Developing end-point methods for absolute binding free energy calculation using the Boltzmann-quasiharmonic model.

Understanding the physical forces underlying receptor-ligand binding requires robust methods for analyzing the binding thermodynamics. In end-point binding free energy methods the binding free energy is naturally decomposable into physically intuitive contributions such as the solvation free energy and configurational entropy that can provide insights. Here we present a new end-point method called EE-BQH (Effective Energy-Boltzmann-Quasiharmonic) which combines the Boltzmann-Quasiharmonic model for configurational entropy with different solvation free energy methods, such as the continuum solvent PBSA model and the integral equation-based 3D-RISM, to estimate the absolute binding free energy. We compare EE-BQH with other treatments of configurational entropy such as Quasiharmonic models in internal coordinates (QHIC) and in Cartesian coordinates (QHCC), and Normal Mode analysis (NMA), by testing them on the octa acids host-guest complexes from the SAMPL8 blind challenge. The accuracies in the calculated absolute binding free energies strongly depend on the configurational entropy and solvation free energy methods used. QHIC and BQH yield the best agreements with the established potential of mean force (PMF) estimates, with R2 of ∼0.7 and mean unsigned error of ∼1.7 kcal mol-1. These results from the end-point calculations are also in similar agreement with experiments. While 3D-RISM in combination with QHIC or BQH lead to reasonable correlations with the PMF results and experiments, the calculated absolute binding free energies are underestimated by ∼5 kcal mol-1. While the binding is accompanied by a significant reduction in the ligand translational/rotational entropy, the change in the torsional entropy in these host-guest systems is slightly positive. Compared with BQH, QHIC underestimates the reduction of configurational entropy because of the non-Gaussian probability distributions in the ligand rotation and a small number of torsions. The study highlights the crucial role of configurational entropy in determining binding and demonstrates the potential of using the new end-point method to provide insights in more complex protein-ligand systems.

Thermodynamic Decomposition of Solvation Free Energies with Particle Mesh Ewald and Long-Range Lennard-Jones Interactions in Grid Inhomogeneous Solvation Theory.

Grid Inhomogeneous Solvation Theory (GIST) maps out solvation thermodynamic properties on a fine meshed grid and provides a statistical mechanical formalism for thermodynamic end-state calculations. However, differences in how long-range nonbonded interactions are calculated in molecular dynamics engines and in the current implementation of GIST have prevented precise comparisons between free energies estimated using GIST and those from other free energy methods such as thermodynamic integration (TI). Here, we address this by presenting PME-GIST, a formalism by which particle mesh Ewald (PME)-based electrostatic energies and long-range Lennard-Jones (LJ) energies are decomposed and assigned to individual atoms and the corresponding voxels they occupy in a manner consistent with the GIST approach. PME-GIST yields potential energy calculations that are precisely consistent with modern simulation engines and performs these calculations at a dramatically faster speed than prior implementations. Here, we apply PME-GIST end-state analyses to 32 small molecules whose solvation free energies are close to evenly distributed from 2 kcal/mol to -17 kcal/mol and obtain solvation energies consistent with TI calculations (R2 = 0.99, mean unsigned difference 0.8 kcal/mol). We also estimate the entropy contribution from the second and higher order entropy terms that are truncated in GIST by the differences between entropies calculated in TI and GIST. With a simple correction for the high order entropy terms, PME-GIST obtains solvation free energies that are highly consistent with TI calculations (R2 = 0.99, mean unsigned difference = 0.4 kcal/mol) and experimental results (R2 = 0.88, mean unsigned difference = 1.4 kcal/mol). The precision of PME-GIST also enables us to show that the solvation free energy of small hydrophobic and hydrophilic molecules can be largely understood based on perturbations of the solvent in a region extending a few solvation shells from the solute. We have integrated PME-GIST into the open-source molecular dynamics analysis software CPPTRAJ.

Exploring the Free-Energy Landscape and Thermodynamics of Protein-Protein Association.

We report the free-energy landscape and thermodynamics of the protein-protein association responsible for the drug-induced multimerization of HIV-1 integrase (IN). Allosteric HIV-1 integrase inhibitors promote aberrant IN multimerization by bridging IN-IN intermolecular interactions. However, the thermodynamic driving forces and kinetics of the multimerization remain largely unknown. Here, we explore the early steps in the IN multimerization by using umbrella sampling and unbiased molecular dynamics simulations in explicit solvent. In direct simulations, the two initially separated dimers spontaneously associate to form near-native complexes that resemble the crystal structure of the aberrant tetramer. Most strikingly, the effective interaction of the protein-protein association is very short-ranged: the two dimers associate rapidly within tens of nanoseconds when their binding surfaces are separated by d ≤ 4.3 Å (less than two water diameters). Beyond this distance, the oligomerization kinetics appears to be diffusion controlled with a much longer association time. The free-energy profile also captured the crucial role of allosteric IN inhibitors in promoting multimerization and explained why several C-terminal domain mutations are remarkably resistant to the drug-induced multimerization. The results also show that at small separation, the protein-protein binding process contains two consecutive phases with distinct thermodynamic signatures. First, interprotein water molecules are expelled to the bulk, resulting in a small increase in entropy, as the solvent entropy gain from the water release is nearly cancelled by the loss of side-chain entropies as the two proteins approach each other. At shorter distances, the two dry binding surfaces adapt to each other to optimize their interaction energy at the expense of further protein configurational entropy loss. Although the binding interfaces feature clusters of hydrophobic residues, overall, the protein-protein association in this system is driven by enthalpy and opposed by entropy.

Combining Alchemical Transformation with a Physical Pathway to Accelerate Absolute Binding Free Energy Calculations of Charged Ligands to Enclosed Binding Sites.

We present a new approach to more accurately and efficiently compute the absolute binding free energy for receptor-ligand complexes. Currently, the double decoupling method (DDM) and the potential of mean force method (PMF) are widely used to compute the absolute binding free energy of biomolecular complexes. DDM relies on alchemically decoupling the ligand from its environments, which can be computationally challenging for large ligands and charged ligands because of the large magnitude of the decoupling free energies involved. In contrast, the PMF method uses a physical pathway to directly transfer the ligand from solution to the receptor binding pocket and thus avoids some of the aforementioned problems in DDM. However, the PMF method has its own drawbacks: because of its reliance on a ligand binding/unbinding pathway that is free of steric obstructions from the receptor atoms, the method has difficulty treating ligands with buried atoms. To overcome the limitation in the standard PMF approach and enable buried ligands to be treated, here we develop a new method called AlchemPMF in which steric obstructions along the physical pathway for binding are alchemically removed. We have tested the new approach on two important drug targets involving charged ligands. One is HIV-1 integrase bound to an allosteric inhibitor; the other is the human telomeric DNA G-quadruplex in complex with a natural product protoberberine buried in the binding pocket. For both systems, the new approach leads to more reliable estimates of absolute binding free energies with smaller error bars and closer agreements with experiments compared with those obtained from the existing methods, demonstrating the effectiveness of the new method in overcoming the hysteresis often encountered in PMF binding free energy calculations of such systems. The new approach could also be used to improve the sampling of water equilibration and resolvation of the binding pocket as the ligand is extracted.

Inclusion of enclosed hydration effects in the binding free energy estimation of dopamine D3 receptor complexes.

Confined hydration and conformational flexibility are some of the challenges encountered for the rational design of selective antagonists of G-protein coupled receptors. We present a set of C3-substituted (-)-stepholidine derivatives as potent binders of the dopamine D3 receptor. The compounds are characterized biochemically, as well as by computer modeling using a novel molecular dynamics-based alchemical binding free energy approach which incorporates the effect of the displacement of enclosed water molecules from the binding site. The free energy of displacement of specific hydration sites is obtained using the Hydration Site Analysis method with explicit solvation. This work underscores the critical role of confined hydration and conformational reorganization in the molecular recognition mechanism of dopamine receptors and illustrates the potential of binding free energy models to represent these key phenomena.

Role of Displacing Confined Solvent in the Conformational Equilibrium of ß-Cyclodextrin.

This study investigates the role of hydration and its relationship to the conformational equilibrium of the host molecule ß-cyclodextrin. Molecular dynamics simulations indicate that the unbound ß-cyclodextrin exhibits two state behavior in explicit solvent due to the opening and closing of its cavity. In implicit solvent, these transitions are not observed, and there is one dominant conformation of ß-cyclodextrin with an open cavity. Based on these observations, we investigate the hypothesis that the expulsion of thermodynamically unfavorable water molecules into the bulk plays an important role in controlling the accessibility of the closed macrostate at room temperature. We compare the results of the molecular mechanics analytical generalized Born plus nonpolar solvation approach to those obtained through grid inhomogeneous solvation theory analysis with explicit solvation to elucidate the thermodynamic forces at play. The work illustrates the use of continuum solvent models to tease out solvation effects related to the inhomogeneity and the molecular nature of water and demonstrates the key role of the thermodynamics of enclosed hydration in driving the conformational equilibrium of molecules in solution.

Ligand Selectivity in the Recognition of Protoberberine Alkaloids by Hybrid-2 Human Telomeric G-Quadruplex: Binding Free Energy Calculation, Fluorescence Binding, and NMR Experiments.

The human telomeric G-quadruplex (G4) is an attractive target for developing anticancer drugs. Natural products protoberberine alkaloids are known to bind human telomeric G4 and inhibit telomerase. Among several structurally similar protoberberine alkaloids, epiberberine (EPI) shows the greatest specificity in recognizing the human telomeric G4 over duplex DNA and other G4s. Recently, NMR study revealed that EPI recognizes specifically the hybrid-2 form human telomeric G4 by inducing large rearrangements in the 5'-flanking segment and loop regions to form a highly extensive four-layered binding pocket. Using the NMR structure of the EPI-human telomeric G4 complex, here we perform molecular dynamics free energy calculations to elucidate the ligand selectivity in the recognition of protoberberines by the human telomeric G4. The MM-PB(GB)SA (molecular mechanics-Poisson Boltzmann/Generalized Born) Surface Area) binding free energies calculated using the Amber force fields bsc0 and OL15 correlate well with the NMR titration and binding affinity measurements, with both calculations correctly identifying the EPI as the strongest binder to the hybrid-2 telomeric G4 wtTel26. The results demonstrated that accounting for the conformational flexibility of the DNA-ligand complexes is crucially important for explaining the ligand selectivity of the human telomeric G4. While the MD-simulated (molecular dynamics) structures of the G-quadruplex-alkaloid complexes help rationalize why the EPI-G4 interactions are optimal compared with the other protoberberines, structural deviations from the NMR structure near the binding site are observed in the MD simulations. We have also performed binding free energy calculation using the more rigorous double decoupling method (DDM); however, the results correlate less well with the experimental trend, likely due to the difficulty of adequately sampling the very large conformational reorganization in the G4 induced by the protoberberine binding.

New Dopamine D3-Selective Receptor Ligands Containing a 6-Methoxy-1,2,3,4-tetrahydroisoquinolin-7-ol Motif.

A series of analogues featuring a 6-methoxy-1,2,3,4-tetrahydroisoquinolin-7-ol unit as the arylamine "head" group of a classical D3 antagonist core structure were synthesized and evaluated for affinity at dopamine D1, D2, and D3 receptors (D1R, D2R, D3R). The compounds generally displayed strong affinity for D3R with very good D3R selectivity. Docking studies at D2R and D3R crystal structures revealed that the molecules are oriented such that their arylamine units are positioned in the orthosteric binding pocket of D3R, with the arylamide "tail" units residing in the secondary binding pocket. Hydrogen bonding between Ser 182 and Tyr 365 at D3R stabilize extracellular loop 2 (ECL2), which in turn contributes to ligand binding by interacting with the "tail" units of the ligands in the secondary binding pocket. Similar interactions between ECL2 and the "tail" units were absent at D2R due to different positioning of the D2R loop region. The presence of multiple H-bonds with the phenol moiety of the headgroup of 7 and Ser192 accounts for its stronger D3R affinity as compared to the 6,7-dimethoxy-1,2,3,4-tetrahydroisoquinoline-containing analogue 8.

Resolving the Ligand-Binding Specificity in c-MYC G-Quadruplex DNA: Absolute Binding Free Energy Calculations and SPR Experiment.

We report the absolute binding free energy calculation and surface plasmon resonance (SPR) experiment for ligand binding with the c-MYC G-quadruplex DNA. The unimolecular parallel DNA G-quadruplex formed in nuclease hypersensitivity element III1 of the c-MYC gene promoter regulates the c-MYC transcription and is recognized as an emerging drug target for cancer therapy. Quindoline derivatives have been shown to stabilize the G-quadruplex and inhibit the c-MYC expression in cancer cells. NMR revealed two binding sites located at the 5' and 3' termini of the G-quadruplex. Questions about which site is more favored and the basis for the ligand-induced binding site formation remain unresolved. Here, we employ two absolute binding free energy methods, the double decoupling and the potential of mean force methods, to dissect the ligand-binding specificity in the c-MYC G-quadruplex. The calculated absolute binding free energies are in general agreement with the SPR result and suggest that quindoline has a slight preference for the 5' site. The flanking residues around the two sites undergo significant reorganization as the ligand unbinds, which provides evidence for ligand-induced binding pocket formation. The results help interpret experimental data and inform rational design of small molecules targeting the c-MYC G-quadruplex.

A combined treatment of hydration and dynamical effects for the modeling of host-guest binding thermodynamics: the SAMPL5 blinded challenge.

As part of the SAMPL5 blinded experiment, we computed the absolute binding free energies of 22 host-guest complexes employing a novel approach based on the BEDAM single-decoupling alchemical free energy protocol with parallel replica exchange conformational sampling and the AGBNP2 implicit solvation model specifically customized to treat the effect of water displacement as modeled by the Hydration Site Analysis method with explicit solvation. Initial predictions were affected by the lack of treatment of ionic charge screening, which is very significant for these highly charged hosts, and resulted in poor relative ranking of negatively versus positively charged guests. Binding free energies obtained with Debye-Hückel treatment of salt effects were in good agreement with experimental measurements. Water displacement effects contributed favorably and very significantly to the observed binding affinities; without it, the modeling predictions would have grossly underestimated binding. The work validates the implicit/explicit solvation approach employed here and it shows that comprehensive physical models can be effective at predicting binding affinities of molecular complexes requiring accurate treatment of conformational dynamics and hydration.

Enthalpic Breakdown of Water Structure on Protein Active-Site Surfaces.

The principles underlying water reorganization around simple nonpolar solutes are well understood and provide the framework for the classical hydrophobic effect, whereby water molecules structure themselves around solutes so that they maintain favorable energetic contacts with both the solute and the other water molecules. However, for certain solute surface topographies, water molecules, due to their geometry and size, are unable to simultaneously maintain favorable energetic contacts with both the surface and neighboring water molecules. In this study, we analyze the solvation of ligand-binding sites for six structurally diverse proteins using hydration site analysis and measures of local water structure, in order to identify surfaces at which water molecules are unable to structure themselves in a way that maintains favorable enthalpy relative to bulk water. These surfaces are characterized by a high degree of enclosure, weak solute-water interactions, and surface constraints that induce unfavorable pair interactions between neighboring water molecules. Additionally, we find that the solvation of charged side chains in an active site generally results in favorable enthalpy but can also lead to pair interactions between neighboring water molecules that are significantly unfavorable relative to bulk water. We find that frustrated local structure can occur not only in apolar and weakly polar pockets, where overall enthalpy tends to be unfavorable, but also in charged pockets, where overall water enthalpy tends to be favorable. The characterization of local water structure in these terms may prove useful for evaluating the displacement of water from diverse protein active-site environments.

Parameterization of an effective potential for protein-ligand binding from host-guest affinity data.

Force field accuracy is still one of the "stalemates" in biomolecular modeling. Model systems with high quality experimental data are valuable instruments for the validation and improvement of effective potentials. With respect to protein-ligand binding, organic host-guest complexes have long served as models for both experimental and computational studies because of the abundance of binding affinity data available for such systems. Binding affinity data collected for cyclodextrin (CD) inclusion complexes, a popular model for molecular recognition, is potentially a more reliable resource for tuning energy parameters than hydration free energy measurements. Convergence of binding free energy calculations on CD host-guest systems can also be obtained rapidly, thus offering the opportunity to assess the robustness of these parameters. In this work, we demonstrate how implicit solvent parameters can be developed using binding affinity experimental data and the binding energy distribution analysis method (BEDAM) and validated using the Grid Inhomogeneous Solvation Theory analysis. These new solvation parameters were used to study protein-ligand binding in two drug targets against the HIV-1 virus and improved the agreement between the calculated and the experimental binding affinities. This work illustrates how benchmark sets of high quality experimental binding affinity data and physics-based binding free energy models can be used to evaluate and optimize force fields for protein-ligand systems. Copyright © 2015 John Wiley & Sons, Ltd.

ff14SB: Improving the Accuracy of Protein Side Chain and Backbone Parameters from ff99SB.

Molecular mechanics is powerful for its speed in atomistic simulations, but an accurate force field is required. The Amber ff99SB force field improved protein secondary structure balance and dynamics from earlier force fields like ff99, but weaknesses in side chain rotamer and backbone secondary structure preferences have been identified. Here, we performed a complete refit of all amino acid side chain dihedral parameters, which had been carried over from ff94. The training set of conformations included multidimensional dihedral scans designed to improve transferability of the parameters. Improvement in all amino acids was obtained as compared to ff99SB. Parameters were also generated for alternate protonation states of ionizable side chains. Average errors in relative energies of pairs of conformations were under 1.0 kcal/mol as compared to QM, reduced 35% from ff99SB. We also took the opportunity to make empirical adjustments to the protein backbone dihedral parameters as compared to ff99SB. Multiple small adjustments of φ and ψ parameters were tested against NMR scalar coupling data and secondary structure content for short peptides. The best results were obtained from a physically motivated adjustment to the φ rotational profile that compensates for lack of ff99SB QM training data in the ß-ppII transition region. Together, these backbone and side chain modifications (hereafter called ff14SB) not only better reproduced their benchmarks, but also improved secondary structure content in small peptides and reproduction of NMR χ1 scalar coupling measurements for proteins in solution. We also discuss the Amber ff12SB parameter set, a preliminary version of ff14SB that includes most of its improvements.

Distinguishing binders from false positives by free energy calculations: fragment screening against the flap site of HIV protease.

Molecular docking is a powerful tool used in drug discovery and structural biology for predicting the structures of ligand-receptor complexes. However, the accuracy of docking calculations can be limited by factors such as the neglect of protein reorganization in the scoring function; as a result, ligand screening can produce a high rate of false positive hits. Although absolute binding free energy methods still have difficulty in accurately rank-ordering binders, we believe that they can be fruitfully employed to distinguish binders from nonbinders and reduce the false positive rate. Here we study a set of ligands that dock favorably to a newly discovered, potentially allosteric site on the flap of HIV-1 protease. Fragment binding to this site stabilizes a closed form of protease, which could be exploited for the design of allosteric inhibitors. Twenty-three top-ranked protein-ligand complexes from AutoDock were subject to the free energy screening using two methods, the recently developed binding energy analysis method (BEDAM) and the standard double decoupling method (DDM). Free energy calculations correctly identified most of the false positives (≥83%) and recovered all the confirmed binders. The results show a gap averaging ≥3.7 kcal/mol, separating the binders and the false positives. We present a formula that decomposes the binding free energy into contributions from the receptor conformational macrostates, which provides insights into the roles of different binding modes. Our binding free energy component analysis further suggests that improving the treatment for the desolvation penalty associated with the unfulfilled polar groups could reduce the rate of false positive hits in docking. The current study demonstrates that the combination of docking with free energy methods can be very useful for more accurate ligand screening against valuable drug targets.

Virtual screening of integrase inhibitors by large scale binding free energy calculations: the SAMPL4 challenge.

As part of the SAMPL4 blind challenge, filtered AutoDock Vina ligand docking predictions and large scale binding energy distribution analysis method binding free energy calculations have been applied to the virtual screening of a focused library of candidate binders to the LEDGF site of the HIV integrase protein. The computational protocol leveraged docking and high level atomistic models to improve enrichment. The enrichment factor of our blind predictions ranked best among all of the computational submissions, and second best overall. This work represents to our knowledge the first example of the application of an all-atom physics-based binding free energy model to large scale virtual screening. A total of 285 parallel Hamiltonian replica exchange molecular dynamics absolute protein-ligand binding free energy simulations were conducted starting from docked poses. The setup of the simulations was fully automated, calculations were distributed on multiple computing resources and were completed in a 6-weeks period. The accuracy of the docked poses and the inclusion of intramolecular strain and entropic losses in the binding free energy estimates were the major factors behind the success of the method. Lack of sufficient time and computing resources to investigate additional protonation states of the ligands was a major cause of mispredictions. The experiment demonstrated the applicability of binding free energy modeling to improve hit rates in challenging virtual screening of focused ligand libraries during lead optimization.

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 ß-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.

The linear interaction energy method for the prediction of protein stability changes upon mutation.

The coupling of protein energetics and sequence changes is a critical aspect of computational protein design, as well as for the understanding of protein evolution, human disease, and drug resistance. To study the molecular basis for this coupling, computational tools must be sufficiently accurate and computationally inexpensive enough to handle large amounts of sequence data. We have developed a computational approach based on the linear interaction energy (LIE) approximation to predict the changes in the free-energy of the native state induced by a single mutation. This approach was applied to a set of 822 mutations in 10 proteins which resulted in an average unsigned error of 0.82 kcal/mol and a correlation coefficient of 0.72 between the calculated and experimental ΔΔG values. The method is able to accurately identify destabilizing hot spot mutations; however, it has difficulty in distinguishing between stabilizing and destabilizing mutations because of the distribution of stability changes for the set of mutations used to parameterize the model. In addition, the model also performs quite well in initial tests on a small set of double mutations. On the basis of these promising results, we can begin to examine the relationship between protein stability and fitness, correlated mutations, and drug resistance.

Improving the description of salt bridge strength and geometry in a Generalized Born model.

The Generalized Born (GB) solvent model is widely used in molecular dynamics simulations because it can be less computationally expensive and it samples conformational changes more efficiently than explicit solvent simulations. Meanwhile, great efforts have been made in the past to improve its precision and accuracy. Previous studies have shown that reducing intrinsic GB radii of some hydrogen atoms would improve AMBER GB-HCT solvent model's accuracy on salt bridges. Here we present our finding that similar correction also shows dramatic improvement for the AMBER GB-OBC solvent model. Potential of mean force and cluster analysis for small peptide replica exchange molecular dynamics simulations suggested that new radii GB simulation with ff99SB/GB-OBC corrected salt bridge strength and achieved significantly higher geometry similarity with TIP3P simulation. Improved performance in 60 ns HIV-1 protease GB simulation further validated this approach for large systems.

Evaluating the performance of the ff99SB force field based on NMR scalar coupling data.

Force-field validation is essential for the identification of weaknesses in current models and the development of more accurate models of biomolecules. NMR coupling and relaxation methods have been used to effectively diagnose the strengths and weaknesses of many existing force fields. Studies using the ff99SB force field have shown excellent agreement between experimental and calculated order parameters and residual dipolar calculations. However, recent studies have suggested that ff99SB demonstrates poor agreement with J-coupling constants for short polyalanines. We performed extensive replica-exchange molecular-dynamics simulations on Ala(3) and Ala(5) in TIP3P and TIP4P-Ew solvent models. Our results suggest that the performance of ff99SB is among the best of currently available models. In addition, scalar coupling constants derived from simulations in the TIP4P-Ew model show a slight improvement over those obtained using the TIP3P model. Despite the overall excellent agreement, the data suggest areas for possible improvement.