Shape representations

Fig. 2. Schematic representation of the most common architectures of metalated container molecules. The open boxes represent active coordination sites at the metal ions. The bowl-shaped representation of the host molecules should not be confused with the one used for the cyclodextrins.

In this chapter some of the physical properties and approximate models used for molecular shape representation will be reviewed. 

In this chapter we shall combine some of the ideas described in Chapters 3 and 4 the applications of topological concepts and methods for the study of various representations of molecular shapes. Among the shape representations molecular contour surfaces have a prominent role, but we shall also consider alternatives, primarily for the purposes of characterizing the large scale shape features of biological macromolecules. 

Special cases are discussed in some detail in the literature [112,197,198], where the shape representation P is chosen as a space curve representing a protein backbone and the topological descriptors Fj(s) on the local tangent plane projections are either graphs or knots defined by the crossing pattern on the planar projection at each tangent plane T(s) of the sphere S. 

An example of this approach is shown in Figure 5.10, where the shape representation P is the space curve representing a protein backbone, and the shape... 

Fused sphere surfaces, such as fused sphere Van der Waals surfaces (VDWS ) are simple approximations to molecular contour surfaces. By specifying the locations of the centers and the radii of formal atomic spheres in a molecule, the fused sphere surface is fully defined as the envelope surface of the fused spheres and can be easily generated by a computer. Although fused sphere VDW surfaces are not capable of representing the fine details of molecular shape, such surfaces are very useful for an approximate shape representation. 

As it has been pointed out in Section 5.2, it is natural to formulate dynamic shape analysis aproaches in terms of the dynamic shape space D described earlier [158]. The reader may recall that the dynamic shape space D is a composition of the nuclear configuration space M, and the space of the parameters involved in the shape representation, for example, the two-dimensional parameter space defined by the possible values of the density threshold a, and the reference curvature parameter b of a given MIDCO surface. 

Isopotential contours of the composite nuclear potentials (NUPCO s, see Chapter 4), provide an inexpensive, approximate shape representation that can be computed easily even for very large molecules. Although NUPCO s only approximate the MIDCO s of molecules, the family of NUPCO s of a molecule describes an important molecular property that has a major effect on the actual molecular shape. Consequently, NUPCO s can be used for direct comparisons between molecules, and similar NUPCO s are likely to be associated with similar molecular shapes. All the shape analysis techniques originally developed for MIDCO s are equally applicable to NUPCO s. The shape groups, the (a,b) parameter maps [where a is the nuclear potential threshold of a NUPCO G(a)], the shape matrices, shape codes, and the shape globe invariance maps of NUPCO s of molecules can serve as inexpensive methods for the detection and evaluation of a particular aspect of molecular similarity. 

The method of (P,W)-similarity assessment [108] is a general scheme for the quantification of molecular similarity in terms of shape representations P (e.g., electronic charge isodensity surface) and shape descriptor W (e.g., molecular shape groups). 

Isopotential contours of the composite nuclear potentials (NUPCO s), provide an inexpensive approximate shape representation that can be computed easily even for very large molecules. 

C. Davatzikos and R. N. Bryan. 1996. Using a deformable surface model to obtain a shape representation of the cortex. IEEE Transactions on Medical Imaging 15(6) 785-795. 

Figure 1.18 Mean curvature H of a non-ionic surfactant film at the water/oil interface as a function of temperature T (a) [26] and composition of the internal interface v,i (b) [90]. The decrease in H with increasing T is mainly due to the shrinking size of the head group, while the decrease in H with increasing v,i is due to the smaller head group area of the alcohol compared to the sugar surfactant. In order to illustrate this behaviour, a wedge-shaped representation has been chosen.

If the shape representation is given in matrix form of the (a,b) map of Betti... 

A simple molecular property that can be calculated easily and can also be used for shape representation is the composite nuclear potential, denoted by V (r). This 3D molecular function was first compared to 3D molecular electronic densities in a systematic study by Parr and Berk. The VnC ") potential can be determined easily for a large number of nuclear arrangements of large molecules, providing a valid shape representation and a tool for shape comparisons of chemical species. 

An important fact has been pointed out by Parr and Berk the bare nuclear potential Vn(r) shows many similarities with the electronic density function p(r). The computed isopotential contours of the composite nuclearpotential VnC lwere remarkably similar to some of the molecular isodensity contours (MIDCOs) of the electronic ground states in several simple molecules. One may regard the composite nuclear potential as the harbinger of electronic density, and isopotential contours of the composite nuclear potential V (r) can serve as surprisingly good approximations of MIDCOs. The nuclear potential contours (NUPCOs) are suitable for an inexpensive, approximate shape representation of molecules. 

A simple, additive fragmentation approach to the molecular electronic density, proposed by the author, can be used for the construction of electronic densities and density-based shape representations for macromolecules. The simplest of these approaches is motivated by Mulliken s population analysis technique,and can be regarded as a natural generalization of Mulliken s approach a formal population analysis without integration. This method, the Mulliken-Mezey approach, is the simplest realization of a more general, additive fuzzy density fragmentation (AFDF) principle. ... 

The Somoyai function is defined in terms of the electronic density function and the composite nuclear potential, providing a 3D shape representation of the bonding pattern within the molecule under study. Some of the topological techniques of molecular shape analysis have been reviewed, with special emphasis on applications to the Somoyai function. A combination of a family of recently introduced ab initio quality macromolecular electronic density computation methods with the electrostatic Hellmann-Feynman theorem provides a new technique for the computation of forces acting on the nuclei of large molecules. This method of force computation offers a new approach to macromolecular geometry optimization. 

To create a virtual representation of machining processes within computers, a description of the movement as well as geometric models of the workpiece and the tool is necessary. To include the change of the shape of the workpiece caused by intersection of the moving tool, it is necessary to use shape representations which allow fast calculation of modihcations. 

The boundary of a shape is composed of surfaces, as well as curves as intersections of surfaces. Complex shapes consist of a large number of curves and surfaces. Both the outside and inside boundaries can be defined on mechanical parts. The following are fundamental goals of the boundary type of shape representations. 

The shape group method (SGM), reviewed in ref. [2], has been proposed for the analysis of three-dimensional shape properties of formal molecular bodies. For example, by choosing the electronic charge isodensity contours G(a) (of various density values a) as the physical property P for shape representation, and by taking the family of Betti numbers b as the topological tool for shape description [2], the similarity of the geometrical shapes of two molecules, A and B, is transformed into an equivalence of their topological shape, expressed as... 

A contact is detected if one pixel is occupied by two or more particles. The overlaps can be accurately and efficiently calculated using the digitized pixels. The time taken to check overlaps is a linear function of the particle number and does not increase with the complexity of particle shapes, which makes this technique preferable compared to other shape representations in terms of computational complexity. A linear contact force as a function of overlapping volume is proposed for this approach of shape representation [45]. This model is yet to be validated against experimental evidence. 

Favier, J.F. Abbaspour-Fard, M.H. Kremmer, M. Raji, A.O. (1999) Shape representation of axisymmetrical, non-spherical particles in discrete element simulation using multielement model particles. Engineering Computations 16,467 80. 

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