Topological frameworks

A special technique, the Alexandrov one-point compactification method, often used by topologists within a differential-topological framework, has been applied in the proof of the Holographic Electron Density Fragment Theorem [159-161],... 

In one extreme case within the topological framework, the two objects can be brought into a perfect correspondence, demonstrating topological equivalence. In a more precise formulation, two topological spaces (Aj, T,) and (X2, T2) are called topologically equivalent or homeomorphic if there exists a function... 

Sets of local coordinate systems describing certain local features of complicated objects are often advantageous when compared to a single, global coordinate system. Within a topological framework, the general theory of sets of local coordinate systems is called manifold theory. Often, the local coordinate systems are interrelated, and these relations can be expressed by continuous, and in the case of differentiable manifolds, by differentiable mappings, called homeomorphisms (see Equation (15)), and diffeomorphisms, respectively. 

The localized product-NBO view also makes clear the importance of the non-planarity of the TS in achieving favorable hyperconjugative stabilizations, because interactions of 7t-planar geometry. Indeed, one can see that favorable vicinal n-a or a-7r overlap should primarily involve one end (hybrid) of each NBO, oriented, if possible, in the anti conformation for maximum stabilization. This viewpoint allows one to recognize the importance of angular and orientational factors that would not be evident in a purely topological framework. 

Application of this topological framework analysis to ion channel modulators yields access to privileged ion channel chemotypes. Conversion of these frameworks into appropriate scaffolds for synthesis allows subsequent building of ion channel focused libraries. 

FIGURE 12.3 Topological frameworks for compounds in the CMC database (numbers indicate frequency of occurrence) Drug molecules are first reduced to a combination of rings and linkers forming the framework of the molecule, then atom and bond type identities are removed. 

Some properties such as overall molecular shape or compactness may depend on the entire topological framework or skeleton. The symmetry or topological equivalence properties of atoms in molecules also depend on the whole molecular structure. Some aspects of symmetry and shape will be discussed later. 

Scaffold abstraction approaches like topological frameworks or reduced graphs were first proposed by Bemis and Murcko [28] and further explored by Xu and Johnson [29]. Groupings into series by some fuzzier similarity are obtained by using different abstraction levels for atom types, bond types, and ring topologies. 

Similar to Bemis and Murcko [18], the whole compound collection was fragmented into topological frameworks. For this purpose, a compound was stripped of its side chain and the resulting molecular framework was reduced to fragments of the topological framework as shown in Figure 21.7. 

Figure 21.7 Side chains (arrow) are defined the topological framework. In the final step, as atoms not connected via a linker to another the topological framework is fragmented by...

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