Shape codes

Fig. 1. Diagram of intermediate filaments (IFs) and their associated proteins. Various IFs on the left are color- and shape-coded and matched with interacting partners on the right. For interactions of nuclear lamins with their associated proteins refer to (Zastrow et al., 2004). This figure is not meant to be comprehensive but is rather a summary of those interactions highlighted in this review.

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],... 

Based on this "ID number" approach, the shape similarity of molecules can be evaluated by numerical comparisons of shape codes [2]. The same technique of similarity evaluation, originally developed for complete molecules, can also be applied, in identical form, to the shape codes of functional groups as discussed in earlier parts of this report. 

We shall use the following notations subscripts and superscripts nil, ni2, and i correspond to the choices of non-interacting RIDCOs GR M (a), noninteracting RIDCOs GR iM (a). and the interacting RIDCOs GR(M )(a), respectively. Accordingly, the shape analysis can be carried out for contour surfaces following either one of the conventions nil, ni2, and i, leading to the shape code matrices... 

Additional detail can be found in the shape code matrices Mt(a.b),Rhg,i and M(a>b)>Rh+g,i themselves their comparisons can be carried out with special focus on low or high density ranges, considering a specific range of the... 

TOPOLOGICAL SHAPE GROUPS, SHAPE CODES, SHAPE GRAPHS, SHAPE MATRICES, AND SHAPE... 

Shape Codes, Shape Graphs, and Shape Matrices... 

We have seen that a simple list of Betti numbers of the shape groups can serve as a numerical shape code for a partitioned molecular surface. Some of the alternative topological shape de.scriptors of molecular surfaces, such as the shape matrices s(a,b) and shape graphs g(a,b), can also serve as 3D topological shape codes 143,109,110,158,199]. In Chapter 6, several examples of shape codes are described and used as numerical shape similarity measures. 

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. 

Molecular Similarity Measures Based on Shape Codes... 

Local Shape Codes and Local Similarity Measures... 

The shape types Tj are usually specified by various algebraic methods, for example, by a shape group or a shape matrix, or by some other algebraic or numerical means. The algebraic invariants or the elements of the matrices are numbers, and these numbers form a shape code. The (P,W)-shape similarity technique provides a nonvisual, algebraic, algorithmic shape description in terms of numerical shape codes, suitable for automatic, computer characterization and comparison of shapes and for the numerical evaluation of 3D shape similarity. 

Such a decreasing sequence is associated with each point of a grid on the (a,b) parameter map. The grid itself can be regarded as a matrix with the sequences of Betti numbers as elements. This matrix is a shape code for the molecule that can be used for storing shape information in molecular data banks and can be retrieved for shape similarity assessment by numerical methods. 

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