Cartesian distance

Each spectrum Is regarded as a point In an N dimensional space. The coordinates of each point are the absorbance values at each wavenumber Interval. Several metrics are widely used.(19) The first Is the N dimensional cartesian distance between points 1 and j. 

FIGURE 4.1 Matrices can be used for describing chemical structures in different fashions. The adjacency matrix (a) of thionyl chloride shows whether are bonded to each other. The distance matrix (b) describes the number of bonds between two elements of a structure. The Cartesian distance matrix (c) contains the real three-dimensional (Euclidean) distances between atoms calculated from Cartesian coordinates of the atom positions. The bond path distance matrix (d) contains the sum of bond length between two atoms and is, in contrast to the Cartesian matrix, independent of the conformation of the molecule. 

FIGURE 5.3 RDF descriptor calculated with Cartesian distances and the partial atomic charge as dynamic property. The charge distribution affects the probability in both the positive and negative direction. The strong negative peaks correspond to atom pairs with charges... 

We have seen before that different types of matrices can be used for characterizing a molecule. Depending on which matrix is used, the distance r j in a radial function can represent either the Cartesian distance, a bond-path distance, or simply the number of bonds between two atoms. Consequently, we yield three groups of RDF descriptors. 

This two-dimensional RDF descriptor is calculated depending on the distance r and an additional property p. In this case, p is a property difference calculated in the same fashion as the Cartesian distance r, in fact, p can be regarded as a property distance. Mnch in the same way as B influences the resolution of the distance dimension, the property-smoothing parameter D affects the resolution — and, thus, the half-peak width — in the property dimension. D is measured in inverse squared units [p l of the property p. ... 

FIGURE 5.17 Two-dimensional RDF descriptor of ethene calculated with Cartesian distances in the first and the partial atomic charge as property for the second dimension. Instead of the one-dimensional descriptor with four peaks, the six distances occurring in ethene are clearly divided into the separate property and distance dimensions. 

RDF. The distance mode dehnes the mode for distance calculation available modes are Cartesian distances, bond-path distances, and topological distances. Descriptors may be calculated on particular atoms. Exclusive mode restricts the calculation to the atom type, and with ignore mode the selected atom type is ignored when calculating the descriptor. In partial-atom mode an atom number has to be given instead of the atom type. The second atom property is available if 2D RDF is selected as code method. 

The dimensions for a descriptor are dehned in two groups — Cartesian distance and 2D property — where the minimum, maximum, and resolution of the vector in the hrst dimension of the descriptor can be dehned. The track bars are adapted automatically to changes for example, resolution is calculated and minimum-maximum dependencies are corrected. When the binary checkbox is clicked, only selections are possible that result in dyadic vector length (i.e., the dimension is a factor of 2"). This feature prevents the complicated adjustment of all settings to gain a binary vector that is necessary for transformations. 

D and 3D autocorrelation vectors [70] represent intramolecular 2D topologies or 3D distances within particular molecules. An autocorrelation coefficient is a sum over all atom pair properties separated by a predefined number of bonds (2D) or distance (3D), while the entire vector represents a series of coefficients for all topological or Cartesian distances. Atomic properties involve hydrophobicity [71], partial atomic charges, hydrogen bonding potential and others. Again, a PCA is often used to reduce the number of variables. 3D autocorrelation vectors of properties based on distances calculated from 3D molecular surfaces [72] have also been applied to visually assess the diversity of different libraries [73]. 

The X, y, and z components of T express the Cartesian distance from Cp to Cm. The orientation can be derived from the rotational sub-matrix of... 

In Image, the pixel coordinates of the center of the melted calibration regions are recorded and the distances between them are calculated using the Cartesian distance formula see Note 8). If several equivalent distances can be calculated between the various calibration points, the variations in these distances are averaged see Note 9). 

The situation is more complicated in molecular mechanics optimizations, which use Cartesian coordinates. Internal constraints are now relatively complicated, nonlinear functions of the coordinates, e.g., a distance constraint between atoms andJ in the system is — AjI" + (Vj — + ( , - and this... 

One way to describe the conformation of a molecule other than by Cartesian or intern coordinates is in terms of the distances between all pairs of atoms. There are N(N - )/ interatomic distances in a molecule, which are most conveniently represented using a N X N S5munetric matrix. In such a matrix, the elements (i, j) and (j, i) contain the distant between atoms i and and the diagonal elements are all zero. Distance geometry explort conformational space by randomly generating many distance matrices, which are the converted into conformations in Cartesian space. The crucial feature about distance geometi (and the reason why it works) is that it is not possible to arbitrarily assign values to ti... 

Dmparison of various methods for searching conformational space has been performed cycloheptadecane (C17H34) [Saunders et al. 1990]. The methods compared were the ematic search, random search (both Cartesian and torsional), distance geometry and ecular dynamics. The number of unique minimum energy conformations found with 1 method within 3 kcal/mol of the global minimum after 30 days of computer processing e determined (the study was performed in 1990 on what would now be considered a / slow computer). The results are shown in Table 9.1. 

A similarity measure is required for quantitative comparison of one strucmre with another, and as such it must be defined before the analysis can commence. Structural similarity is often measured by a root-mean-square distance (RMSD) between two conformations. In Cartesian coordinates the RMS distance dy between confonnation i and conformation j of a given molecule is defined as the minimum of the functional... 

The conformational distance does not have to be defined in Cartesian coordinates. Eor comparing polypeptide chains it is likely that similarity in dihedral angle space is more important than similarity in Cartesian space. Two conformations of a linear molecule separated by a single low barrier dihedral torsion in the middle of the molecule would still be considered similar on the basis of dihedral space distance but will probably be considered very different on the basis of their distance in Cartesian space. The RMS distance is dihedral angle space differs from Eq. (12) because it has to take into account the 2n periodicity of the torsion angle. 

It is up to the researcher to decide whether to use a Cartesian similarity measure or a dihedral measure and what elements to include in the summation [29]. It should be stressed that while the RMS distances perfonn well and are often used, there are no restrictions against other similarity measures. Eor example, similarity measures that emphasize chemical interactions, hydrophobicity, or the relative orientation of large molecular domains rather than local geometry may serve well if appropriately used. 

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