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.     

Blind Prediction of Charged Ligand Binding Affinities in a Model Binding Site

Predicting absolute protein–ligand binding affinities remains a frontier challenge in ligand discovery and design. This becomes more difficult when ionic interactions are involved because of the large opposing solvation and electrostatic attraction energies. In a blind test, we examined whether alchemical free-energy calculations could predict binding affinities of 14 charged and 5 neutral compounds previously untested as ligands for a cavity binding site in cytochrome c peroxidase. In this simplified site, polar and cationic ligands compete with solvent to interact with a buried aspartate. Predictions were tested by calorimetry, spectroscopy, and crystallography. Of the 15 compounds predicted to bind, 13 were experimentally confirmed, while 4 compounds were false negative predictions. Predictions had a root-mean-square error of 1.95 kcal/mol to the experimental affinities, and predicted poses had an average RMSD of 1.7 Å to the crystallographic poses. This test serves as a benchmark for these thermodynamically rigorous calculations at predicting binding affinities for charged compounds and gives insights into the existing sources of error, which are primarily electrostatic interactions inside proteins. Our experiments also provide a useful set of ionic binding affinities in a simplified system for testing new affinity prediction methods.     

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.     

Quantifying the Entropy of Binding for Water Molecules in Protein Cavities by Computing Correlations

Protein structural analysis demonstrates that water molecules are commonly found in the internal cavities of proteins. Analysis of experimental data on the entropies of inorganic crystals suggests that the entropic cost of transferring such a water molecule to a protein cavity will not typically be greater than 7.0 cal/mol/K per water molecule, corresponding to a contribution of approximately +2.0 kcal/mol to the free energy. In this study, we employ the statistical mechanical method of inhomogeneous fluid solvation theory to quantify the enthalpic and entropic contributions of individual water molecules in 19 protein cavities across five different proteins. We utilize information theory to develop a rigorous estimate of the total two-particle entropy, yielding a complete framework to calculate hydration free energies. We show that predictions from inhomogeneous fluid solvation theory are in excellent agreement with predictions from free energy perturbation (FEP) and that these predictions are consistent with experimental estimates. However, the results suggest that water molecules in protein cavities containing charged residues may be subject to entropy changes that contribute more than +2.0 kcal/mol to the free energy. In all cases, these unfavorable entropy changes are predicted to be dominated by highly favorable enthalpy changes. These findings are relevant to the study of bridging water molecules at protein-protein interfaces as well as in complexes with cognate ligands and small-molecule inhibitors.     

Accurate PDZ/Peptide Binding Specificity with Additive and Polarizable Free Energy Simulations

PDZ domains contain 80–100 amino acids and bind short C-terminal sequences of target proteins. Their specificity is essential for cellular signaling pathways. We studied the binding of the Tiam1 PDZ domain to peptides derived from the C-termini of its Syndecan-1 and Caspr4 targets. We used free energy perturbation (FEP) to characterize the binding energetics of one wild-type and 17 mutant complexes by simulating 21 alchemical transformations between pairs of complexes. Thirteen complexes had known experimental affinities. FEP is a powerful tool to understand protein/ligand binding. It depends, however, on the accuracy of molecular dynamics force fields and conformational sampling. Both aspects require continued testing, especially for ionic mutations. For six mutations that did not modify the net charge, we obtained excellent agreement with experiment using the additive, AMBER ff99SB force field, with a root mean square deviation (RMSD) of 0.37 kcal/mol. For six ionic mutations that modified the net charge, agreement was also good, with one large error (3 kcal/mol) and an RMSD of 0.9 kcal/mol for the other five. The large error arose from the overstabilization of a protein/peptide salt bridge by the additive force field. Four of the ionic mutations were also simulated with the polarizable Drude force field, which represents the first test of this force field for protein/ligand binding free energy changes. The large error was eliminated and the RMS error for the four mutations was reduced from 1.8 to 1.2 kcal/mol. The overall accuracy of FEP indicates it can be used to understand PDZ/peptide binding. Importantly, our results show that for ionic mutations in buried regions, electronic polarization plays a significant role.     

Halogenated Benzenes Bound within a Non-polar Cavity in T4 Lysozyme Provide Examples of I?S and I?Se Halogen-bonding

We showed earlier that the mutation of Leu99 to alanine in bacteriophage T4 lysozyme creates an internal cavity of volume ∼ 150 Å³ that binds benzene and a variety of other ligands. As such, this cavity provides an excellent target to study protein-ligand interaction. Here, we use low-temperature crystallography and related techniques to analyze the binding of halogen-incorporated benzenes typified by C₆F₅X, where X = H, F, Cl, Br or I, and C₆H₅X, where X = H or I was also studied. Because of the increased electron density of fluorine relative to hydrogen, the geometry of binding of the fluoro compounds can often be determined more precisely than their hydrogen-containing analogs. All of the ligands bind in essentially the same plane but the center of the phenyl ring can translate by up to 1.2 Å. In no case does the ligand rotate freely within the cavity. The walls of the cavity consist predominantly of hydrocarbon atoms, and in several cases it appears that van der Waals interactions define the geometry of binding. In comparing the smallest with the largest ligand, the cavity volume increases from 181 Å³ to 245 Å³. This shows that the protein is flexible and adapts to the size and shape of the ligand. There is a remarkably close contact of 3.0 Å between the iodine atom on C₆F₅I and the sulfur or selenium atom of Met or SeMet102. This interaction is 1.0 Å less than the sum of the van der Waals radii and is a clear example of a so-called halogen bond. Notwithstanding this close approach, the increase in binding energy for the halogen bond relative to a van der Waals contact is estimated to be only about 0.5-0.7 kcal/mol.     

Theoretical study of the ligand-CYP2B4 complexes: effect of structure on binding free energies and heme spin state

The molecular origins of temperature-dependent ligand-binding affinities and ligand-induced heme spin state conversion have been investigated using free energy analysis and DFT calculations for substrates and inhibitors of cytochrome P450 2B4 (CYP2B4), employing models of CYP2B4 based on CYP2C5(3LVdH)/CYP2C9 crystal structures, and the results compared with experiment. DFT calculations indicate that large heme-ligand interactions (ca. -15 kcal/mol) are required for inducing a high to low spin heme transition, which is correlated with large molecular electrostatic potentials (approximately -45 kcal/mol) at the ligand heteroatom. While type II ligands often contain oxygen and nitrogen heteroatoms that ligate heme iron, DFT results indicate that BP and MF heme complexes, with weak substrate-heme interactions (ca. -2 kcal/mol), and modest MEPS minima (>-35 kcal/mol) are high spin. In contrast, heme complexes of the CYP2B4 inhibitor, 4PI, the product of benzphetamine metabolism, DMBP, and water are low spin, have substantial heme-ligand interaction energies (<-15 kcal/mol) and deep MEPS minima (<-45 kcal/mol) near their heteroatoms. MMPBSA analysis of MD trajectories were made to estimate binding free energies of these ligands at the heme binding site of CYP2B4. In order to initially assess the realism of this approach, the binding free energy of 4PI inhibitor was computed and found to be a reasonable agreement with experiment: -7.7 kcal/mol [-7.2 kcal/mol (experiment)]. BP was determined to be a good substrate [-6.3 kcal/mol (with heme-ligand water), -7.3 kcal/mol (without ligand water)/-5.8 kcal/mol (experiment)], whereas the binding of MF was negligible, with only marginal binding binding free energy of -1.7 kcal/mol with 2-MF bound [-3.8 kcal/mol (experiment)], both with and without retained heme-ligand water. Analysis of the free energy components reveal that hydrophobic/nonpolar contributions account for approximately 90% of the total binding free energy of these substrates and are the source of their differential and temperature-dependent CYP2B4 binding. The results indicate the underlying origins of the experimentally observed differential binding affinities of BP and MF, and indicate the plausibility of the use of models derived from moderate sequence identity templates in conjunction with approximate free energy methods in the estimation of ligand-P450 binding affinities.     

Protein surface roughness accounts for binding free energy of Plasmepsin II‐ligand complexes

The calculation of absolute binding affinities for protein‐inhibitor complexes remains as one of the main challenges in computational structure‐based ligand design. The present work explored the calculations of surface fractal dimension (as a measure of surface roughness) and the relationship with experimental binding free energies of Plasmepsin II complexes. Plasmepsin II is an attractive target for novel therapeutic compounds to treat malaria. However, the structural flexibility of this enzyme is a drawback when searching for specific inhibitors. Concerning that, we performed separate explicitly solvated molecular dynamics simulations using the available high‐resolution crystal structures of different Plasmepsin II complexes. Molecular dynamics simulations allowed a better approximation to systems dynamics and, therefore, a more reliable estimation of surface roughness. This constitutes a novel approximation in order to obtain more realistic values of fractal dimension, because previous works considered only x‐ray structures. Binding site fractal dimension was calculated considering the ensemble of structures generated at different simulation times. A linear relationship between binding site fractal dimension and experimental binding free energies of the complexes was observed within 20 ns. Previous studies of the subject did not uncover this relationship. Regression model, coined FD model, was built to estimate binding free energies from binding site fractal dimension values. Leave‐one‐out cross‐validation showed that our model reproduced accurately the absolute binding free energies for our training set (R² = 0.76; <|error|> =0.55 kcal/mol; SD_error = 0.19 kcal/mol). The fact that such a simple model may be applied raises some questions that are addressed in the article.     

Structure-activity relationship for extracellular block of K+ channels by tetraalkylammonium ions

External tetraalkylammonium ion binding to potassium channels is studied using microscopic molecular modelling methods and the experimental structure of the KcsA channel. Relative binding free energies of the KcsA complexes with Me4N+, Et4N+, and n-Pr4N+ are calculated with the molecular dynamics free energy perturbation approach together with automated ligand docking. The four-fold symmetry of the entrance cavity formed by the Tyr82 residues is found to provide stronger binding for the D2d than for the S4 conformation of the ligands. In agreement with experiment the Et4N+ blocker shows several kcal/mol better binding than the other tetraalkylammonium ions.     

Binding energetics of phosphorus-containing inhibitors of thermolysin

The importance of a specific hydrogen bond between thermolysin and a phosphonamidate inhibitor, Z-NHCH2-PO(O-)-Leu-Leu (1) [Bartlett, P. A., & Marlowe, C. K. (1987) Science (Washington D.C.) 235, 569-571], has been reevaluated. We have determined the inhibition constants (binding free energies) for thermolysin of phosphonamidate n-hexyl-P(O)(O-)-Leu-Trp-NHMe (4), phosphonate n-hexyl-P-(O)(O-)OCH(iBu)CO-Trp-NHMe (5), and phosphinates n-hexyl-P(O)(O-)CH2CH(iBu)CO-Trp-NHMe (6) and Z-NHCH2PO(O-)CH2CH(iBu)CO-Leu (3). Replacement of the P-NH group by P-CH2 (1----3 and 4----6) weakens the overall binding free energy by about 1.5 kcal/mol. A negligible difference in solvation energy has been measured for these pairs, and the basicity of the P-O- ligand for zinc in each pair remains nearly unchanged as determined by pH titration of their 31P NMR resonances. Therefore, this value of 1.5 kcal/mol can be assigned to the specific hydrogen bond known to exist between the P-NH of 1 and thermolysin [Tronrud, D. E., Holden, H. M., & Matthews, B. W. (1987) Science (Washington, D.C.) 235, 871-574] and inferred to exist between 4 and the enzyme. Substitution of P-O for P-NH (1----2 [Bartlett, P. A., & Marlowe, C. K. (1987) Science (Washington, D.C.) 235, 569-571] and 4----5) weakens the overall binding free energy by 4.1 kcal/mol for each pair as the basicity of the P-O- ligand decreases by about 1.6 pH units. The measured solvation energy difference between 4 and 5 (and by inference between 1 and 2) is negligible.(ABSTRACT TRUNCATED AT 250 WORDS)     

Theoretical study on the interaction of titanocene dichloride with deoxyguanosine monophosphate

The completely hydrolyzed titanocene dichloride, [Cp₂Ti(H₂O)₂]²⁺ binding to guanine (G) and phosphate group sites of DNA were investigated by DFT method, with using deoxyguanosine monophosphate (dGMP) as incoming ligand. In the first substitutions, the calculations reveal that the diaquated titanocene binding to O6 shows the lowest activation free energy with 17.9 kcal/mol, closely followed by N7 is 20.5 kcal/mol and the O of phosphate group is 26.3 kcal/mol, respectively. It was also found that all the titanation processes are mildly endothermic. In addition, for the Ti-B(dGMP) in all separated products, the bond dissociation free energies (BDFE) of Ti-O(P, P = phosphate) is higher than those of Ti-N7/O6. In the second substitutions, the reactions leading to the didentate adducts are considered. For bidentate-bridging N7, O6 binding mode, the path of the metal Ti binding to O6 has the lower activation free energy (11.3 kcal/mol) than that of the metal Ti binding to N7 (15.3 kcal/mol). For the bidentate-bridging N7, O(P) binding mode, the path of the metal Ti binding to O(P) has the lower activation free energies (25.3 kcal/mol) than that of the metal Ti binding to N7 (26.2 kcal/mol).     

Absolute binding free energy calculations using molecular dynamics simulations with restraining potentials

The absolute (standard) binding free energy of eight FK506-related ligands to FKBP12 is calculated using free energy perturbation molecular dynamics (FEP/MD) simulations with explicit solvent. A number of features are implemented to improve the accuracy and enhance the convergence of the calculations. First, the absolute binding free energy is decomposed into sequential steps during which the ligand-surrounding interactions as well as various biasing potentials restraining the translation, orientation, and conformation of the ligand are turned "on" and "off." Second, sampling of the ligand conformation is enforced by a restraining potential based on the root mean-square deviation relative to the bound state conformation. The effect of all the restraining potentials is rigorously unbiased, and it is shown explicitly that the final results are independent of all artificial restraints. Third, the repulsive and dispersive free energy contribution arising from the Lennard-Jones interactions of the ligand with its surrounding (protein and solvent) is calculated using the Weeks-Chandler-Andersen separation. This separation also improves convergence of the FEP/MD calculations. Fourth, to decrease the computational cost, only a small number of atoms in the vicinity of the binding site are simulated explicitly, while all the influence of the remaining atoms is incorporated implicitly using the generalized solvent boundary potential (GSBP) method. With GSBP, the size of the simulated FKBP12/ligand systems is significantly reduced, from approximately 25,000 to 2500. The computations are very efficient and the statistical error is small ( approximately 1 kcal/mol). The calculated binding free energies are generally in good agreement with available experimental data and previous calculations (within approximately 2 kcal/mol). The present results indicate that a strategy based on FEP/MD simulations of a reduced GSBP atomic model sampled with conformational, translational, and orientational restraining potentials can be computationally inexpensive and accurate.     

Estimation of the binding affinities of FKBP12 inhibitors using a linear response method.

A series of non-immunosuppressive inhibitors of FK506 binding protein (FKBP12) are investigated using Monte Carlo statistical mechanics simulations. These small molecules may serve as scaffolds for chemical inducers of protein dimerization, and have recently been found to have FKBP12-dependent neurotrophic activity. A linear response model was developed for estimation of absolute binding free energies based on changes in electrostatic and van der Waals energies and solvent-accessible surface areas, which are accumulated during simulations of bound and unbound ligands. With average errors of 0.5 kcal/mol, this method provides a relatively rapid way to screen the binding of ligands while retaining the structural information content of more rigorous free energy calculations.     

Reciprocal cooperative effects of multiple ligand binding to pyruvate kinase

The formation of multiple ligand complexes with muscle pyruvate kinase was measured in terms of dissociation constants and the standard free energies of formation were calculated. The binding of Mn2+ to the enzyme (KA = 55 +/- 5 X 10(-6) M; deltaF degrees = -5.75 +/- 0.05 kcal/mol) and to the enzyme saturated with phosphoenolpyruvate (conditional free energy) KA' = 0.8 +/- 0.4 X 10(-6) M; deltaF degrees = -8.22 +/- 0.34 kcal/mol) has been measured under identical conditions giving a free energy of coupling, delta(deltaF degrees) = -2.47 +/- 0.34 kcal/mol. Such a large negative free energy of coupling is diagnostic of a strong positively cooperative effect in ligand binding. The binding of the substrate phosphoenolpyruvate to free enzyme and the enzyme-Mn2+ complex was, by necessity, measured by different methods. The free energy of phosphoenolpyruvate binding to free enzyme (KS = 1.58 +/- 0.10 X 10(-4)M; deltaF degrees = -5.13 +/- 0.04 kcal/mol) and to the enzyme-Mn2+ complex (K3 = 0.75 +/- 0.10 X 10(-6)M; deltaF degrees = -8.26 +/- 0.07 kcal/mol) also gives a large negative free energy of coupling, delta(deltaF degrees) = -3.16 +/- 0.08 kcal/mol. Such a large negative value confirms reciprocal binding effects between the divalent cation and the substrate phosphoenolpyruvate. The binding of Mn2+ to the enzyme-ADP complex was also investigated and a free energy of coupling, delta(deltaF degrees) = -0.08 +/- 0.08 kcal/mol, was measured, indicative of little or no cooperativity in binding. The free energy of coupling with Mn2+ and pyruvate was measured as -1.52 +/- 0.14 kcal/mol, showing a significant amount of cooperativity in ligand binding but a substantially smaller effect than that observed for phosphoenolpyruvate binding. The magnitude of the coupling free energy may be related to the role of the divalent cation in the formation of the enzyme-substrate complexes. In the absence of the activating monovalent cation, the coupling free energies for phosphoenolpyruvate and pyruvate binding decrease by 40-60% and 25%, respectively, substantiating a role for the monovalent cation in the formation of enzyme-substrate complexes with phosphoenolpyruvate and with pyruvate.     

Glycogen phosphorylase inhibitors: a free energy perturbation analysis of glucopyranose spirohydantoin analogues

GP catalyzes the phosphorylation of glycogen to Glc-1-P. Because of its fundamental role in the metabolism of glycogen, GP has been the target for a systematic structure-assisted design of inhibitory compounds, which could be of value in the therapeutic treatment of type 2 diabetes mellitus. The most potent catalytic-site inhibitor of GP identified to date is spirohydantoin of glucopyranose (hydan). In this work, we employ MD free energy simulations to calculate the relative binding affinities for GP of hydan and two spirohydantoin analogues, methyl-hydan and n-hydan, in which a hydrogen atom is replaced by a methyl- or amino group, respectively. The results are compared with the experimental relative affinities of these ligands, estimated by kinetic measurements of the ligand inhibition constants. The calculated binding affinity for methyl-hydan (relative to hydan) is 3.75 +/- 1.4 kcal/mol, in excellent agreement with the experimental value (3.6 +/- 0.2 kcal/mol). For n-hydan, the calculated value is 1.0 +/- 1.1 kcal/mol, somewhat smaller than the experimental result (2.3 +/- 0.1 kcal/mol). A free energy decomposition analysis shows that hydan makes optimum interactions with protein residues and specific water molecules in the catalytic site. In the other two ligands, structural perturbations of the active site by the additional methyl- or amino group reduce the corresponding binding affinities. The computed binding free energies are sensitive to the preference of a specific water molecule for two well-defined positions in the catalytic site. The behavior of this water is analyzed in detail, and the free energy profile for the translocation of the water between the two positions is evaluated. The results provide insights into the role of water molecules in modulating ligand binding affinities. A comparison of the interactions between a set of ligands and their surrounding groups in X-ray structures is often used in the interpretation of binding free energy differences and in guiding the design of new ligands. For the systems in this work, such an approach fails to estimate the order of relative binding strengths, in contrast to the rigorous free energy treatment.     

Theoretical calculation of the binding free energies for pyruvate dehydrogenase E1 binding with ligands

We have tested a computational protocol based on molecular mechanics-Poisson-Boltzmann surface area (MM-PBSA) free-energy calculations to examine the detailed microscopic structures and binding free energies for the pyruvate dehydrogenase multienzyme complex (PDHc) E1 binding with its ligands (cofactor and inhibitors). The calculated binding free energies are all in good agreement with available experimental data, with an average absolute deviation of approximately 0.7 kcal/mol, suggesting that the computational protocol tested may be valuable in future rational design of new, more potent inhibitors of PDHc E1.     

Analysis of the binding energies of testosterone, 5alpha-dihydrotestosterone,androstenedione and dehydroepiandrosterone sulfate with an antitestosterone antibody

Molecular dynamics simulations and molecular mechanics-Poisson-Boltzmann surface area (MM-PBSA) free energy calculations were used to study the binding of testosterone (TES), 5alpha-dihydrotestosterone (5ADHT), androstenedione (AND), and dehydroepiandrosterone sulfate (DHEAS) to the monoclonal antitestosterone antibody 3-C(4)F(5). The relative binding free energy of TES and AND was also calculated with free energy perturbation (FEP) simulations. The antibody 3-C(4)F(5) has a relatively high affinity (3 x 10(8) M(-1)) and on overall good binding profile for testosterone but its cross-reactivity with DHEAS has been the main reason for the failure to use this antibody in clinical immunoassays. The relative binding free energies obtained with the MM-PBSA method were 1.5 kcal/mol for 5ADHT, 3.8 kcal/mol for AND, and 4.3 kcal/mol for DHEAS, as compared to TES. When a water molecule of the ligand binding site, observed in the antibody-TES crystal structure, was explicitly included in MM-PBSA calculations, the relative binding energies were 3.4, 4.9, and 5.4 kcal/mol for 5ADHT, AND, and DHEAS, respectively. The calculated numbers are in correct order but larger than the corresponding experimental energies of 1.3, 1.5, and 2.6 kcal/mol, respectively. The fact that the MM-PBSA method reproduced the relative binding free energies of DHEAS, a steroid having a negatively charged sulfate group, and the neutrally charged TES, 5ADHT, and AND in satisfactory agreement with experiment shows the robustness of the method in predicting relative binding affinities. The 800-ps FEP simulations predicted that the antibody 3-C(4)F(5) binds TES 1.3 kcal/mol tighter than AND. Computational mutagenesis of selected amino acid residues of the ligand binding site revealed that the lower affinities of AND and DHEAS as compared to TES are due to a combined effect of several residues, each contributing a small fraction to the tighter binding of TES. An exception to this is Tyr99H, whose mutation to Ala lowered the binding of DHEAS 0.7 kcal/mol more than the binding of TES. This is probably due to the hydrogen bonding interaction formed between the OH group of Tyr99H and the sulfate group of DHEAS. Computational mutagensis data also showed that the affinity of the steroids to the antitestosterone antibody 3-C(4)F(5) would be enhanced if Trp47H were repositioned so that it would make more extensive contacts with the bound ligands. In addition, the binding of steroids to antitestosterone, antiprogesterone, and antiestradiol antibodies is discussed.     

The effect of multiple binding modes on empirical modeling of ligand docking to proteins.

A popular approach to the computational modeling of ligand/receptor interactions is to use an empirical free energy like model with adjustable parameters. Parameters are learned from one set of complexes, then used to predict another set. To improve these empirical methods requires an independent way to study their inherent errors. We introduce a toy model of ligand/receptor binding as a workbench for testing such errors. We study the errors incurred from the two state binding assumption--the assumption that a ligand is either bound in one orientation, or unbound. We find that the two state assumption can cause large errors in free energy predictions, but it does not affect rank order predictions significantly. We show that fitting parameters using data from high affinity ligands can reduce two state errors; so can using more physical models that do not use the two state assumption. We also find that when using two state models to predict free energies, errors are more severe on high affinity ligands than low affinity ligands. And we show that two state errors can be diagnosed by systematically adding new binding modes when predicting free energies: if predictions worsen as the modes are added, then the two state assumption in the fitting step may be at fault.     

What determines the van der Waals coefficient beta in the LIE (linear interaction energy) method to estimate binding free energies using molecular dynamics simulations?

Recently a semiempirical method has been proposed by Aqvist et al. to calculate absolute and relative binding free energies. In this method, the absolute binding free energy of a ligand is estimated as deltaGbind = alpha + beta, where Vel(bound) and Vvdw(bound) are the electrostatic and van der Waals interaction energies between the ligand and the solvated protein from an molecular dynamics (MD) trajectory with ligand bound to protein and Vel(free) and Vel(free) and Vvdw(free) are the electrostatic and van der Waals interaction energies between the ligand and the water from an MD trajectory with the ligand in water. A set of values, alpha = 0.5 and beta = 0.16, was found to give results in good agreement with experimental data. Later, however, different optimal values of beta were found in studies of compounds binding to P450cam and avidin. The present work investigates how the optimal value of beta depends on the nature of binding sites for different protein-ligand interactions. By examining seven ligands interacting with five proteins, we have discovered a linear correlation between the value of beta and the weighted non-polar desolvation ratio (WNDR), with a correlation coefficient of 0.96. We have also examined the ability of this correlation to predict optimal values of beta for different ligands binding to a single protein. We studied twelve neutral compounds bound to avidin. In this case, the WNDR approach gave a better estimate of the absolute binding free energies than results obtained using the fixed value of beta found for biotin-avidin. In terms of reproducing the relative binding free energy to biotin, the fixed-beta value gave better results for compounds similar to biotin, but for compounds less similar to biotin, the WNDR approach led to better relative binding free energies.     

Binding of ionic ligands to polyelectrolytes.

Ionic ligands can bind to polyelectrolytes such as DNA or charged polysaccharides. We develop a Poisson-Boltzmann treatment to compute binding constants as a function of ligand charge and salt concentration in the limit of low ligand concentration. For flexible chain ligands, such as oligopeptides, we treat their conformations using lattice statistics. The theory predicts the salt dependence and binding free energies, of Mg(2+) ions to polynucleotides, of hexamine cobalt(III) to calf thymus DNA, of polyamines to T7 DNA, of oligolysines to poly(U) and poly(a), and of tripeptides to heparin, a charged polysaccharide. One parameter is required to obtain absolute binding constants, the distance of closest separation of the ligand to the polyion. Some, but not all, of the binding entropies and enthalpies are also predicted accurately by the model.