Cartesian RDF

The Cartesian RDF uses the distances r calculated from the Cartesian coordinates 

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. 

---

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. 

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. 

FIGURE 5.5 Typical pattern of a sulfide-bridge in the cat pheromone Felinine. (Cartesian RDF, 128 components). Two typical distances, 1.84 and 2.75 A, are a characteristic feature for the presence of the sulfide bridge. 

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). 

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. 

FIGURE 5.23 Overlay of the Djo transformed (above) and the normal Cartesian RDF descriptors (below) of Cholesterol (dark) and Cholesterol chloroacetate (light). Discontinuities (framed area) in the Cartesian RDF are revealed by the high-pass filtered transform, leading to a significant decrease in the correlation coefficient. 

FIGURE 5.24 Comparison of the low-pass filtered Djo transformed RDF (128 components) of the original Cartesian RDF (256 components) with a Cartesian RDF of half the resolution (128) components. Remarkable differences are indicated in bold. 

Having the three-dimensional coordinates of atoms in the molecules, we can convert these into Cartesian RDF descriptors of 128 components (B = 100 A ). To simplify the descriptor we can exclude hydrogen atoms, which do not essentially contribute to the skeleton structure. Finally, a wavelet transform can be applied using a Daubechies wavelet with 20 filter coefficients (D20) to compress the descriptor. A low-pass filter on resolution level 1 results in vectors containing 64 components. These descriptors can be encoded in binary format to allow fast comparison during descriptor search. 

Cartesian RDF descriptors of 128 components (B = 100 A ) are calculated for each structure without hydrogen atoms. 

The query compound is considered as unknown that is, only infrared spectrum is used for prediction. The prediction of a molecule is performed by a search for the most similar descriptors in a binary descriptor database. The database contains compressed low-pass filtered D20 transformed RDF descriptors of 64 components each. The descriptors originally used for training (Cartesian RDF, 128 components) were compressed in the same way before the search process. 

FIGURE 6.6 Benzene derivatives predicted by a CPG neural network (low-pass Z>2o Cartesian RDF, 128 components). The 2D images of the eight best matching structures from the descriptor database are shown together with the correlation coefficients between their descriptor and the one predicted from the neural network. 

We have seen that RDF descriptors are one-dimensional representations of the 3D structure of a molecule. A classification of molecular structures containing characteristic structural features shows how the descriptor preserves effectively the 3D structure information. For this experiment, Cartesian RDF descriptors were calculated for a mixed data set of 100 benzene derivatives and 100 cyclohexane derivatives. Each compound was assigned to one of these classes, and a Kohonen neural network was trained with these data. The task for the Kohonen network was to classify the compounds according to their Cartesian RDF descriptors. 

FIGURE 6.10 Results of the classifieation with 100 benzene derivatives (dark) and 100 cycloaliphatic compounds (light). The topological map shows a clear distinction between compounds with planar benzene ring systems and nonplanar cyclic systems (rectangular network, Cartesian RDF, 128 components). 

The next example shows how the complexity of an RDF descriptor might influence the classification. The compiled data set consisted of benzene derivatives, phosphorous compounds, and amines. The Cartesian RDF descriptors were calculated once including all atoms and a second time without hydrogen atoms. The left-hand image in Figure 6.11 shows the classification with normal RDF descriptors. Two remarkable situations occur ... 

FIGURE 6.11 Results of the classification with Cartesian RDF descriptors for 24 benzene derivatives, 20 phosphorous compounds, and 11 amines, calculated including (left) and ignoring (right) hydrogen atoms (rectangular network, Cartesian RDF, 256 components). 

The reduction in descriptor size and resolution of Cartesian RDF descriptors leads to a significant decrease of the quality of classification. The high-pass D20 transformed descriptors — although half the size of the Cartesian RDF — are suited for classification even down to extremely short vectors with a resolution of just 0.8 A (B = 1.5625 A ) of the original descriptor. 

FIGURE 6.16 Correlation between calculated and predicted molecular polarizability for 50 benzene derivatives encoded with one-stage filtered D2Q transformed Cartesian RDF (128 components). The standard deviation of the prediction error is 0.6 A. ... 

The next example illustrates the different representation of normal and wavelet-transformed RDF descriptors. The first 50 training compounds were selected from a set of 100 benzene derivatives. The remaining 50 compounds were used for the test set. Compounds were encoded as low-pass filtered D20 Cartesian RDF descriptors, each of a length of 64 components, and were divided linearly into eight classes of mean molecular polarizability between 10 and 26 AT... 

---