Structures Graphs. Reduced

Fischer, J.R. and Rarey, M. (2007) SwiFT an index structure for reduced graph descriptors in virtual screening and clustering./. Chem. Inf. Model., 47, 1341-1353. 

Figure 14-1. Atomic structure and reduced representation in terms of a graph of peaks of a protein chain (PDB file 1B69). The peaks are coloiu-coded per amino-acid type. Visualizations of the atomic and reduced representations were obtained on combining the Swiss PDB Viewer (http //www.expasy.org/ spdbv) and the ray-tracing POV-Ray (http //www.povray.org) programs (see color plate section)...

Proof. If ip0 had 2 extensions ip" Spec (R) X, then apply the definition of scheme to ip1 and ip" and it follows that ip1 = ip". This takes care of uniqueness. Next apply the definition of proper to cp Spec (R) -> Y and to the closed subset of X xy Spec (R) obtained by taking the closure Z of (ipo,i) [Spec K]. Give Z the structure of reduced closed subschemes. Then p2(Z) is a closed subset of Spec (R) containing its generic point, hence equal to Spec (R). 1 claim that p2 Z - Spec (R) is an isomorphism, hence Z is the graph of the ip we are looking for. 

In the ring/non-ring reduced graph, therefore, the components of the structure are reduced to nodes of which there are two basic t5q>es, the ring node and the non-ring node. This is readily illustrated for a specific structure in Figure 2. 

Fig. 8.9(b), ZnV20g is able to crystallize with the brannerite structure whose theoretical bond valences, calculated from the network equations ((3.3) and (3.4)) and shown in Fig. 8.9(b), already predict an out-of-centre distortion for the ion. ZnV20g thus adopts a bond graph that supports the electronically induced distortion. In this case the adoption of a lower symmetry bond graph is favoured because it is able to reduce the bond strain. 

Molecular Connectivity Indexes and Graph Theory. Perhaps the chief obstacle to developing a general theory for quantification of physical properties is not so much in the understanding of the underlying physical laws, but rather the inability to solve the requisite equations. The plethora of assumptions and simplifications in the statistical mechanics and group contribution sections of this article provide examples of this. Computational procedures are simplified when the number of parameters used to describe the salient features of a problem is reduced. Because many properties of molecules correlate well with structures, parameters have been developed which grossly quantify molecular structural characteristics. These parameters, or connectivity indexes, are usually based on the numbers and orientations of atoms and bonds in the molecule. 

Gillet, V.J., Downs, G.M., Holliday, J.D., Lynch, M.F., Dethlefsen, W. Computer storage and retrieval of generic chemical structures in patents. 13. Reduced graph generation. Journal of Chemical Information and Computer Science 1991, 31, 260-270. 

Fig. 14. Example of the reduced-graph expression of a chemical structure...

Using the decay factors in eqs 3 and 4, one can search for the lowest order contributions to TDA. In this way, a macromolecule electronic structure problem is reduced to the relatively simple task of finding the minimum distance in a graph. The minimum-distance problem is well-known and can be solved in a reasonable amount of time. Thus, a given protein structure defines the connectivity and decay factors between... 

Figure 1.11 Examples of chemical structures corresponding to the same GWB reduced graph of type R/F (shown in center). (Adapted from ref 224.)...

Figure 4.1 Ligand-based virtual screening methods. The figure shows different computational methods for screening compound databases that take either a local or a global view on molecular structure. Molecular similarity methods that operate on molecular descriptors, histogram representations, superposition or (reduced) molecular graphs evaluate molecular structure globally. By contrast, local structural features are explored by substructure and pharmacophore searching or QSAR modeling.

Takahashi, Y, Sukekawa, M. and Sasaki, S.I. (1992). Automatic Identification Molecular Similarity Using Reduced-Graph Representation of Chemical Structure. J.Chem.Inf.Comput.ScL, 32, 639-643. 

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