Shape group methods

An important concern is the efficient detection of local shape changes introduced by chemical changes in remote locations of a molecule. One simple approach [20] applied a truncation method, compatible with the truncation process already used within the shape group methods for molecular shape analysis [41-44]. 

For both types of FIDCO surfaces, the usual Shape Group method [2] of electron density shape analysis is applicable. The additional formal domain boundaries AD i(Ga b(3)) and AD i(GA(B)(a)) introduce one additional index -1, which can be treated the same way as relative curvature indices. The one-dimensional homology groups obtained by truncations using all possible index combinations are the shape groups of FIDCO surfaces. The (a,b)-parameter maps and shape codes are generated the same way as for complete molecules [2],... 

The complete, three-dimensional shape of fuzzy electron densities can be described in detail using the Shape Group Methods, (SGM), reviewed in detail in a recent monograph [27], Here, only a brief summary of this method will be given. 

The standard Shape Group Method is applicable for the analysis of the entire series of non-interacting RIDCOs, for a whole range of density thresholds a, with the provision of an additional domain type representing the connection of region R to the rest of the molecule within the actual RM system. This additional domain type D i is defined as... 

Whereas framework groups are more informative than point symmetry groups, they are able to describe only a rather restricted aspect of molecular shape. Alternative group theoretical methods, notably, the Shape Group Methods (SGM) of molecular topology, are more suitable for a detailed shape characterization. The shape groups will be discussed in Chapter 5 of this book. 

The Shape Group Method (SGM) for the Analysis of Molecular Shapes... 

It should be emphasized that the above shape group methods combine the advantages of geometry and topology. The truncation of the MIDCO s is defined in terms of a geometrical classification of points of the surfaces, and the truncated surfaces are characterized topologically by the shape groups. 

The nondifferentiability of these surfaces at the seams of interpenetrating spheres as well as the local nondifferentiability of solvent accessible surfaces or union surfaces, are a technical disadvantage. Local nondifferentiability limits the application of the shape group methods in their original form that requires second derivatives for curvature analysis. For example, at every point r of a VDWS where two or more atomic spheres interpenetrate one another, the surface is not smooth and is not differentiable. For such nondifferentiable molecular surfaces, alternative shape descriptors and shape codes have been introduced. 

Consider a family of solid objects with well-defined boundaries. Their size can be characterized numerically. The topological shape group methods and related techniques are suitable for an algorithmic determination of their dominant shape features. However, this characterization becomes more complicated if the objects have no proper boundaries, for example, if fuzzy charge density clouds are compared. Nevertheless, the advantages of algorithmic techniques and automated... 

The QShAR (Quantitative Shape-Activity Relations) method, combined with the integrated main and side effect modeling of bioactive molecules, forms the conceptual basis of the approaches described in this chapter. The density scalable FSGH method for a simple representation of molecular bodies, in combination with the Shape Group Method and various other shape code approaches for quantitative shape analysis, as well as the multiple shape ranking methods for integrated main and side effect analysis, are the components of a computational implementation of the basic concepts. 

The integrated main effect and multiple side-effect analysis by multiple shape ranking of bioactive molecules, based on the density scalable FSGH and Shape Group Methods, are suitable for shape analysis of large molecules. 

Subsequently, these topological methods have been adopted and modified to the significantly simpler, three-dimensional molecular shape problem, where the shape of the molecule is the quantum mechanical shape of the electron density cloud [13-19], This has led to the development of the shape group methods, where the ranks of homology groups describing relative convexity domains of the complete set of all isodensity surfaces of the molecule, the so-called Shape Group Betti numbers, provided a detailed, numerical shape code for the quantum chemical electron density [13-19]. 

THE TOPOLOGICAL MOLECULAR SHAPE AND SIMILARITY ANALYSIS THE SHAPE GROUP METHOD... 

The main tool for a systematic, topological shape and similarity analysis of molecules is the shape group analysis of molecular electron density clouds [13-25]. The shape group methods are not restricted to molecular electron densities however, in the present context, we shall phrase our brief review of these techniques in terms of electron densities. 

The topological Shape Group Methods and their various applications have been reviewed extensively in the literature.Here only a brief summary of the main features of these methods will be given. 

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