Multiple protonation equilibria in electrostatics of protein-protein binding

All proteins contain groups capable of exchanging protons with their environment. We present here an approach, based on a rigorous thermodynamic cycle and the partition functions for energy levels characterizing protonation states of the associating proteins and their complex, to compute the electrostatic pH-dependent contribution to the free energy of protein-protein binding. The computed electrostatic binding free energies include the pH of the solution as the variable of state, mutual "polarization" of associating proteins reflected as changes in the distribution of their protonation states upon binding and fluctuations between available protonation states. The only fixed property of both proteins is the conformation; the structure of the monomers is kept in the same conformation as they have in the complex structure. As a reference, we use the electrostatic binding free energies obtained from the traditional Poisson-Boltzmann model, computed for a single macromolecular conformation fixed in a given protonation state, appropriate for given solution conditions. The new approach was tested for 12 protein-protein complexes. It is shown that explicit inclusion of protonation degrees of freedom might lead to a substantially different estimation of the electrostatic contribution to the binding free energy than that based on the traditional Poisson-Boltzmann model. This has important implications for the balancing of different contributions to the energetics of protein-protein binding and other related problems, for example, the choice of protein models for Brownian dynamics simulations of their association. Our procedure can be generalized to include conformational degrees of freedom by combining it with molecular dynamics simulations at constant pH. Unfortunately, in practice, a prohibitive factor is an enormous requirement for computer time and power. However, there may be some hope for solving this problem by combining existing constant pH molecular dynamics algorithms with so-called accelerated molecular dynamics algorithms.     

Treating entropy and conformational changes in implicit solvent simulations of small molecules

Implicit solvent models are increasingly popular for estimating aqueous solvation (hydration) free energies in molecular simulations and other applications. In many cases, parameters for these models are derived to reproduce experimental values for small molecule hydration free energies. Often, these hydration free energies are computed for a single solute conformation, neglecting solute conformational changes upon solvation. Here, we incorporate these effects using alchemical free energy methods. We find significant errors when hydration free energies are estimated using only a single solute conformation, even for relatively small, simple, rigid solutes. For example, we find conformational entropy (TDeltaS) changes of up to 2.3 kcal/mol upon hydration. Interestingly, these changes in conformational entropy correlate poorly (R2 = 0.03) with the number of rotatable bonds. The present study illustrates that implicit solvent modeling can be improved by eliminating the approximation that solutes are rigid.     

Molecular Mechanics Investigation of an Adenine-Adenine Non-Canonical Pair Conformational Change

Conformational changes are important in RNA for binding and catalysis and understanding these changes is important for understanding how RNA functions. Computational techniques using all-atom molecular models can be used to characterize conformational changes in RNA. These techniques are applied to an RNA conformational change involving a single base pair within a nine base pair RNA duplex. The Adenine-Adenine (AA) non-canonical pair in the sequence 5'GGUGAAGGCU3' paired with 3'PCCGAAGCCG5', where P is Purine, undergoes conformational exchange between two conformations on the timescale of tens of microseconds, as demonstrated in a previous NMR solution structure [Chen, G., et al., Biochemistry, 2006. 45: 6889-903]. The more populated, major, conformation was estimated to be 0.5 to 1.3 kcal/mol more stable at 30 °C than the less populated, minor, conformation. Both conformations are trans-Hoogsteen/sugar edge pairs, where the interacting edges on the adenines change with the conformational change. Targeted Molecular Dynamics (TMD) and Nudged Elastic Band (NEB) were used to model the pathway between the major and minor conformations using the AMBER software package. The adenines were predicted to change conformation via intermediates in which they are stacked as opposed to hydrogen-bonded. The predicted pathways can be described by an improper dihedral angle reaction coordinate. Umbrella sampling along the reaction coordinate was performed to model the free energy profile for the conformational change using a total of 1800 ns of sampling. Although the barrier height between the major and minor conformations was reasonable, the free energy difference between the major and minor conformations was the opposite of that expected based on the NMR experiments. Variations in the force field applied did not improve the misrepresentation of the free energies of the major and minor conformations. As an alternative, the Molecular Mechanics Poisson-Boltzmann Surface Area (MMPBSA) approximation was applied to predict free energy differences between the two conformations using a total of 800 ns of sampling. MM-PBSA also incorrectly predicted the major conformation to be higher in free energy than the minor conformation.     

The large scale conformational change of the human DPP III-substrate prefers the "closed" form

Human dipeptidyl peptidase III (DPP III) is a two domain metallo-peptidase from the M49 family. The wide interdomain cleft and broad substrate specificity suggest that this enzyme could experience significant conformational change. Long (>100 ns) molecular dynamics (MD) simulations of DPP III revealed large range conformational changes of the protein, suggesting the pre-existing equilibrium model for a substrate binding. The binding free energy calculations revealed tighter binding of the preferred synthetic substrate Arg-Arg-2-naphtylamide to the "closed" than to the "open" DPP III conformation. Our assumption that Asp372 plays a crucial role in the large scale interdomain closure was proved by the MD simulations of the Asp372Ala variant. During the same simulation time, the variant remained more "open" than the wild type protein. Apparently, Ala was not as efficient as Asp in establishing the interdomain interactions. According to the MM-PBSA calculations, the electrostatic component of the free energy of solvation turned out to be higher for the "closed" protein than for its less compact form. However, the gain in entropy due to water released from the interdomain cleft nicely balanced this negative effect.     

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.     

PEARLS: program for energetic analysis of receptor-ligand system

Analysis of the energetics of small molecule ligand-protein, ligand-nucleic acid, and protein-nucleic acid interactions facilitates the quantitative understanding of molecular interactions that regulate the function and conformation of proteins. It has also been extensively used for ranking potential new ligands in virtual drug screening. We developed a Web-based software, PEARLS (Program for Energetic Analysis of Ligand-Receptor Systems), for computing interaction energies of ligand-protein, ligand-nucleic acid, protein-nucleic acid, and ligand-protein-nucleic acid complexes from their 3D structures. AMBER molecular force field, Morse potential, and empirical energy functions are used to compute the van der Waals, electrostatic, hydrogen bond, metal-ligand bonding, and water-mediated hydrogen bond energies between the binding molecules. The change in the solvation free energy of molecular binding is estimated by using an empirical solvation free energy model. Contribution from ligand conformational entropy change is also estimated by a simple model. The computed free energy for a number of PDB ligand-receptor complexes were studied and compared to experimental binding affinity. A substantial degree of correlation between the computed free energy and experimental binding affinity was found, which suggests that PEARLS may be useful in facilitating energetic analysis of ligand-protein, ligand-nucleic acid, and protein-nucleic acid interactions. PEARLS can be accessed at http://ang.cz3.nus.edu.sg/cgi-bin/prog/rune.pl.     

Prediction of SAMPL3 host-guest affinities with the binding energy distribution analysis method (BEDAM)

BEDAM calculations are described to predict the free energies of binding of a series of anaesthetic drugs to a recently characterized acyclic cucurbituril host. The modeling predictions, conducted as part of the SAMPL3 host-guest affinity blind challenge, are generally in good quantitative agreement with the experimental measurements. The correlation coefficient between computed and measured binding free energies is 70% with high statistical significance. Multiple conformational stereoisomers and protonation states of the guests have been considered. Better agreement is obtained with high statistical confidence under acidic modeling conditions. It is shown that this level of quantitative agreement could have not been reached without taking into account reorganization energy and configurational entropy effects. Extensive conformational variability of the host, the guests and their complexes is observed in the simulations, affecting binding free energy estimates and structural predictions. A conformational reservoir technique is introduced as part of the parallel Hamiltonian replica exchange molecular dynamics BEDAM protocol to fully capture conformational variability. It is shown that these advanced computational strategies lead to converged free energy estimates for these systems, offering the prospect of utilizing host-guest binding free energy data for force field validation and development.     

Quantitative analysis of energy transfer between fluorescent proteins in CFP-GBP-YFP and its response to Ca2+

This article reports the full characterisation of the optical properties of a biosynthesised protein consisting of fused cyan fluorescent protein, glucose binding protein and yellow fluorescent protein. The cyan and yellow fluorescent proteins act as donors and acceptors for intramolecular fluorescence resonance energy transfer. Absorption, fluorescence, excitation and fluorescence decays of the compound protein were measured and compared with those of free fluorescent proteins. Signatures of energy transfer were identified in the spectral intensities and fluorescence decays. A model describing the fluorescence properties including energy transfer in terms of rate equations is presented and all relevant parameters are extracted from the measurements. The compound protein changes conformation on binding with calcium ions. This is reflected in a change of energy transfer efficiency between the fluorescent proteins. We track the conformational change and the kinetics of the calcium binding reaction from fluorescence intensity and decay measurements and interpret the results in light of the rate equation model. This visualisation of change in protein conformation has the potential to serve as an analytical tool in the study of protein structure changes in real time, in the development of biosensor proteins and in characterizing protein-drug interactions.     

Simple estimation of absolute free energies for biomolecules

One reason that free energy difference calculations are notoriously difficult in molecular systems is due to insufficient conformational overlap, or similarity, between the two states or systems of interest. The degree of overlap is irrelevant, however, if the absolute free energy of each state can be computed. We present a method for calculating the absolute free energy that employs a simple construction of an exactly computable reference system which possesses high overlap with the state of interest. The approach requires only a physical ensemble of conformations generated via simulation and an auxiliary calculation of approximately equal central-processing-unit cost. Moreover, the calculations can converge to the correct free energy value even when the physical ensemble is incomplete or improperly distributed. As a "proof of principle," we use the approach to correctly predict free energies for test systems where the absolute values can be calculated exactly and also to predict the conformational equilibrium for leucine dipeptide in implicit solvent.     

Model for proton transport coupled to protein conformational change: application to proton pumping in the bacteriorhodopsin photocycle

A modeling method is presented for protein systems in which proton transport is coupled to conformational change, as in proton pumps and in motors driven by the proton-motive force. Previously developed methods for calculating pKa values in proteins using a macroscopic dielectric model are extended beyond the equilibrium case to a master-equation model for the time evolution of the system through states defined by ionization microstate and a discrete set of conformers. The macroscopic dielectric model supplies free energy changes for changes of protonation microstate, while the method for obtaining the energetics of conformational change and the relaxation rates, the other ingredients needed for the master equation, are system dependent. The method is applied to the photoactivated proton pump, bacteriorhodopsin, using conformational free energy differences from experiment and treating relaxation rates through three adjustable parameters. The model is found to pump protons with an efficiency relatively insensitive to parameter choice over a wide range of parameter values, and most of the main features of the known photocycle from very early M to the return to the resting state are reproduced. The boundaries of these parameter ranges are such that short-range proton transfers are faster than longer-range ones, which in turn are faster than conformational changes. No relaxation rates depend on conformation. The results suggest that an "accessibility switch", while not ruled out, is not required and that vectorial proton transport can be achieved through the coupling of the energetics of ionization and conformational states.     

Free Energy Level Correction by Monte Carlo Resampling with Weighted Histogram Analysis Method

Free energy calculations may provide vital information for studying various chemical and biological processes. Quantum mechanical methods are required to accurately describe interaction energies, but their computations are often too demanding for conformational sampling. As a remedy, level correction schemes that allow calculating high level free energies based on conformations from lower level simulations have been developed. Here, we present a variation of a Monte Carlo (MC) resampling approach in relation to the weighted histogram analysis method (WHAM). We show that our scheme can generate free energy surfaces that can practically converge to the exact one with sufficient sampling, and that it treats cases with insufficient sampling in a more stable manner than the conventional WHAM-based level correction scheme. It can also provide a guide for checking the uncertainty of the level-corrected surface and a well-defined criterion for deciding the extent of smoothing on the free energy surface for its visual improvement. We demonstrate these aspects by obtaining the free energy maps associated with the alanine dipeptide and proton transfer network of the KillerRed protein in explicit water, and exemplify that the MC resampled WHAM scheme can be a practical tool for producing free energy surfaces of realistic systems.     

Conformational energy penalties of protein-bound ligands

The conformational energies required for ligands to adopt their bioactive conformations were calculated for 33 ligand–protein complexes including 28 different ligands. In order to monitor the force field dependence of the results, two force fields, MM3 and AMBER, were employed for the calculations. Conformational analyses were performed in vacuo and in aqueous solution by using the generalized Born/solvent accessible surface (GB/SA) solvation model. The protein-bound conformations were relaxed by using flat-bottomed Cartesian constraints. For about 70% of the ligand–protein complexes studied, the conformational energies of the bioactive conformations were calculated to be 3 kcal/mol. It is demonstrated that the aqueous conformational ensemble for the unbound ligand must be used as a reference state in this type of calculations. The calculations for the ligand–protein complexes with conformational energy penalties of the ligand calculated to be larger than 3 kcal/mol suffer from uncertainties in the interpretation of the experimental data or limitations of the computational methods. For example, in the case of long-chain flexible ligands (e.g. fatty acids), it is demonstrated that several conformations may be found which are very similar to the conformation determined by X-ray crystallography and which display significantly lower conformational energy penalties for binding than obtained by using the experimental conformation. For strongly polar molecules, e.g. amino acids, the results indicate that further developments of the force fields and of the dielectric continuum solvation model are required for reliable calculations on the conformational properties of this type of compounds.     

Converging free energies of binding in cucurbit[7]uril and octa-acid host–guest systems from SAMPL4 using expanded ensemble simulations

Molecular containers such as cucurbit[7]uril (CB7) and the octa-acid (OA) host are ideal simplified model test systems for optimizing and analyzing methods for computing free energies of binding intended for use with biologically relevant protein–ligand complexes. To this end, we have performed initially blind free energy calculations to determine the free energies of binding for ligands of both the CB7 and OA hosts. A subset of the selected guest molecules were those included in the SAMPL4 prediction challenge. Using expanded ensemble simulations in the dimension of coupling host–guest intermolecular interactions, we are able to show that our estimates in most cases can be demonstrated to fully converge and that the errors in our estimates are due almost entirely to the assigned force field parameters and the choice of environmental conditions used to model experiment. We confirm the convergence through the use of alternative simulation methodologies and thermodynamic pathways, analyzing sampled conformations, and directly observing changes of the free energy with respect to simulation time. Our results demonstrate the benefits of enhanced sampling of multiple local free energy minima made possible by the use of expanded ensemble molecular dynamics and may indicate the presence of significant problems with current transferable force fields for organic molecules when used for calculating binding affinities, especially in non-protein chemistries.     

Converging free energy estimates: MM-PB(GB)SA studies on the protein-protein complex Ras-Raf

Estimating protein-protein interaction energies is a very challenging task for current simulation protocols. Here, absolute binding free energies are reported for the complex H-Ras/C-Raf1 using the MM-PB(GB)SA approach, testing the internal consistency and model dependence of the results. Averaging gas-phase energies (MM), solvation free energies as determined by Generalized Born models (GB/SA), and entropic contributions calculated by normal mode analysis for snapshots obtained from 10 ns explicit-solvent molecular dynamics in general results in an overestimation of the binding affinity when a solvent-accessible surface area-dependent model is used to estimate the nonpolar solvation contribution. Applying the sum of a cavity solvation free energy and explicitly modeled solute-solvent van der Waals interaction energies instead provides less negative estimates for the nonpolar solvation contribution. When the polar contribution to the solvation free energy is determined by solving the Poisson-Boltzmann equation (PB) instead, the calculated binding affinity strongly depends on the atomic radii set chosen. For three GB models investigated, different absolute deviations from PB energies were found for the unbound proteins and the complex. As an alternative to normal-mode calculations, quasiharmonic analyses have been performed to estimate entropic contributions due to changes of solute flexibility upon binding. However, such entropy estimates do not converge after 10 ns of simulation time, indicating that sampling issues may limit the applicability of this approach. Finally, binding free energies estimated from snapshots of the unbound proteins extracted from the complex trajectory result in an underestimate of binding affinity. This points to the need to exercise caution in applying the computationally cheaper "one-trajectory-alternative" to systems where there may be significant changes in flexibility and structure due to binding. The best estimate for the binding free energy of Ras-Raf obtained in this study of -8.3 kcal mol(-1) is in good agreement with the experimental result of -9.6 kcal mol(-1), however, further probing the transferability of the applied protocol that led to this result is necessary.     

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.     

Determination of conformational free energies of peptides by multidimensional adaptive umbrella sampling

We improve the multidimensional adaptive umbrella sampling method for the computation of conformational free energies of biomolecules. The conformational transition between the alpha-helical and beta-hairpin conformational states of an alanine decapeptide is used as an example. Convergence properties of the weighted-histogram-analysis-based adaptive umbrella sampling can be improved by using multiple replicas in each adaptive iteration and by using adaptive updating of the bounds of the umbrella potential. Using positional root-mean-square deviations from structures of the alpha-helical and beta-hairpin reference states as reaction coordinates, we obtained well-converged free energy surfaces of both the in-vacuum and in-solution decapeptide systems. From the free energy surfaces well-converged relative free energies between the two conformational states can be derived. Advantages and disadvantages of different methods for obtaining conformational free energies as well as implications of our results in studying conformational transitions of proteins and in improving force field are discussed.