From secondary structure to three-dimensional structure: Improved dihedral angle probability distribution function for use with energy searches for native structures of polypeptides and proteins

An improved scheme to help in the prediction of protein structure is presented. This procedure generates improved starting conformations of a protein suitable for energy minimization. Trivariate gaussian distribution functions for the π, ψ, and χ¹ dihedral angles have been derived, using conformational data from high resolution protein structures selected from the Protein Data Bank (PDB). These trivariate probability functions generate initial values for the π, ψ, and χ¹ dihedral angles which reflect the experimental values found in the PDB. These starting structures speed the search of the conformational space by focusing the search mainly in the regions of native proteins. The efficiency of the new trivariate probability distributions is demonstrated by comparing the results for the α-class polypeptide fragment, the mutant Antennapedia (C39 → S) homeodomain (2HOA), with those from two reference probability functions. The first reference probability function is a uniform or flat probability function and the second is a bivariate probability function for π and ψ. The trivariate gaussian probability functions are shown to search the conformational space more efficiently than the other two probability functions. The trivariate gaussian probability functions are also tested on the binding domain of Streptococcal protein G (2GB1), an α/β class protein. Since presently available energy functions are not accurate enough to identify the most native-like energy-minimized structures, three selection criteria were used to identify a native-like structure with a 1.90-Å rmsd from the NMR structure as the best structure for the Antennapedia fragment. Each individual selection criterion (ECEPP/3 energy, ECEPP/3 energy-plus-free energy of hydration, or a knowledge-based mean field method) was unable to identify a native-like structure, but simultaneous application of more than one selection criterion resulted in a successful identification of a native-like structure for the Antennapedia fragment. In addition to these tests, structure predictions are made for the Antennapedia polypeptide, using a Pattern Recognition-based Importance-Sampling Minimization (PRISM) procedure to predict the backbone conformational state of the mutant Antennapedia homeodomain. The ten most probable backbone conformational state predictions were used with the trivariate and bivariate gaussian dihedral angle probability distributions to generate starting structures (i.e., dihedral angles) suitable for energy minimization. The final energy-minimized structures show that neither the trivariate nor the bivariate gaussian probability distributions are able to overcome the inaccuracies in the backbone conformational state predictions to produce a native-like structure. Until highly accurate predictions of the backbone conformational states become available, application of these dihedral angle probability distributions must be limited to problems, such as homology modeling, in which only a limited portion of the backbone (e.g., surface loops) must be explored. © 1996 John Wiley & Sons, Inc.     

Standard-geometry chains fitted to X-ray derived structures: Validation of the rigid-geometry approximation. II. Systematic searches for short loops in proteins: Applications to bovine pancreatic ribonuclease A and human lysozyme

A rigid-geometry approach to protein conformational searches has been used to calculate stable structures for localized regions of the molecules bovine pancreatic ribonuclease A and human lysozyme. The search method is essentially an application of the local deformation algorithm of Gō and Scheraga [Macromolecules, 3 , 178–187 (1970)]. A series of local chain deformations is produced in the polypeptide chain. The deformations are screened to eliminate structures having serious atomic overlaps or energetically unreasonable backbone dihedral angles. The remaining structures are refined by energy minimization and the rms deviations of the energy-minimized structures, relative to the native structures, are calculated. The correlation between low rms deviation and low energy is reasonably good, indicating that this method should be useful in generating a small number of candidate structures for further energy refinement. Further applications to proteins with an unknown structure, such as homology-based modeling applications, should now be amenable to this type of procedure.     

LEAP: Highly accurate prediction of protein loop conformations by integrating coarse‐grained sampling and optimized energy scores with all‐atom refinement of backbone and side chains

Prediction of protein loop conformations without any prior knowledge (ab initio prediction) is an unsolved problem. Its solution will significantly impact protein homology and template‐based modeling as well as ab initio protein‐structure prediction. Here, we developed a coarse‐grained, optimized scoring function for initial sampling and ranking of loop decoys. The resulting decoys are then further optimized in backbone and side‐chain conformations and ranked by all‐atom energy scoring functions. The final integrated technique called loop prediction by energy‐assisted protocol achieved a median value of 2.1 Å root mean square deviation (RMSD) for 325 12‐residue test loops and 2.0 Å RMSD for 45 12‐residue loops from critical assessment of structure‐prediction techniques (CASP) 10 target proteins with native core structures (backbone and side chains). If all side‐chain conformations in protein cores were predicted in the absence of the target loop, loop‐prediction accuracy only reduces slightly (0.2 Å difference in RMSD for 12‐residue loops in the CASP target proteins). The accuracy obtained is about 1 Å RMSD or more improvement over other methods we tested. The executable file for a Linux system is freely available for academic users at http://sparks‐lab.org . © 2013 Wiley Periodicals, Inc.     

Predicting helical hairpins from sequences by Monte Carlo simulations

The key problem in polypeptide‐structure prediction is with regard to thermodynamics. Two factors limit prediction in ab initio computer simulations. First, the thermodynamically dominant conformations must be found from an extremely large number of possible conformations. Second, these low‐energy forms must deviate little from the experimental structures. Here, we report on the application of the diffusion‐controlled Monte Carlo approach to predict four α‐helical hairpins with 34–38 residues by global optimization, using an energy optimized on other supersecondary structures. A total of seven simulations is carried out for each protein starting from fully extended conformations. Three proteins are correctly folded (within 3.0 Å rms from the experimental structures), but the fourth protein cannot distinguish between several equienergetic conformations. Possible improvement of the energy model is suggested. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 582–589, 2000     

Loop problem in proteins: Developments on Monte Carlo simulated annealing approach

Calculations of loop segments in bovine pancreatic trypsin inhibitor starting from random conformations are more efficient, reproducible, and reliable due to several program enhancements. Monte Carlo simulated annealing (MCSA) calculations of a five-residue α-helix N-terminus segment (H5) and β-strand segment (B5) in this study are compared to the corresponding loop calculations in our previous study. Characteristics of the calculations are: the lowest final total energy conformations (LECs) are within 5 kcal/mol; the average backbone deviations of the computed segments from the native X-ray conformations are 0.43 ± 0.15 Å for H5 and 0.68 ± 0.20 Å for B5; and all the native backbone-backbone hydrogen bonds (H bonds) are present in the best LECs. Compared to the previous study, the H5 and B5 calculations are about 3 and 24 times more efficient, respectively. In the analysis of the best H5 simulated annealing run, backbone-backbone H bonds appear between RT = 4 and 70 kcal/mol, where RT is the Boltzmann temperature factor. H bonds that involve side chains appear in the RT = 1–10 kcal/mol range. Program enhancements implemented are varying main chain versus side chain dihedral angle selection rate, varying ϕ/ψ and χ₁/χ₂ dihedral angles in pairs, the use of main chain and side chain rotamer libraries, and varying the location of the segment origin. © 1996 by John Wiley & Sons, Inc.     

Protein side chain modeling with orientation-dependent atomic force fields derived by series expansions

We describe the development of new force fields for protein side chain modeling called optimized side chain atomic energy (OSCAR). The distance‐dependent energy functions (OSCAR‐d) and side‐chain dihedral angle potential energy functions were represented as power and Fourier series, respectively. The resulting 802 adjustable parameters were optimized by discriminating the native side chain conformations from non‐native conformations, using a training set of 12,000 side chains for each residue type. In the course of optimization, for every residue, its side chain was replaced by varying rotamers, whereas conformations for all other residues were kept as they appeared in the crystal structure. Then, the OSCAR‐d were multiplied by an orientation‐dependent function to yield OSCAR‐o. A total of 1087 parameters of the orientation‐dependent energy functions (OSCAR‐o) were optimized by maximizing the energy gap between the native conformation and subrotamers calculated as low energy by OSCAR‐d. When OSCAR‐o with optimized parameters were used to model side chain conformations simultaneously for 218 recently released protein structures, the prediction accuracies were 88.8% for χ₁, 79.7% for χ_{1 + 2}, 1.24 Å overall root mean square deviation (RMSD), and 0.62 Å RMSD for core residues, respectively, compared with the next‐best performing side‐chain modeling program which achieved 86.6% for χ₁, 75.7% for χ_{1 + 2}, 1.40 Å overall RMSD, and 0.86 Å RMSD for core residues, respectively. The continuous energy functions obtained in this study are suitable for gradient‐based optimization techniques for protein structure refinement. A program with built‐in OSCAR for protein side chain prediction is available for download at http://sysimm.ifrec.osaka‐u.ac.jp/OSCAR/ . © 2011 Wiley Periodicals, Inc. J Comput Chem 2011     

β-diketone interactions: Part 4. The structures and properties of 1,3-diphenyl-2-methylpropane-1,3-dione,C16H14O2, and 1,3-diphenyl-2-(4-methoxyphenyl)propane-1,3-dione,C22H18O3

The X-ray crystal structures of 1,3-diphenyl-2-methylpropane-1,3-dione and 1,3-diphenyl-2-(4-methoxyphenyl)propane-1,3-dione show them both to adopt cis-diketo (Z,Z) conformations with carbonyl—carbonyl dihedral angles of 89.0(3)° (2-methyl derivative), and 85.5(4)° and 77.7(4)° for the two molecules in the asymmetric unit of the 2-(4-methoxyphenyl) derivative. These are the first acyclic β-diketones with an α-hydrogen to be reported which do not have an enol configuration in the solid state.     

Effective protein conformational sampling based on predicted torsion angles

Protein structure prediction is a long‐standing problem in molecular biology. Due to lack of an accurate energy function, it is often difficult to know whether the sampling algorithm or the energy function is the most important factor for failure of locating near‐native conformations of proteins. This article examines the size dependence of sampling effectiveness by using a perfect “energy function”: the root‐mean‐squared distance from the target native structure. Using protein targets up to 460 residues from critical assessment of structure prediction techniques (CASP11, 2014), we show that the accuracy of near native structures sampled is relatively independent of protein sizes but strongly depends on the errors of predicted torsion angles. Even with 40% out‐of‐range angle prediction, 2 Å or less near‐native conformation can be sampled. The result supports that the poor energy function is one of the bottlenecks of structure prediction and predicted torsion angles are useful for overcoming the bottleneck by restricting the sampling space in the absence of a perfect energy function. © 2015 Wiley Periodicals, Inc.     

Comparison between self-guided Langevin dynamics and molecular dynamics simulations for structure refinement of protein loop conformations

This article presents a comparative analysis of two replica‐exchange simulation methods for the structure refinement of protein loop conformations, starting from low‐resolution predictions. The methods are self‐guided Langevin dynamics (SGLD) and molecular dynamics (MD) with a Nosé–Hoover thermostat. We investigated a small dataset of 8‐ and 12‐residue loops, with the shorter loops placed initially from a coarse‐grained lattice model and the longer loops from an enumeration assembly method (the Loopy program). The CHARMM22 + CMAP force field with a generalized Born implicit solvent model (molecular‐surface parameterized GBSW2) was used to explore conformational space. We also assessed two empirical scoring methods to detect nativelike conformations from decoys: the all‐atom distance‐scaled ideal‐gas reference state (DFIRE‐AA) statistical potential and the Rosetta energy function. Among the eight‐residue loop targets, SGLD out performed MD in all cases, with a median of 0.48 Å reduction in global root‐mean‐square deviation (RMSD) of the loop backbone coordinates from the native structure. Among the more challenging 12‐residue loop targets, SGLD improved the prediction accuracy over MD by a median of 1.31 Å, representing a substantial improvement. The overall median RMSD for SGLD simulations of 12‐residue loops was 0.91 Å, yielding refinement of a median 2.70 Å from initial loop placement. Results from DFIRE‐AA and the Rosetta model applied to rescoring conformations failed to improve the overall detection calculated from the CHARMM force field. We illustrate the advantage of SGLD over the MD simulation model by presenting potential‐energy landscapes for several loop predictions. Our results demonstrate that SGLD significantly outperforms traditional MD in the generation and populating of nativelike loop conformations and that the CHARMM force field performs comparably to other empirical force fields in identifying these conformations from the resulting ensembles. Published 2011 Wiley Periodicals, Inc. J Comput Chem, 2011     

Force field studies of cholesterol and cholesteryl acetate crystals and cholesterol–cholesterol intermolecular interactions

To model the physical properties of sterols and related species, an all-atom Class II force field has been derived based on the recently reported CFF93 force field for hydrocarbons. It has been tested using both energy minimization and molecular dynamics (MD) simulations of the low-temperature neutron-diffraction structure of cholesteryl acetate crystals and the X-ray diffraction crystal structure of cholesterol. Thus these studies test the techniques and limitations of high-accuracy crystal simulations as well. Employing energy minimization, all cell vectors and volumes were reproduced to within 2.4% of experimental values. For cholesteryl acetate, the root mean square (rms) deviations between the calculated and experimental bond lengths, angles, and torsions of nonhydrogen atoms are 0.013 Å, 1.2°, and 2.4°, respectively. The corresponding maximum deviations are also very small: 0.027 Å for bond length, 3.2° for angle, and 7.6° for torsion. For cholesterol, good agreement between the calculated and experimental structures was found only when the comparison was limited to atoms with relatively small thermal factors (B_eq < 15 Ų). It was found that for both systems, the MD averaged structures were in better agreement with the experimental ones than the energy minimized structures, since the rms deviations in atom positions are smaller for the MD-averaged structures (0.064 Å for cholesteryl acetate and 0.152 Å for cholesterol) than those for the minimized structures (0.178 Å for cholesteryl acetate and 0.189 Å for cholesterol). The force field was then applied to isolated molecules focusing on the rigidity of the cholesteryl ring and cholesterol–cholesterol interaction energies. It is concluded that the cholesteryl ring is fairly rigid since no major conformational change was observed during an MD simulation of a single cholesterol molecule in vacuo at 500 K, in agreement with condensed phase experiments. Calculations of cholesterol–cholesterol pairs suggest that there are only four low-energy configurations and that it is more useful to describe each molecule as having a plane (flat face) and two grooves rather than as having two (one flat and one rough) faces. This provides some insight into the equilibrium crystal structures. Limited results from a modified Class I (CVFF) force field are presented for comparison. © 1995 by John Wiley & Sons, Inc.     

Polypeptide folding using Monte Carlo sampling, concerted rotation, and continuum solvation

An efficient concerted rotation algorithm for use in Monte Carlo statistical mechanics simulations is applied to fold three polypeptides, U(1-17)T9D, alpha(1), and trpzip2, which exhibit native beta-hairpin and alpha-helix folds. The method includes flexible bond and dihedral angles, and a Gaussian bias is applied with driver bond and dihedral angles to optimize the sampling efficiency. Solvation in water is implemented with the generalized Born (GBSA) model. The computed lowest-energy manifolds for the folded structures of the two beta-hairpins agree closely with the corresponding NMR structures. In the case of the alpha(1) peptide, the folded alpha-helical state, which is observed as oligomers in concentrated solution and crystals, is not stable in isolation. The computed preference for random coil structures is in agreement with NMR experiments at low concentration. The fact that native states can be located on high dimensional energy surfaces starting from extended conformations shows that the present methodology samples all relevant parts of the conformational space. The OPLS-AA force field with the GBSA solvent model was also found to perform well in leading to clear energetic separation of the correctly folded structures from misfolded structures for the two peptides that form beta-turns.     

A new method for side-chain conformation prediction using a Hopfield network and reproduced rotamers

We present a new side-chain prediction method based on energy minimization using a Hopfield network, focusing on the buried residues of proteins. In this method, the network is composed of automata assigned to each rotamer to restrict side-chain conformational space. We reproduced a rotamer library that enabled us to more widely cover the space for side-chain conformations than those previously produced. The accuracy of the side-chain modeling was estimated by three standards: root mean square deviations (rmsds) between the modeled and the crystal structures, the percentages of correctly predicted side-chain torsion angles, and the percentages of correctly predicted hydrogen bonds. The average rmsd for buried side chains of 21 proteins was 1.10 Å. The value was almost always improved relative to the previous works. The percentage of side-chain X₁ angles for buried residues was 87.3%. By considering the hydrogen bond energy, the average percentage of correctly predicted hydrogen bonds rose from 33% without hydrogen bond energy to 52% with the bond energy. We applied this method to homology modeling, where the protein backbone used to predict side-chain conformations deviates from the correct conformation, and could predict side-chain conformations as correctly as those using the correct backbones. © 1996 by John Wiley & Sons, Inc.     

Quantum mechanical calculations of tryptophan and comparison with conformations in native proteins

We report a detailed analysis of the potential energy surface of N-acetyl-l-tryptophan-N-methylamide, (NATMA) both in the gas phase and in solution. The minima are identified using the density-functional-theory (DFT) with the 6-31g(d) basis set. The full potential energy surface in terms of torsional angles is spanned starting from various initial configurations. We were able to locate 77 distinct L-minima. The calculated energy maps correspond to the intrinsic conformational propensities of the individual NATMA molecule. We show that these conformations are essentially similar to the conformations of tryptophan in native proteins. For this reason, we compare the results of DFT calculations in the gas and solution phases with native state conformations of tryptophan obtained from a protein library. In native proteins, tryptophan conformations have strong preferences for the beta sheet, right-handed helix, tight turn, and bridge structures. The conformations calculated by DFT, the solution-phase results in particular, for the single tryptophan residue are in agreement with native state values obtained from the Protein Data Bank.     

Finite-state and reduced-parameter representations of protein backbone conformations

This article studies the representation of protein backbone conformations using a finite number of values for the backbone dihedral angles. We develop a combinatorial search algorithm that guarantees finding the global minima of functions over the configuration space of discrete protein conformations, and use this procedure to fit finite-state models to the backbones of globular proteins. It is demonstrated that a finite-state representation with a reasonably small number of states yields either a small root-mean-square error or a small dihedral angle deviation from the native structure, but not both at the same time. The problem can be resolved by introducing limited local optimization in each step of the combinatorial search. In addition, it is shown that acceptable approximation is achieved using a single dihedral angle as an independent variable in local optimization. Results for 11 proteins demonstrate the advantages and shortcomings of both the finite-state and reduced-parameter approximations of protein backbone conformations. © 1994 by John Wiley & Sons, Inc.     

Protein structure prediction using a combination of sequence homology and global energy minimization: II. Energy functions

A protein energy surface is constructed. Validation is through applications of global energy minimization to surface loops of protein crystal structures. For 9 of 10 predictions, the native backbone conformation is identified correctly. Electrostatic energy is modeled as a pairwise sum of interactions between anisotropic atomic charge densities. Model repulsion energy has a softness similar to that seen in ab initio data. Intrinsic torsional energy is modeled as a sum over pairs of adjacent torsion angles of 2-dimensional Fourier series. Hydrophobic energy is that of a hydration shell model. The remainder of hydration free energy is obtained as the energetic effect of a continuous dielectric medium. Parameters are adjusted to reproduce the following data: a complete set of ab initio energy surfaces, meaning one for each pair of adjacent torsion angles of each blocked amino acid; experimental crystal structures and sublimation energies for nine model compounds; ab initio energies over 1014 conformations of 15 small-molecule dimers; and experimental hydration free energies for 48 model compounds. All ab initio data is at the Hartree–Fock/6–31G* level. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 548–573, 1998     

Synthesis,Spectral Characterization,and Crystal Structure of Two Polymorphs of o,o’-Dichloro Dibenzyl Disulfide

2-Chlorophenyl methanethiol undergoes air oxidation catalyzed by different selenides and yields the corresponding disulfide 1 in two polymorphic forms, 1a and 1b. In the molecular structures of the two new polymorphs of o,o′-dichloro dibenzyl disulfide, the dihedral angles between the dibenzyl groups are 82.0(1)°, (1a), and 73.7(4)°, (1b), respectively [(1a): P-1, a = 8.424(2) Å, b = 8.838(2) Å, c = 10.5823(19) Å, α = 90.122(18)°, β = 112.19(2)°, γ = 95.19(2)°, V = 725.9(3) ų; (1b): P2₁/n, a = 10.5888(10) Å, b = 9.1590(6) Å, c = 15.2489(14) Å, β = 103.072(9)°, V = 1440.6(2) ų]. MOPAC computational studies yield dihedral angles of 89.6(5)° and 71.9(9)°. Crystal packing is stabilized by weak π-ring (C?H···Cg) intermolecular interactions in both 1a and 1b and by additional weak Cg ··· Cg intermolecular interactions in 1b, which influence the bond distances, bond angles, and torsion angles of the dibenzyl groups in each polymorph. Additional characterization of each polymorph has been carried out by TEM, IR, ¹H and ¹³C NMR spectroscopy, microanalysis, and by FAB mass spectrometry. TEM studies of a sample of 1a show that it contains cigar-shaped crystallites.

Supplemental materials are available for this article. Go to the publisher's online edition of Phosphorus, Sulfur, and Silicon and the Related Elements to view the free supplemental file.     

Closed loop folding units from structural alignments: Experimental foldons revisited

Nonoverlapping closed loops of around 25–35 amino acids formed via nonlocal interactions at the loop ends have been proposed as an important unit of protein structure. This hypothesis is significant as such short loops can fold quickly and so would not be bound by the Leventhal paradox, giving insight into the possible nature of the funnel in protein folding. Previously, these closed loops have been identified either by sequence analysis (conservation and autocorrelation) or studies of the geometry of individual proteins. Given the potential significance of the closed loop hypothesis, we have explored a new strategy for determining closed loops from the insertions identified by the structural alignment of proteins sharing the same overall fold. We determined the locations of the closed loops in 37 pairs of proteins and obtained excellent agreement with previously published closed loops. The relevance of NMR structures to closed loop determination is briefly discussed. For cytochrome c, cytochrome b₅₆₂ and triosephophate isomerase, independent folding units have been determined on the basis of hydrogen exchange experiments and misincorporation proton‐alkyl exchange experiments. The correspondence between these experimentally derived foldons and the theoretically derived closed loops indicates that the closed loop hypothesis may provide a useful framework for analyzing such experimental data. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2010     

Accelerated molecular dynamics simulations of protein folding

Folding of four fast‐folding proteins, including chignolin, Trp‐cage, villin headpiece and WW domain, was simulated via accelerated molecular dynamics (aMD). In comparison with hundred‐of‐microsecond timescale conventional molecular dynamics (cMD) simulations performed on the Anton supercomputer, aMD captured complete folding of the four proteins in significantly shorter simulation time. The folded protein conformations were found within 0.2–2.1 Å of the native NMR or X‐ray crystal structures. Free energy profiles calculated through improved reweighting of the aMD simulations using cumulant expansion to the second‐order are in good agreement with those obtained from cMD simulations. This allows us to identify distinct conformational states (e.g., unfolded and intermediate) other than the native structure and the protein folding energy barriers. Detailed analysis of protein secondary structures and local key residue interactions provided important insights into the protein folding pathways. Furthermore, the selections of force fields and aMD simulation parameters are discussed in detail. Our work shows usefulness and accuracy of aMD in studying protein folding, providing basic references in using aMD in future protein‐folding studies. © 2015 Wiley Periodicals, Inc.