Topological drawings

Establish logieal topology drawings and confignration speeilications. 

As an example, in a cube one has v = 8,e = 12, f = 6, and hence 8 — 12-1-6 = 2. The 2 in the right-hand side of Eq. (6.128) is called the Euler invariant. It is a topological characteristic. Topology draws attention to properties of surfaces, which are not affected when surfaces are stretched or deformed, as one can do with objects made of rubber or clay. Topology is thus not concerned with regular shapes, and in this sense seems to be completely outside our subject of symmetry yet, as we intend to show in this section, there is in fact a deep connection, which also carries over to molecular properties. The surface to which the 2 in the theorem refers is the surface of a sphere. A convex polyhedron is indeed a polyhedron which can be embedded or mapped on the surface of a sphere. Group theory, and in particular the induction of representations, provides the tools to understand this invariant. To this end, each of the terms in the Euler equation is replaced by an induced representation, which is based on the particular nature of the corresponding structural element. In Fig. 6.8 we illustrate the results for the case of the tetrahedron. 

A molecule editor can draw a chemical structure and save it, for example as a Molfile. Although it is possible to include stereochemical properties in the drawing as wedges and hashed bonds, or even to assign a stereocenter/stereogroup with its identifiers R/S or E/Z), the connection table of the Molfile only represents the constitution (topology) of the molecule. 

Fig. 46. Characteristics of the packing arrangement in unhydrated 1 imidazole with a separate schematics emphasizing the central loop topology lu) (H-bonds are indicated as broken lines backbone H atoms are omitted O atoms dotted N atoms hatched the hatched segments in the schematic drawing signify the imidazole rings)...

Figure 1. Schematic drawings of graphite and WS2 nanoclusters. Note that in both cases the surface energy, which destabilizes the planar topology of the nanocluster, is concentrated in the prismatic edges parallel to the c axis (lie) (3).

When n > 2, one can draw the reducible contributions made up of sequences of binary kernels and where states k = 0 between these kernels exist. Thus, the class associated with the skeleton of Fig. 3b contains a state k = 0 and contributes, not to Eq. (56), but to Eq. (70). In the following we shall need the relation which expresses Yg,- n) as the difference between ) and the ensemble of reducible contributions to (70) (of the type of Fig. 3b for n = 3, for example). It is necessary for us now to study systematically the points k = 0 of Eq. (70) so as to extract the reducible contributions. A study of the selection rules will permit us to solve this problem. We shall associate the appearance of the points k = 0 with the structure of the skeletons that we have introduced we shall see that the reduci-bility will be a dynamical translation of certain topological properties of the equilibrium clusters. 

Nonetheless, the topological and stoichiometric analysis of metabolic networks is probably the most powerful computational approach to large-scale metabolic networks that is currently available. Stoichiometric analysis draws upon extensive work on the structure of complex reaction systems in physical chemistry in the 1970s and 1980s [59], and can be considered as one of the few theoretically mature areas of Systems Biology. While the variety and amount of applications of stoichiometric analysis prohibit any comprehensive summary, we briefly address some essential aspects in the following. 

Fig. 3. (a) Copper site in plastocyanin. (b) Ribbon drawing of the plastocyanin backbone, (c and d) Schematic of plastocyanin topology. 

Fig. 7. (a) Ribbon drawing of immunoglobulin domain, (b and c) Schematic of the folding topology of the immunoglobulin domain. 

In the third chapter, Elinor T. Adman presents a comprehensive view of the structures of copper-containing proteins. This view includes the topological folding of many of these proteins, as shown by ribbon drawings, as well as details of copper-ligand interactions. 

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