Topology Cartesian coordinates

To be able to calculate 3D descriptors, amino acids have to be characterized by (x, y, z) Cartesian coordinates. Being the a-carbon present in all coded amino acids, the Cartesian coordinates of that atom are selected as the coordinates of the whole amino acid. From the peptide topological representation and/or the corresponding geometrical representation (using only a-carbon spatial coordinates), several constitutional, topological and geometrical descriptors can be calculated. 

The Monte Carlo version of minimal steric difference (denoted as MCD) improves the computation of non-overlapping volumes in the standard-ligand superposition, translating thus the topological MSD parameter into the (3D) metric context (Mojoc et al. 1975 Ciubotariu et al. 1983). In order to calculate the MCD, the molecules are described by the Cartesian coordinates and vdW radii of their atoms. The atomic coordinates implicitly specify the way one achieves the superposition all molecules of the series are represented in the same Cartesian coordinate system. The mathematical method used in the MCD-technique for computation of nonoverlapping volumes is the Monte Carlo method (Demidovich and Maron 1987). 

The values of the diverse geometrical descriptors depend only on the type of atoms and on the Cartesian coordinates of these atoms in the minimum energy conformer. The type of a particular atom can be identified by the tabulated values of the atomic number, mass, or van der Waals radius. Often included in this category are descriptors whose value depends on other characteristics of atoms. For instance, topological indices are sometimes considered geometrical descriptors. However, topological indices can only be calculated after the identification of the way in which atoms are connected. 

To summarize, the relaxation times (or eigenvalues) of a rather complex system such as a 3-D topologically-regular network end-Unked from Rouse chains were determined analytically. In fact, one can do even better it is possible to construct all of the eigenfunctions of the network analytically (which amounts to the transformation from Cartesian coordinates to normal coordinates). Briefly, to construct the normal mode transformation, see Eqs. 84 and 85, one has to combine the Langevin equations of motion of a network jimction, Eq. 80, and the boundary conditions in the network junctions, Eqs. 87 to 92. After some algebra one finds [25,66] ... 

The spatial arrangement of the atoms constituting a material is specified completely by the topology and geometry. Topology is simply the pattern of interconnections between atoms. It is often expressed in the form of a connectivity table. Geometry also encompasses the coordinates of the atoms, usually in Cartesian (x, y and z) coordinates but sometimes in alternative coordinate systems such as spherical, cylindrical or internal coordinates. 

Figure 2 shows a fuller representation of the information in Fig. 1, namely, a 3D view of the so-called gradient vector field (i.e. the collection of gradient paths) of HC=N. This type of picture brings out a main feature of the topological atom it is a subspace bounded in Cartesian space but unbounded in the space of the natural coordinates. These coordinates are the familiar 0,(p) angles of the spherical polar coordinates (centred on the nucleus) and a path parameter s, which spans the full... 

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