Relative shape descriptors

By contrast, relative shape analysis and relative shape descriptors can change for each molecule, depending on the other molecule used for comparison. For n molecules there are n(n-l)/2 molecule pairs, hence n(n-l)/2 families of relative shape descriptors of the given type. Consequently, in the study of shape similarities in large molecular families, the quadratic dependence of the number of relative shape descriptors on the number of molecules is a disadvantage and the use of relative shape descriptors is often impractical. Shape similarity measures based on relative shape descriptors are called similarity measures of the second kind. 

A relative shape descriptor is a d-dimensional function associated with a pair of molecules (in their p-dimensional models). These functions measure relative molecular similarity instead of the absolute (i.e., nonrelative) shape of the corresponding model. In this category we find, among others, root-mean-square (rms) deviations, °-2 quantum similarity measures, and some polymer compactness measures. 

Relative shape descriptors are defined in terms of a reference structure, which may be an experimental X-ray or NMR structure. Similarly, we can use a molecular mechanics conformational energy minimum as the initial structure for a dynamics simulation. The reference can also be a target conformation with desired shape features—for example, being maximally compact i -aie or... 

The USR (Ultrafast Shape Recognition) Method. This method was reported by Ballester and Richards (53) for compound database search on the basis of molecular shape similarity. It was reportedly capable of screening billions of compounds for similar shapes on a single computer. The method is based on the notion that the relative position of the atoms in a molecule is completely determined by inter-atomic distances. Instead of using all inter-atomic distances, USR uses a subset of distances, reducing the computational costs. Specifically, the distances between all atoms of a molecule to each of four strategic points are calculated. Each set of distances forms a distribution, and the three moments (mean, variance, and skewness) of the four distributions are calculated. Thus, for each molecule, 12 USR descriptors are calculated. The inverse of the translated and scaled Manhattan distance between two shape descriptors is used to measure the similarity between the two molecules. A value of 1 corresponds to maximum similarity and a value of 0 corresponds to minimum similarity. 

A number of methods have been proposed for particle shape analysis these include verbal description, various shape coefficients and shape factors, curvature signatures, moment invariants, solid shape descriptors, the octal chain code and mathematical functions like Fourier series expansion or fractal dimensions. As in particle size analysis, here one can also detect intense preoccupation with very detailed and accurate description of particle shape, and yet efforts to relate the shape-describing parameters to powder bulk behaviour are relatively scarce.10... 

The kappa shape indexes are relatively new descriptors, and few studies have been reported to date. It is expected that the kappa indexes will generally be used along with other topological indexes in QSAR equations rather than by themselves as in most of the examples above. Shape is not the sole determinant... 

The discussion in this section has dealt with descriptors of absolute shape, which are very discriminating functions. There are also numerous descriptors of relative shape (e.g., deviations between conformations), which characterize flexibility. We deal briefly with these descriptors in the next section. 

Several other techniques for 2D surface matching and relative comparison have been proposed. A detailed discussion is beyond the goal of this chapter on basic molecular shape descriptors. An overview of different alternatives can be gathered by comparing the work in Refs. 2, 155, 240, and 241. 

Throughout the realm of molecular modeling, the concept of molecular shape arises over and over in one form or another. Just what do scientists mean by a molecule s shape, and how can one use three-dimensional shape in modeling. In Chapter 5, Professor Gustavo A. Arteca examines these issues and delineates the hierarchical levels of molecular shape and shape descriptors. He explains molecular shape in terms of mathematical descriptors of nuclear geometry, connectivity, and molecular surfaces. Of special note are his comments on shape dynamics of flexible molecules and descriptors of relative shape. 

The adoption of a particular structure type is found to depend on the relative importance of different intermo-lecular interactions—part of the molecule being likely to result in stack formation (C-. C interactions) and part in the formation of glides (C- H interactions). In these simple molecules, some degree of prediction was found to be possible based on the assignment of a stacking ability or "glide-forming" ability to each atom in the molecule equivalently, an overall molecular size/shape descriptor can be defined. 

The solubility of a compound is thus affected by many factors the state of the solute, the relative aromatic and aliphatic degree of the molecules, the size and shape of the molecules, the polarity of the molecule, steric effects, and the ability of some groups to participate in hydrogen bonding. In order to predict solubility accurately, all these factors correlated with solubility should be represented numerically by descriptors derived from the structure of the molecule or from experimental observations. 

A from the center of a positive ionizable group was identified. However, its predictive performance on a test set consisted of eight structurally similar compounds was relatively poor. To achieve a computational model with greater predictability, a descriptor-based QSPR model was also developed. Descriptors related to molecular hydrophobicity as well as hydrogen bond donor, shape and charge features contributed to explain hOCTl inhibitor properties of the analyzed compounds. 

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