Molecular properties from conformational ensembles. 1. Dipole moments of molecules with multiple internal rotations

We present a novel method for constructing the stable conformational space of small molecules with many rotatable bonds that uses our iterative stochastic elimination (ISE) algorithm, a robust stochastic search method capable of finding ensembles of best solutions for large combinatorial problems. To validate the method, we show that ISE reproduces the best conformers found in a fully exhaustive search, as well as compare computed dipole moments to experimental values, based on molecular ensembles and their Boltzmann distributions. Results were also compared to the alternative molecular dynamics and simulated annealing methods. Our results clarify that many low energy conformations may be required to reproduce molecular properties, while single low energy conformers or ensembles of low energy conformers cannot account for the experimental properties of flexible molecules. Whereas ISE well reproduces conformations that are not separated by very large energy barriers, it has not been successful in reproducing conformations of strained molecules.     

A comparison of a direct search method and a genetic algorithm for conformational searching

We present results from the application of two conformational searching methods: genetic algorithms (GA) and direct search methods for finding low energy conformations of organic molecules. GAs are in a class of biologically motivated optimization methods that evolve a population of individuals in which individuals who are more “fit” have a higher probability of surviving into subsequent generations. The parallel direct search method (PDS) is a type of pattern search method that uses an adaptive grid to search for minima. Both methods found energies equal to or lower than the energy of the relaxed crystal structure in all cases, at a relatively small cost in CPU time. We suggest that either method would be a good candidate to find 3-D conformations in a large scale screening application. © 1996 by John Wiley & Sons, Inc.     

Conformational energy downward driver (CEDD): Characterization and calibration of the method

Summary A method has been developed that allows one to drive a molecule to conformations of lowest energy given the starting conformation, the identity of the rotatable bonds and the step size. This method has proved useful in our hands in the drug design arena where it is frequently more important to get low-energy conformers of a molecule that match some other (e.g. pharmacophoric or enzyme pocket) requirements than to exhaustively enumerate all possible low-energy conformations for each of the molecules to be studied. The method has been shown to work in the test cases studied to date. Furthermore, so far it has been shown to be sufficiently fast to be used for molecules containing up to 70 rotatable bonds.     

Generating conformer ensembles using a multiobjective genetic algorithm

The task of generating a nonredundant set of low-energy conformations for small molecules is of fundamental importance for many molecular modeling and drug-design methodologies. Several approaches to conformer generation have been published. Exhaustive searches suffer from the exponential growth of the search space with increasing degrees of conformational freedom (number of rotatable bonds). Stochastic algorithms do not suffer as much from the exponential increase of search space and provide a good coverage of the energy minima. Here, the use of a multiobjective genetic algorithm in the generation of conformer ensembles is investigated. Distance geometry is used to generate an initial conformer, which is then subject to geometric modifications encoded by the individuals of the genetic algorithm. The geometric modifications apply to torsion angles about rotatable bonds, stereochemistry of double bonds and tetrahedral chiral centers, and ring conformations. The geometric diversity of the evolving conformer ensemble is preserved by a fitness-sharing mechanism based on the root-mean-square distance of the atomic coordinates. Molecular symmetry is taken into account in the distance calculation. The geometric modifications introduce strain into the structures. The strain is relaxed using an MMFF94-like force field in a postprocessing step that also removes conformational duplicates and structures whose strain energy remains above a predefined window from the minimum energy value found in the set. The implementation, called Balloon, is available free of charge on the Internet ( http://www.abo.fi/~mivainio/balloon/).     

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.     

An exploration of a novel strategy for superposing several flexible molecules

Summary This paper describes a computational strategy for the superposition of a set of flexible molecules. The combinatorial problems of searching conformational space and molecular matching are reduced drastically by the combined use of simulated annealing methods and cluster analysis. For each molecule, the global minimum of the conformational energy is determined by annealing and the search trajectory is retained in a history file. All the significantly different low-energy conformations are extracted by cluster analysis of data in the history file. Each pair of molecules, in each of their significantly different conformations, is then matched by simulated annealing, using the difference-distance matrix as the objective function. A set of match statistics is then obtained, from which the best match taken from all different conformations can be found. The molecules are then superposed either by reference to a base molecule or by a consensus method. This strategy ensures that as wide a range of conformations as possible is considered, but at the same time that the smallest number of significantly different conformations is used. The method has been tested on a set of six angiotensin II antagonists with between 7–11 rotatable bonds.     

Identification of symmetries in molecules and complexes

An algorithm is presented that quickly detects local and global symmetries of single molecules and complexes. Based upon the Morgan Naming Algorithm, the algorithm involves traversing the molecule from a starting atom and building up a molecule name based upon the names of the atoms encountered along the traversal. Additional molecule names are generated from other starting atoms, and name-name matches are identified as corresponding to symmetry operations. A number of enhancements relative to prior methods yield increased efficiency and extended functionality. In particular, the present method detects not only global symmetries but also local symmetries associated with bond rotations as well as symmetries that are only apparent when alternate resonance forms are considered. Importantly, the present method works not only for single molecules but also for multimolecular complexes. As a consequence, it is well, and perhaps uniquely, suited to applications in supramolecular and host-guest chemistry. Applications include filtering out redundant conformations during conformational searching and free energy calculations; accelerating ligand-receptor docking calculations by reducing the sampling ranges of rotatable bonds linked to locally symmetric groups, such as phenyls; and automating the calculation of symmetry numbers for thermochemical applications.     

Enhanced sampling method in molecular simulations using genetic algorithm for biomolecular systems

We propose a molecular simulation method using genetic algorithm (GA) for biomolecular systems to obtain ensemble averages efficiently. In this method, we incorporate the genetic crossover, which is one of the operations of GA, to any simulation method such as conventional molecular dynamics (MD), Monte Carlo, and other simulation methods. The genetic crossover proposes candidate conformations by exchanging parts of conformations of a target molecule between a pair of conformations during the simulation. If the candidate conformations are accepted, the simulation resumes from the accepted ones. While conventional simulations are based on local update of conformations, the genetic crossover introduces global update of conformations. As an example of the present approach, we incorporated genetic crossover to MD simulations. We tested the validity of the method by calculating ensemble averages and the sampling efficiency by using two kinds of peptides, ALA3 and (AAQAA)₃. The results show that for ALA3 system, the distribution probabilities of backbone dihedral angles are in good agreement with those of the conventional MD and replica-exchange MD simulations. In the case of (AAQAA)₃ system, our method showed lower structural correlation of α-helix structures than the other two methods and more flexibility in the backbone ψ angles than the conventional MD simulation. These results suggest that our method gives more efficient conformational sampling than conventional simulation methods based on local update of conformations. © 2018 Wiley Periodicals, Inc.     

A genetic algorithm for flexible molecular overlay and pharmacophore elucidation

Summary A genetic algorithm (GA) has been developed for the superimposition of sets of flexible molecules. Molecules are represented by a chromosome that encodes angles of rotation about flexible bonds and mappings between hydrogen-bond donor proton, acceptor lone pair and ring centre features in pairs of molecules. The molecule with the smallest number of features in the data set is used as a template, onto which the remaining molecules are fitted with the objective of maximising structural equivalences. The fitness function of the GA is a weighted combination of: (i) the number and the similarity of the features that have been overlaid in this way; (ii) the volume integral of the overlay; and (iii) the van der Waals energy of the molecular conformations defined by the torsion angles encoded in the chromosomes. The algorithm has been applied to a number of pharmacophore elucidation problems, i.e., angiotensin II receptor antagonists, Leu-enkephalin and a hybrid morphine molecule, 5-HT_1D agonists, benzodiazepine receptor ligands, 5-HT₃ antagonists, dopamine D₂ antagonists, dopamine reuptake blockers and FKBP12 ligands. The resulting pharmacophores are generated rapidly and are in good agreement with those derived from alternative means.     

Energy-directed tree search: an efficient systematic algorithm for finding the lowest energy conformation of molecules

We present a new systematic algorithm, energy-directed tree search (EDTS), for exploring the conformational space of molecules. The algorithm has been designed to reliably locate the global minimum (or, in the worst case, a structure within 4 kJ mol(-1) of this species) at a fraction of the cost of a full conformational search, and in this way extend the range of chemical systems for which accurate thermochemistry can be studied. The algorithm is inspired by the build-up approach but is performed on the original molecule as a whole, and objectively determines the combinations of torsional angles to optimise using a learning process. The algorithm was tested for a set of 22 large molecules, including open- and closed-shell species, stable structures and transition structures, and neutral and charged species, incorporating a range of functional groups (such as phenyl rings, esters, thioesters and phosphines), and covering polymers, peptides, drugs, and natural products. For most of the species studied the global minimum energy structure was obtained; for the rest the EDTS algorithm found conformations whose total electronic energies are within chemical accuracy from the true global minima. When the conformational space is searched at a resolution of 120 degrees , the cost of the EDTS algorithm (in its worst-case scenario) scales as 2(N) for large N (where N is the number of rotatable bonds), compared with 3(N) for the corresponding systematic search.     

TFD: Torsion Fingerprints as a new measure to compare small molecule conformations

Advantages like intuitive interpretation, objectivity, general applicability, and its easy, automated calculation make the rmsd (root-mean-squared deviation) the measure of choice for the investigation of the accuracy of conformational model generators. For comparing conformations of a single molecule this is a clearly superior method. Single molecule analysis is, however, a rare scenario. Typically, conformations are generated for huge corporate or external vendor databases of high diversity which are then further investigated with high-throughput computational methods like docking or pharmacophore searching, in virtual screening campaigns. Representative subsets for accuracy investigations of computational methods need to mimic this diversity. Averaged rmsd values over these data sets are frequently used to assess the accuracy of the methods. There are, however, significant weaknesses in rmsd comparisons for such kind of data sets. The interpretation is for example no longer intuitive because what can be expected in terms of good or bad rmsd values crucially depends on the data set composition like size or number of rotatable bonds of the underlying molecules. Further, rmsd lacks normalization which might result in very high averaged rmsd values for highly flexible molecules and thus might completely skew results. We have developed a novel measure to compare conformations of molecules called Torsion Fingerprint Deviation (TFD). It extracts, weights, and compares Torsion Fingerprints from a query molecule and generated conformations under consideration of acyclic bonds as well as ring systems. TFD is alignment-free and overcomes major limitations of rmsd while retaining its advantages.     

Homodromic hydrogen bonds in low-energy conformations of single molecule cyclodextrins

Low-energy conformations of β-cyclodextrin under anhydrous conditions in the gas phase were investigated by DFT calculations. In these conformations, two homodromic hydrogen bond rings are formed with very short hydrogen bonds at the narrow side of the cyclodextrin ring and a second one at the wider side. These hypothetical conformations are not comparable to those conformations, which have been studied experimentally, forming inclusion complexes with small and medium-sized guest molecules, but their energy is significantly lower than the open conformations (ΔE = 10 kcal/mol).     

Analysis of the genetic algorithm method of molecular conformation determination

Many important problems in chemistry require knowledge of the 3-D conformation of a molecule. A commonly used computational approach is to search for a variety of low-energy conformations. Here, we study the behavior of the genetic algorithm (GA) method as a global search technique for finding these low-energy conformations. Our test molecule is cyclic hexaglycine. The goal of this study is to determine how to best utilize GAs to find low-energy populations of conformations given a fixed amount of CPU time. Two measures are presented that help monitor the improvement in the GA populations and their loss of diversity. Different hybrid methods that combine coarse GA global search with local gradient minimization are evaluated. We present several specific recommendations about trade-offs when choosing GA parameters such as population size, number of generations, rate of interaction between subpopulations, and combinations of GA and gradient minimization. In particular, our results illustrate why approaches that emphasize convergence of the GA can actually decrease its effectiveness as a global conformation search method. © John Wiley & Sons, Inc.     

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     

SODOCK: swarm optimization for highly flexible protein-ligand docking

Protein-ligand docking can be formulated as a parameter optimization problem associated with an accurate scoring function, which aims to identify the translation, orientation, and conformation of a docked ligand with the lowest energy. The parameter optimization problem for highly flexible ligands with many rotatable bonds is more difficult than that for less flexible ligands using genetic algorithm (GA)-based approaches, due to the large numbers of parameters and high correlations among these parameters. This investigation presents a novel optimization algorithm SODOCK based on particle swarm optimization (PSO) for solving flexible protein-ligand docking problems. To improve efficiency and robustness of PSO, an efficient local search strategy is incorporated into SODOCK. The implementation of SODOCK adopts the environment and energy function of AutoDock 3.05. Computer simulation results reveal that SODOCK is superior to the Lamarckian genetic algorithm (LGA) of AutoDock, in terms of convergence performance, robustness, and obtained energy, especially for highly flexible ligands. The results also reveal that PSO is more suitable than the conventional GA in dealing with flexible docking problems with high correlations among parameters. This investigation also compared SODOCK with four state-of-the-art docking methods, namely GOLD 1.2, DOCK 4.0, FlexX 1.8, and LGA of AutoDock 3.05. SODOCK obtained the smallest RMSD in 19 of 37 cases. The average 2.29 A of the 37 RMSD values of SODOCK was better than those of other docking programs, which were all above 3.0 A.     

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.     

Conformations of laulimalide in DMSO-d6

Laulimalide is one of the newest naturally occurring macrolides known to act as a microtubule stabilizing agent with properties similar to Taxol. It also stands as being one of the most flexible with 18 rotatable bonds. This large number of rotatable bonds allows for approximately 3(18) potential conformers. To examine the conformational energy surface of laulimalide, we have performed an NAMFIS deconvolution analysis for laulimalide in DMSO-d6. The latter has been supplemented with a post-NAMFIS energy analysis at the Becke3LYP/6-31G level that examines the opposing effects of internal hydrogen bonding and syn-pentane interactions. In this way, we have identified 15 laulimalide conformations that can be classified into 5 different families: Supine, Convex, Cobra, Stretch, and Concave motifs.     

“Crystal” conformations of 2-methyl-2,4-pentanediol: Dipole moment calculations and molecular structure optimizations

The structures of eight symmetrically independent molecules of 2-methyl-2,4-pentanediol (MPD) in six crystal substances are studied based on the data retrieved from the Cambridge Structural Database (CSD). Coordinates of the most part of hydrogen atoms in MPD molecules were not determined experimentally or not presented in CSD, however, O...O distances provide the conclusion about the formation of intramolecular hydrogen bonds in four molecules. To perform quantum chemical calculations the absent hydrogen atoms were added. The choice of H atomic positions in hydroxyl groups are based on the analysis of possible formation of intra- and intermolecular hydrogen bonds by MPD molecules in the respective crystals. The DFT method with the B3PW91 functional and the 6-31G(d,p) basis set is used to carry out for the first time: 1) the calculation of dipole moments and energies for MPD molecules in “crystal” conformations; 2) the optimization of the structure of these molecules with the calculation of dipole moments for the conformations corresponding to the local energy minima. It is found that among the molecules with the experimental geometric parameters one of the conformations without intramolecular hydrogen bonds is most favorable (μ = 0.56 D). As a result of the energy minimization of eight “crystal” conformations in vacuum, five energetically different conformers are obtained. Among them the conformer with the intramolecular hydrogen bond has the lowest energy (μ = 3.53 D). Four variants of the molecular structure correspond to it in the considered crystals, out of which two are R-enantiomers and two S-enantiomers.     

Docking flexible molecules: A case study of three proteins

A genetic algorithm (GA) conformation search method is used to dock a series of flexible molecules into one of three proteins. The proteins examined are thermolysin (tmn), carboxypeptidase A (cpa), and dihydrofolate reductase (dfr). In the latter two proteins, the crystal ligand was redocked. For thermolysin, we docked eight ligands into a protein conformation derived from a single crystal structure. The bound conformations of the other ligands in tmn are known. In the cpa and dfr cases, and in seven of the eight tmn ligands, the GA docking method found conformations within 1.6 Å root mean square (rms) of the relaxed crystal conformation. © 1995 John Wiley & Sons, Inc.