Bond profiles for cuboctahedron and twist cuboctahedron

Abstract

rnUsing the information on the interatomic separations in a molecule, one can construct structural invariants that are the components of a molecular profile. The entries in the profile are derived by averaging different powers of the interatomic separations, suitably normalized so that the increasing powers do not dominate the sequence. Although only a few hundreds of structures have been so analyzed, no two different chemical structures were found to be characterized by the same sequences. A critical test for the conjecture that molecular profiles are unique is to consider structurally closely related systems that are very similar and have several similar properties. In this contribution we investigated the cuboctahedron and the accompanying polyhedron obtained by rotating half of the cuboctahedron against the other half, resulting in the so-called twist cutoctahedron. We show that even this pair of closely related structures has different profiles. We have also examined the generalized molecular profiles obtained by inserting n additional points along each edge of the polyhedra. The convergence of the profiles as n increases is discussed. It appears thus that these generalized molecular profiles, called line profiles or bond profiles, are likely to lead to a unique characterization of structures in which not only the geometry of atoms is recorded but also the geometry of the connectivity of the structure. 0 1996 John Wiley 19 Sons, Inc. ent constitution. Such characterization would allow one to compare different conformations of a molecule and determine which conformation is more similar to a desired standard. In contrast to a representation of molecules by codes that require canonical rules for labeling of atoms, a list of invariants is independent of atomic labels. Hence characterization of a molecule by invariants is independent of ordering of atoms and of rigid-body

On representation of proteins by star-like graphs

“…Because matrix elements of D/D matrix are smaller than one, or at most equal to one, when considering higher powers of their elements there is no need to introduce appropriate normalizing factor to ensure convergence of its eigenvalues. The k D/ k D matrices were also applied to non-chain structures, such as smaller molecules, ring systems, polyhedra and molecular surfaces [53][54][55]. In these systems, the leading eigenvalue can be interpreted as a measure of the compactness of a system, because in more compact systems the Euclidean distances will be relatively small and consequently the bounds on the leading eigenvalue will be decreased, implying thus that smaller magnitudes for the leading eigenvalue will characterize more compact systems.…”

On Characterization of Chemical Structure

“…Instead of the adjacency matrix of molecular graph one considers geometric matrix for a structure rigidly embedded in 3-D. However, in contrast to the main theme of Crippen's Distance Geometry , which is concerned with the constraints imposed by interatomic distances and how to minimize the inconsistencies that originate with experimental data, in the mathematical chemistry the emphasis is on structural invariants of the geometrical distance matrices. − …”

“…Hence, we arrive at 3-D connectivity indices 117-119 and 3-D Wiener index. − More recently a scheme was outlined that generates mathematical descriptors, i.e., structural invariants, for 3-D structure, which are not only sufficiently general and apply to molecules of arbitrary geometrical forms but also offer a sizable set of structurally related invariants. These novel 3-D descriptors have been referred to as molecular profile. − Importantly, the approach has the necessary flexibility that it can be extended to molecules having heteroatoms, although at present only carbon molecular skeletal forms have been considered.…”

“…These novel 3-D descriptors have been referred to as molecular profile. [129][130][131][132][133][134][135][136] Importantly, the approach has the necessary flexibility that it can be extended to molecules having heteroatoms, although at present only carbon molecular skeletal forms have been considered.…”

On Characterization of Dose Variations of 2-D Proteomics Maps by Matrix Invariants

“…In this way we obtain the so-called D / D matrix for the embedded zigzag line. The D / D matrix has been initially designed for characterization of conformations (i.e., the geometrical shape) of chainlike structures. ,− Once the D / D matrix is obtained by use of standard matrix algebra, one can extract various matrix invariants to serve as the map descriptor and generate additional structurally related map matrixes.…”