Standard Binding Free Energy Research Articles

Blinded predictions of host-guest standard free energies of binding in the SAMPL5 challenge.

In the context of the SAMPL5 blinded challenge standard free energies of binding were predicted for a dataset of 22 small guest molecules and three different host molecules octa-acids (OAH and OAMe) and a cucurbituril (CBC). Three sets of predictions were submitted, each based on different variations of classical molecular dynamics alchemical free energy calculation protocols based on the double annihilation method. The first model (model A) yields a free energy of binding based on computed free energy changes in solvated and host-guest complex phases; the second (model B) adds long range dispersion corrections to the previous result; the third (model C) uses an additional standard state correction term to account for the use of distance restraints during the molecular dynamics simulations. Model C performs the best in terms of mean unsigned error for all guests (MUE [Formula: see text]-95% confidence interval) for the whole data set and in particular for the octa-acid systems (MUE [Formula: see text]). The overall correlation with experimental data for all models is encouraging ([Formula: see text]). The correlation between experimental and computational free energy of binding ranks as one of the highest with respect to other entries in the challenge. Nonetheless the large MUE for the best performing model highlights systematic errors, and submissions from other groups fared better with respect to this metric.

Binding of Capsaicin to the TRPV1 Ion Channel.

Transient receptor potential (TRP) ion channels constitute a notable family of cation channels involved in the ability of an organisms to detect noxious mechanical, thermal, and chemical stimuli that give rise to the perception of pain, taste, and changes in temperature. One of the most experimentally studied agonist of TRP channels is capsaicin, which is responsible for the burning sensation produced when chili pepper is in contact with organic tissues. Thus, understanding how this molecule interacts and regulates TRP channels is essential to high impact pharmacological applications, particularly those related to pain treatment. The recent publication of a three-dimensional structure of the vanilloid receptor 1 (TRPV1) in the absence and presence of capsaicin from single particle electron cryomicroscopy experiments provides the opportunity to explore these questions at the atomic level. In the present work, molecular docking and unbiased and biased molecular dynamics simulations were employed to generate a structural model of the capsaicin-channel complex. In addition, the standard free energy of binding was estimated using alchemical transformations coupled with conformational, translational, and orientational restraints on the ligand. Key binding modes consistent with previous experimental data are identified, and subtle but essential dynamical features of the binding site are characterized. These observations shed some light into how TRPV1 interacts with capsaicin, and may help to refine design parameters for new TRPV1 antagonists, and potentially guide further developments of TRP channel modulators.

Molecular dynamics study of binding of µ-conotoxin GIIIA to the voltage-gated sodium channel Na(v)1.4.

Homology models of mammalian voltage-gated sodium (NaV) channels based on the crystal structures of the bacterial counterparts are needed to interpret the functional data on sodium channels and understand how they operate. Such models would also be invaluable in structure-based design of therapeutics for diseases involving sodium channels such as chronic pain and heart diseases. Here we construct a homology model for the pore domain of the NaV1.4 channel and use the functional data for the binding of µ-conotoxin GIIIA to NaV1.4 to validate the model. The initial poses for the NaV1.4–GIIIA complex are obtained using the HADDOCK protein docking program, which are then refined in molecular dynamics simulations. The binding mode for the final complex is shown to be in broad agreement with the available mutagenesis data. The standard binding free energy, determined from the potential of mean force calculations, is also in good agreement with the experimental value. Because the pore domains of NaV1 channels are highly homologous, the model constructed for NaV1.4 will provide an excellent template for other NaV1 channels.

Energetics of Hydrophilic Protein-Protein Association and the Role of Water.

Hydrophilic protein-protein interfaces constitute a major part of all protein-protein interfaces and are thus of great importance. However, the quantitative characterization of their association is still an ongoing challenge and the driving force behind their association remains poorly characterized. Here, we have addressed the association of hydrophilic proteins and the role of water by means of extensive molecular dynamics simulations in explicit water using three well studied protein complexes; Barnase-Barstar, cytochrome c-cytochrome c peroxidase, and the N-terminal domain of enzyme I-histidine-containing phosphocarrier. The one-dimensional free energy profiles obtained from umbrella sampling simulations are downhill or, in other words, barrierless. Using these one-dimensional free energy profiles, the computed standard free energies of binding are -12.7 ± 1.1 kcal/mol, -9.4 ± 0.7 kcal/mol, and -8.4 ± 1.9 kcal/mol that are in reasonable to very good agreement with the experimental values of -19.6 kcal/mol, -8.8 kcal/mol, and -7.8 kcal/mol. As expected, analysis of the confined water between the hydrophilic complex partners shows that the density and the orientational order parameter deviate noticeably from the bulk values, especially at close separations of the confining proteins.

Indolicidin Binding Induces Thinning of a Lipid Bilayer

We use all-atom molecular dynamics simulations on a massive scale to compute the standard binding free energy of the 13-residue antimicrobial peptide indolicidin to a lipid bilayer. The analysis of statistical convergence reveals systematic sampling errors that correlate with reorganization of the bilayer on the microsecond timescale and persist throughout a total of 1.4 ms of sampling. Consistent with experimental observations, indolicidin induces membrane thinning, although the simulations significantly overestimate the lipophilicity of the peptide.

Computation of Standard Binding Free Energies of Polar and Charged Ligands to the Glutamate Receptor GluA2

Accurate calculation of the binding affinity of small molecules to proteins has the potential to become an important tool in rational drug design. In this study, we use the free energy perturbation (FEP) method with restraints to calculate the standard binding free energy of five ligands (ACPA, AMPA, CNQX, DNQX, and glutamate) to the glutamate receptor GluA2, which plays an essential role in synaptic transmission. To deal with the convergence problem in FEP calculations with charged ligands, we use a protocol where the ligand is coupled in the binding site while it is decoupled in bulk solution simultaneously. The contributions from the conformational, rotational, and translational entropies to the standard binding free energy are determined by applying/releasing respective restraints to the ligand in bulk/binding site. We also employ the confine-and-release approach, which helps to resolve convergence problems in FEP calculations. Our results are in good agreement with the experimental values for all five ligands, including the charged ones which are often problematic in FEP calculations. We also analyze the different contributions to the binding free energy of each ligand to GluA2 and discuss the nature of these interactions.

Overcoming dissipation in the calculation of standard binding free energies by ligand extraction.

This article addresses calculations of the standard free energy of binding from molecular simulations in which a bound ligand is extracted from its binding site by steered molecular dynamics (MD) simulations or equilibrium umbrella sampling (US). Host-guest systems are used as test beds to examine the requirements for obtaining the reversible work of ligand extraction. We find that, for both steered MD and US, marked irreversibilities can occur when the guest molecule crosses an energy barrier and suddenly jumps to a new position, causing dissipation of energy stored in the stretched molecule(s). For flexible molecules, this occurs even when a stiff pulling spring is used, and it is difficult to suppress in calculations where the spring is attached to the molecules by single, fixed attachment points. We, therefore, introduce and test a method, fluctuation-guided pulling, which adaptively adjusts the spring's attachment points based on the guest's atomic fluctuations relative to the host. This adaptive approach is found to substantially improve the reversibility of both steered MD and US calculations for the present systems. The results are then used to estimate standard binding free energies within a comprehensive framework, termed attach-pull-release, which recognizes that the standard free energy of binding must include not only the pulling work itself, but also the work of attaching and then releasing the spring, where the release work includes an accounting of the standard concentration to which the ligand is discharged.

Efficient Determination of Protein–Protein Standard Binding Free Energies from First Principles

Characterizing protein-protein association quantitatively has been a longstanding challenge for computer simulations. Here, a theoretical framework is put forth that addresses this challenge on the basis of detailed all-atom molecular dynamics simulations with explicit solvent. The proposed methodology relies upon independent potential of mean force (PMF) free-energy calculations carried out sequentially, wherein the biological objects are restrained in the conformation, position and orientation of the bound state, using adequately chosen biasing potentials. These restraints systematically narrow down the configurational entropy available to the system and effectively guarantee that the relevant network of interactions is properly sampled as the two proteins reversibly associate. Decomposition of the binding process into consecutive, well-delineated stages, for both the protein complex and the individual, unbound partners, offers a rigorous definition of the standard state, from which the absolute binding free energy can be determined. The method is applied to the difficult case of the extracellular ribonuclease barnase binding to its intracellular inhibitor barstar. The calculated binding free energy is -21.0 ± 1.4 kcal/mol, which compares well with the experimental value of -19.0 ± 0.2 kcal/mol. The relatively small statistical error reflects the precision and convergence afforded by the PMF-based simulation methodology. In addition to providing an accurate reproduction of the standard binding free energy, the proposed strategy offers a detailed picture of the protein-protein interface, illuminating the thermodynamic forces that underlie reversible association. The application of the present formal framework to barnase:barstar binding provides a foundation for tackling nearly any protein-protein complex.

Accelerating Convergence in Molecular Dynamics Simulations of Solutes in Lipid Membranes by Conducting a Random Walk along the Bilayer Normal.

All molecular dynamics simulations are susceptible to sampling errors, which degrade the accuracy and precision of observed values. The statistical convergence of simulations containing atomistic lipid bilayers is limited by the slow relaxation of the lipid phase, which can exceed hundreds of nanoseconds. These long conformational autocorrelation times are exacerbated in the presence of charged solutes, which can induce significant distortions of the bilayer structure. Such long relaxation times represent hidden barriers that induce systematic sampling errors in simulations of solute insertion. To identify optimal methods for enhancing sampling efficiency, we quantitatively evaluate convergence rates using generalized ensemble sampling algorithms in calculations of the potential of mean force for the insertion of the ionic side chain analog of arginine in a lipid bilayer. Umbrella sampling (US) is used to restrain solute insertion depth along the bilayer normal, the order parameter commonly used in simulations of molecular solutes in lipid bilayers. When US simulations are modified to conduct random walks along the bilayer normal using a Hamiltonian exchange algorithm, systematic sampling errors are eliminated more rapidly and the rate of statistical convergence of the standard free energy of binding of the solute to the lipid bilayer is increased 3-fold. We compute the ratio of the replica flux transmitted across a defined region of the order parameter to the replica flux that entered that region in Hamiltonian exchange simulations. We show that this quantity, the transmission factor, identifies sampling barriers in degrees of freedom orthogonal to the order parameter. The transmission factor is used to estimate the depth-dependent conformational autocorrelation times of the simulation system, some of which exceed the simulation time, and thereby identify solute insertion depths that are prone to systematic sampling errors and estimate the lower bound of the amount of sampling that is required to resolve these sampling errors. Finally, we extend our simulations and verify that the conformational autocorrelation times estimated by the transmission factor accurately predict correlation times that exceed the simulation time scale-something that, to our knowledge, has never before been achieved.

Large scale affinity calculations of cyclodextrin host-guest complexes: Understanding the role of reorganization in the molecular recognition process.

Host-guest inclusion complexes are useful models for understanding the structural and energetic aspects of molecular recognition. Due to their small size relative to much larger protein-ligand complexes, converged results can be obtained rapidly for these systems thus offering the opportunity to more reliably study fundamental aspects of the thermodynamics of binding. In this work, we have performed a large scale binding affinity survey of 57 β-cyclodextrin (CD) host guest systems using the binding energy distribution analysis method (BEDAM) with implicit solvation (OPLS-AA/AGBNP2). Converged estimates of the standard binding free energies are obtained for these systems by employing techniques such as parallel Hamitionian replica exchange molecular dynamics, conformational reservoirs and multistate free energy estimators. Good agreement with experimental measurements is obtained in terms of both numerical accuracy and affinity rankings. Overall, average effective binding energies reproduce affinity rank ordering better than the calculated binding affinities, even though calculated binding free energies, which account for effects such as conformational strain and entropy loss upon binding, provide lower root mean square errors when compared to measurements. Interestingly, we find that binding free energies are superior rank order predictors for a large subset containing the most flexible guests. The results indicate that, while challenging, accurate modeling of reorganization effects can lead to ligand design models of superior predictive power for rank ordering relative to models based only on ligand-receptor interaction energies.

Standard Binding Free Energies from Computer Simulations: What Is the Best Strategy?

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

End-to-end attraction of duplex DNA

Recent experiments [Nakata, M. et al., End-to-end stacking and liquid crystal condensation of 6 to 20 basepair DNA duplexes. Science 2007; 318:1276–1279] have demonstrated spontaneous end-to-end association of short duplex DNA fragments into long rod-like structures. By means of extensive all-atom molecular dynamic simulations, we characterized end-to-end interactions of duplex DNA, quantitatively describing the forces, free energy and kinetics of the end-to-end association process. We found short DNA duplexes to spontaneously aggregate end-to-end when axially aligned in a small volume of monovalent electrolyte. It was observed that electrostatic repulsion of 5′-phosphoryl groups promoted the formation of aggregates in a conformation similar to the B-form DNA double helix. Application of an external force revealed that rupture of the end-to-end assembly occurs by the shearing of the terminal base pairs. The standard binding free energy and the kinetic rates of end-to-end association and dissociation processes were estimated using two complementary methods: umbrella sampling simulations of two DNA fragments and direct observation of the aggregation process in a system containing 458 DNA fragments. We found the end-to-end force to be short range, attractive, hydrophobic and only weakly dependent on the ion concentration. The relation between the stacking free energy and end-to-end attraction is discussed as well as possible roles of the end-to-end interaction in biological and nanotechnological systems.

Accelerating the Rate of Convergence of Umbrella Sampling Simulations in Lipid Bilayers

All molecular dynamics simulations are susceptible to sampling errors, which degrade the accuracy and precision of observed values. Systems containing atomistic lipid bilayers are particularly susceptible to sampling errors because bilayer conformational autocorrelation times can exceed hundreds of nanoseconds. To identify optimal methods for enhancing sampling efficiency, we quantitatively evaluate convergence rates using generalized ensemble sampling algorithms in calculations of the potential of mean force for the insertion of a side chain analog of arginine in a lipid bilayer. Umbrella sampling (US) is used to restrain solute insertion depth along the bilayer normal, the order parameter commonly used in simulations of molecular solutes in lipid bilayers. The rate of statistical convergence of the standard free energy of binding of the solute to the lipid bilayer is increased three-fold when US simulations are modified to conduct random walks along the bilayer normal using Hamiltonian exchange algorithms. We introduce a new metric, computed from simulations conducting random walks along the bilayer normal, to detect sampling barriers in degrees of freedom orthogonal to the US order parameter, which often result in systematic sampling errors but usually remain hidden. This new metric is used to evaluate the height of hidden free energy barriers in order to identify solute insertion depths which are prone to systematic sampling errors. Accordingly, we demonstrate that applying random walks in temperature at these selected insertion depths leads to further increases in the rate of convergence of the binding free energy. Finally, we apply an enhanced US protocol combining random walks in insertion depth and in temperature to quantify the influence of embedded arginine on lipid flip-flop and on the formation of transient water pores, effects that are likely to contribute to the activity of arginine-rich antimicrobial peptides.

Conformational Transitions and Convergence of Absolute Binding Free Energy Calculations

The Binding Energy Distribution Analysis Method (BEDAM) is employed to compute the standard binding free energies of a series of ligands to a FK506 binding protein (FKBP12) with implicit solvation. Binding free energy estimates are in reasonably good agreement with experimental affinities. The conformations of the complexes identified by the simulations are in good agreement with crystallographic data, which was not used to restrain ligand orientations. The BEDAM method is based on λ -hopping Hamiltonian parallel Replica Exchange (HREM) molecular dynamics conformational sampling, the OPLS-AA/AGBNP2 effective potential, and multi-state free energy estimators (MBAR). Achieving converged and accurate results depends on all of these elements of the calculation. Convergence of the binding free energy is tied to the level of convergence of binding energy distributions at critical intermediate states where bound and unbound states are at equilibrium, and where the rate of binding/unbinding conformational transitions is maximal. This finding mirrors similar observations in the context of order/disorder transitions as for example in protein folding. Insights concerning the physical mechanism of ligand binding and unbinding are obtained. Convergence for the largest FK506 ligand is achieved only after imposing strict conformational restraints, which however require accurate prior structural knowledge of the structure of the complex. The analytical AGBNP2 model is found to underestimate the magnitude of the hydrophobic driving force towards binding in these systems characterized by loosely packed protein-ligand binding interfaces. Rescoring of the binding energies using a numerical surface area model corrects this deficiency. This study illustrates the complex interplay between energy models, exploration of conformational space, and free energy estimators needed to obtain robust estimates from binding free energy calculations.

Statistical Convergence of Equilibrium Properties in Simulations of Molecular Solutes Embedded in Lipid Bilayers.

In recent years, atomistic molecular simulations have become a method of choice for studying the interaction of small molecules, peptides, and proteins with biological membranes. Here, we critically examine the statistical convergence of equilibrium properties in molecular simulations of two amino acid side-chain analogs, leucine and arginine, in the presence of a hydrated phospholipid bilayer. To this end, the convergence of the standard binding free energy for the reversible insertion of the solutes in the bilayer is systematically assessed by evaluating dozens of separate sets of umbrella sampling calculations for a total simulation time exceeding 400 μs. We identify rare and abrupt transitions in bilayer structure as a function of solute insertion depth. These transitions correspond to the slow reorganization of ionic interactions involving zwitterionic phospholipid headgroups when the solutes penetrate the lipid-water interface and when arginine is forced through the bilayer center. These rare events are shown to constitute hidden sampling barriers that limit the rate of convergence of equilibrium properties and result in systematic sampling errors. Our analysis demonstrates that the difficulty of attaining convergence for lipid bilayer-embedded solutes has, in general, been drastically underestimated. This information will assist future studies in improving accuracy by selecting a more appropriate reaction coordinate or by focusing computational resources on those regions of the reaction coordinate that exhibit slow convergence of equilibrium properties.

On the Role of Water Models in Quantifying the Binding Free Energy of Highly Conserved Water Molecules in Proteins: The Case of Concanavalin A.

The ability of ligands to displace conserved water molecules in protein binding sites is of significant interest in drug design and is particularly pertinent in the case of glycomimetic drugs. This concept was explored in previous work [ Clarke et al. J. Am. Chem. Soc. 2001 , 123 , 12238 - 12247 and Kadirvelraj et al. J. Am. Chem. Soc. 2008 , 130 , 16933 - 16942 ] for a highly conserved water molecule located in the binding site of the prototypic carbohydrate-binding protein Concanavalin A (Con A). A synthetic ligand was designed with the aim of displacing such water. While the synthetic ligand bound to Con A in an analogous manner to that of the natural ligand, crystallographic analysis demonstrated that it did not displace the conserved water. In order to quantify the affinity of this particular water for the Con A surface, we report here the calculated standard binding free energy for this water in both ligand-bound and free Con A, employing three popular water models: TIP3P, TIP4P, and TIP5P. Although each model was developed to perform well in simulations of bulk-phase water, the computed binding energies for the isolated water molecule displayed a high sensitivity to the model. Both molecular dynamics simulation and free energy results indicate that the choice of water model may greatly influence the characterization of surface water molecules as conserved (TIP5P) or not (TIP3P) in protein binding sites, an observation of considerable significance to rational drug design. Structural and theoretical aspects at the basis of the different behaviors are identified and discussed.

Complete reconstruction of an enzyme-inhibitor binding process by molecular dynamics simulations

The understanding of protein-ligand binding is of critical importance for biomedical research, yet the process itself has been very difficult to study because of its intrinsically dynamic character. Here, we have been able to quantitatively reconstruct the complete binding process of the enzyme-inhibitor complex trypsin-benzamidine by performing 495 molecular dynamics simulations of free ligand binding of 100ns each, 187 of which produced binding events with an rmsd less than 2Å compared to the crystal structure. The binding paths obtained are able to capture the kinetic pathway of the inhibitor diffusing from solvent (S0) to the bound (S4) state passing through two metastable intermediate states S2 and S3. Rather than directly entering the binding pocket the inhibitor appears to roll on the surface of the protein in its transition between S3 and the final binding pocket, whereas the transition between S2 and the bound pose requires rediffusion to S3. An estimation of the standard free energy of binding gives ΔG°=-5.2±0.4kcal/mol (cf. the experimental value -6.2kcal/mol), and a two-states kinetic model k(on)=(1.5±0.2)×10(8)M(-1)s(-1) and k(off)=(9.5±3.3)×10(4)s(-1) for unbound to bound transitions. The ability to reconstruct by simple diffusion the binding pathway of an enzyme-inhibitor binding process demonstrates the predictive power of unconventional high-throughput molecular simulations. Moreover, the methodology is directly applicable to other molecular systems and thus of general interest in biomedical and pharmaceutical research.

Optimized Potential of Mean Force Calculations for Standard Binding Free Energies

The prediction of protein-ligand binding free energies is an important goal of computational biochemistry, yet accuracy, reproducibility, and cost remain a problem. Nevertheless, these are essential requirements for computational methods to become standard binding prediction tools in discovery pipelines. Here, we present the results of an extensive search for an optimal method based on an ensemble of umbrella sampling all-atom molecular simulations tested on the phosphorylated tetrapeptide, pYEEI, binding to the SH2 domain, resulting in an accurate and converged binding free energy of -9.0 ± 0.5 kcal/mol (compared to an experimental value of -8.0 ± 0.1 kcal/mol). We find that a minimum of 300 ns of sampling is required for every prediction, a target easily achievable using new generation accelerated MD codes. Convergence is obtained by using an ensemble of simulations per window, each starting from different initial conformations, and by optimizing window-width, orthogonal restraints, reaction coordinate harmonic potentials, and window-sample time. The use of uncorrelated initial conformations in neighboring windows is important for correctly sampling conformational transitions from the unbound to bound states that affect significantly the precision of the calculations. This methodology thus provides a general recipe for reproducible and practical computations of binding free energies for a class of semirigid protein-ligand systems, within the limit of the accuracy of the force field used.

Uni‐molecular detection and quantification of selected β‐lactam antibiotics with a hybrid α‐hemolysin protein pore

Single-nanopores have recently been used to electrically detect a wide range of analytes. Similarly, using electrophysiology, we demonstrate how a system comprised of an ion channel formed by α-hemolysin (α-HL) and single-cyclic γ-cyclodextrin (γ-CD) molecule permits the detection of, and differentiation between three different antibiotics from the β-lactam family. Specifically, histograms of the time between the successive binding events, and the residence time distributions of the antibiotic in the γ-CD molecular adapter vary with the antibiotic type. The results show that the association times of amoxicillin, azlocillin, and ampicillin are τ(on) = 2.1 ± 0.2, 2.2 ± 0.3, and 3.1 ± 0.4 ms, respectively. Interestingly, we found that the residence time of the bulkier and negatively charged azlocillin (τ(off) = 0.008 ± 0.0005 ms) is much less than that of ampicillin (τ(off) = 0.07 ± 0.005 ms) and amoxicillin (τ(off) = 0.1 ± 0.02 ms), even though the γ-CD-α-HL complex is anionic selective. The data were also used to estimate the standard free energy of binding between ampicillin to γ-CDs binding (-12 kJ mol(-1) [corrected]). The difference in association times might be due to γ-CDs-imposed steric hindrance or an energetically more expensive desolvation step for the antibiotics to gain access to the binding site in the CD. We suggest that this technique may be used to detect other analytes used in pharmaceutical applications.

Binding Energy Distribution Analysis Method (BEDAM) for Estimation of Protein−Ligand Binding Affinities

The Binding Energy Distribution Analysis Method (BEDAM) for the computation of receptor-ligand standard binding free energies with implicit solvation is presented. The method is based on a well established statistical mechanics theory of molecular association. It is shown that, in the context of implicit solvation, the theory is homologous to the test particle method of solvation thermodynamics with the solute-solvent potential represented by the effective binding energy of the protein-ligand complex. Accordingly, in BEDAM the binding constant is computed by means of a weighted integral of the probability distribution of the binding energy obtained in the canonical ensemble in which the ligand is positioned in the binding site but the receptor and the ligand interact only with the solvent continuum. It is shown that the binding energy distribution encodes all of the physical effects of binding. The balance between binding enthalpy and entropy is seen in our formalism as a balance between favorable and unfavorable binding modes which are coupled through the normalization of the binding energy distribution function. An efficient computational protocol for the binding energy distribution based on the AGBNP2 implicit solvent model, parallel Hamiltonian replica exchange sampling and histogram reweighting is developed. Applications of the method to a set of known binders and non-binders of the L99A and L99A/M102Q mutants of T4 lysozyme receptor are illustrated. The method is able to discriminate without error binders from non-binders, and the computed standard binding free energies of the binders are found to be in good agreement with experimental measurements. Analysis of the results reveals that the binding affinities of these systems reflect the contributions from multiple conformations spanning a wide range of binding energies.