Predicting absolute ligand binding free energies to a simple model site

A central challenge in structure-based ligand design is the accurate prediction of binding free energies. Here we apply alchemical free energy calculations in explicit solvent to predict ligand binding in a model cavity in T4 lysozyme. Even in this simple site, there are challenges. We made systematic improvements, beginning with single poses from docking, then including multiple poses, additional protein conformational changes, and using an improved charge model. Computed absolute binding free energies had an RMS error of 1.9 kcal/mol relative to previously determined experimental values. In blind prospective tests, the methods correctly discriminated between several true ligands and decoys in a set of putative binders identified by docking. In these prospective tests, the RMS error in predicted binding free energies relative to those subsequently determined experimentally was only 0.6 kcal/mol. X-ray crystal structures of the new ligands bound in the cavity corresponded closely to predictions from the free energy calculations, but sometimes differed from those predicted by docking. Finally, we examined the impact of holding the protein rigid, as in docking, with a view to learning how approximations made in docking affect accuracy and how they may be improved.     

Predicting Ligand Binding Affinity with Alchemical Free Energy Methods in a Polar Model Binding Site

We present a combined experimental and modeling study of organic ligand molecules binding to a slightly polar engineered cavity site in T4 lysozyme (L99A/M102Q). For modeling, we computed alchemical absolute binding free energies. These were blind tests performed prospectively on 13 diverse, previously untested candidate ligand molecules. We predicted that eight compounds would bind to the cavity and five would not; 11 of 13 predictions were correct at this level. The RMS error to the measurable absolute binding energies was 1.8 kcal/mol. In addition, we computed “relative” binding free energies for six phenol derivatives starting from two known ligands: phenol and catechol. The average RMS error in the relative free energy prediction was 2.5 kcal/mol (phenol) and 1.1 kcal/mol (catechol). To understand these results at atomic resolution, we obtained x-ray co-complex structures for nine of the diverse ligands and for all six phenol analogs. The average RMSD of the predicted pose to the experiment was 2.0 Å (diverse set), 1.8 Å (phenol-derived predictions), and 1.2 Å (catechol-derived predictions). We found that predicting accurate affinities and rank-orderings required near-native starting orientations of the ligand in the binding site. Unanticipated binding modes, multiple ligand binding, and protein conformational change all proved challenging for the free energy methods. We believe that these results can help guide future improvements in physics-based absolute binding free energy methods.     

X-ray structural and simulation analysis of a protein mutant: the value of a combined approach

The effect of the mutation Arg 96 to His on the stability of bacteriophage T4 lysozyme has been previously studied by calorimetric experiments, X-ray crystallography, and free energy simulation techniques. The experimental and calculated values for the difference between the free energy of denaturation of the mutant and the wild type are in reasonable agreement. However, the two approaches led to different explanations for the loss in stability. To analyze the differences, a series of refinements based on the crystallographic data were performed, a number of aspects of the simulations were reexamined, and continuum electrostatic calculations were done to complement the latter. The results of those comparisons provide a better understanding of the origin of the free energy difference in this mutant. Furthermore, they show the importance of the combined use of simulations and crystallography for interpreting the effects of mutations on the energetics of the system.     

Lattice models, packing density, and Boltzmann-like distribution of cavities in proteins

A model reproducing the experimental Boltzmann-like distribution of empty cavity sizes in proteins is introduced. Proteins are represented by lattices of different dimensionalities, corresponding to different numbers of nearest neighbor contacts. Small cavities emerge and join into larger ones in a random process that can be related to random mutations. Simulations of cavity creation are performed under the constraint of a limiting total packing density. Cavities sufficiently large (20 A(3) or more), that they might accommodate at least one additional methyl group produced by a mutation, are counted and compared to the distribution of cavities according to their sizes from protein statistics. The distributions calculated with this very simple model within a realistic range of packing densities are in good agreement with the empirical cavity distribution. The results suggest that the Boltzmann-like distribution of cavities in proteins might be affected by a mechanism controlled by limiting packing density and maximum allowed protein destabilization. This supports an earlier suggestion that the agreement between the free energies of cavity formation from the mutational experiments and from the statistics of the empty cavity distribution in X-ray protein structures is nonfortuitous. A possible relation of the suggested model to the Boltzmann hypothesis is discussed.     

A molecular dynamics study of thermodynamic and structural aspects of the hydration of cavities in proteins.

The structure and activity of a protein molecule are strongly influenced by the extent of hydration of its cavities. This is, in turn, related to the free energy change on transfer of a water molecule from bulk solvent into a cavity. Such free energy changes have been calculated for two cavities in a sulfate-binding protein. One of these cavities contains a crystallographically observed water molecule while the other does not. Thermodynamic integration and perturbation methods were used to calculate free energies of hydration for each of the cavities from molecular dynamics simulations of two separate events: the removal of a water molecule from pure water, and the introduction of a water molecule into each protein cavity. From the simulations for the pure water system, the excess chemical potential of water was computed to be -6.4 +/- 0.4 kcal/mol, in accord with experiment and with other recent theoretical calculations. For the protein cavity containing an experimentally observed water molecule, the free energy change on hydrating it with one water molecule was calculated as -10.0 +/- 1.3 kcal/mol, indicating the high probability that this cavity is occupied by a water molecule. By contrast, for the cavity in which no water molecules were experimentally observed, the free energy change on hydrating it with one water molecule was calculated as 0.2 +/- 1.5 kcal/mol, indicating its low occupancy by water. The agreement of these results with experiment suggests that thermodynamic simulation methods may become useful for the prediction and analysis of internal hydration in proteins.     

Thermodynamics of water mediating protein-ligand interactions in cytochrome P450cam: a molecular dynamics study.

Ordered water molecules are observed by crystallography and nuclear magnetic resonance to mediate protein-ligand interactions. Here, we examine the energetics of hydrating cavities formed at protein-ligand interfaces using molecular dynamics simulations. The free energies of hydrating two cavities in the active site of two liganded complexes of cytochrome P450cam were calculated by multiconfigurational thermodynamic integration. The complex of cytochrome P450cam with 2-phenyl-imidazole contains a crystallographically well defined water molecule mediating hydrogen bonds between the protein and the inhibitor. We calculate that this water molecule is stabilized by a binding free energy of -11.6 +/- kJ/mol. The complex of cytochrome P450cam with its natural substrate, camphor, contains a cavity that is empty in the crystal structure although a water molecule in it could make a hydrogen bond to camphor. Here, solvation of this cavity is calculated to be unfavorable by +15.8 +/- 5.0 kJ/mol. The molecular dynamics simulations can thus distinguish a hydrated interfacial cavity from an empty one. They also provide support for the notion that protein-ligand complexes can accommodate empty interfacial cavities and that such cavities are likely to be unhydrated unless more than one hydrogen bond can be made to a water molecule in the cavity.     

The investigation of hydrocarbon cracking reaction energetics with composite energy methods

Hydrocarbo cracking reactions are one of the most commonly encountered reactions in the petroleum industry, and the energetics of the reactions are crucial in understanding the reaction mechanisms and predicting reaction rates. In this work, a modified composite energy method (CBS-RAD(MP2)) is created as a version of the CBS-RAD method which gives accurate energetics for hydrocarbon free radical reactions. It replaces the time consuming QCISD(fc)/6-31g* method in the geometry optimization and frequency calculation steps with MP2(full)/6-31g* level calculations. The accuracy of the new CBS-RAD(MP2) method is compared with the widely used G2, G3 and CBS-QB3 composite methods for predicting heats of reaction and activation barriers of 14 hydrocarbon cracking reactions. We find that the new CBS-RAD(MP2) method has the second least RMS error of 1.22 kcal/mol for heats of reaction calculations. For activation energy calculations, the new CBS-RAD(MP2) method has the least RMS error of 1.37 kcal/mol. Moreover, the CBS-RAD(MP2) method was found to require only 81% of the computational time required compared to the CBS-QB3 method, 32% of G3 and 15% of the G2 method, making it an attractive alternative for predicting hydrocarbon cracking reaction energetics.     

Direct comparison of experimental and calculated folding free energies for hydrophobic deletion mutants of chymotrypsin inhibitor 2: free energy perturbation calculations using transition and denatured states from molecular dynamics simulations of unfolding

Previous molecular dynamics (MD) simulations of thermal denaturation of chymotrypsin inhibitor 2 (CI2) have provided transition-state models in good agreement with experiment. Unfortunately, however, the comparisons have been necessarily indirect. The simulations have provided detailed structural information but not energetics, while from experiment, structure is inferred from a ratio of free energy changes upon mutation (Phi values). Here, direct comparison with experimental free energies is obtained by performing free energy perturbation calculations of hydrophobic deletion mutants of CI2 using transition- and denatured-state structures from various denaturation MD simulations. The agreement between the calculated and experimental DeltaDeltaG and Phi values is quite good (R = 0.8-0.9). In addition, given the availability of realistic atomic models for the denatured protein, the common approach of using small peptides to represent the denatured state in stability calculations can now be evaluated. To this end, two different extended tripeptide models were used: one using the sequence from the protein with the residue to be mutated in the center and the other with this residue surrounded by Ala residues. The results for the two peptides agree neither with one another nor with the different full-length denatured-state models, which do provide results in good agreement with experiment. This finding is noteworthy because the denatured state of CI2 is very disrupted with little residual structure, such that the peptides might have been expected to serve as reasonable models. Overall the calculations presented here validate our previous MD-generated transition- and denatured-state models and therefore the simulated unfolding pathways and their relevance to refolding.     

Quantification of helix-helix binding affinities in micelles and lipid bilayers

A theoretical approach for estimating association free energies of alpha-helices in nonpolar media has been developed. The parameters of energy functions have been derived from DeltaDeltaG values of mutants in water-soluble proteins and partitioning of organic solutes between water and nonpolar solvents. The proposed approach was verified successfully against three sets of published data: (1) dissociation constants of alpha-helical oligomers formed by 27 hydrophobic peptides; (2) stabilities of 22 bacteriorhodopsin mutants, and (3) protein-ligand binding affinities in aqueous solution. It has been found that coalescence of helices is driven exclusively by van der Waals interactions and H-bonds, whereas the principal destabilizing contributions are represented by side-chain conformational entropy and transfer energy of atoms from a detergent or lipid to the protein interior. Electrostatic interactions of alpha-helices were relatively weak but important for reproducing the experimental data. Immobilization free energy, which originates from restricting rotational and translational rigid-body movements of molecules during their association, was found to be less than 1 kcal/mole. The energetics of amino acid substitutions in bacteriorhodopsin was complicated by specific binding of lipid and water molecules to cavities created in certain mutants.     

Molecular Dynamics Free Energy Calculations to Assess the Possibility of Water Existence in Protein Nonpolar Cavities

Are protein nonpolar cavities filled with water molecules? Although many experimental and theoretical investigations have been done, particularly for the nonpolar cavity of IL-1β, the results are still conflicting. To study this problem from the thermodynamic point of view, we calculated hydration free energies of four protein nonpolar cavities by means of the molecular dynamics thermodynamic integration method. In addition to the IL-1β cavity (69 ų), we selected the three largest nonpolar cavities of AvrPphB (81 ų), Trp repressor (87 ų), and hemoglobin (108 ų) from the structural database, in view of the simulation result from another study that showed larger nonpolar cavities are more likely to be hydrated. The calculations were performed with flexible and rigid protein models. The calculated free energy changes were all positive; hydration of the nonpolar cavities was energetically unfavorable for all four cases. Because hydration of smaller cavities should happen more rarely, we conclude that existing protein nonpolar cavities are not likely to be hydrated. Although a possibility remains for much larger nonpolar cavities, such cases are not found experimentally. We present a hypothesis to explain this: hydrated nonpolar cavities are quite unstable and the conformation could not be maintained.     

Incorporating protein conformational flexibility into the calculation of pH-dependent protein properties.

A method for combining calculations of residue pKa's with changes in the position of polar hydrogens has been developed. The Boltzmann distributions of proton positions in hydroxyls and neutral titratable residues are found in the same Monte Carlo sampling procedure that determines the amino acid ionization states at each pH. Electrostatic, Lennard-Jones potentials, and torsion angle energies are considered at each proton position. Many acidic and basic residues are found to have significant electrostatic interactions with either a water- or hydroxyl-containing side chain. Protonation state changes are coupled to reorientation of the neighboring hydroxyl dipoles, resulting in smaller free energy differences between neutral and ionized residues than when the protein is held rigid. Multiconformation pH titration gives better agreement with the experimental pKa's for triclinic hen egg lysozyme than conventional rigid protein calculations. The hydroxyl motion significantly increases the protein dielectric response, making it sensitive to the composition of the local protein structure. More than one conformer per residue is often found at a given pH, providing information about the distribution of low-energy lysozyme structures.     

Molecular mechanisms of resistance: free energy calculations of mutation effects on inhibitor binding to HIV-1 protease

The changes in the inhibitor binding constants due to the mutation of isoleucine to valine at position 84 of HIV-1 protease are calculated using molecular dynamics simulations. The calculations are done for three potent inhibitors--KNI-272, L-735,524 (indinavir or MK-639), and Ro 31-8959 (saquinavir). The calculations agree with the experimental data both in terms of an overall trend and in the magnitude of the resulting free energy change. HIV-1 protease is a homodimer, so each mutation causes two changes in the enzyme. The decrease in the binding free energy from each mutated side chain differs among the three inhibitors and correlates well with the size of the cavities induced in the protein interior near the mutated residue. The cavities are created as a result of a mutation to a smaller side chain, but the cavities are less than would be predicted from the wild-type structures, indicating that there is significant relaxation to partially fill the cavities.     

Development, validation, and application of adapted PEOE charges to estimate pKa values of functional groups in protein-ligand complexes

For routine pK(a) calculations of protein-ligand complexes in drug design, the PEOE method to compute partial charges was modified. The new method is applicable to a large scope of proteins and ligands. The adapted charges were parameterized using experimental free energies of solvation of amino acids and small organic ligands. For a data set of 80 small organic molecules, a correlation coefficient of r(2) = 0.78 between calculated and experimental solvation free energies was obtained. Continuum electrostatics pK(a) calculations based on the Poisson-Boltzmann equation were carried out on a validation set of nine proteins for which 132 experimental pK(a) values are known. In total, an overall RMSD of 0.88 log units between calculated and experimentally determined data is achieved. In particular, the predictions of significantly shifted pK(a) values are satisfactory, and reasonable estimates of protonation states in the active sites of lysozyme and xylanase could be obtained. Application of the charge-assignment and pK(a)-calculation procedure to protein-ligand complexes provides clear structural interpretations of experimentally observed changes of protonation states of functional groups upon complex formation. This information is essential for the interpretation of thermodynamic data of protein-ligand complex formation and provides the basis for the reliable factorization of the free energy of binding in enthalpic and entropic contributions. The modified charge-assignment procedure forms the basis for future automated pK(a) calculations of protein-ligand complexes.     

On the attribution of binding energy in antigen-antibody complexes McPC 603, D1.3, and HyHEL-5

Using X-ray coordinates of antigen-antibody complexes McPC 603, D1.3, and HyHEL-5, we made semiquantitative estimates of Gibbs free energy changes (delta G) accompanying noncovalent complex formation of the McPC 603 Fv fragment with phosphocholine and the D1.3 or HyHEL-5 Fv fragments with hen egg white lysozyme. Our empirical delta G function, which implicitly incorporates solvent effects, has the following components: hydrophobic force, solvent-modified electrostatics, changes in side-chain conformational entropy, translational/overall rotational entropy changes, and the dilutional (cratic) entropy term. The calculated delta G ranges matched the experimentally determined delta G of McPC 603 and D1.3 complexes and overestimated it (i.e., gave a more negative value) in the case of HyHEL-5. Relative delta G contributions of selected antibody residues, calculated for HyHEL-5 complexes, agreed with those determined independently in site-directed mutagenesis experiments. Analysis of delta G attribution in all three complexes indicated that only a small number of amino acids probably contribute actively to binding energetics. These form a subset of the total antigen-antibody contact surface. In the antibodies, the bottom part of the antigen binding cavity dominated the energetics of binding whereas in lysozyme, the energetically most important residues defined small (2.5-3 nm2) "energetic" epitopes. Thus, a concept of protein antigenicity emerges that involves the active, attractive contributions mediated by the energetic antigenic epitopes and the passive surface complementarity contributed by the surrounding contact area. The D1.3 energetic epitope of lysozyme involved Gly 22, Gly 117, and Gln 121; the HyHEL-5 epitope consisted of Arg 45 and Arg 68. These are also the essential antigenic residues determined experimentally. The above positions belong to the most protruding parts of the lysozyme surface, and their backbones are not exceptionally flexible. Least-squares analysis of six different antibody binding regions indicated that the geometry of the VH-VL interface beta-barrel is well conserved, giving no indication of significant changes in domain-domain contacts upon complex formation.     

Comparison of intramolecular and intermolecular reactions in protein folding

The intrinsic component of the standard free energy change for the formation of a disulfide bond in a protein molecule is compared to that for an analogous chemical reaction. The former reaction, which represents theintramolecular formation of a disulfide bond in a protein molecule from a cysteine group containing a mixed disulfide bond with glutathione, and a free cysteine residue, is a unimolecular reaction. In contrast, its chemical analogue is a bimolecular reaction, and corresponds to theintermolecular disulfide interchange between a mixed disulfide-bonded compound between a cysteine residue and glutathione, and a free cysteine molecule. The difference in the intrinsic free energy of the above two reactions is estimated by two different approaches. First, a theoretical estimate of the magnitude of the difference in free energy of the two reactions (for a standard state of 1 M) is obtained using a gas-phase statistical thermodynamic approach, which indicates that the intramolecular reaction is energetically favored over its intermolecular counterpart by as much as 15.6 kcal/mole. For comparison, an experimentally derived value is also obtained, using experimental data from a study by Konishi et al. of the regeneration of the protein ribonuclease A (RNase A) from its reduced form by reduced and oxidized glutathiones. The intrinsic component of the free energy change of the intramolecular reaction, as it occurs in the protein molecule, is obtained from such experimental data by accounting explicitly for the free energy change (assumed to be solely an entropy change) pertaining to the conformational changes (ring closure) that the protein molecule undergoes in the course of the reaction. On the basis of the value derived from such an experimental approach, the intramolecular reaction is also energetically more favorable as compared to its intermolecular analogue, but only by a difference of 2.3 kcal/mole (for a standard state of 1 M). The large apparent discrepancy between the two values estimated from the theoretical and experimental approaches is rationalized by the postulation of several additional factors not inherent in the gas-phase theoretical estimate, such as dehydration and intramolecular hydrogen-bonding effects, which can largely compensate for the otherwise favorable energetics of the intramolecular reaction.     

Protein-protein recognition and interaction hot spots in an antigen-antibody complex: free energy decomposition identifies "efficient amino acids"

The molecular mechanics Poisson-Boltzmann surface area (MM/PBSA) method was applied to the study of the protein-protein complex between a camelid single chain variable domain (cAb-Lys3) and hen egg white lysozyme (HEL), and between cAb-Lys3 and turkey egg white lysozyme (TEL). The electrostatic energy was estimated by solving the linear Poisson-Boltzmann equation. A free energy decomposition scheme was developed to determine binding energy hot spots of each complex. The calculations identified amino acids of the antibody that make important contributions to the interaction with lysozyme. They further showed the influence of small structural variations on the energetics of binding and they showed that the antibody amino acids that make up the hot spots are organized in such a way as to mimic the lysozyme substrate. Through further analysis of the results, we define the concept of "efficient amino acids," which can provide an assessment of the binding potential of a particular hot spot interaction. This information, in turn, can be useful in the rational design of small molecules that mimic the antibody. The implications of using free energy decomposition to identify regions of a protein-protein complex that could be targeted by small molecules inhibitors are discussed.     

Energetics of protein folding

The energetics of protein folding determine the 3D structure of a folded protein. Knowledge of the energetics is needed to predict the 3D structure from the amino acid sequence or to modify the structure by protein engineering. Recent developments are discussed: major factors are reviewed and auxiliary factors are discussed briefly. Major factors include the hydrophobic factor (burial of non-polar surface area) and van der Waals interactions together with peptide hydrogen bonds and peptide solvation. The long-standing model for the hydrophobic factor (free energy change proportional to buried non-polar surface area) is contrasted with the packing-desolvation model and the approximate nature of the proportionality between free energy and apolar surface area is discussed. Recent energetic studies of forming peptide hydrogen bonds (gas phase) are reviewed together with studies of peptide solvation in solution. Closer agreement is achieved between the 1995 values for protein unfolding enthalpies in vacuum given by Lazaridis-Archontis-Karplus and Makhatadze-Privalov when the solvation enthalpy of the peptide group is taken from electrostatic calculations. Auxiliary factors in folding energetics include salt bridges and side-chain hydrogen bonds, disulfide bridges, and propensities to form alpha-helices and beta-structure. Backbone conformational entropy is a major energetic factor which is discussed only briefly for lack of knowledge.     

Extracting knowledge from protein structure geometry

Protein structure prediction techniques proceed in two steps, namely the generation of many structural models for the protein of interest, followed by an evaluation of all these models to identify those that are native‐like. In theory, the second step is easy, as native structures correspond to minima of their free energy surfaces. It is well known however that the situation is more complicated as the current force fields used for molecular simulations fail to recognize native states from misfolded structures. In an attempt to solve this problem, we follow an alternate approach and derive a new potential from geometric knowledge extracted from native and misfolded conformers of protein structures. This new potential, Metric Protein Potential (MPP), has two main features that are key to its success. Firstly, it is composite in that it includes local and nonlocal geometric information on proteins. At the short range level, it captures and quantifies the mapping between the sequences and structures of short (7‐mer) fragments of protein backbones through the introduction of a new local energy term. The local energy term is then augmented with a nonlocal residue‐based pairwise potential, and a solvent potential. Secondly, it is optimized to yield a maximized correlation between the energy of a structural model and its root mean square (RMS) to the native structure of the corresponding protein. We have shown that MPP yields high correlation values between RMS and energy and that it is able to retrieve the native structure of a protein from a set of high‐resolution decoys. Proteins 2013. © 2012 Wiley Periodicals, Inc.     

Predicting the structure of protein cavities created by mutation

To assist in the efficient design of protein cavities, we have developed a minimization strategy that can predict with accuracy the fate of cavities created by mutation. We first modelled, under different conditions, the structures of six T4 lysozyme and cytochrome c peroxidase mutants of known crystal structure (where long, hydrophobic, buried side chains have been replaced by shorter ones) by minimizing the virtual structures derived from the corresponding wild-type co-ordinates. An unconstrained pathway together with an all-atom atom representation and a steepest descent minimization yielded modelled structures with lower root mean square deviations (r.m.s.d) from the crystal structures than other conditions. To test whether the method developed was generally applicable to other mutations of the kind, we have then modelled eighteen additional T4 lysozyme, barnase and cytochrome c peroxidase mutants of known crystal structure. The models of both cavity expanding and cavity collapsing mutants closely fit their crystal structures (average r.m.s.d. 0.33 +/- 0.25 A, with only one poorer prediction: L121A). The structure of protein cavities generated by mutation can thus be confidently simulated by energy minimization regardless of the tendency of the cavity to collapse or to expand. We think this is favoured by the fact that the typical response observed in these proteins to cavity-creating mutations is to experience only a limited rearrangement.