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.     

A new force field (ECEPP-05) for peptides, proteins, and organic molecules

Parametrization and testing of a new all-atom force field for organic molecules and peptides with fixed bond lengths and bond angles are described. The van der Waals parameters for both the organic molecules and the peptides were taken from J. Phys. Chem. B 2003, 107, 7143 and J. Phys. Chem. B 2004, 108, 12181. First, the values of the 1-4 nonbonded and electrostatic scale factors appropriate to the new force field were determined by computing the conformational energies of six model molecules, namely, ethanol, ethylamine, propanol, propylamine, 1,2-ethanediol, and 1,3-propanediol with different values of these factors. The partial atomic charges of these molecules were obtained by fitting to the electrostatic potentials calculated with the HF/6-31G quantum-mechanical method. Two different charge models (single- and multiple-conformation-derived) were also considered. We demonstrated that the charge model has a stronger effect on the conformational energies than the 1-4 scaling. The choice of a charge model affected the conformational energies of even the smallest molecules considered, whereas the effect of the 1-4 electrostatic or nonbonded scaling was apparent only for 1,3-propanediol. The best agreement with high-level ab initio data was obtained with the multiple-conformation-derived charges and with no scaling of the 1-4 nonbonded or electrostatic interactions (scale factors of 1.0). Next, the torsional parameters of a large number of neutral and charged organic molecules, assumed to be models of the side chains of the 20 naturally occurring amino acids, were computed by fitting to rotational energy profiles obtained from ab initio MP2/6-31G calculations. The quality of the fits was high with average errors for torsional profiles of less than 0.2 kcal/mol. To derive the torsional parameters for the peptide backbone, the partial atomic charges of the 20 neutral and charged amino acids were obtained by fitting to the electrostatic potentials of terminally blocked amino acids using the HF/6-31G quantum-mechanical method. Then, the phi-psi energy maps of Ac-Ala-NMe and Ac-Gly-NMe were computed using MP2/6-31G//HF/6-31G quantum-mechanical methods. The phi-psi energy map of Ac-Ala-NMe was used for refinement of the nonbonded parameters for the backbone nitrogen and hydrogen bonded to it. Subsequently, the main-chain torsional parameters were obtained by fitting the molecular mechanics energies to the phi-psi energy maps of Ac-Ala-NMe and Ac-Gly-NMe. The transferability of the entire force field was demonstrated by reproducing the main energy minima of terminally blocked Ala3 from the literature. The performance of the force field was also evaluated by simulating crystal structures of small peptides. By comparison of simulated and experimental data, examination of the torsional-angle and atom-positional root-mean-square deviations of the energy-minimized crystal structures from the corresponding X-ray model structures demonstrated high accuracy of the force field.     

Molecular mechanics models for tetracycline analogs

Tetracyclines (Tcs) are an important family of antibiotics that bind to the ribosome and several proteins. To model Tc interactions with protein and RNA, we have developed a molecular mechanics force field for 12 tetracyclines, consistent with the CHARMM force field. We considered each Tc variant in its zwitterionic tautomer, with and without a bound Mg(2+). We used structures from the Cambridge Crystallographic Data Base to identify the conformations likely to be present in solution and in biomolecular complexes. A conformational search by simulated annealing was undertaken, using the MM3 force field, for tetracycline, anhydrotetracycline, doxycycline, and tigecycline. Resulting, low-energy structures were optimized with an ab initio method. We found that Tc and its analogs all adopt an extended conformation in the zwitterionic tautomer and a twisted one in the neutral tautomer, and the zwitterionic-extended state is the most stable in solution. Intermolecular force field parameters were derived from a standard supermolecule approach: we considered the ab initio energies and geometries of a water molecule interacting with each Tc analog at several different positions. The final, rms deviation between the ab initio and force field energies, averaged over all forms, was 0.35 kcal/mol. Intramolecular parameters were adopted from either the standard CHARMM force field, the ab initio structure, or the earlier, plain Tc force field. The model reproduces the ab initio geometry and flexibility of each Tc. As tests, we describe MD and free energy simulations of a solvated complex between three Tcs and the Tet repressor protein.     

Consistent free energy landscapes and thermodynamic properties of small proteins based on a single all-atom force field employing an implicit solvation

We have attempted to improve the PARAM99 force field in conjunction with the generalized Born (GB) solvation model with a surface area correction for more consistent protein folding simulations. For this purpose, using an extended alphabeta training set of five well-studied molecules with various folds (alpha, beta, and betabetaalpha), a previously modified version of PARAM99/GBSA is further refined, such that all native states of the five training species correspond to their lowest free energy minimum states. The resulting modified force field (PARAM99MOD5/GBSA) clearly produces reasonably acceptable conformational free energy surfaces of the training set with correct identifications of their native states in the free energy minimum states. Moreover, due to its well-balanced nature, this new force field is expected to describe secondary structure propensities of diverse folds in a more consistent manner. Remarkably, temperature dependent behaviors simulated with the current force field are in good agreement with the experiment. This agreement is a significant improvement over the existing standard all-atom force fields. In addition, fundamentally important thermodynamic quantities, such as folding enthalpy (DeltaH) and entropy (DeltaS), agree reasonably well with the experimental data.     

Blind prediction of SAMPL4 cucurbit[7]uril binding affinities with the mining minima method

Accurate methods for predicting protein–ligand binding affinities are of central interest to computer-aided drug design for hit identification and lead optimization. Here, we used the mining minima (M2) method to predict cucurbit[7]uril binding affinities from the SAMPL4 blind prediction challenge. We tested two different energy models, an empirical classical force field, CHARMm with VCharge charges, and the Poisson–Boltzmann surface area solvation model; and a semiempirical quantum mechanical (QM) Hamiltonian, PM6-DH+, coupled with the COSMO solvation model and a surface area term for nonpolar solvation free energy. Binding affinities based on the classical force field correlated strongly with the experiments with a correlation coefficient (R²) of 0.74. On the other hand, binding affinities based on the QM energy model correlated poorly with experiments (R² = 0.24), due largely to two major outliers. As we used extensive conformational search methods, these results point to possible inaccuracies in the PM6-DH+ energy model or the COSMO solvation model. Furthermore, the different binding free energy components, solute energy, solvation free energy, and configurational entropy showed significant deviations between the classical M2 and quantum M2 calculations. Comparison of different classical M2 free energy components to experiments show that the change in the total energy, i.e. the solute energy plus the solvation free energy, is the key driving force for binding, with a reasonable correlation to experiment (R² = 0.56); however, accounting for configurational entropy further improves the correlation.     

Calculation of free energy surfaces using the methods of thermodynamic perturbation theory

A molecular dynamics method for determining the free energy difference between systems separated in configuration space has been developed. With this new approach, which is based on thermodynamic perturbation techniques, potentials of mean force for conformational changes may be calculated. As a test of the method, the potential of mean force and radial distribution function for liquid argon have been computed. The results are in good agreement with those obtained from an ordinary simulation.     

Peptide conformational equilibria computed via a single-stage shifting protocol

We study the conformational equilibria of two peptides using a novel statistical mechanics approach designed for calculating free energy differences between highly dissimilar conformational states. Our results elucidate the contrasting roles of entropy in implicitly solvated leucine dipeptide and decaglycine. The method extends earlier work by Voter and overcomes the notorious "overlap" problem in free energy computations by constructing a mathematically equivalent calculation with high conformational similarity. The approach requires only equilibrium simulations of the two states of interest, without the need for sampling transition states. We discuss possible extensions and optimizations of the approach.     

Simulation of multiphase systems utilizing independent force fields to control intraphase and interphase behavior

Fixed‐charge empirical force fields have been developed and widely used over the past three decades for all‐atom molecular simulations. Most simulation programs providing these methods enable only one set of force field parameters to be used for the entire system. Whereas this is generally suitable for single‐phase systems, the molecular environment at the interface between two phases may be sufficiently different from the individual phases to require a different set of parameters to be used to accurately represent the system. Recently published simulations of peptide adsorption to material surfaces using the CHARMM force field have clearly demonstrated this issue by revealing that calculated values of adsorption free energy substantially differ from experimental results. Whereas nonbonded parameters could be adjusted to correct this problem, this cannot be done without also altering the conformational behavior of the peptide in solution, for which CHARMM has been carefully tuned. We have developed a dual‐force‐field approach (Dual‐FF) to address this problem and implemented it in the CHARMM simulation package. This Dual‐FF method provides the capability to use two separate sets of nonbonded force field parameters within the same simulation: one set to represent intraphase interactions and a separate set to represent interphase interactions. Using this approach, we show that interfacial parameters can be adjusted to correct errors in peptide adsorption free energy without altering peptide conformational behavior in solution. This program thus provides the capability to enable both intraphase and interphase molecular behavior to be accurately and efficiently modeled in the same simulation. © 2012 Wiley Periodicals, Inc.     

ICFF: a new method to incorporate implicit flexibility into an internal coordinate force field

We introduce a new method to accurately "project" a Cartesian force field onto an internal coordinate molecular model with fixed-bond geometry. The algorithm automatically generates the Internal Coordinate Force Field (ICFF), which is a close approximation of the "source" Cartesian force field. The ICFF method reduces the number of free variables in a model by at least 10-fold and facilitates the fast convergence of geometry optimizations, an advantage that is critical for many applications such as the docking of flexible ligands or conformational modeling of macromolecules. Although covalent geometry is fixed in an ICFF model, implicit flexibility is incorporated into the force field parameters in the following two ways. First, we formulate an empirical torsion energy term in ICFF as a sixfold Fourier series and develop a procedure to calculate the Fourier coefficients from the conformational energy profiles of the fully flexible Cartesian model. The ICFF torsion parameters thus represent not only torsion component of the source force field, but also bond bending, bond stretching, and "1-4" van der Waals interactions. Second, we use a soft polynomial repulsion function for "1-5" and "1-6" interactions to mimic the flexibility of bonds, connecting these atoms. Also, we suggest a way to use a local part of the Cartesian force field to automatically generate fixed covalent geometries, compatible with the ICFF energy function. Here, we present an implementation of the ICFF algorithm, which employs the MMFF94s Cartesian force field as a "source." Extensive benchmarking of ICFF with a representative set of organic molecules demonstrates that the implicit flexibility model accurately reproduces MMFF94s equilibrium conformational energy differences (RMSD approximately 0.64 kcal) and, most importantly, detailed torsion energy profiles (RMSD approximately 0.37 kcal). This accuracy is characteristic of the method, because all the ICFF parameters (except one scaling factor in the "1-5,1-6" repulsion term) are derived directly from the source Cartesian force field and do not depend on any particular molecular set. In contrast, the rigid geometry model with the MMFF94s energy function yields highly biased estimations in this test with the RMSD exceeding 1.2 kcal for the equilibrium energy comparisons and approximately 3.4 kcal for the torsion energy profiles.     

Analyzing the robustness of the MM/PBSA free energy calculation method: application to DNA conformational transitions

The ability to predict and characterize free energy differences associated with conformational equilibria or the binding of biomolecules is vital to understanding the molecular basis of many important biological functions. As biological studies focus on larger molecular complexes and properties of the genome, proteome, and interactome, the development and characterization of efficient methods for calculating free energy becomes increasingly essential. The aim of this study is to examine the robustness of the end-point free energy method termed the molecular mechanics Poisson-Boltzmann solvent accessible surface area (MM/PBSA) method. Specifically, applications of MM/PBSA to the conformational equilibria of nucleic acid (NA) systems are explored. This is achieved by comparing A to B form DNA conformational free energy differences calculated using MM/PBSA with corresponding free energy differences determined with a more rigorous and time-consuming umbrella sampling algorithm. In addition, the robustness of NA MM/PBSA calculations is also evaluated in terms of the sensitivity towards the choice of force field and the choice of solvent model used during conformational sampling. MM/PBSA calculations of the free energy difference between A-form and B-form DNA are shown to be in very close agreement with the PMF result determined using an umbrella sampling approach. Further, it is found that the MM/PBSA conformational free energy differences were also in agreement using either the CHARMM or AMBER force field. The influence of ionic strength on conformational stability was particularly insensitive to the choice of force field. Finally, it is also shown that the use of a generalized Born implicit solvent during conformational sampling results in free energy estimates that deviate slightly from those obtained using explicitly solvated MD simulations in these NA systems.     

Calibration of force-field dependency in free energy landscapes of peptide conformations by quantum chemical calculations

The free energy landscapes of peptide conformations were calibrated by ab initio quantum chemical calculations, after the enhanced conformational diversity search using the multicanonical molecular dynamics simulations. Three different potentials of mean force for an isolated dipeptide were individually obtained by the multicanonical molecular dynamics simulations using the conventional force fields, AMBER parm94, AMBER parm96, and CHARMm22. Each potential of mean force was then calibrated based upon the umbrella sampling algorithm from the adiabatic energy map that was calculated separately by the ab initio molecular orbital method, and all of the calibrated potentials of mean force coincided well. The calibration method was also applied to the simulations of a peptide dimer in explicit water models, and it was shown that the calibrated free energy landscapes did not depend on the force field used in the classical simulations, as far as the conformational space was sampled well. The current calibration method fuses the classical free energy calculation with the quantum chemical calculation, and it should generally make simulations for biomolecular systems much more reliable when combining with enhanced conformational sampling.     

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.     

Absolute free energies of biomolecules from unperturbed ensembles

Computing the absolute free energy of a macromolecule's structural state, F, is a challenging problem of high relevance. This study presents a method that computes F using only information from an unperturbed simulation of the macromolecule in the relevant conformational state, ensemble, and environment. Absolute free energies produced by this method, dubbed V aluation of L ocal C onfiguration I ntegral with D ynamics (VALOCIDY), enable comparison of alternative states. For example, comparing explicitly solvated and vaporous states of amino acid side‐chain analogs produces solvation free energies in good agreement with experiments. Also, comparisons between alternative conformational states of model heptapeptides (including the unfolded state) produce free energy differences in agreement with data from μs molecular‐dynamics simulations and experimental propensities. The potential of using VALOCIDY in computational protein design is explored via a small design problem of stabilizing a β‐turn structure. When VALOCIDY‐based estimation of folding free energy is used as the design metric, the resulting sequence folds into the desired structure within the atomistic force field used in design. The VALOCIDY‐based approach also recognizes the distinct status of the native sequence regardless of minor details of the starting template structure, in stark contrast with a traditional fixed‐backbone approach. © 2013 Wiley Periodicals, Inc.     

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.     

Reciprocating motion on the nanoscale

This paper analyzes the confined motion of a Brownian particle fluctuating between two conformational states with different potential profiles and different position-dependent rate constants of the transitions, the fluctuations arising from both thermal (equilibrium) and external (nonequilibrium) noise. The model illustrates a mechanism to transduce, on the nanoscale, the energy of nonequilibrium fluctuations into mechanical energy of reciprocating motion. Expressions for the reciprocating velocity and the efficiency of energy conversion are derived. These expressions are treated in more detail in the slow-fluctuation (quasi-equilibrium) regime, by simple perturbation theory arguments, and in the fast fluctuation limit, in terms of the potential of mean force. A notable observation is that the generalized driving force of the reciprocating motion is caused by two sources: the energy contribution due to the difference between the potential profiles of the states and the entropic contribution due to the difference between the position-dependent rate constants. Two illustrative examples are presented, where one of the two sources can be ignored and an exact solution is allowed. Among other aspects, we also discuss the ways to construct a molecular motor based on the reciprocating engine.     

New-generation amber united-atom force field

We have developed a new-generation Amber united-atom force field for simulations involving highly demanding conformational sampling such as protein folding and protein-protein binding. In the new united-atom force field, all hydrogens on aliphatic carbons in all amino acids are united with carbons except those on Calpha. Our choice of explicit representation of all protein backbone atoms aims at minimizing perturbation to protein backbone conformational distributions and to simplify development of backbone torsion terms. Tests with dipeptides and solvated proteins show that our goal is achieved quite successfully. The new united-atom force field uses the same new RESP charging scheme based on B3LYP/cc-pVTZ//HF/6-31g** quantum mechanical calculations in the PCM continuum solvent as that in the Duan et al. force field. van der Waals parameters are empirically refitted starting from published values with respect to experimental solvation free energies of amino acid side-chain analogues. The suitability of mixing new point charges and van der Waals parameters with existing Amber covalent terms is tested on alanine dipeptide and is found to be reasonable. Parameters for all new torsion terms are refitted based on the new point charges and the van der Waals parameters. Molecular dynamics simulations of three small globular proteins in the explicit TIP3P solvent are performed to test the overall stability and accuracy of the new united-atom force field. Good agreements between the united-atom force field and the Duan et al. all-atom force field for both backbone and side-chain conformations are observed. In addition, the per-step efficiency of the new united-atom force field is demonstrated for simulations in the implicit generalized Born solvent. A speedup around two is observed over the Duan et al. all-atom force field for the three tested small proteins. Finally, the efficiency gain of the new united-atom force field in conformational sampling is further demonstrated with a well-known toy protein folding system, an 18 residue polyalanine in distance-dependent dielectric. The new united-atom force field is at least a factor of 200 more efficient than the Duan et al. all-atom force field for ab initio folding of the tested peptide.     

The theory of viscoelastic characteristics of a highly stretched macromolecule in single molecule AFM

The theory for the deformation of a model macromolecule stretched by its ends under the action of high constant and low periodic forces is constructed. The macromolecule is composed of monomer units in three conformational states. The proposed theory describes the regime of a severe stretching of a macromolecule extended to a length close to its contour length, when its extension proceeds via conformational transitions between different states of monomer units. The structural parameters of the monomer unit are found to correlate with viscoelastic characteristics, which are calculated from the experimental results on the deformation of an individual macromolecule obtained by the frequency atomic force microscopy. For a monomer unit with three conformations, the force dependences of viscoelastic characteristics (effective coefficients of elasticity and friction) can show one or two minima. When the experimental dependences of the above parameters show two minima, the monomer unit can have three or more equilibrium states. With the knowledge of the viscoelastic characteristics of a macromolecule, it is possible to unequivocally estimate all structural parameters of a monomer unit for its three-state conformational model. When the force dependence of viscoelastic characteristics show only one minimum, the monomer unit can have two or more states and analysis of the corresponding viscoelastic characteristics at the minimum makes it possible to select between two- and three-state conformational models. Then, for the three-state model, experimental data allow the prediction of only equilibrium parameters of the monomer unit (position of the minima and energy); dynamic parameters (positions and height of barriers between equilibrium states) remain indeterminate. The proposed theory is used for the interpretation of the viscoelastic characteristics of dextran obtained by single-molecule AFM experiments. The three-state conformational model of a dextran unit is shown to agree better with the experimental data than with the two-state conformational model.     

Optimization of the GB/SA solvation model for predicting the structure of surface loops in proteins

Implicit solvation models are commonly optimized with respect to experimental data or Poisson-Boltzmann (PB) results obtained for small molecules, where the force field is sometimes not considered. In previous studies, we have developed an optimization procedure for cyclic peptides and surface loops in proteins based on the entire system studied and the specific force field used. Thus, the loop has been modeled by the simplified solvation function E(tot) = E(FF) (epsilon = 2r) + Sigma(i) sigma(i)A(i), where E(FF) (epsilon = nr) is the AMBER force field energy with a distance-dependent dielectric function, epsilon = nr, A(i) is the solvent accessible surface area of atom i, and sigma(i) is its atomic solvation parameter. During the optimization process, the loop is free to move while the protein template is held fixed in its X-ray structure. To improve on the results of this model, in the present work we apply our optimization procedure to the physically more rigorous solvation model, the generalized Born with surface area (GB/SA) (together with the all-atom AMBER force field) as suggested by Still and co-workers (J. Phys. Chem. A 1997, 101, 3005). The six parameters of the GB/SA model, namely, P(1)-P(5) and the surface area parameter, sigma (programmed in the TINKER package) are reoptimized for a "training" group of nine loops, and a best-fit set is defined from the individual sets of optimized parameters. The best-fit set and Still's original set of parameters (where Lys, Arg, His, Glu, and Asp are charged or neutralized) were applied to the training group as well as to a "test" group of seven loops, and the energy gaps and the corresponding RMSD values were calculated. These GB/SA results based on the three sets of parameters have been found to be comparable; surprisingly, however, they are somewhat inferior (e.g, of larger energy gaps) to those obtained previously from the simplified model described above. We discuss recent results for loops obtained by other solvation models and potential directions for future studies.