General shape descriptor

Let us denote the average of a general shape descriptor P( r, ) over all accessible molecular configurations by (P). If the analysis is restricted to conformational changes, then atomic connectivity is maintained. For a given type of connectivity (e.g., linear chains), the dependence of (P) with the number of atoms n relates to shape features, especially shape flexibility. Later in the chapter, we discuss shape dynamics in a broader context. Here, we can illustrate this relation with a simple example. 

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). 

A number of - shape descriptors are defined in terms of principal moments of inertia. Moreover, principal moments of inertia are used to provide a unique reference framework for the calculation of the - shadow indices, and in general are used as - alignment rules of the molecules. They are encountered among - WHIM descriptors and used by the CoMMA method. 

Several shape descriptors are defined within more general approaches to - molecular descriptors. This is the case of - Kier shape descriptors, -> shape profiles, -> shadow indices, -> WHIM shape descriptors, - Sterimol shape parameters L/Bj and B1/B5, molecular - periphery codes, and -> centric indices. Other approaches to the study of molecular surface and shape are Mezey 3D shape analysis and Hopfinger - molecular shape analysis. -> Triangular descriptors have also been used to characterize molecular shape to search for similarities among molecules. 

The generality of this new type of shape-activity correlation is demonstrated for five receptor/substrate systems trypsin/arylammonium inhibitors the D2-dop-amine receptor/dopamine derivative agonists trypsin/organophosphate inhibitors acetylcholinesterase/organophosphates and butyrylcholinesterase/organo-phosphates. The correlations were obtained both for active-site induced chiral conformers and for inherently chiral inhibitors. Interestingly, for some of these cases the correlation of activity with structure is hidden when classical parameters, such as chain length, are taken, but is revealed with this shape descriptor. 

As this table shows, the F-values are larger and show a wider variation than the standard deviations. They generally correlated with the standard deviation values, but in some cases (cluster 2 in Table II), there were discrepancies. Because of their wider span, and because they generally accorded with graphical comparison of the structures, the F-values were u in the correlations with biological activity and for comparison with other shape descriptors. 

In general, the various different shape descriptors in this study were not intercorrelated (R values 0.7 or less). Simple correlations of the SIMCA F-value descripmrs with the various biologictd response variables yielded rather poor results. 

This chapter has presented an introduction to the general problem of selecting and using a molecular shape descriptor. It should be apparent that there is an enormous range of descriptors among which we can choose. However, our first decision in shape analysis is not the choice of descriptor. As we have seen, the nature of the descriptor can change with the molecular properties under study, and their model representations. Therefore, our first true choice must be the selection of a molecular model relevant to the problem. 

A number of studies have used an alternative approach to assess descriptor quality for diversity profiling. In these studies, descriptors were ranked by their ability to discriminate active and inactive compounds within a number of medicinal chemistry project data sets. In the work of Brown and Martin, this discrimination involved the ability to separate one class of compounds from a general pool of compounds. The approach put forward by Patterson et al. i (see also Refs. 132 and 133) introduced the concept of neighborhood behavior that is, compounds close in biological space should have a small difference in descriptor values. In these studies, it was suggested that 2-D fingerprints and simple shape descriptors make better descriptors than other alternatives... 

The Jones polynomials of the knots Kn, obtained by the switches specified by vectors (vn), are in general different from the polynomial of the actual, original knot Kq. Consequently, they provide a more detailed characterization of the projection. We shall use the complete set of knots (Kn) as a shape descriptor, following a formal vector notation knot vector K) ... 

On equating the atomic radius to a characteristic atomic radius, r, a single curve of d vs D describes homonuclear covalent interaction, irrespective of bond order. Practical use of the formulae requires definition of a complex set of characteristic radii, which could be derived empirically [1] and was used subsequently to calculate molecular shape descriptors [2] and as the basis of a generalized Heitler-London procedure, valid for all pairwise covalent interactions [3,4], In all of these applications, interaction is correctly described by the dimensionless curves of Fig. 1. 

For a general review, see Arteca, G. A. Molecular Shape Descriptors, in Reviews in Computational Chemistry, 1996, 9, pp. 191-253, edited by K. B. Lipkowitz and D. B. Boyd, VCH, Weinheim. For other samples of the rich literature Arteca, G. A. Mezey, P. Shape characterization of some molecular model surfaces, J. Comp. Chem. 1988, 9, 554-563 (topological method) Shoichet, B. 

As a general descriptor of the molecular shape of 1, one may consider the dihedral angle between the chemically bound naphthalene moieties. Data of this parameter are listed in Table 20 for the studied inclusion compounds of 1 combined with a scheme that... 

Molecular weight is often taken as the size descriptor of choice, mainly because it is easy to calculate and is generally in the chemist s mind. However, other size and shape properties are equally very simple to calculate and may offer a better guide to estimate potential for permeability. As yet, no systematic studies have been reported which investigate this in detail. Cross-sectional area (Ad, obtained from surface activity measurements) has been reported as being a useful size descriptor to discriminate compounds which can access the brain (Ad < 80 A2) from those that are too large to cross the BBB [62]. Similar studies have been performed to define a cut-off for oral absorption [63]. 

The optimized geometries of (4.55a)-(4.55d) are shown in Fig. 4.17 and selected geometrical parameters are summarized in Table 4.13. The molecular shapes and NBO descriptors (not presented) generally agree with the idealized Lewis-like sd/x picture for Os(CH2)2, HW(CH2)(CH), and W(CH2)3. However, theC—W—C angle (42°) in the ground state of W(CH)2 is much smaller than expected for idealized sd1 geometry, and the optimal NBO description corresponds to a metallacyclopropene,... 

This similarity indicator, in fact, precedes Parr and Bartolotti s introduction of the shape function terminology [59]. In general, it seems that the shape function is preferred to the electron density as a descriptor of molecular similarity whenever one is interested in chemical similarity. Similarity measures that use the electron density will typically predict that fluorine resembles chlorine less than it resembles sodium, oxygen, or neon using the shape function helps one to avoid conflating similarity of electron number with chemical similarity [53,57]. 

Combination of several descriptors believed to be important for oral absorption have been used in various multivariate analysis studies [26]. The general trend is that a combination of size/shape and a hydrogen bond descriptor, sometimes in combination with log D, has good predictive value. At present such models do not account for the biological function of the membrane, such as P-gp-mediated efflux. 

Traditional 2D-QSAR descriptors are generally considered to be the characteristics of a molecule, as a chemist would perceive the molecules. The molecules are described by their physical properties, subdivided surface area (86), atom counts and bonds, Kier and Hall connectivity and kappa shape indices... 

A general theory of quantitatively comparing molecular shapes using common overlap steric volume(33-36) and, more recently, descriptors derived from superimposed molecular potential energy fields of pairs of molecules(37) has been derived and tested. This theory allows a "marriage between Hansch analysis and conformational analysis. 

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