Quantifying polypeptide conformational space: sensitivity to conformation and ensemble definition

Quantifying the density of conformations over phase space (the conformational distribution) is needed to model important macromolecular processes such as protein folding. In this work, we quantify the conformational distribution for a simple polypeptide (N-mer polyalanine) using the cumulative distribution function (CDF), which gives the probability that two randomly selected conformations are separated by less than a "conformational" distance and whose inverse gives conformation counts as a function of conformational radius. An important finding is that the conformation counts obtained by the CDF inverse depend critically on the assignment of a conformation's distance span and the ensemble (e.g., unfolded state model): varying ensemble and conformation definition (1 --> 2 A) varies the CDF-based conformation counts for Ala(50) from 10(11) to 10(69). In particular, relatively short molecular dynamics (MD) relaxation of Ala(50)'s random-walk ensemble reduces the number of conformers from 10(55) to 10(14) (using a 1 A root-mean-square-deviation radius conformation definition) pointing to potential disconnections in comparing the results from simplified models of unfolded proteins with those from all-atom MD simulations. Explicit waters are found to roughen the landscape considerably. Under some common conformation definitions, the results herein provide (i) an upper limit to the number of accessible conformations that compose unfolded states of proteins, (ii) the optimal clustering radius/conformation radius for counting conformations for a given energy and solvent model, (iii) a means of comparing various studies, and (iv) an assessment of the applicability of random search in protein folding.     

Amyloid Polymorphism in the Protein Folding and Aggregation Energy Landscape

Protein folding involves a large number of steps and conformations in which the folding protein samples different thermodynamic states characterized by local minima. Kinetically trapped on‐ or off‐pathway intermediates are metastable folding intermediates towards the lowest absolute energy minima, which have been postulated to be the natively folded state where intramolecular interactions dominate, and the amyloid state where intermolecular interactions dominate. However, this view largely neglects the rich polymorphism found within amyloid species. We review the protein folding energy landscape in view of recent findings identifying specific transition routes among different amyloid polymorphs. Observed transitions such as twisted ribbon→crystal or helical ribbon→nanotube, and forbidden transitions such helical ribbon?crystal, are discussed and positioned within the protein folding and aggregation energy landscape. Finally, amyloid crystals are identified as the ground state of the protein folding and aggregation energy landscape.     

Kinetics of cooperative protein folding involving two separate conformational families

The master equation that describes the kinetics of protein folding is solved numerically for a portion of Staphylococcal Protein A by a Laplace transformation. The calculations are carried out with 50 local-minimum conformations belonging to two conformational families. The master equation allows for transitions among all the 50 conformations in the evolution toward the final folded equilibrium distribution of conformations. It is concluded that the native protein folds in a fast cooperative process. The global energy minimum of a native protein can be reached after a sufficiently long folding time regardless of the initial state and the existence of a large number of local energy minima. Conformations representing non-native states of the protein can transform to the native state even if they do not belong to the native conformational family. Given a starting conformation, the protein molecule can fold to its final conformation through different paths. Finally, when the folding reaches the equilibrium distribution, the protein molecule adopts a set of conformations in which the global minimum has the largest average probability.     

Intrinsic energy landscapes of amino acid side-chains

Amino acid side-chain conformational properties influence the overall structural and dynamic properties of proteins and, therefore, their biological functions. In this study, quantum mechanical (QM) potential energy surfaces for the rotation of side-chain χ(1) and χ(2) torsions in dipeptides in the alphaR, beta, and alphaL backbone conformations were calculated. The QM energy surfaces provide a broad view of the intrinsic conformational properties of each amino acid side-chain. The extent to which intrinsic energetics dictates side-chain orientation was studied through comparisons of the QM energy surfaces with χ(1) and χ(2) free energy surfaces from probability distributions obtained from a survey of high resolution crystal structures. In general, the survey probability maxima are centered in minima of the QM surfaces as expected for sp(3) (or sp(2) for χ(2) of Asn, Phe, Trp, and Tyr) atom centers with strong variations between amino acids occurring in the energies of the minima indicating intrinsic differences in rotamer preferences. High correlations between the QM and survey data were found for hydrophobic side-chains except Met, suggesting minimal influence of the protein and solution environments on their conformational distributions. Conversely, low correlations for polar or charged side-chains indicate a dominant role of the environment in stabilizing conformations that are not intrinsically favored. Data also link the presence of off-rotamers in His and Trp to favorable interactions with the backbone. Results also suggest that the intrinsic energetics of the side-chains of Phe and Tyr may play important roles in protein folding and stability. Analyses on whether intrinsic side-chain energetics can influence backbone preference identified a strong correlation for residues in the alphaL backbone conformation. It is suggested that this correlation reflects the intrinsic instability of the alphaL backbone such that assumption of this backbone conformation is facilitated by intrinsically favorable side-chain conformations. Together our results offer a broad overview of the conformational properties of amino acid side-chains and the QM data may be used as target data for force field optimization.     

Comparative analysis of protein-bound ligand conformations with respect to catalyst's conformational space subsampling algorithms

We examined the quality of Catalyst's conformational model generation algorithm via a large scale study based on the crystal structures of a sample of 510 pharmaceutically relevant protein-ligand complexes extracted from the Protein Data Bank (PDB). Our results show that the tested algorithms implemented within Catalyst are able to produce high quality conformers, which in most of the cases are well suited for in silico drug research. Catalyst-specific settings were analyzed, such as the method used for the conformational model generation (FAST vs BEST) and the maximum number of generated conformers. By setting these options for higher fitting quality, the average RMS values describing the similarity of experimental and simulated conformers were improved from an RMS of 1.06 with max. 50 FAST generated conformers to an RMS of 0.93 with max. 255 BEST generated conformers, which represents an improvement by 12%. Each method provides best fitting conformers with an RMS value<1.50 in more than 80% of all cases. We analyzed the computing time/quality ratio of various conformational model generation settings and examined ligands in high energy conformations. Furthermore, properties of the same ligands in various proteins were investigated, and the fitting qualities of experimental conformations from the PDB and the Cambridge Structural Database (CSD) were compared. One of the most important conclusions of former studies, the fact that bioactive conformers often have energy high above that of global minima, was confirmed.     

Efficient algorithm for conformational search of macrocyclic molecules

A new algorithm, complementarity, is developed for conformational search of macrocyclic molecules. The algorithm scans a large number of candidate conformations and energy-minimizes only the promising ones. These candidates can be generated by two operators that construct new conformations from known minima. The candidates have similar bonded-interaction energy as the known minima and possibly lower nonbonded interaction energy. This algorithm is 9 to 11 times faster than the existing methods when tested on two large rings, cycloheptadecane and rifamycin SV. © 1997 by John Wiley & Sons, Inc.     

Open science grid study of the coupling between conformation and water content in the interior of a protein

Computational grids are a promising resource for modeling complex biochemical processes such as protein folding, penetration of gases or water into proteins, or protein structural rearrangements coupled to ligand binding. We have enabled the molecular dynamics program CHARMM to run on the Open Science Grid. The implementation is general, flexible, easily modifiable for use with other molecular dynamics programs and other grids and automated in terms of job submission, monitoring, and resubmission. The usefulness of grid computing was demonstrated through the study of hydration of the Glu-66 side chain in the interior of protein staphylococcal nuclease. Multiple simulations started with and without two internal water molecules shown crystallographically to be associated with the side chain of Glu-66 yielded two distinct populations of rotameric states of Glu-66 that differed by as much as 20%. This illustrates how internal water molecules can bias protein conformations. Furthermore, there appeared to be a temporal correlation between dehydration of the side chain and conformational transitions of Glu-66. This example demonstrated how difficult it is to get convergence even in the relatively simple case of a side chain oscillating between two conformations. With grid computing, we also benchmarked the self-guided Langevin dynamics method against the Langevin dynamics method traditionally used for temperature control in molecular dynamics simulations and showed that the two methods yield comparable results.     

The effects of biasing torsional mutations in a conformational GA

This paper describes the effects of incorporating torsional bias into a conformational Genetic Algorithm (GA) such as that found in the GASP program. Several major conclusions can be drawn. Biasing torsional angles toward values associated with local energy minima increases the rate of convergence of the fitness function (consisting of energy, steric, and pharmacophoric compatibility terms) for a set of molecules, but a definite tradeoff exists between total model energy and the steric and pharmacophoric compatibility terms in the fitness score. Biasing torsions in favor of sets of angles drawn from low-energy conformations does not guarantee low total energy, but biased torsional sampling does generally produce less strained models than does the uniform torsional sampling in classical GASP. Overall, torsionally biased sampling produces good models comprised of energetically favorable ligand conformations.     

Exploring the similarities between potential smoothing and simulated annealing

Simulated annealing and potential function smoothing are two widely used approaches for global energy optimization of molecular systems. Potential smoothing as implemented in the diffusion equation method has been applied to study partitioning of the potential energy surface (PES) for N‐Acetyl‐Ala‐Ala‐N‐Methylamide (CDAP) and the clustering of conformations on deformed surfaces. A deformable version of the united‐atom OPLS force field is described, and used to locate all local minima and conformational transition states on the CDAP surface. It is shown that the smoothing process clusters conformations in a manner consistent with the inherent structure of the undeformed PES. Smoothing deforms the original surface in three ways: structural shifting of individual minima, merging of adjacent minima, and energy crossings between unrelated minima. A master equation approach and explicit molecular dynamics trajectories are used to uncover similar features in the equilibrium probability distribution of CDAP minima as a function of temperature. Qualitative and quantitative correlations between the simulated annealing and potential smoothing approaches to enhanced conformational sampling are established. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 531–552, 2000     

New optimization method for conformational energy calculations on polypeptides: Conformational space annealing

A new optimization method is presented to search for the global minimum-energy conformations of polypeptides. The method combines essential aspects of the build-up procedure and the genetic algorithm, and it introduces the important concept of “conformational space annealing.” Instead of considering a single conformation, attention is focused on a population of conformations while new conformations are obtained by modifying a “seed conformation.” The annealing is carried out by introducing a distance cutoff, D_cut, which is defined in the conformational space; D_cut effectively divides the whole conformational space of local minima into subdivisions. The value of D_cut is set to a large number at the beginning of the algorithm to cover the whole conformational space, and annealing is achieved by slowly reducing it. Many distinct local minima designed to be distributed as far apart as possible in conformational space are investigated simultaneously. Therefore, the new method finds not only the global minimum-energy conformation but also many other distinct local minima as by-products. The method is tested on Met-enkephalin, a 24-dihedral angle problem. For all 100 independent runs, the accepted global minimum-energy conformation was obtained after about 2600 minimizations on average. © 1997 John Wiley & Sons, Inc. J Comput Chem 18: 1222–1232     

Overcoming entropic barrier with coupled sampling at dual resolutions

An enhanced sampling method is proposed for ab initio protein folding simulations. The new method couples a high-resolution model for accuracy and a low-resolution model for efficiency. It aims to overcome the entropic barrier found in the exponentially large protein conformational space when a high-resolution model, such as an all-atom molecular mechanics force field, is used. The proposed method is designed to satisfy the detailed balance condition so that the Boltzmann distribution can be generated in all sampling trajectories in both high and low resolutions. The method was tested on model analytical energy functions and ab initio folding simulations of a beta-hairpin peptide. It was found to be more efficient than replica-exchange method that is used as its building block. Analysis with the analytical energy functions shows that the number of energy calculations required to find global minima and to converge mean potential energies is much fewer with the new method. Ergodic measure shows that the new method explores the conformational space more rapidly. We also studied imperfect low-resolution energy models and found that the introduction of errors in low-resolution models does decrease its sampling efficiency. However, a reasonable increase in efficiency is still observed when the global minima of the low-resolution models are in the vicinity of the global minimum basin of the high-resolution model. Finally, our ab initio folding simulation of the tested peptide shows that the new method is able to fold the peptide in a very short simulation time. The structural distribution generated by the new method at the equilibrium portion of the trajectory resembles that in the equilibrium simulation starting from the crystal structure.     

Different approaches to conformational analysis: A comparison of completeness,efficiency, and reliability based on the study of a nine-membered lactam

Different methods such as molecular dynamics, systematic, or stochastic search and a special “generic shape” algorithm have been employed in the conformational analysis of a nine-membered lactam. Furthermore, crystal data were used to generate conformations of the compound under consideration. The various methods are compared in terms of their efficiency and completeness in the search for conformations with an energy content of up to 60 kJ/mol above the global minimum. Additionally, the generated conformations have been optimized by different techniques, molecular mechanics and quantum chemical calculations, to compare the number of existing local minima and their relative energies and geometries.     

Modification and optimization of the united-residue (UNRES) potential energy function for canonical simulations. I. Temperature dependence of the effective energy function and tests of the optimization method with single training proteins

We report the modification and parametrization of the united-residue (UNRES) force field for energy-based protein structure prediction and protein folding simulations. We tested the approach on three training proteins separately: 1E0L (beta), 1GAB (alpha), and 1E0G (alpha + beta). Heretofore, the UNRES force field had been designed and parametrized to locate native-like structures of proteins as global minima of their effective potential energy surfaces, which largely neglected the conformational entropy because decoys composed of only lowest-energy conformations were used to optimize the force field. Recently, we developed a mesoscopic dynamics procedure for UNRES and applied it with success to simulate protein folding pathways. However, the force field turned out to be largely biased toward -helical structures in canonical simulations because the conformational entropy had been neglected in the parametrization. We applied the hierarchical optimization method, developed in our earlier work, to optimize the force field; in this method, the conformational space of a training protein is divided into levels, each corresponding to a certain degree of native-likeness. The levels are ordered according to increasing native-likeness; level 0 corresponds to structures with no native-like elements, and the highest level corresponds to the fully native-like structures. The aim of optimization is to achieve the order of the free energies of levels, decreasing as their native-likeness increases. The procedure is iterative, and decoys of the training protein(s) generated with the energy function parameters of the preceding iteration are used to optimize the force field in a current iteration. We applied the multiplexing replica-exchange molecular dynamics (MREMD) method, recently implemented in UNRES, to generate decoys; with this modification, conformational entropy is taken into account. Moreover, we optimized the free-energy gaps between levels at temperatures corresponding to a predominance of folded or unfolded structures, as well as to structures at the putative folding-transition temperature, changing the sign of the gaps at the transition temperature. This enabled us to obtain force fields characterized by a single peak in the heat capacity at the transition temperature. Furthermore, we introduced temperature dependence to the UNRES force field; this is consistent with the fact that it is a free-energy and not a potential energy function. beta     

Torsion angle preference and energetics of small-molecule ligands bound to proteins

Small organic molecules can assume conformations in the protein-bound state that are significantly different from those in solution. We have analyzed the conformations of 21 common torsion motifs of small molecules extracted from crystal structures of protein-ligand complexes and compared them with their torsion potentials calculated by an ab initio DFT method. We find a good correlation between the potential energy of the torsion motifs and their conformational distribution in the protein-bound state: The most probable conformations of the torsion motifs agree well with the calculated global energy minima, and the lowest torsion-energy state becomes increasingly dominant as the torsion barrier height increases. The torsion motifs can be divided into 3 groups based on torsion barrier heights: high (>4 kcal/mol), medium (2-4 kcal/mol), and low (<2 kcal/mol). The calculated torsion energy profiles are predictive for the most preferred bound conformation for the high and medium barrier groups, the latter group common in druglike molecules. In the high-barrier group of druglike ligands, >95% of conformational torsions occur in the energy region <4 kcal/mol. The conformations of the torsion motifs in the protein-bound state can be modeled by a Boltzmann distribution with a temperature factor much higher than room temperature. This high-temperature factor, derived by fitting the theoretical model to the experimentally observed conformation occurrence of torsions, can be interpreted as the perturbation that proteins inflict on the conformation of the bound ligand. Using this model, it is calculated that the average strain energy of a torsion motif in ligands bound to proteins is approximately 0.6 kcal/mol, a result which can be related to the lower binding efficiency of larger ligands with more rotatable bonds. The above results indicate that torsion potentials play an important role in dictating ligand conformations in both the free and the bound states.     

The conformational free energy landscape of beta-D-glucopyranose. Implications for substrate preactivation in beta-glucoside hydrolases

Using ab initio metadynamics we have computed the conformational free energy landscape of beta-D-glucopyranose as a function of the puckering coordinates. We show that the correspondence between the free energy and the Stoddard's pseudorotational itinerary for the system is rather poor. The number of free energy minima (9) is smaller than the number of ideal structures (13). Moreover, only six minima correspond to a canonical conformation. The structural features, the electronic properties, and the relative stability of the predicted conformers permit the rationalization of the occurrence of distorted sugar conformations in all the available X-ray structures of beta-glucoside hydrolase Michaelis complexes. We show that these enzymes recognize the most stable distorted conformers of the isolated substrate and at the same time the ones better prepared for catalysis in terms of bond elongation/shrinking and charge distribution. This suggests that the factors governing the distortions present in these complexes are largely dictated by the intrinsic properties of a single glucose unit.     

Stochastic protein folding simulation in the three-dimensional HP-model

We present results from three-dimensional protein folding simulations in the HP-model on ten benchmark problems. The simulations are executed by a simulated annealing-based algorithm with a time-dependent cooling schedule. The neighbourhood relation is determined by the pull-move set. The results provide experimental evidence that the maximum depth D of local minima of the underlying energy landscape can be upper bounded by D相似文献   

Conformational analysis of a stereochemically complete set of cis-enediol peptide analogues

A conformational analysis of a stereochemically complete set of peptide analogues based on a cis-enediol unit is presented. The cis-enediol unit, which can replace a two or a three amino acid segment of a peptide, contains two "side chains", four asymmetrical carbon atoms, and six free dihedral angles. To determine the accessible conformational space, the molecules are divided into three fragments, each containing two free dihedral angles. The energy surfaces are computed for all dihedral angle values, and the possible conformations of the cis-enediol unit analogues are built using all combinations of the surface minima. Such a "build-up" procedure, which is very fast, is able to reproduce 75% of the minima obtained from a full dihedral angle exploration of the conformational space. The cis-enediol unit minima are compared with the corresponding di- and tripeptide minima; all peptide minima can be closely matched by a cis-enediol unit minimum of low energy (less than 2.2 kcal/mol above the lowest energy conformer). However, there are low energy minima of the cis-enediol unit that have no corresponding minima in peptides. The results are shown to depend strongly on the chirality of the analogues. The ability of each of the stereoisomers to mimic natural peptides, evaluated by the present approach, is correlated with its experimental activity in a renin inhibition assay.