Conformational searching using a population-based incremental learning algorithm

A new population-based incremental learning algorithm for conformational searching of molecules is presented. This algorithm is particularly effective at determining, by relatively small number of energy minimizations, global energy minima of large flexible molecules. The algorithm is also able to find a large set of low energy conformations of more rigid small molecules. The performance of the algorithm is relation to other algorithm is examined via the test molecules: C(18) H(38) , C(39)H(80) , cycloheptadecane and a set of five drug-like molecules.     

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     

Global energy minimization by rotational energy embedding

Given a sufficiently good empirical potential function for the internal energy of molecules, prediction of the preferred conformations is nearly impossible for large molecules because of the enormous number of local energy minima. Energy embedding has been a promising method for locating extremely good local minima, if not always the global minimum. The algorithm starts by locating a very good local minimum when the molecule is in a high-dimensional Euclidean space, and then it gradually projects down to three dimensions while allowing the molecule to relax its energy throughout the process. Now we present a variation on the method, called rotational energy embedding, where the descent into three dimensions is carried out by a sequence of internal rotations that are the multidimensional generalization of varying torsion angles in three dimensions. The new method avoids certain kinds of difficulties experienced by ordinary energy embedding and enables us to locate conformations very near the native for avian pancreatic polypeptide and apamin, given only their amino acid sequences and a suitable potential function.     

The ellipsoid algorithm as a method for the determination of polypeptide conformations from experimental distance constraints and energy minimization

A new method for constrained nonlinear optimization known as the ellipsoid algorithm is evaluated as a means of determining and refining the conformations of peptides. Advantages of the ellipsoid algorithm over conventional optimization methods include that it avoids many local minima that other methods would be trapped by, and that it is sometimes able to find optimum solutions in which the constraints are satisfied exactly. The dihedral angles about single bonds were used as variables to keep the dimensionality low (the rate of convergence decreases rapidly with increasing dimensionality of the problem). The method is evaluated on problems involving distance constraints, and for minimization of conformational energy functions. In an initial application, conformations consistent with an experimental set of NMR distance constraints were obtained in a problem involving 48 variable dihedral angles.     

A strategy to find minimal energy nanocluster structures

An unbiased strategy to search for the global and local minimal energy structures of free standing nanoclusters is presented. Our objectives are twofold: to find a diverse set of low lying local minima, as well as the global minimum. To do so, we use massively the fast inertial relaxation engine algorithm as an efficient local minimizer. This procedure turns out to be quite efficient to reach the global minimum, and also most of the local minima. We test the method with the Lennard–Jones (LJ) potential, for which an abundant literature does exist, and obtain novel results, which include a new local minimum for LJ₁₃, 10 new local minima for LJ₁₄, and thousands of new local minima for . Insights on how to choose the initial configurations, analyzing the effectiveness of the method in reaching low‐energy structures, including the global minimum, are developed as a function of the number of atoms of the cluster. Also, a novel characterization of the potential energy surface, analyzing properties of the local minima basins, is provided. The procedure constitutes a promising tool to generate a diverse set of cluster conformations, both two‐ and three‐dimensional, that can be used as an input for refinement by means of ab initio methods. © 2013 Wiley Periodicals, Inc.     

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     

Systematic stepsize variation: Efficient method for searching conformational space of polypeptides

A new and efficient method for overcoming the multiple minima problem of polypeptides, the systematic stepsize variation (SSV) method, is presented. The SSV is based on the assumption that energy barriers can be passed over by sufficiently large rotations about rotatable bonds: randomly chosen dihedral angles are updated starting with a small stepsize (i.e., magnitude of rotation). A new structure is accepted only if it possesses a lower energy than the precedent one. Local minima are passed over by increasing the stepsize systematically. When no new structures are found any longer, the simulation is continued with the starting structure, but other trajectories will be followed due to the random order in updating the torsional angles. First, the method is tested with Met-enkephalin, a peptide with a known global minimum structure; in all runs the latter is found at least once. The global minimum conformations obtained in the simulations show deviations of ±0.0004 kcal/mol from the reference structure and, consequently, are perfectly superposable. For comparison, Metropolis Monte Carlo simulated annealing (MMC-SA) is performed. To estimate the efficiency of the algorithm depending on the complexity of the optimization problem, homopolymers of Ala and Gly of different lengths are simulated, with both the SSV and the MMC-SA method. The comparative simulations clearly reveal the higher efficiency of SSV compared with MMC-SA. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 1470–1481, 1998     

A Dynamic Study of the Stereoisomerization of Pentaphenylethane by Molecular Mechanics Methods

Two conformations, 1 and 2 , of pentaphenylethane are compared. The ground state conformation 2 results from an earlier computational work by force fields procedures [1], whereas 1 has been more recently observed in the crystalline state by X-ray diffraction methods. The strain energy of 1 minimizes very close to the value computed for 2 . These conformations belong to two distinct minima of the potential energy surface and are at the most separated by a barrier of about 7 kcal/mol. The pathway converting 1 into its enantiomer is shown to run over a barrier of only 1.5 kcal/mol.     

Conformational analysis and molecular dynamics simulation of cellobiose and larger cellooligomers

The conformational behavior of cellobiose (D -glc-ß(1→4)-D -glc), cellotetraose, and cellooctaose was studied by a combination of energy minimization and molecular dynamics simulations in vacuo at 400 K. These diand oligosaccharide models have considerable flexibility and exhibit a variety of different motions in glycosidic and exocyclic torsions. The glycosidic ?, ψ torsions moved frequently between two local minima on the cellobiose energy surface in the region of known crystal structures. Transitions of the hydroxymethyl side chain were observed between gt,gg, and tg conformations accompanied by changes in intramolecular hydrogen bonding patterns. A reasonable fit to the experimental optical rotation and nuclear magnetic resonance vicinal coupling data of cellobiose in solution required a distribution of its conformations. The oligomers, although generally extended, assumed a more coiled or twisted shape than is observed in the crystalline state of cellulose and exhibited considerable backbone motion due to local ring rotations about the glycosidic bonds. Long-lived transitions to structures having torsion angles 180° from the major minima (ring flips) introduced kinks and bends into the tetramer and octamer. While the glucose rings of the structures remained primarily in the ⁴C₁ conformation, twist and boat structures were also observed in the tetramer and octamer structures. Reducing the simulation temperature to 300 K eliminated some of the transitions seen at 400 K. © 1993 John Wiley & Sons, Inc.     

Revised algorithms for the build-up procedure for predicting protein conformations by energy minimization

The build-up procedure for predicting low-energy conformations of polypeptides has been extended to cover the case of peptides in aqueous solutions. The revised procedure consists of five steps to be applied to each stage of the build-up. I. All low-energy minima of each of the two fragments to be joined are combined as starting points for energy minimization of the enlarged fragment, and those minima of the enlarged fragment within a certain upper bound of the lowest energy are retained. II. Whenever one of the combinations in Step I leads to an atomic overlap, the minimization is started again using a pseudoenergy function which remains finite everywhere and becomes equal to the standard energy function when no atoms overlap. III. The minima generated in Steps I and II are culled by ignoring side-chain conformations and retaining only those minima whose backbone conformations differ significantly. IV. The rotameric states of the side chains are optimized, by testing their energy of interaction with the rest of the molecule, and subjecting the whole molecule to a further round of energy minimization if the test indicates that this would reduce the energy. V. The energies of all minima are recomputed with inclusion of a term for solvation and with a smaller upper bound as the criterion for retention. The original build-up procedure consisted of Steps I and III only. Examples are presented showing the effectiveness of the new Steps II and IV in locating low-energy minima, and the problems that remain to be solved, chiefly concerning Step V, are discussed.     

Monte Carlo temperature basin paving with effective fragment potential: an efficient and fast method for finding low-energy structures of water clusters (H2O)20 and (H2O)25

Determining low-energy structures of large water clusters is a challenge for any optimization algorithm. In this work, we have developed a new Monte Carlo (MC)-based method, temperature basin paving (TBP), which is related to the well-known basin hopping method. In the TBP method, the Boltzmann weight factor used in MC methods is dynamically modified based on the history of the simulation. The states that are visited more are given a lower probability by increasing their temperatures and vice versa. This allows faster escapes from the states frequently visited in the simulation. We have used the TBP method to find a large number of low-energy minima of water clusters of size 20 and 25. We have found structures energetically same to the global minimum structures known for these two clusters. We have compared the efficiency of this method to the basin-hopping method and found that it can locate the minima faster. Statistical efficiency of the new method has been investigated by running a large number of trajectories. The new method can locate low-energy structures of both the clusters faster than some of the reported algorithms for water clusters and can switch between high energy and low-energy structures multiple times in a simulation illustrating its efficiency. The large number of minima obtained from the simulations is used to get both general and specific features of the minima. The distribution of minima for these two clusters based on the similarity of their oxygen frames shows that the (H(2)O)(20) can have different variety of structures, but for (H(2)O)(25), low-energy structures are mostly cagelike. Several (H(2)O)(25) structures are found with similar energy but with different cage architectures. Noncage structures of (H(2)O)(25) are also found but they are 6-7 kcal/mol higher in energy from the global minimum. The TBP method is likely to play an important role for exploring the complex energy landscape of large molecules.     

Stereodynamics of triethylamine: Molecular mechanics calculations on the direct rotational racemization of the C3-symmetric conformers

The direct interconversion of the two C₃-symmetric enantiomeric conformations of triethylamine, via C? N bond rotation, has been studied by molecular mechanics (MM 2) calculations. The MM 2 calculations have been used to characterize the minima (equilibrium geometries) and first-order saddle points (transition states) for this process. For one interconversion, there are five saddle points and six minima. The highest energy saddle point results from the uncoupled rotation of one ethyl group to eclipse the lone pair. Two of the barriers result from coupled rotation of two ethyl groups in close passage.     

Generic shapes for the conformation analysis of macrocyclic structures

A new algorithm for the systematic generation of conformations of macrocyclic systems is presented. The procedure is based on the concept of generic shapes that are found in such structures. These shapes are characterized by a selection of harmonics which occur in an approximate Fourier representation of the atomic coordinates of the rings. Following a fixed protocol, a limited set of in-plane and out-of-plane circular harmonics is used to define an ensemble of generic ring shapes. These generic shapes are used as start structures for energy minimizations by a given force-field method. To account for the possibility of having several final conformations originating from the same generic shape, the corresponding initial structure is taken several times and subjected to a randomization step before minimization. The resulting conformations that fall within a preset low-energy band are collected and screened for duplicates and enantiomers. The efficiency of this procedure (ratio between the number of accepted conformations and the total number of energy minimizations) depends on the flexibility of the macrocyclic system. The efficiency is generally quite high for very flexible rings. According to the proposed protocol, the number of generic shapes used as start structures grows as the square of N(lnN), where N is the ring size. The algorithm lends itself to conformational analyses of medium-size and large rings as well as of loops spanned between fixed structural units.     

Inherent structure versus geometric metric for state space discretization

Inherent structure (IS) and geometry‐based clustering methods are commonly used for analyzing molecular dynamics trajectories. ISs are obtained by minimizing the sampled conformations into local minima on potential/effective energy surface. The conformations that are minimized into the same energy basin belong to one cluster. We investigate the influence of the applications of these two methods of trajectory decomposition on our understanding of the thermodynamics and kinetics of alanine tetrapeptide. We find that at the microcluster level, the IS approach and root‐mean‐square deviation (RMSD)‐based clustering method give totally different results. Depending on the local features of energy landscape, the conformations with close RMSDs can be minimized into different minima, while the conformations with large RMSDs could be minimized into the same basin. However, the relaxation timescales calculated based on the transition matrices built from the microclusters are similar. The discrepancy at the microcluster level leads to different macroclusters. Although the dynamic models established through both clustering methods are validated approximately Markovian, the IS approach seems to give a meaningful state space discretization at the macrocluster level in terms of conformational features and kinetics. © 2016 Wiley Periodicals, Inc.     

A genetic algorithm encoded with the structural information of amino acids and dipeptides for efficient conformational searches of oligopeptides

The genetic algorithm (GA) is an intelligent approach for finding minima in a highly dimensional parametric space. However, the success of GA searches for low energy conformations of biomolecules is rather limited so far. Herein an improved GA scheme is proposed for the conformational search of oligopeptides. A systematic analysis of the backbone dihedral angles of conformations of amino acids (AAs) and dipeptides is performed. The structural information is used to design a new encoding scheme to improve the efficiency of GA search. Local geometry optimizations based on the energy calculations by the density functional theory are employed to safeguard the quality and reliability of the GA structures. The GA scheme is applied to the conformational searches of Lys, Arg, Met‐Gly, Lys‐Gly, and Phe‐Gly‐Gly representative of AAs, dipeptides, and tripeptides with complicated side chains. Comparison with the best literature results shows that the new GA method is both highly efficient and reliable by providing the most complete set of the low energy conformations. Moreover, the computational cost of the GA method increases only moderately with the complexity of the molecule. The GA scheme is valuable for the study of the conformations and properties of oligopeptides. © 2016 Wiley Periodicals, Inc.     

Protein structure prediction using basin-hopping

Associative memory Hamiltonian structure prediction potentials are not overly rugged, thereby suggesting their landscapes are like those of actual proteins. In the present contribution we show how basin-hopping global optimization can identify low-lying minima for the corresponding mildly frustrated energy landscapes. For small systems the basin-hopping algorithm succeeds in locating both lower minima and conformations closer to the experimental structure than does molecular dynamics with simulated annealing. For large systems the efficiency of basin-hopping decreases for our initial implementation, where the steps consist of random perturbations to the Cartesian coordinates. We implemented umbrella sampling using basin-hopping to further confirm when the global minima are reached. We have also improved the energy surface by employing bioinformatic techniques for reducing the roughness or variance of the energy surface. Finally, the basin-hopping calculations have guided improvements in the excluded volume of the Hamiltonian, producing better structures. These results suggest a novel and transferable optimization scheme for future energy function development.     

Density functional calculations on cyclodextrins

Conformations of α-, β-, and γ-CDs under anhydrous conditions in the gas phase were investigated by a density functional method, B3LYP/6-31G(d,p). These calculations resulted in several symmetric conformations with different energies. The lowest energy conformations contain two rings of homodromic hydrogen bonds and are referred to “one-gate-closed” conformations. Different orientations of hydrogen bonds lead to four minima. Other conformational minima were found for “open” conformations which correspond to some extent to experimentally determined structures.     

Ab initio studies of structural features not easily amenable to experiment. 30. Conformational analysis and molecular structures of propanal and butanal

The geometries of several conformations of propanal and butanal have been refined by geometrically unconstrained ab initio gradient relaxation on the 4-21G level. Both compounds possess energy minima at O? C? C? C torsional angles of 0° and in the 120° region, and energy maxima in the 70° region and at 180°. The structure of the aldehyde functional group is found to be relatively invariant both when different systems or when different conformations of the same system are compared. Conformationally dependent geometrical trends in propanal and butanal are discussed and found to be subtle yet noticeable.     

Distributed data parallel coupled-cluster algorithm: Application to the 2-hydroxypyridine/2-pyridone tautomerism

The recently developed parallel coupled-cluster algorithm of Rendell, Lee, and Lindh [Chem. Phys. Lett., 194 , 84 (1992)] is extended to allow four-indexed quantities containing one or two indices in the virtual orbital space to be stored across the global memory of distributed-memory parallel processors. Quantities such as the double-excitation amplitudes can now be distributed over multiple nodes, with blocks of data retrieved from remote nodes by the use of interrupt handlers. As an application of the new code, we have investigated the potential energy surface of the 2-hydroxypyridine/2-pyridone tautomers. Using large basis sets, the structure of each tautomer and the transition state connecting the two minima has been determined at the SCF level. The relative energy difference and the activation energy were then redetermined using the MP2, CCSD, and CCSD(T) methods. All calculations have been performed on Intel distributed-memory supercomputers. The largest coupled-cluster calculations contained over 2 million double-excitation amplitudes. © John Wiley & Sons, Inc.     

Investigation of the Potential Surface of Tetraalanine-Peptide by Sequentially Locking the Molecular Dynamic Trajectory of the System in Attraction Basins

A new approach to investigation of finite polyatomic systems such as clusters and biomolecules is suggested. The molecular dynamic trajectory of the system is sequentially locked in attraction basins on the potential energy surface (PES) subject to a certain strategy of system dislocation over the surface. The approach is illustrated by investigating the PES of the tetraalanine-peptide N-acetyl-(Ala)₃-methylamide in aqueous solution. It is indicated that the PES of the system consists of a great number of local minima united in clusters, which correspond to different trans/cis conformations of the molecule and are separated by barriers with a height of 14 kcal/mole or more.