Cartesian descriptor

Using static atomic properties allows controlling the effect on the peak height depending on the chosen property. Calculating a Cartesian descriptor with atomic volume as static property allows, for instance, emphasizing chlorine atoms in the descriptor. The descriptor can become an indicator for almost any property that can be attributed to an atom. 

FIGURE 5.7 Comparison of Cartesian and bond-path RDF descriptor (256 components each) of a cyclohexanedione derivative. The bond-path descriptor exhibits sharper peaks in particular single bond-distance patterns and is generally larger than the corresponding Cartesian descriptor. 

Cartesian Descriptors use the distances calculated from the Cartesian coordinates in a three-dimensional structure. 

Molecules are usually represented as 2D formulas or 3D molecular models. WhOe the 3D coordinates of atoms in a molecule are sufficient to describe the spatial arrangement of atoms, they exhibit two major disadvantages as molecular descriptors they depend on the size of a molecule and they do not describe additional properties (e.g., atomic properties). The first feature is most important for computational analysis of data. Even a simple statistical function, e.g., a correlation, requires the information to be represented in equally sized vectors of a fixed dimension. The solution to this problem is a mathematical transformation of the Cartesian coordinates of a molecule into a vector of fixed length. The second point can... 

A descriptor for the 3D arrangement of atoms in a molceulc can be derived in a similar manner. The Cartesian coordinates of the atoms in a molecule can be calculated by semi-empirical quantum mechanical or molecular mechanics (force field) methods, For larger data sets, fast 3D structure generators are available that combine data- and rule-driven methods to calculate Cartesian coordinates from the connection table of a molecule (e.g., CORINA [10]). 

Further prerequisites depend on the chemical problem to be solved. Some chemical effects have an undesired influence on the structure descriptor if the experimental data to be processed do not account for them. A typical example is the conformational flexibility of a molecule, which has a profound influence on a 3D descriptor based on Cartesian coordinates. In particular, for the application of structure descriptors with structure-spectrum correlation problems in... 

Once the protein interaction pattern is translated from Cartesian coordinates into distances from the reactive center of the enzyme and the structure of the ligand has been described with similar fingerprints, both sets of descriptors can be compared [25]. The hydrophobic complementarity, the complementarity of charges and H-bonds for the protein and the substrates are all computed using Carbo similarity indices [26]. The prediction of the site of metabolism (either in CYP or in UGT) is based on the hypothesis that the distance between the reactive center on the protein (iron atom in the heme group or the phosphorous atom in UDP) and the interaction points in the protein cavity (GRID-MIF) should correlate to the distance between the reactive center of the molecule (i.e. positions of hydrogen atoms and heteroatoms) and the position of the different atom types in the molecule [27]. 

Other simple geometrical descriptors are interatomic distances between pairs of atoms s and t. Interatomic distances are devided into intramolecular interatomic distances, i.e. distances between any pair of atoms (s, t) within the molecule, and intermo-lecular interatomic distances, i.e. distances between atoms of a molecule and atoms of a receptor structure, a reference compound or another molecule. While classical computational chemistry describes molecular geometry in terms of three-dimensional Cartesian coordinates or internal coordinates, the -> distance geometry (DG) method takes the interatomic distances as the fundamental coordinates of molecules, exploiting their close relationship to experimental quantities and molecular energies. 

Distance matrices may also be calculated for real three-dimensional (Euclidean) distances between atoms using the Cartesian coordinates of the atom positions (Figure 4.1c). These matrices allow the calculation of descriptors that account for the shape and conformation of atoms. If conformation is not required or desired, the bond... 

Let us summarize the three important prerequisites for a 3D structure descriptor It should be (1) independent of the number of atoms, that is, the size of a molecule (2) unambiguous regarding the three-dimensional arrangement of the atoms and (3) invariant against translation and rotation of the entire molecule. Further prerequisites depend on the chemical problem to be solved. Some chemical effects may have an undesired influence on the structure descriptor if the experimental data to be processed do not account for them. A typical example is the conformational flexibility of a molecule, which has a profound influence on a 3D descriptor based on Cartesian coordinates. The application in the field of structure-spectrum correlation problems in vibrational spectroscopy requires that a descriptor contains physicochemical information related to vibration states. In addition, it would be helpful to gain the complete 3D structure from the descriptor or at least structural information (descriptor decoding). 

D Molecular Descriptors are molecule representations based on Cartesian coordinates and Euclidean distances. 

Rotational Invariance describes the independency of a molecular descriptor from partial or complete rotation of the molecule — independent of absolute Cartesian coordinates of the atoms. 

Thus, with increasing B, the resolution increases and the step size for an RDF descriptor decreases. Figure 5.2 shows the differences in a Cartesian RDF descriptor calculated between 1 and 2 A with a smoothing parameter between 25 A and 1000 A . With the corresponding resolution between 0.1 A and about 0.032 A the halfpeak width of an intense maximum in the Euclidean L -normalized RDF lies around 0.05 and 0.2 A the maximum width is about 0.2-0.4 A. 

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... 

Some chemical structures exhibit typical distances that occur independently of secondary features, which mainly affect the intensity distribution. In particular, aromatic systems contain at least a distance pattern of ortho-, meta-, and para-carbon atoms in the aromatic ring. A monocyclic aromatic system shows additionally a typical frequency distribution. Consequently, Cartesian RDF descriptors for benzene, toluene, and xylene isomers show a typical pattern for the three C-C distances of ortho-, meta-, and para-position (1.4, 2.4, and 2.8 A, respectively) within a benzene ring. This pattern is unique and indicates a benzene ring. Additional patterns occur for the substituted derivatives (3.8 and 4.3 A) that are also typical for phenyl systems. The increasing distance of the methyl groups in meta- and para-Xylene is indicated by a peak shift at 5.1 and 5.8 A, respectively. These types of patterns are primarily used in rule bases for the modeling of structures explained in detail in the application for structure prediction with infrared spectra. 

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. 

These functions map real three-dimensional information onto a one-dimensional function. This type of function strongly depends on the exact and consistent calcnla-tion of the Cartesian coordinates of the atoms and the conformational flexibility of a molecule. When r is measured in A, the unit of a smoothing parameter B is The previously shown descriptors rely on this distance measure. 

The recognition of differences in molecular structures — the characterization of structural similarity — is a special feature of RDF descriptors. Changes in the constitution of a molecule will generally lead to changes in peak positions. For instance, a typical Cartesian RDF descriptor of a linear alkane shows periodic peaks — essentially the sum of the C-C distances. Small changes in the structure can lead to a series of changes in the descriptor. Some of the typical effects on a Cartesian RDF descriptor are as follows ... 

Some of the effects previously described are valuable for automatic RDF interpretation. In fact, this sensitivity is an elementary prerequisite in a rule base for descriptor interpretation. However, since many molecular properties are independent of the conformation, the sensitivity of RDF descriptors can be an undesired effect. Conformational changes occur through several effects, such as rotation, inversion, configuration interchange, or pseudo-rotation, and almost all of these effects occur more or less intensely in Cartesian RDF descriptors. If a descriptor needs to be insensitive to changes in the conformation of the molecule, bond-path descriptors or topological bond-path descriptors are more appropriate candidates. Figure 5.7 shows a comparison of the Cartesian and bond-path descriptors. 

A typical feature of Cartesian RDF descriptors is a (at least virtual) decrease in characteristic information with increasing distance. The influence of the short distance range (in particular, the bond information) dominates the shape of a Cartesian RDF. In contrast to that, the bond-path descriptor is generally simpler it exhibits... 

Cartesian RDF descriptors cover the three-dimensional arrangement of atoms these descriptors are suited to represent steric differences that may affect different behavior in chemical reactions. Whereas the initial bond distance range is similar. 

FIGURE 5.9 (a) Molecular Cartesian RDF descriptors for the stereoisomers shown in... 

The Cartesian RDF seems to represent the biological activity of the Ruthenium complex. In any case, the descriptor is qnite complex and cannot be compared easily with other molecnles of similar ligand arrangement and with similar biologic potency. Another approach is based on a local descriptor that specifies the chemical environment of the reaction center the Rntheninm atom. 

Local, or atomic, RDF descriptors are snitable to characterize an individual atom in its chemical environment. They are nsnally not appropriate for investigations of diverse data sets, since each A-atomic molecnle can have N local descriptors. A typical application of local descriptors is the characterization of steric hindrance at reaction centers. This can be performed nsing a conseqnent numbering of the atoms of the reactants. In the following experiment, the Rnthenium atom of each conformer shown in Figure 5.8 was the first atom in the data file, and the local RDF descriptors for atom 1 (Ru) were calculated. Figures 5.10a through 5.10c show the results for the Cartesian RDF descriptors. 

The local Cartesian RDF descriptors of the stereoisomers are generally more similar among each other than the molecnlar ones. They exhibit particularly two patterns that describe the different ligand sphere of the stereoisomers in the distance... 

FIGURE 5.10 (a) Local Cartesian RDF descriptors calculated on the Ruthenium atom in the... 

In addition, three distance modes — Cartesian, bond-path, and topological-path distances — are compared. Cartesian RDF descriptors are usually quite sensitive to small constitntional changes in the molecule. The bond-path descriptors exhibit less sensitivity, whereas topological bond-path descriptors only indicate extreme changes in the entire molecnle or in the size of the molecule. 

Whereas the skewness of Cartesian RDF descriptors reacts qnite insensitively to changes in the dataset (except in hydrazine, 14), significant changes occnr in bond-path descriptors when the molecnle becomes more compact (e.g., the sequence 2-1-3-4) and when the freqnency of side chains changes (e.g., 7, 9 and 8, 10). 

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. 

FIGURE 5.19 Comparison of the coarse-filtered D20 transformed RDF (128 components) with the original Cartesian RDF (256 components). The transformed RDF represents a smoothed descriptor containing all the valuable information in a vector half the size of the original RDF descriptor. 

RDF descriptors can either be transformed completely or partially. If only a one-stage >20 transform (i.e., = 1) is applied, the resulting descriptor can reveal discontinuities — and, thus, differences between the two molecules — that are not seen in the normal RDF descriptor. This is shown with an example of RDF descriptors of cholesterol and cholesterol-chloroacetate (Figure 5.22) that were encoded with a Cartesian RDF and a one-stage high-pass filtered D20 transform (Figure 5.23). 

The Cartesian RDF exhibits the differences between the two molecules, but the overall shape is quite similar, leading to a high correlation coefficient of 0.96. The transformed and high-pass filtered RDF emphasizes discontinuities — in particular, opposite slopes — of the nontransformed descriptor and leads to a strongly decreased correlation coefficient of 0.83. 

---