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.     

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.     

Conformational sampling by self-organization

A new stochastic algorithm for conformational sampling is described. The algorithm generates molecular conformations that are consistent with a set of geometric constraints, which include interatomic distance bounds and chiral volumes derived from the molecular connectivity table. The algorithm repeatedly selects individual geometric constraints at random and updates the respective atomic coordinates toward satisfying the chosen constraint. When compared to a conventional distance geometry algorithm based on the same set of geometric constraints, our method is faster and generates conformations that are more diverse and more energetically favorable.     

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.     

Use of augmented Lagrangians in the calculation of molecular conformations by distance geometry

Distance geometry is a technique widely used to find atomic coordinates that agree with given upper and lower bounds on the interatomic distances. It is successful because it chooses at random some relatively good "trial coordinates" that take into account the whole molecule and all constraints at once. Customarily, these trial coordinates must be refined by minimizing a penalty function until the structure agrees with the original bounds. Here we present an alternative to minimizing the penalty function, which has the advantage of more precisely satisfying the bounds, showing more clearly when the bounds are mutually contradictory, and simultaneously optimizing an objective function subject to precise satisfaction of the bounds.     

Small-amplitude protein conformational dynamics: Second-order analytic relation between cartesian coordinates and dihedral angles

An analytic expression for protein atomic displacements in Cartesian coordinate space (CCS) against small changes in dihedral angles is derived. To study time-dependent dynamics of a native protein molecule in CCS from dynamics in the internal coordinate space (ICS), it is necessary to convert small changes of internal coordinate variables to Cartesian coordinate variables. When we are interested in molecular motion, six degrees of freedom for translational and rotational motion of the molecule must be eliminated in this conversion, and this conversion is achieved by requiring the Eckart condition to hold. In this article, only dihedral angles are treated as independent internal variables (i.e., bond angles and bond lengths are fixed), and Cartesian coordinates of atoms are given analytically by a second-order Taylor expansion in terms of small deviations of variable dihedral angles. Coefficients of the first-order terms are collected in the K matrix obtained previously by Noguti and Go (1983) (see ref. 2). Coefficients of the second-order terms, which are for the first time derived here, are associated with the (newly termed) L matrix. The effect of including the resulting quadratic terms is compared against the precise numerical treatment using the Eckart condition. A normal mode analysis (NMA) in the dihedral angle space (DAS) of the protein bovine pancreatic trypsin inhibitor (BPTI) has been performed to calculate shift of mean atomic positions and mean square fluctuations around the mean positions. The analysis shows that the second-order terms involving the L matrix have significant contributions to atomic fluctuations at room temperature. This indicates that NMA in CCS involves significant errors when applied for such large molecules as proteins. These errors can be avoided by carrying out NMA in DAS and by considering terms up to second order in the conversion of atomic motion from DAS to CCS. © 1995 by John Wiley & Sons, Inc.     

Initial conformations of macrocyclic compounds with rotational symmetry generated from a molecular fragment

An algorithm is described for generating atomic Cartesian coordinates of conformations of macrocyclic molecules possessing exact rotational or rotation-reflection symmetries. A fragment representing the asymmetric unit of the molecule is suitably oriented in space, and then a symmetry operator is applied to generate the initial coordinates of the molecule. An empirical force field of interatomic interactions is used to generate equilibrium conformations. Results of calculations performed on two cyclic polylactones and one crown ether using this approach are given. They reveal that symmetric conformations of these molecules are often preferred. Since the latter conformations are probably responsible for the specialized properties of these molecules, this method should facilitate doing theoretical studies on these kinds of compounds.     

Self-organizing superimposition algorithm for conformational sampling

A novel self-organizing algorithm for conformational sampling is introduced, in which precomputed conformations of rigid fragments are used as templates to enforce the desired geometry. Starting from completely random coordinates, the algorithm repeatedly superimposes the templates to adjust the positions of the atoms, thereby gradually refining the conformation of the molecule. Combined with pair-wise adjustments of the atoms to resolve steric clashes, conformations that satisfy all geometric constraints can be generated from this procedure. The algorithm is demonstrated to achieve good performance and promises potential applications on more challenging modeling problems.     

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.     

Improvements to the distance geometry algorithm for conformational sampling of cyclic structures

Modifications to the distance geometry algorithm as embodied in the program DGEOM have been made to improve sampling capabilities. Specifically, torsion angle sampling replaces distance sampling for 1,4 atomic relationships and correlated distance sampling is disabled. The effects of these modifications are illustrated by comparing the different sets of conformations produced for butane. In addition, these changes are shown to increase the conformational sampling of two medium-sized rings, cycloheptadecane and caprylolactam. The current results for these molecules are compared to those of other conformational searching methods.     

Quasi-Hamiltonian equations of motion for internal coordinate molecular dynamics of polymers

Conventional molecular dynamics simulations of macromolecules require long computational times because the most interesting motions are very slow compared to the fast oscillations of bond lengths and bond angles that limit the integration time step. Simulation of dynamics in the space of internal coordinates, that is, with bond lengths, bond angles, and torsions as independent variables, gives a theoretical possibility of eliminating all uninteresting fast degrees of freedom from the system. This article presents a new method for internal coordinate molecular dynamics simulations of macromolecules. Equations of motion are derived that are applicable to branched chain molecules with any number of internal degrees of freedom. Equations use the canonical variables and they are much simpler than existing analogs. In the numerical tests the internal coordinate dynamics are compared with the traditional Cartesian coordinate molecular dynamics in simulations of a 56 residue globular protein. For the first time it was possible to compare the two alternative methods on identical molecular models in conventional quality tests. It is shown that the traditional and internal coordinate dynamics require the same time step size for the same accuracy and that in the standard geometry approximation of amino acids, that is, with fixed bond lengths, bond angles, and rigid aromatic groups, the characteristic step size is 4 fs, which is 2 times higher than with fixed bond lengths only. The step size can be increased up to 11 fs when rotation of hydrogen atoms is suppressed. © 1997 by John Wiley & Sons, Inc. J Comput Chem 18 : 1354–1364, 1997     

The choice of internal coordinates in complex chemical systems

This article presents several considerations for the appropriate choice of internal coordinates in various complex chemical systems. The appropriate and black box recognition of internal coordinates is of fundamental importance for the extension of internal coordinate algorithms to all fields where previously Cartesian coordinates were the preferred means of geometry manipulations. Such fields range from local and global geometry optimizations to molecular dynamics as applied to a wide variety of chemical systems. We present a robust algorithm that is capable to quickly determine the appropriate choice of internal coordinates in a wide range of atomic arrangements. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2010     

Using redundant internal coordinates to optimize equilibrium geometries and transition states

A redundant internal coordinate system for optimizing molecular geometries is constructed from all bonds, all valence angles between bonded atoms, and all dihedral angles between bonded atoms. Redundancies are removed by using the generalized inverse of the G matrix; constraints can be added by using an appropriate projector. For minimizations, redundant internal coordinates provide substantial improvements in optimization efficiency over Cartesian and nonredundant internal coordinates, especially for flexible and polycyclic systems. Transition structure searches are also improved when redundant coordinates are used and when the initial steps are guided by the quadratic synchronous transit approach. © 1996 by John Wiley & Sons, Inc.     

Calculation of intramolecular force fields from second-derivative tensors

A practical procedure (FUERZA) to obtain internal force constants from Cartesian second derivatives (Hessians) is presented and discussed. It allows a systematic analysis of pair atomic interactions in a molecular system, and it is fully invariant to the choice of internal coordinates of the molecule. Force constants for bonds or for any pair of atoms in general are defined by means of the eigenanalysis of their pair interaction matrix. Force constants for the angles are obtained from their corresponding two-pair interaction matrices of the two bonds or distances forming the angle, and the dihedral force constants are similarly obtained using their corresponding three-pair interaction matrices. © 1996 John Wiley & Sons, Inc.     

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.     

General methods of analysing molecular vibrations

An acute need may arise to develop for the complete analysis of molecular vibrations practically convenient general methods based on coordinates other than “chemical coordinates”. One reason is the proven proposition: Among independent internal coordinates corresponding to a molecule, there cannot be one which describes a small displacement of a chemical group as a whole relative to a certain molecular plane, provided this group contains more than two linearly or three non-linearly arranged atoms. Two methods are presented in some detail. The first is based on the use of X⁰_δ coordinates which are components of “bond vectors” in the “sown” (for each “bond”) Cartesian coordinate system. The second method utilizes X⁰ coordinates, i.e. the components of atomic displacements in the “down” (for each atom) Cartesian coordinate system. Computation of the torsional vibration of transdichloroethane is given as an example illustrating the first method. The Mayants treatment of the symmetry of a molecule, proceeding from elementary considerations which do not use the group theory explicitly and are valid for any coordinates, is expounded in a somewhat improved version. The peculiarities arising when considering the mean-square amplitude matrix, Σ, in X⁰_δ and X⁰ coordinates are also discussed.     

First and second derivative matrix elements for the stretching,bending, and torsional energy

Matrix elements for the first and second derivatives of the internal coordinates with respect to Cartesian coordinates are reported for stretching, linear, nonlinear, and out-of-plane bending and torsional motion. Derivatives of the energy with respect to the Cartesian coordinates are calculated with the chain rule. Derivatives of the energy with respect to the internal coordinates are straightforward, but the calculation of the derivatives of the internal coordinates with respect to the Cartesian coordinates can be simplified by the following two steps outlined in this article. First, the number of terms in the analytical functions can be reduced or will vanish when the derivatives of the bond length, bond angle, and torsion angle are reported in a local coordinate system in which one bond lies on an axis and an adjacent bond lies in the plane of two axes or is projected onto perpendicular planes for linear and out-of-plane bending motion. Second, a simple rotation transforms these derivatives to the appropriate orientation in the space-fixed molecular coordinate system. Functions of the internal coordinates are invariant with respect to translation and rotation. The translational invariance and the symmetry of the second derivatives for a system with L atoms are used to select L-1- and L(L-1)/2-independent first and second derivatives, respectively, of which approximately half of the latter vanish in the local coordinate system. The rotational invariance permits the transformation of the simplified derivatives in the local coordinate system to any orientation in space. The approach outlined in this article simplifies the formulas by expressing them in a local coordinate system, identifies the most convenient independent elements to compute, from which the dependent ones are calculated, and defines a transformation to the space-fixed molecular coordinate system.     

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