Shape matrix

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

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

If a different reference curvature value b is chosen, then the shape matrix may be different, although the matrix is invariant within small enough intervals of the b values. As examples, the three different shape matrices s(0.0l,0), s(0.01,0.005), and s(0.01,-0.008) of the three shape domain partitionings of the ally I alcohol MIDCO G(O.OI) shown in Figure 5.6 are given below. The index... 

If the entire range of curvature parameter b is considered, then a list of the finite number of distinct shape matrices and those curvature values bj where a change of the shape matrix occurs, gives a detailed, numerical shape characterization of the MIDCO surface G(a). In the most general case of variations in the two parameters a and b, as well as in the nuclear configuration K, one can study the dynamic shape space invariance domains, the (a,b)-maps, and various projections of the invariance domains of shape matrices, following the principles [158] applied for the shape group invariance domains of the dynamic shape space D. 

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. 

If the two molecules A and B turn out to be dissimilar by a given (P,W)-shape similarity criterion [i.e., if they do not fulfill the equivalence relation A (P,W) B], then the differences between their numerical shape descriptors can serve as a dissimilarity measure. That is, for a (P,W)-dissimilar molecule pair A and B, the (P,W)-similarity concept allows one to quantify how different their topological invariants are. A simple and straightforward approach is based on a simple vector comparison of the lists of Betti numbers of the shape group technique, or on the numerical comparison of shape matrices. 

One of the most useful shape codes is based on shape matrices. As we have seen in Chapter 5, the N-neighbor relation N(D j, D j ) of various curvature domains and, given by Equation (5.8), leads to a shape matrix... 

Also note that concatenation of the upper off-diagonal triangle of the shape matrix by columns instead of rows ensures that shape matrices of different dimensions can be easily compared if a k-th digit is present in both of the resulting binary numbers, then they correspond to the same pair of row and column indices in the two shape matrices. Evidently, this feature is of importance when comparing... 

More detailed shape comparison is possible if the decoded elements of the two vectors C(Mi) and C(M2) are compared directly. For example, by taking the number of matches along the diagonals and within the off-diagonal upper triangles of the two shape matrices s(a,b,Mi) and s(a,b,M2), divided by n(n-i-l)/2, where n is the dimension of the larger of the two matrices, an elementary similarity measure s(a,b) is obtained, characteristic to the point (a,b) of the parameter map. Clearly,... 

The general methods applied for shape codes based on shape matrices are esf)ecially suitable for developing local shap>e codes. 

The very same procedure that has been used to construct the global C(Mi) codes for the global shape matrices s(a,b,M]) along the parameter map (a,b) can also be applied to the set of local shape matrices Ib(a,b,Mi) along the parameter map (a,b), resulting in a local shape code vector... 

When searching for local similarities of two molecules, the decoded local shape matrices Ib(a,b,Mi) of molecule M) are compared to various diagonal blocks of the global shape matrix s(a,b,M2) of molecule M2. In the most general case, the local shape matrix Ib(a,b,Mi) is used as a template, and it is compared to k-dimensional blocks of s(a,b,M2) obtained by all possible simultaneous row and column permutations. If the size ordering is considered important then only those permutations are taken which preserve the monotonicity of size ordering in the permuted diagonal block that is compared to the template. A local similarity measure... 

For most practical applications, CIMM is used within the framework of local measures. These measures are based on local shape matrices or on the shape groups of local moieties, defined either by the density domain approach mentioned earlier, or by alternative conditions, such as the simple truncation condition replacing the "remainder" of the molecule by a generic domain [192], For proper complementarity, identity or close similarity of the patterns of the matched domains is an advantage, hence the parts Cl HM)) and Cl KM2) of the corresponding local shape codes are compared directly. For shape complementarity only the specified density range [bq - Aa, Bq + Aa] and a specified curvature range of the (a,b) parameter maps is considered. A local shape complementarity measure, denoted by... 

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. 

Shape codes [43,109,196,351,408]. The simplest topological shape codes derived from the shape group approach are the (a,b) parameter maps, where a is the isodensity contour value and b is a reference curvature against which the molecular contour surface is compared. Alternative shape codes and local shape codes are derived from shape matrices and the Density Domain Approach to functional groups [262], as well as from Shape Globe Invariance Maps (SGIM). 

Rhodifuse lode was designed to deliver sodium iodide in a source of drinking water continuously over a one-year period. It can be placed in a well to release a therapeutic supply of iodine (100 pg / day / individual) at a nearly constant rate. It is a modular matrix system which is composed of three polypropylene baskets, each of them containing three matrices (Fig. 10). The cylinder-shaped matrices are loaded with 30% Nal by weight. They are prepared by molding a dispersion of Nal powder in a two-component silicone RTV (platinum catalyzed). 

Figure 2.3 IgG levels after administration of drug delivery systems in rats. Controlled-delivery systems for antibody class IgG. The insert figures show the release of antibody from the delivery system during incubation in buffered saline. The panel (a) inset shows release from poly(lactic acid) microspheres these spherical particles were 10-100/rm in diameter. The panel (b) inset shows release from a poly[ethylene-co-(vinyl acetate)] matrix these disk-shaped matrices were 1 cm in diameter and 1 mm thick. In both cases, molecules of IgG were dispersed throughout the solid polymer phase. Although the amount of IgG released during the initial 1-2 days is greater for the matrix, the delivery systems have released comparable amounts after day 5. (a) Comparison of plasma IgG levels after direct injection of IgG (open circles) or subcutaneous injection of the IgG-releasing polymeric microspheres characterized in the inset (filled circles). The delivery system produces sustained IgG concentrations in the blood [3]. (b) Comparison of plasma IgG levels after direct intracranial injection of IgG (open squares) or implantation of an IgG-releasing matrix (filled squares) [4]. The influence of the delivery is less dramatic in this situation, probably because the rate of IgG movement from the brain into the plasma controls the kinetics of the overall process.

For two molecules identical shape matrices indicate a rather strong similarity of their shapes. If the two matrices are not identical, but if they can be converted into identical forms by simultaneous row and column permutations, then the essential shape patterns of the distribution of convex, concave, and saddle-type domains agree, but the sizes of these domains are sufficiently different, so they do not follow the same order. This latter case implies a somewhat weaker similarity between the two molecules. The complexity (e.g., the number of inversions) of the actual permutation required serves as a further qualification of their similarity. 

The most natural, and informative choice for reference curvature is b = 0. However, one may be interested in much finer details of shape than the simplest classification of surface domains into concave, convex, and saddle-type domains. A more detailed description is possible by utilizing the fact that for a different reference curvature value b the shape matrix may be different, although it is invariant within certain intervals of b values. By considering an entire range of curvature parameter b, and by listing the finite number of distinct shape matrices and the curvature values b, where a change in the shape matrix occurs, a detailed, numerical shape characterization of the molecular surface G a) can be given. 

Griiner, B., Mikulasek, L., Baca, J. eL al. 2005. Cobalt bis(dicarbollides)(l-) covalently attached to the calix[4] arene platform— The first combination of organic bowl shaped matrices and inorganic metallaborane cluster anions. J. Organomet. Chem. 2005 2022-2039. 

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