Conformational sampling by a general linearized embedding algorithm

Linearized embedding is a variant on the usual distance geometry methods for finding atomic Cartesian coordinates given constraints on interatomic distances. Instead of dealing primarily with the matrix of interatomic distances, linearized embedding concentrates on properties of the metric matrix, the matrix of inner products between pairs of vectors defining local coordinate systems within the molecule. We developed a pair of general computer programs that first convert a given arbitrary conformation of any covalent molecule from atomic Cartesian coordinates representation to internal local coordinate systems enforcing rigid valence geometry and then generate a random sampling of conformers in terms of atomic Cartesian coordinates that satisfy the rigid local geometry and a given list of interatomic distance constraints. We studied the sampling properties of this linearized embedding algorithm vs. a standard metric matrix embedding program, DGEOM, on cyclohexane, cycloheptane, and a cyclic pentapeptide. Linearized embedding always produces exactly correct bond lengths, bond angles, planarities, and chiralities; it runs at least two times faster per structure generated, and is successful as much as four times as often at refining these structures to full agreement with the constraints. It samples the full range of allowed conformations broadly, although not perfectly uniformly. Because local geometry is rigid, linearized embedding's sampling in terms of torsion angles is more restricted than that of DGEOM, but it finds in some instances conformations missed by DGEOM. © 1992 by John Wiley & Sons, Inc.     

Conformational analysis of nucleic acid molecules with flexible furanose rings in dihedral angle space

The computational algorithm that works in the coordinate space of dihedral angles (i.e., bond lengths and bond angles are kept fixed and only rotatable dihedral angles are treated as independent variables) is extended to deal with the pseudorotational m otion of furanose rings by introducing a variable of pseudorotation. Then, this algorithm is applied to a distance geometry calculation that generates three-dimensional (3D) structures that are consistent with given constraints of interatomic distances. This method efficiently generates 3D structures of an RNA hairpin loop which satisfy a set of experimental NMR data. © 1996 by John Wiley & Sons, Inc.     

An analytical computation of Christoffel symbols for reaction coordinate and trajectory treatments under internal coordinates

We examine the Hessian matrix of the potential energy under internal coordinates. We report all Christoffel symbols which exist for molecules if we use the known coordinates such as bond distances, bond angles, torsion angles, and out-of-plane angles. We use as an example triatomic HCN in an extended geometry.     

Beiträge zur Chemie des Schwefels. 115. Kristall- und MolekülStruktur von Benzopentathiepin

Contributions to the Chemistry of Sulfur. 115. Crystal and Molecular Structure of Benzopentathiepine The structure of benzopentathiepine has been determined from three-dimensional single-crystal X-ray data. The crystal are monoclinic, space group P2₁/c, with 4 molecules in the unit cell of dimensions a = 6.942(4) Å, b = 10.223(10) Å, c = 13.015(8) Å, β = 102,15(10)°. The conformation of the seven-membered ring is the chair form. Details of the conformation are discussed. Bond distances, bond angles, and dihedral angles are compared with those in other molecules, which have similar structure.     

Crystal structures of synthetic analgetics. IV. Dextropropoxyphene.

The molecular and cystal structure of dextropropoxyphene has been determined by X-ray methods. The crystals are monoclinic, space group P21, with unit cell dimensions a=9.257(2) A; b=9.048(3) A; c=12.074(7) A; beta=93.01(4)degrees. The phase problem was solved by direct methods and the model refined to an R-value of 0.038 for 1799 observed reflections. E.s.d's are, in average, 0.004 A and O.3 degrees in interatomic distances and angles, respectively. The propylamine chain in nearly fully extended, the dihedral angel C4-C5-C7-N being -174.2 degrees. The conformation of this side chain is similar to that in the hydrochloride of the title compound. Thus the proposed bioactive conformation is not preferred by propoxyphene in the crystalline state, as was the case for the free base of methadone.     

Phenomenological Theory of Cluster Orbitals and Electronic Structures of Molecules

A method for calculation of the energy structures of molecules is developed. The method is based on construction of a matrix of interactions between the atoms in the molecules with allowance made for their symmetric arrangement in space. Matrices are parametrized by comparison of the matrix eigenvalues with the experimental values for reference molecules (methane, benzene, ammonia, water, etc.). Interaction matrices are given for molecules with different types of bonding. Eigenvalues of the molecular energy, eigenfunctions of all molecular orbitals, bond angles, atomic orbital charges, etc. are determined. The dependence of the energy levels of MH₂ molecules (M = O, S, Se, and Te) on the interatomic interaction parameter is given. The dependences of the molecular parameters on the nature of atom M are analyzed.     

Conformational analysis by energy embedding

The distance geometry approach to conformational calculation has been shown to be very effective at producing large molecular structures satisfying many given, long-range constraints on the interatomic distances. I now present a significant extension of the method that handles strictly geometric constraints as well as before while also locating conformers of very low energy. The main feature of the algorithm is a projection of the molecule from a low energy conformation in a high dimensional space to three dimensions in such a way as to perturb the energy as little as possible. Tests of the method on very small systems with simple energy functions completely explored by independent means show that the global minimum of energy is sometimes attained. In every case the final energy is very low, and geometric constraints are completely satisfied.     

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.     

Crystal and molecular structure of 1-benzyl-1,2,2-trimethyl azetidinium bromide

The structure of the compound, 1-benzyl-1,2,2-trimethyl azetidinium bromide has been determined by a single crystal x-ray diffraction study. The compound crystallizes in the space group Pnam with four formula weights in a unit cell of dimensions a = 14.36, b = 13.73, c = 6.90₆ Å. The compound is partially disordered with the benzyl part ordered and the remainder of the ring disordered about the mirror plane. The azetidinium ring bears a close resemblance in bond distances, bond angles and conformation to the β-lactam ring found in penicillin. The disordering is discussed in detail and a comparison of the effects of fusing a second ring to the azetidinium moiety are explored. A final value of R = 0.09₄ for the 432 independent reflections was obtained.     

Analysis of liquid structure without construction of any structure models by the X-ray scattering method.

A simple approach for determining a liquid structure using X-ray scattering data, in which a liquid structure is uniquely evaluated without construction of any plausible structure models, has been applied to liquid acetonitrile, acetone and cyclohexane. For a pair of molecules, a given point within a molecule is located at the origin with a given molecular orientation. The site of the given point of another molecule is defined by the polar coordinates and the molecular orientation is treated by three Eulerian angles. These parameters are optimized by a non-linear least-squares calculation applied to X-ray scattering data. The reliability of the method was examined by determining the liquid structure of polar acetonitrile and the obtained intermolecular interatomic distances are in good agreement with the previously reported values. The method was then successfully applied to the determination of the liquid structure of acetone and cyclohexane. Especially for nonpolar cyclohexane, the construction of a variety of plausible structural models is very difficult. It was revealed that acetone has an ordered liquid arrangement similar to that found in its crystal, although the intermolecular distances in liquid acetone are different from those in the crystal. On the other hand, the liquid structure of cyclohexane is disordered.     

Conformational analysis by scaled energy embedding

Distance geometry has been used for some years to find conformations of molecules consistent with given bounds on the interatomic distances. A recent extension of the method, called energy embedding, enforces the geometric constraints as before, but also biases the results toward low-energy structures. Now a significant improvement on energy embedding is presented, which is less dependent on local optimization for satisfying the geometric constraints. Tests on very small systems with well understood energy functions show that the global minimum energy value is often attained, and geometric constraints are always satisfied.     

双-[N(4-取代苯基)-水杨醛亚胺]合锌(Ⅱ)的晶体结构

为进一步研究过渡金属离子与席夫碱配合和的结构与性能的关系,本文报道双-[N-(4-取代苯基)-水杨醛亚胺]合锌的晶体结构.     

Linearized embedding: A new metric matrix algorithm for calculating molecular conformations subject to geometric constraints

There are many methods in the literature for calculating conformations of a molecule subject to geometric constraints, such as those derived from two-dimensional NMR experiments. One of the most general ones is the EMBED algorithm, based on distance geometry, where all constraints except chirality are converted into upper and lower bounds on interatomic distances. Here we propose a variation on this where the molecule is assumed to have fixed bond lengths, vicinal bond angles and chiral centers; and these holonomic constraints are enforced separately from the experimental constraints by being built into the mathematical structure of the problem. The advantages of this approach are: (1) for molecules having large rigid groups of atoms, there are substantially fewer variables in the problem than all the atomic coordinates; (2) rigid groups achieve in the end more accurate local geometry (e.g., planar aromatic rings are truly planar, chiral centers always have their correct absolute chirality); (3) it is easier to detect inconsistencies between the holonomic and the experimental constraints; and (4) when generating a random sampling of conformers consistent with all constraints, the probability of achieving satisfactory structures tends to be greater.     

On the quadratic reaction path evaluated in a reduced potential energy surface model and the problem to locate transition states*

Knowledge of the location of saddle points is crucial to the study the chemical reactivity. Using a path following method defined in a reduced potential energy surface, and starting at either the reactant or product region, we propose an algorithm that locates the corresponding saddle point. The reduced potential energy surface is defined by the set of molecular geometry parameters, namely bond distances, bond angles, and dihedral angles that undergo the largest change for the reaction under consideration; the rest of the coordinates are forced to have a null gradient. Consequently, the proposed method can be seen as a new formulation of the distinguished coordinate method. The method is based on a quadratic model; consequently, it only requires the calculation of the energy and the gradient. The Hessian matrix is normally updated except in the first step and the steps where the resulting updated Hessian matrix is not adequate. Some examples are presented and analyzed. © 2001 John Wiley & Sons, Inc. J Comput Chem 22: 387–406, 2001     

1,2-oxazine chemistry—II : The crystal and molecular structure of N-(p-carboxybenzyl)-tetrahydro-1,2-oxazine

The crystal structure of the title compound has been determined from X-ray diffractometer data by direct methods, and refined by full matrix least squares techniques to R = 0·057 for 1231 reflections. The crystals are monoclinic, space group C2/C, cell dimensions a = 1665, b = 987, c = 1443 pm, β = 107·37° and Z = 8. The conformation of the tetrahydro-1,2-oxazine ring is a chair with the N-substituent equatorial. There is evidence of significantly greater torsional angles around the N and O atoms than around the ring C atoms, showing the ring to be more puckered than cyclohexane. The hydrogen bond is between the acid group on one molecule and the ring nitrogen on its neighbour.     

Compare-Conformer: a program for the rapid comparison of molecular conformers based on interatomic distances and torsion angles.

A computer program for comparison of the conformations of a number of related molecular structures is described. The comparisons are performed on either interatomic distances or torsion angles. The comparisons are accomplished on ordered pairs of distances or torsion angles, and the distance comparisons can be performed in a manner that allows permutation of the distance pairs being compared. The algorithm utilizes bit-string Boolean operations that allow the comparisons to be performed rapidly. The program should be useful for computer-assisted molecular modeling studies in which the viable conformers of bioactive analogues are compared in order to locate those conformers that place key substituents in the same spatial orientation.     

The Lagrange multiplier method for manipulating geometries. Implementation and applications in molecular mechanics

It is shown that a Lagrange multiplier method to constrain one or several internal coordinates, or averages and combinations of these, is easily implemented in a molecular mechanics computer program that uses Newton–Raphson (NR ) minimization. Results are given for constraints on nonbonded distances and torsion angles. When a potential energy surface is to be explored, it is much better to constrain the average of three torsion angles around a bond than to constrain a single torsion angle. Certain conversions can only be achieved when averages of torsion angles around different bonds are constrained. Combinations of constraints have been applied to evaluate differences between calculated and observed geometries and to obtain transition states for relatively large molecules from results for smaller molecules at relatively low costs. The efficieny of the combination of the Lagrange multiplier method and NR minimization in terms of computing time can be rated as good.     

Systematic drug receptor mapping: A new approach to the analysis of conformational energy calculations of flexible molecules with application to dopaminergic and adrenergic agonists

The computer program PRODIS is used to find low energy conformations of flexible molecules by searching the potential energy surface(s) of one or more torsion angles via rigid rotation. The n-dimensional grid of energy versus torsion angles is then converted to a Boltzman probability distribution, with the probability being represented not as a function of torsion angle, but rather a distance between two atoms. These atoms are chosen by comparison with a known, active analogue in which certain atoms have previously been determined as requirements for drug activity. PRODIS produces a list of low energy conformations, their corresponding interatomic distances and the Boltzman probability for each distance ±0.125, as well as the total probability for each conformation. The user also specifies a target interatomic distance and range (usually derived from a more rigid analogue) for which PRODIS lists all conformations and their Boltzman probability that meet this distance.