties of association that they compare to the experimental values previously determined. They finish by giving some structural features of the insertion and outer-sphere complexes at the Gibbs free energy minimum.

Protein-Ligand Binding Affinity Predictions by Implicit Solvent Simulations: A Tool for Lead Optimization?

Continuum electrostatics is combined with rigorous free-energy calculations in an effort to deliver a reliable and efficient method for in silico lead optimization. The methodology is tested by calculation of the relative binding free energies of a set of inhibitors of neuraminidase, cyclooxygenase2, and cyclin-dependent kinase 2. The calculated free energies are compared to the results obtained with explicit solvent simulations and empirical scoring functions. For cyclooxygenase2, deficiencies in the continuum electrostatics theory are identified and corrected with a modified simulation protocol. For neuraminidase, it is shown that a continuum representation of the solvent leads to markedly different protein-ligand interactions compared to the explicit solvent simulations, and a reconciliation of the two protocols is problematic. Cyclin-dependent kinase 2 proves more challenging, and none of the methods employed in this study yield high quality predictions. Despite the differences observed, for these systems, the use of an implicit solvent framework to predict the ranking of congeneric inhibitors to a protein is shown to be faster, as accurate or more accurate than the explicit solvent protocol, and superior to empirical scoring schemes.

Evaluating the Molecular Mechanics Poisson−Boltzmann Surface Area Free Energy Method Using a Congeneric Series of Ligands to p38 MAP Kinase

The recently described molecular mechanics Poisson-Boltzmann surface area (MM-PBSA) method for calculating free energies is applied to a congeneric series of 16 ligands to p38 MAP kinase whose binding constants span approximately 2 orders of magnitude. These compounds have previously been used to test and compare other free energy calculation methods, including thermodynamic integration (TI), OWFEG, ChemScore, PLPScore, and Dock Energy Score. We find that the MM-PBSA performs relatively poorly for this set of ligands, yielding results much inferior to those from TI or OWFEG, inferior to Dock Energy Score, and not appreciably better than ChemScore or PLPScore but at an appreciably larger computational cost than any of these other methods. This suggests that one should be selective in applying the MM-PBSA method and that for systems that are amenable to other free energy approaches, these other approaches may be preferred. We also examine the single simulation approximation for MM-PBSA, whereby the required ligand and protein trajectories are extracted from a single MD simulation rather than two separate MD runs. This assumption, sometimes used to speed the MM-PBSA calculation, is found to yield significantly inferior results with only a moderate net percentage reduction in total simulation time.

Efficient use of nonequilibrium measurement to estimate free energy differences for molecular systems

A promising method for calculating free energy differences DeltaF is to generate nonequilibrium data via "fast-growth" simulations or by experiments--and then use Jarzynski's equality. However, a difficulty with using Jarzynski's equality is that DeltaF estimates converge very slowly and unreliably due to the nonlinear nature of the calculation--thus requiring large, costly data sets. The purpose of the work presented here is to determine the best estimate for DeltaF given a (finite) set of work values previously generated by simulation or experiment. Exploiting statistical properties of Jarzynski's equality, we present two fully automated analyses of nonequilibrium data from a toy model, and various simulated molecular systems. Both schemes remove at least several k(B)T of bias from DeltaF estimates, compared to direct application of Jarzynski's equality, for modest sized data sets (100 work values), in all tested systems. Results from one of the new methods suggest that good estimates of DeltaF can be obtained using 5-40-fold less data than was previously possible. Extending previous work, the new results exploit the systematic behavior of bias due to finite sample size. A key innovation is better use of the more statistically reliable information available from the raw data.

Free energy of solvation from molecular dynamics simulations for low dielectric solvents

Using molecular dynamics simulation, we present new results for the free energy of solvation for solvents with low dielectric constants (CCl(4), CHCl(3), benzene). The solvation free energy is computed as the sum of three contributions originated at the cavitation of the solute by the solvent, the solute-solvent repulsion and dispersion interactions, and the electrostatic solvation of the solute. The cavitational contribution has been obtained from the Claverie-Pierotti model applied to excluded volumes obtained from distances for nearest neighbor configurations between the solute's atoms and a spherical solvent description. An electrostatic continuum model has been adapted for the computation of the electrostatic free energy of solvation, whereas the van der Waals contribution has been calculated directly from the intermolecular interactions defined by the force fields applied to the simulations. For each solvent, a large set of solute molecules containing most of the chemically interesting functionalities has been treated. The simulated solvation free energies are in very good agreement with experimental data, although a small systematical overestimation of the free energy of solvation indicates a failure of the spherical approach to the solvent molecules in the case of benzene.

Improving the efficiency and reliability of free energy perturbation calculations using overlap sampling methods

A challenge in free energy calculation for complex molecular systems by computer simulation is to obtain a reliable estimate within feasible computational time. In this study, we suggest an answer to this challenge by exploring a simple method, overlap sampling (OS), for producing reliable free-energy results in an efficient way. The formalism of the OS method is based on ensuring sampling of important overlapping phase space during perturbation calculations. This technique samples both forward and reverse free energy perturbation (FEP) to improve the free-energy calculation. It considers the asymmetry of the FEP calculation and features an ability to optimize both the precision and the accuracy of the measurement without affecting the simulation process itself. The OS method is tested at two optimization levels: no optimization (simple OS), and full optimization (equivalent to Bennett's method), and compared to conventional FEP techniques, including the widely used direct FEP averaging method, on three alchemical mutation systems: (a) an anion transformation in water solution, (b) mutation between methanol and ethane, and (c) alchemical change of an adenosine molecule. It is consistently shown that the reliability of free-energy estimates can be greatly improved using the OS techniques at both optimization levels, while the performance of Bennett's method is particularly striking. In addition, the efficiency of a calculation can be significantly improved because the method is able to (a) converge to the right answer quickly, and (b) work for large perturbations. The basic two-stage OS method can be extended to admit additional stages, if needed. We suggest that the OS method can be used as a general perturbation technique for computing free energy differences in molecular simulations.

New approach to free energy of solvation applying continuum models to molecular dynamics simulation

A new approach to the calculation of the free energy of solvation from trajectories obtained by molecular dynamics simulation is presented. The free energy of solvation is computed as the sum of three contributions originated at the cavitation of the solute by the solvent, the solute-solvent nonpolar (repulsion and dispersion) interactions, and the electrostatic solvation of the solute. The electrostatic term is calculated based on ideas developed for the broadly used continuum models, the cavitational contribution from the excluded volume by the Claverie-Pierotti model, and the Van der Waals term directly from the molecular dynamics simulation. The proposed model is tested for diluted aqueous solutions of simple molecules containing a variety of chemically important functions: methanol, methylamine, water, methanethiol, and dichloromethane. These solutions were treated by molecular dynamics simulations using SPC/E water and the OPLS force field for the organic molecules. Obtained free energies of solvation are in very good agreement with experimental data.

Free energy grids: a practical qualitative application of free energy perturbation to ligand design using the OWFEG method

Traditional window-based free energy calculations can precisely determine the free energy corresponding to a molecular change of interest. However, calculations performed in this fashion are typically slow and resource-intensive, which renders them less than ideal for drug design. To circumvent this drawback, a new approximate free energy method, OWFEG, has been developed and tested. OWFEG replaces the exact free energy calculation for a single change with a set of approximate calculations for a grid of possible changes surrounding a molecule. One of the key features of OWFEG is that a floating independent reference frame (FIRF) is used, so that each grid point moves with the region of the molecule to which it is closest. In this way, this approach has been made applicable to flexible molecules. OWFEG is applied to two model systems and then to the FKBP-12.FK506 protein-ligand complex. On the basis of the results of these tests, this approximate method shows promise as a predictive tool for drug design.

Assignment of side-chain conformation using adiabatic energy mapping, free energy perturbation, and molecular dynamic simulations

NMR spectroscopic analysis of the C-terminal Kunitz domain fragment (alpha3(VI)) from the human alpha3-chain of type VI collagen has revealed that the side chain of Trp21 exists in two unequally populated conformations. The major conformation (M) is identical to the conformation observed in the X-ray crystallographic structure, while the minor conformation (m) cannot structurally be resolved in detail by NMR due to insufficient NOE data. In the present study, we have applied: (1) rigid and adiabatic mapping, (2) free energy simulations, and (3) molecular dynamic simulations to elucidate the structure of the m conformer and to provide a possible pathway of the Trp21 side chain between the two conformers. Adiabatic energy mapping of conformations of the Trp21 side chain obtained by energy minimization identified two energy minima: One corresponding to the conformation of Trp21 observed in the X-ray crystallographic structure and solution structure of alpha3(VI) (the M conformation) and the second corresponding to the m conformation predicted by NMR spectroscopy. A transition pathway between the M and m conformation is suggested. The free-energy difference between the two conformers obtained by the thermodynamic integration method is calculated to 1.77+/-0.7 kcal/mol in favor of the M form, which is in good agreement with NMR results. Structural and dynamic properties of the major and minor conformers of the alpha3(VI) molecule were investigated by molecular dynamic. Essential dynamics analysis of the two resulting 800 ps trajectories reveals that when going from the M to the m conformation only small, localized changes in the protein structure are induced. However, notable differences are observed in the mobility of the binding loop (residues Thr13-Ile18), which is more flexible in the m conformation than in the M conformation. This suggests that the reorientation of Trp2 might influence the inhibitory activity against trypsin, despite the relative large distance between the binding loop and Trp21.

The nature of trypsin-pancreatic trypsin inhibitor binding: free energy calculation of Tyr39-->Phe39 mutation in trypsin

The main goal of this work is the detailed study of the binding interactions in the trypsin-pancreatic trypsin inhibitor (PTI) complex and, here, we present how meaningful the Tyr39-Ile19 interaction is to the stability of that particular complex using free energy methods. This knowledge should be very important in the design of new inhibitors for trypsin and enzymes homologous to it. In particular, it could help to decide whether it is possible to produce selective inhibitors for these enzymes by appropriate mutations of residues in the contact region of PTI.

A technique to study molecular recognition in drug design: preliminary application of free energy derivatives to inhibition of a malarial cysteine protease

We present molecular dynamics studies on model complexes of inhibitors of a malarial cysteine protease. The initial model for such complexes came from the model building of the protein using its homology with other cysteine proteases and calculations using DOCK to generate new lead compounds. Some of the initial model-built structures were quite stable for 100 psec of dynamics; others moved significantly from their model-built orientation. We also calculated the free energy derivatives at each atom in the inhibitor, both in water and in the binding site. The results of these calculations suggest directions for the design of new, more potent enzyme inhibitors and agree qualitatively with some of the experimental findings. Nonetheless, we stress that we have only used this methodology in an interpretive rather than a predictive manner.

Parametric sensitivity analysis of avian pancreatic polypeptide (APP)

Computer simulations utilizing a classical force field have been widely used to study biomolecular properties. It is important to identify the key force field parameters or structural groups controlling the molecular properties. In the present paper the sensitivity analysis method is applied to study how various partial charges and solvation parameters affect the equilibrium structure and free energy of avian pancreatic polypeptide (APP). The general shape of APP is characterized by its three principal moments of inertia. A molecular dynamics simulation of APP was carried out with the OPLS/Amber force field and a continuum model of solvation energy. The analysis pinpoints the parameters which have the largest (or smallest) impact on the protein equilibrium structure (i.e., the moments of inertia) or free energy. A display of the protein with its atoms colored according to their sensitivities illustrates the patterns of the interactions responsible for the protein stability. The results suggest that the electrostatic interactions play a more dominant role in protein stability than the part of the solvation effect modeled by the atomic solvation parameters.

Free energy simulations: the meaning of the individual contributions from a component analysis

A theoretical analysis is made of the decomposition into contributions from individual interactions of the free energy calculated by thermodynamic integration. It is demonstrated that such a decomposition, often referred to as "component analysis," is meaningful, even though it is a function of the integration path. Moreover, it is shown that the path dependence can be used to determine the relation of the contribution of a given interaction to the state of the system. To illustrate these conclusions, a simple transformation (Cl- to Br- in aqueous solution) is analyzed by use of the Reference Interaction Site Model-Hypernetted Chain Closure integral equation approach; it avoids the calculational difficulties of macromolecular simulation while retaining their conceptual complexity. The difference in the solvation free energy between chloride and bromide is calculated, and the contributions of the Lennard-Jones and electrostatic terms in the potential function are analyzed by the use of suitably chosen integration paths. The model is also used to examine the path dependence of individual contributions to the double free energy differences (delta delta G or delta delta A) that are often employed in free energy simulations of biological systems. The alchemical path, as contrasted with the experimental path, is shown to be appropriate for interpreting the effects of mutations on ligand binding and protein stability. The formulation is used to obtain a better understanding of the success of the Poisson-Boltzmann continuum approach for determining the solvation properties of polar and ionic systems.

Computer modeling of substrate binding to lipases from Rhizomucor miehei, Humicola lanuginosa, and Candida rugosa

The substrate-binding sites of the triacyl glyceride lipases from Rhizomucor miehei, Humicola lanuginosa, and Candida rugosa were studied by means of computer modeling methods. The space around the active site was mapped by different probes. These calculations suggested 2 separate regions within the binding site. One region showed high affinity for aliphatic groups, whereas the other region was hydrophilic. The aliphatic site should be a binding cavity for fatty acid chains. Water molecules are required for the hydrolysis of the acyl enzyme, but are probably not readily accessible in the hydrophobic interface, in which lipases are acting. Therefore, the hydrophilic site should be important for the hydrolytic activity of the enzyme. Lipases from R. miehei and H. lanuginosa are excellent catalysts for enantioselective resolutions of many secondary alcohols. We used molecular mechanics and dynamics calculations of enzyme-substrate transition-state complexes, which provided information about molecular interactions important for the enantioselectivities of these reactions.