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.     

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.     

An analysis of the structural and energetic properties of deoxyribose by potential energy methods

We discuss the three fundamental issues of a computational approach in structure prediction by potential energy minimization, and analyze them for the nucleic acid component deoxyribose. Predicting the conformation of deoxyribose is important not only because of the molecule's central conformational role in the nucleotide backbone, but also because energetic and geometric discrepancies from experimental data have exposed some underlying uncertainties in potential energy calculations. The three fundamental issues examined here are: (i) choice of coordinate system to represent the molecular conformation; (ii) construction of the potential energy function; and (iii) choice of the minimization technique. For our study, we use the following combination. First, the molecular conformation is represented in cartesian coordinate space with the full set of degrees of freedom. This provides an opportunity for comparison with the pseudorotation approximation. Second, the potential energy function is constructed so that all the interactions other than the nonbonded terms are represented by polynomials of the coordinate variables. Third, two powerful Newton methods that are globally and quadratically convergent are implemented: Gill and Murray's Modified Newton method and a Truncated Newton method, specifically developed for potential energy minimization. These strategies have produced the two experimentally-observed structures of deoxyribose with geometric data (bond angles and dihedral angles) in very good agreement with experiment. More generally, the application of these modeling and minimization techniques to potential energy investigations is promising. The use of cartesian variables and polynomial representation of bond length, bond angle and torsional potentials promotes efficient second-derivative computation and, hence, application of Newton methods. The truncated Newton, in particular, is ideally suited for potential energy minimization not only because the storage and computational requirements of Newton methods are made manageable, but also because it contains an important algorithmic adaptive feature: the minimization search is diverted from regions where the function is nonconvex and is directed quickly toward physically interesting regions.     

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.     

Energy minimization of rigid-geometry polypeptides with exactly closed disulfide loops

A method has been developed for minimizing the energy of a polypeptide with rigid geometry while keeping all disulfide loops closed exactly. Exact closure of disulfide loops implies that some dihedral angles become implicit functions of the remaining dihedral angles in the polypeptide; the efficacy of the method is related to the manner in which the implicitly defined dihedral angles are chosen. The method has been used to find minimum-energy conformations of bovine pancreatic trypsin inhibitor, ribonuclease A, crambin, the defensin HNP3 dimer, and ω-conotoxin. For the first two proteins, the starting conformations for energy minimization had been derived previously from crystal structures using pseudopotentials to keep the disulfide loops almost closed. Starting conformations for the remaining three proteins were derived from their crystal or NMR structures by similar procedures. In all cases, the energy-minimized structures had a significantly and, in some cases, substantially, lower energy than the starting structures. The RMS deviations between the exactly closed energy- minimized structures and the crystal or NMR structures from which they were derived ranged from 0.9 Å to 1.9 Å, suggesting that the computed structures can serve as “regularized” native structures for these proteins. The energy of a ribonuclease derivative lacking the 65–72 disulfide bridge was minimized using the procedure; the result showed that this derivative has a low-energy structure with a conformation very close to that of native ribonuclease, and is consistent with its postulated role in the folding of ribonuclease. These results offer strong support for the validity of the rigid-geometry model in the studies of the conformational energy of proteins. © 1997 by John Wiley & Sons, Inc.     

Docking small ligands in flexible binding sites

A novel procedure for docking ligands in a flexible binding site is presented. It relies on conjugate gradient minimization, during which nonbonded interactions are gradually switched on. Short Monte Carlo minimization runs are performed on the most promising candidates. Solvation is implicitly taken into account in the evaluation of structures with a continuum model. It is shown that the method is very accurate and can model induced fit in the ligand and the binding site. The docking procedure has been successfully applied to three systems. The first two are the binding of progesterone and 5β-androstane-3,17-dione to the antigen binding fragment of a steroid binding antibody. A comparison of the crystal structures of the free and the two complexed forms reveals that any attempt to model binding must take protein rearrangements into account. Furthermore, the two ligands bind in two different orientations, posing an additional challenge. The third test case is the docking of N^α-(2-naphthyl-sulfonyl-glycyl)-D -para-amidino-phenyl-alanyl-piperidine (NAPAP) to human α-thrombin. In contrast to steroids, NAPAP is a very flexible ligand, and no information of its conformation in the binding site is used. All docking calculations are started from X-ray conformations of proteins with the uncomplexed binding site. For all three systems the best minima in terms of free energy have a root mean square deviation from the X-ray structure smaller than 1.5 Å for the ligand atoms. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 21–37, 1998     

LigDockCSA: protein-ligand docking using conformational space annealing

Protein–ligand docking techniques are one of the essential tools for structure‐based drug design. Two major components of a successful docking program are an efficient search method and an accurate scoring function. In this work, a new docking method called LigDockCSA is developed by using a powerful global optimization technique, conformational space annealing (CSA), and a scoring function that combines the AutoDock energy and the piecewise linear potential (PLP) torsion energy. It is shown that the CSA search method can find lower energy binding poses than the Lamarckian genetic algorithm of AutoDock. However, lower‐energy solutions CSA produced with the AutoDock energy were often less native‐like. The loophole in the AutoDock energy was fixed by adding a torsional energy term, and the CSA search on the refined energy function is shown to improve the docking performance. The performance of LigDockCSA was tested on the Astex diverse set which consists of 85 protein–ligand complexes. LigDockCSA finds the best scoring poses within 2 Å root‐mean‐square deviation (RMSD) from the native structures for 84.7% of the test cases, compared to 81.7% for AutoDock and 80.5% for GOLD. The results improve further to 89.4% by incorporating the conformational entropy. © 2011 Wiley Periodicals, Inc. J Comput Chem, 2011     

The application of molecular mechanics to the structures of carbohydrates

A study is reported of the accuracy with which the geometries of pyranose and methyl pyranoside molecules are predicted by molecular mechanics. Calculations of the conformational energies of the model compounds dihydroxymethane, methoxymethanol, and dimethoxymethane, made with the program MMI, produced results that compare well with previous ab initio molecular orbital calculations. This indicates that MMI gives a satisfactory account of the energetic and conformational aspects of the anomeric effect, a conclusion further supported by calculations on 2-methoxytetrahydropyran. The prediction of the observed preferred conformations of the primary alcohol group in aldohexopyranoses appears to be less satisfactory. MMI-CARB, a version of MMI with changes in some of the equilibrium C? O bond lengths of the program, has been used to calculate the geometries of 13 pyranose and methyl pyranoside molecules, the crystal structures of which have been studied by neutron diffraction. When the C? C? O? H torsion angles are constrained to approximately the values observed in the crystal structures, good agreement is obtained between the theoretical and experimental molecular geometries. The rms deviation for C? C and C? O bonds, excluding those significantly affected by thermal motion in the crystal structure determinations, is 0.005 Å. Corresponding figures for the valence angles that do not involve hydrogen atoms and for the ring torsion angles are 1.2° and 2.0°, respectively. The Cremer and Pople puckering parameters for the pyranose rings are reproduced within 0.026 Å in Q and 5.4° in θ.     

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     

Evaluation of a knowledge‐based potential of mean force for scoring docked protein–ligand complexes

The Biomolecular Ligand Energy Evaluation Protocol (BLEEP) is a knowledge‐based potential derived from high‐resolution X‐ray structures of protein–ligand complexes. The performance of this potential in ranking the hypothetical structures resulting from a docking study has been evaluated using fifteen protein–ligand complexes from the Protein Data Bank. In the majority of complexes BLEEP was successful in identifying the native (experimental) binding mode or an alternative of low rms deviation (from the native) as the lowest in energy. Overall BLEEP is slightly better than the DOCK energy function in discriminating native‐like modes. Even when alternative binding modes rank lower than the native structure, a reasonable energy is assigned to the latter. Breaking down the BLEEP scores into the atom–atom contributions reveals that this type of potential is grossly dominated by longer range interactions (>5 Å), which makes it relatively insensitive to small local variations in the binding site. However, despite this limitation, the lack, at present, of accurate protein–ligand potentials means that BLEEP is a promising approach to improve the filtering of structures resulting from docking programs. Moreover, BLEEP should improve with the continuously increasing number of complexes available in the PDB. © 2001 John Wiley & Sons, Inc. J Comput Chem 22: 673–688, 2001     

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.     

The structural modeling of EF-hand motifs in parvalbumin

Parvalbumin (Parv) is a typical protein with EF-hand motifs that play an important role in many physiological processes. We present a novel free energy to model the skeletal C\(_\alpha \) chain of the protein from the basic principle of mathematics and physics. Starting from the crystal structure of Parv (PDB code 2PVB), we first analyze the profile of the C\(_\alpha \) bond and torsion angles over the segment that contains the secondary structures. Then the parameters in the energy function are evaluated for the helix ABCD fragment that contains two EF-hand domains in Parv. Meanwhile an eight-soliton configuration at the energy minimum is constructed to model the conformation of ABCD fragment. The deviation of the conformation constructed from the model away from the crystal structure is as small as 1.28 Å. The structural modeling stems from the physical energy, which is a benefit relative to the statistics-based or knowledge-based technologies.     

Identification of ligands for RNA targets via structure-based virtual screening: HIV-1 TAR

Binding of the Tat protein to TAR RNA is necessary for viral replication of HIV-1. We screened the Available Chemicals Directory (ACD) to identify ligands to bind to a TAR RNA structure using a four-step docking procedure: rigid docking first, followed by three steps of flexible docking using a pseudobrownian Monte Carlo minimization in torsion angle space with progressively more detailed conformational sampling on a progressively smaller list of top-ranking compounds. To validate the procedure, we successfully docked ligands for five RNA complexes of known structure. For ranking ligands according to binding avidity, an empirical binding free energy function was developed which accounts, in particular, for solvation, isomerization free energy, and changes in conformational entropy. System-specific parameters for the function were derived on a training set of RNA/ligand complexes with known structure and affinity. To validate the free energy function, we screened the entire ACD for ligands for an RNA aptamer which binds l-arginine tightly. The native ligand ranked 17 out of ca. 153,000 compounds screened, i.e., the procedure is able to filter out >99.98% of the database and still retain the native ligand. Screening of the ACD for TAR ligands yielded a high rank for all known TAR ligands contained in the ACD and suggested several other potential TAR ligands. Eight of the highest ranking compounds not previously known to be ligands were assayed for inhibition of the Tat-TAR interaction, and two exhibited a CD₅₀ of ca. 1 M.     

A pose prediction approach based on ligand 3D shape similarity

Molecular docking predicts the best pose of a ligand in the target protein binding site by sampling and scoring numerous conformations and orientations of the ligand. Failures in pose prediction are often due to either insufficient sampling or scoring function errors. To improve the accuracy of pose prediction by tackling the sampling problem, we have developed a method of pose prediction using shape similarity. It first places a ligand conformation of the highest 3D shape similarity with known crystal structure ligands into protein binding site and then refines the pose by repacking the side-chains and performing energy minimization with a Monte Carlo algorithm. We have assessed our method utilizing CSARdock 2012 and 2014 benchmark exercise datasets consisting of co-crystal structures from eight proteins. Our results revealed that ligand 3D shape similarity could substitute conformational and orientational sampling if at least one suitable co-crystal structure is available. Our method identified poses within 2 Å RMSD as the top-ranking pose for 85.7 % of the test cases. The median RMSD for our pose prediction method was found to be 0.81 Å and was better than methods performing extensive conformational and orientational sampling within target protein binding sites. Furthermore, our method was better than similar methods utilizing ligand 3D shape similarity for pose prediction.     

Improved pose and affinity predictions using different protocols tailored on the basis of data availability

The D3R 2015 grand drug design challenge provided a set of blinded challenges for evaluating the applicability of our protocols for pose and affinity prediction. In the present study, we report the application of two different strategies for the two D3R protein targets HSP90 and MAP4K4. HSP90 is a well-studied target system with numerous co-crystal structures and SAR data. Furthermore the D3R HSP90 test compounds showed high structural similarity to existing HSP90 inhibitors in BindingDB. Thus, we adopted an integrated docking and scoring approach involving a combination of both pharmacophoric and heavy atom similarity alignments, local minimization and quantitative structure activity relationships modeling, resulting in the reasonable prediction of pose [with the root mean square deviation (RMSD) values of 1.75 Å for mean pose 1, 1.417 Å for the mean best pose and 1.85 Å for the mean all poses] and affinity (ROC AUC = 0.702 at 7.5 pIC50 cut-off and R = 0.45 for 180 compounds). The second protein, MAP4K4, represents a novel system with limited SAR and co-crystal structure data and little structural similarity of the D3R MAP4K4 test compounds to known MAP4K4 ligands. For this system, we implemented an exhaustive pose and affinity prediction protocol involving docking and scoring using the PLANTS software which considers side chain flexibility together with protein–ligand fingerprints analysis assisting in pose prioritization. This protocol through fares poorly in pose prediction (with the RMSD values of 4.346 Å for mean pose 1, 4.69 Å for mean best pose and 4.75 Å for mean all poses) and produced reasonable affinity prediction (AUC = 0.728 at 7.5 pIC50 cut-off and R = 0.67 for 18 compounds, ranked 1st among 80 submissions).     

Quantum-Chemical and semiclassical calculations of intermolecular interactions of phospholipids

By use of empirical 0–1–6–12 atom–atom potential functions and the PCILOCC method intra- and intermolecular interactions of glycero–phosphoryl–ethanolamine model head groups in a planar layer crystal were calculated. Starting from investigations of the two-dimensional energy-contour diagrams the minima of energy as a function of all head group torsion angles were calculated using a gradient procedure. Within an interval of 15 kcal/mol above the energy of the global minimum we obtained about 30 local minima. These results demonstrate a high flexibility of the investigated phosphorylethanolamine head group in agreement with experiment. The ethanolamine moiety exists in enantiomeric conformations. With the torsion angles of the 0–1–6–12 energy minimization procedure PCILOCC calculations were carried out. These calculations yield the x-ray conformation as the most stable one (unit-cell stabilization energy = ?36.3 kcal/mol). The PCILOCC as well as the potential function calculations show that the conformation of phospholipid head groups in layer crystals is determined by intramolecular as well as by intermolecular interactions with neighboring phospholipid molecules.     

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     

The MM2QM tool for combining docking,molecular dynamics,molecular mechanics,and quantum mechanics†

The use of the MM2QM tool in a combined docking + molecular dynamics (MD) + molecular mechanics (MM) + quantum mechanical (QM) binding affinity prediction study is presented, and the tool itself is discussed. The system of interest is Mycobacterium tuberculosis (MTB) pantothenate synthetase in complexes with three highly similar sulfonamide inhibitors, for which crystal structures are available. Starting from the structure of MTB pantothenate synthetase in the “open” conformation and following the combined docking + MD + MM + QM procedure, we were able to capture the closing of the enzyme binding pocket and to reproduce the position of the ligands with an average root mean square deviation of 1.6 Å. Protein–ligand interaction energies were reproduced with an average error lower than 10%. The discussion on the MD part and a protein flexibility importance is carried out. The presented approach may be useful especially for finding analog inhibitors or improving drug candidates. © 2012 Wiley Periodicals, Inc.     

Flexible docking simulations: Scaled collective variable Monte Carlo minimization approach using Bezier splines,and comparison with a standard Monte Carlo algorithm

An algorithm for docking a flexible ligand onto a flexible or rigid receptor, using the scaled‐collective‐variables Monte Carlo with energy minimization approach, is presented. Energy minimization is shown to be one of the best techniques for distinguishing between native‐ and nonnative‐generated conformations. Incorporation of this technique into a Monte Carlo procedure enables one to distinguish the native conformation directly during the conformational search. It avoids the generation of a large number of ligand conformers for which more sophisticated energy evaluation tools would have had to be applied to identify the nativelike conformations. The efficiency of the Monte Carlo minimization was greatly improved by incorporating a new grid‐based energy evaluation technique using Bezier splines for which the energy function, as well as all of its derivatives, can be deduced from the values at the grid points. Comparison between our ECEPP/3‐based algorithm and the Monte Carlo algorithm presented elsewhere (Hart, T. N.; Read, R. J. Prot Struct Funct Genet 1992, 13, 206–222) has been made for docking NH₂ D Phe Pro Arg COOH, the noncovalent analog of NH₂ D Phe Pro Arg chloromethylketone (PPACK), onto the active site of human α‐thrombin. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 244–252, 1999     

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.