Many chemical phenomena are influenced simultaneously by a whole series of factors and depend on many parameters. These parameters can be seen as coordinates in a multidimensional space. Individual observations then represent points in this multidimensional space. To visualize the structure of this multivariate information in order to make visible the essential relationships between the individual data points, the dimensionality of the space has to be reduced to such a degree that it can be represented graphically and thus analyzed by the human eye. Such a mapping, say onto a two-dimensional plane, should preserve the essential relationships between the individual data points as far as possible.
In Section 4.3 we saw how a three-dimensional space, the surface of a sphere, could be mapped by a Kohonen network onto the surface of a torus, and the neighborhood relationships between the points on the sphere were kept intact. In the following sections we present two examples for the mapping of multidimensional chemical information by a Kohonen network.
10.1. Chemical Reactivity
We wish once more to examine the data set introduced in Section 7.1, which classified a series of single bonds in aliphatic molecules according to whether they were easy or difficult to break heterolytically  (cf. Figure 31). Each of the bonds was characterized by seven electronic and energetic parameters, such as difference in charge and polarizability, and bond dissociation energy (cf. Figure 30). The rupture of a bond is thus represented by a point in a seven-dimensional space.
The question now is, are the points with reactive bonds in this seven-dimensional space separated from those of the nonreactive bonds? Moreover, if this is the case, can this separation be retained in a map onto a two-dimensional surface by means of a Kohonen network?
A Kohonen network consisting of 11 x 11 neurons was trained with the 149 bonds; the values of the seven electronic and energetic factors were used as input data (see Fig. 30).
The results are shown in Figure 46. Indeed the reactive bonds do find themselves predominantly in certain neurons, and the nonreactive bonds in different neurons. Additionally the neurons of the reactive bonds form a contiguous part of the Kohonen network .
Fig. 46. A Kohonen network that maps polar bond rupture characterized by seven electronic and energetic parameters. + indicates a reactive bond, - a nonreactive bond, and * a bond whose reactivity was not decided.
It may be concluded from this that the chosen parameters characterize the heterolysis of a bond well (because reactive and nonreactive bonds are separate) and that a Kohonen network is able to preserve this separation even when mapping it onto a surface. It should be emphasized once again that the Kohonen network learns without supervision, and therefore that the information on the reactivity of a bond was not used during the learning process.
Yet another conclusion may be drawn from Figure 46. Bond cleavages that activate the same neuron ought also to contain more or less the same reactivity information; the interaction of the seven parameters ought to produce a similar net effect in reactivity. Accordingly, if the relationships between reactivity and the electronic and energetic parameters are to be investigated further, it is sufficient to choose a single bond out of the many that activate the same neuron. At the same time, at least one bond should be chosen from each occupied neuron in order to cover as much as possible of the reactivity spectrum. In fact the data set that was learned by the back-propagation algorithm (see Section 7.1) was chosen with these considerations in mind. Thus two different neural network models were used: a Kohonen network for the selection of the data set and a multilayered neural network, trained by the back-propagation algorithm, for the task of classification .
The Kohonen method can therefore be used to make balanced selections of data sets for investigations with statistical or pattern-recognition methods, or even used with other neural networks.
10.2. Electrostatic Potential
The electrostatic potential surrounding a molecule has a decisive influence on many physical, chemical, and biological properties. Electrostatic potentials are analyzed in detail, especially in investigations into the interactions between substrate and receptor, but also in studies into chemical reactivity.
If one moves a probe charge, for example a positive unit charge in the form of a point, around a molecule, for every point in space the electrostatic potential may be determined by quantum mechanical or classical electrostatic methods.
Scheme 4 shows a three-dimensional molecular model of 3-chloro-3-methylbutan-1-ol, and Figure 47 the electrostatic potential that a positive point charge experiences on the van der Waals surface of this molecule.
Scheme 4. Molecular model of 3-chloro-3-methylbutan-1-ol.
Fig. 47. Electrostatic potential on the van der Waals surface of 3-chloro-3-methylbutan-1-ol. Red regions possess a negative potential, and thus attract a positive charge; blue and violet regions repel this charge.
The magnitude of the electrostatic potential is translated into a color code: Strongly negative values of the electrostatic potential (positions to which a positive charge is attracted - that is, nucleophilic positions) are represented by red. Strongly positive values (positions from which a positive charge is repelled - that is, electrophilic positions) are marked in blue or violet. The intermediate values are represented by continuous blends of color. The electrostatic potential was calculated in the classical way by summing the Coulomb interactions of the probe charge with the atomic charges as calculated by the PEOE procedure .
Figure 47 represents a parallel projection of the distribution of the electrostatic potential on the van der Waals surface of the molecule onto the plane of the graphics screen. Of course, only that part of the electrostatic potential that happens to be visible to the observer can be shown. Thus for example, the chlorine atom can barely be seen in Figure 47. A complete view of the distribution of the electrostatic potential can only be obtained by taking a series of such mappings from different observation points. The more complex the form of the van der Waals surface and the variation in electrostatic potential, the more mappings are required. For this reason it will be very difficult to obtain a complete impression of the distribution of the electrostatic potential and the relationships between all electrophilic and nucleophilic centers.
The inadequacies of a parallel projection onto a plane (monitor screen) led us to look for other projection methods that are capable of representing the essential aspects of the potentials on a molecular surface in a single mapping . A Kohonen network is one such projection method.
Figure 48 shows the projection of the electrostatic potential of 3-chloro-3-methylbutan-1-ol from Figure 47 onto a Kohonen network. To obtain this mapping, 20000 points were chosen at random from the molecular surface and a Kohonen network of 60 x 60 neurons was trained with the x, y, and z coordinates of each point. After each point had been passed through the network the neurons were examined to see which points were assigned to which neurons.
Fig. 48. Kohonen network of the electrostatic potential in Figure 47.
Points of the same or similar potential values were indeed found to be located in the same or neighboring neurons. The electrostatic potential on the van der Waals surface of a molecule had therefore been mapped onto the Kohonen map, and the neighborhood relationships on the van der Waals surface were largely preserved intact.
We remind the reader once more that in the case of a Kohonen network the mapping takes place onto a torus, that is onto a surface without beginning or end (cf. Figure 24). The map in Figure 47 can therefore be shifted up, down, to the left or to the right.
A comparison between Kohonen maps of electrostatic potentials of different molecules reflects the essential similarities of the electronic properties on the surface of the molecules, that is, at the places where molecules come into direct contact with their environment. The electrostatic potential on the surface of a molecule is a decisive factor in the interaction of a substrate with a biological receptor. Kohonen maps of electrostatic potentials of different molecules that are bound to the same receptor should therefore indicate certain similarities.
Indeed, it could be shown that Kohonen maps of the electrostatic potential of compounds that bind to the muscarinic receptor have common characteristics. Similarities are also perceived in the Kohonen maps of electrostatic potentials of molecules that bind to the nicotinic receptor, but these characteristics are different from those for molecules bound to the muscarinic receptor .
Kohonen maps show in a single picture essential characteristics of electrostatic potentials. These characteristics are apparently responsible for the binding of a substrate to a biological receptor.