Topological map

Figure 6.10 The topological map of an idealized mountain represented by the circular contours of constant height on a topological map. Two gradient paths or lines of steepest ascent (a) are shown, together with a path (b) that is not a line of steepest ascent but is an easier route up the mountain. The lines of steepest ascent—gradient paths—cross the contours at right angles.

The microscopic world of atoms is difficult to imagine, let alone visualize in detail. Chemists and chemical engineers employ different molecular modelling tools to study the structure, properties, and reactivity of atoms, and the way they bond to one another. Richard Bader, a chemistry professor at McMaster University, has invented an interpretative theory that is gaining acceptance as an accurate method to describe molecular behaviour and predict molecular properties. According to Dr. Bader, shown below, small molecules are best represented using topological maps, where contour lines (which are commonly used to represent elevation on maps) represent the electron density of molecules. 

Seal RP, Leighton BH, Amara SG. 1998. Transmembrane topology mapping using biotin-containing sulfhydryl reagents. Methods Enzymol 296 318-331. 

Burello, E. and Rothenberg, G. (2005) Topological mapping of bidentate ligands a fast approach for screening homogeneous catalysts. Adv. Synth. Catal, 347, 1969. 

For more complex spectra, the data are usually presented as a series of contours just as hills and valleys are represented on a topological map. We see this representation of the data in the right part of Figure 5.6. Projections of the data are often included in 2-D spectra, which is equivalent to shining a light on the peak to reveal its shadow, which is obviously 1-D. Often these projections are replaced with actual 1-D spectra that have been acquired separately. So long as there are no negative peaks (e.g., phase sensitive COSY, not covered in this book), we use this method without comment. 

Burritt JB, Quinn MT, Jutila MA, Bond CW, Jesaitis AJ, Topological mapping of neutrophil cytochrome b epitopes with phage-display libraries, J. Biol. Chem., 270(28) 16974-16980, 1995. 

From the notion that animals form an overall representation of an experienced spatial environment, at least three implications have been drawn. One is that animals use multiple landmarks to locate important places by computing their distance and direction from these landmarks. A second implication is that animals can use the cognitive map to infer new routes or shortcuts through space that would be to their advantage. The third implication to be discussed is the suggestion that by exploring a spatial environment, an animal can form a topological map of that environment. 

Kast C, Gros P (1997) Topology mapping of the amino-terminal half of multidrug resistance-associated protein by epitope insertion and immunofluorescence. J Biol Chem 272 26479-26487... 

When an ISNet is matched to its cluster center, it may output a number of topological mappings as illustrated in Fig. 7. The best mapping will be used for the fuzzy graph similarity calculation. 

Figure 1.15 contains a composite of several planes from an electron density map of a protein. By continually increasing the final coordinate by Az, the electron density map is built up from the series of two-dimensional planes. The individual sections are plotted on some transparent material after contour lines have been drawn around areas within certain density limits. The result is a topological map of the electron density presented on sequential planes of the unit cell as a series of contour levels. When the individual planes are stacked in consecutive order, a three-dimensional electron density image is created. This is discussed in more detail in Chapter 10. Currently, however, the presentation of the electron density is considerably more sophisticated. We use automated computer graphics systems to present detailed density images in three-dimensional space as in Figure 1.16. 

FIGURE 10.2 A small area of a raw electron density map of a protein, directly from the Fourier calculation as it comes off the computer. The location of each number on the plane corresponds to a specific x, y, z fractional coordinate in the unit cell. In general, one of the three coordinates will be constant for the entire plane, and rows and columns will correspond to the other two coordinates. The value of the number at each position is p(x, y, z), the electron density at that point. Contours are incrementally drawn around areas having p(x, y, z) greater than certain values. This yields a topological map of the electron density on each plane of the unit cell. 

An enhanced concept of Kohonen networks is the CPG nenral network, hrst introduced by Hecht-Nielsen [64], The CPG network can be established by nsing basically a Kohonen layer and an additional ontpnt layer. The inpnt layer contains the input objects (e.g., molecular descriptors). The output layer contains the variables to be predicted, such as a one- or mnltidimensional property of the corresponding molecules. Additionally, a topological map layer [65,66] may be added that contains classes for the individnal test cases (Fignre 4.15). 

FIGURE 4.15 Scheme of a multilayer Kohonen network including input layer, output layer, and a topological map layer for classification of input data. The dimension of input and output vectors is equal to the dimension of the corresponding neurons in the network. 

The iterative training procedure adapts the network in a way that similar input objects are also situated close together on the topological map. The network s topological layer can be seen as a two-dimensional grid, which is folded and distorted into the -dimensional input space to preserve the original structure as well as possible. Clearly, any attempt to represent an n-dimensional space in two dimensions will result in loss of detail however, the technique is useful to visualize data that might otherwise be hard to understand. 

Once the network is trained, the topological map represents a classification sheet. Some or all of the units in the topological map may be labeled with class names. If the distance is small enough, then the case is assigned to the class. A new vector presented to the Kohonen network ends up in its central neuron in the topological map layer. The central neuron points to the corresponding neuron in the output layer. A CPG neural network is able to evaluate the relationships between input and output information and to make predictions for missing output information. 

Once class or properties are available in the training set, the training parameters can be selected net dimension and number of epochs, the learn radius and learn rate, and the initialization parameters. Neurons can be arranged in a rectangular or quadratic network, as well as in a toroidal mode that is, the left and right side as well as the upper and lower sides of the topological map are connected to a closed toroidal plane. 

When the training is finished, a topological map layer is generated and colored in the order of classes as defined by the user. By clicking on a colored square in the topological map, the contents can be investigated in a separate window. This window contains a preview of the input (Kohonen) layer, the output layer (if a property vector was used), and the three-dimensional model of the corresponding molecules for which the descriptor has been calculated. 

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 topological map layer after training of a Kohonen network shows a reasonable clustering, a first hint that a reliable prediction can be achieved. Figure 6.16 displays the correlation between predicted and experimental values for molecular polarizability for the set of 50 test compounds. 

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