Shape description

With this aim, we developed the TIP probe, a pseudo-MIF representing the local curvature of the molecular surface. The values provided by this probe have the ability to identify any sharp change in the local curvature, like those produced by substituents or present at the molecule ends (see Fig. 6.8), which are highlighted as a set of relevant nodes. Even if this description is not formally a MIF, the probe-like characterization allows a seamless integration of the description into the GRIND. A detailed description of the computations involved as well as some examples of the advantages of its incorporation was published recently [35]. 

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Mak L, Grandison S, Morris RJ (2008) An extension of spherical harmonics to region-based rotationally invariant descriptors for molecular shape description and comparison. J Mol Graph Model 26 1035-1045... 

In order to allow for a variation of interfacial area from that of an equivalent sphere, the eccentricity of the ellipsoidal drop must be taken into account. The area ratio of Eq. (44) does not exceed unity by a serious amount until an eccentricity of 1.5 is attained. An experimental plot of eccentricity as ordinate vs. equivalent spherical drop diameter as abscissa may result in a straight line (G7, Kl, K3, S12). A parameter is yet to be developed by which the lines can be predicted without recourse to experiment. Eccentricity is not an accurate shape description of violently oscillating drops and should therefore be used only for drop size below the peak diameter (region B of Fig. 5). 

Figure 1. Illustration of the local shape description of non-interacting and interacting functional groups. See text for definitions of symbols.

The absence of a precise shape description, based on the GRID molecular interaction fields, prompted the development of a new procedure, called PathFinder, that is described here. 

The minimum paths provide a molecular shape description, coded as a two- vay matrix table. The matrix may then be unfolded into a single array, which describes the molecular shape, as illustrated in Fig. 5.3. 

The resulting matrix is then unfolded into a one-dimensional vector, which can be merged with the shape description, and is suitable for multivariate statistics analysis such as principal component analysis (PGA) and partial least squares (PLS). 

In the above, qualitative description of the expected trends of interrelations of changes in molecular shape and electronic state, an important element was missing a quantitative description of molecular shape and a numerical measure of shape changes. A precise, quantitative molecular shape description is also needed in the study of most other problems of chemistry, as well as in various related subjects, such as biochemistry, pharmacology, medicinal chemistry, and drug design. 

An ideal molecular shape description method S is expected to fulfill several criteria. An ideal method S... 

In the following chapters we shall discuss the fundamental physical basis of the molecular shape concept, and describe several of the computational methods of molecular shape description, fulfilling some or all of the above criteria. 

It is natural to imagine molecular shape properties as they would appear to an observer moving about a sphere enclosing the molecule. If the observer is able to characterize all possible views, this characterization can provide a detailed shape description. 

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. 

Fig. 5 When calibrating a sieve mesh with trapped irregularly shaped grains, a subset of powder grains is obtained that can be used to generate a shape description of the powder grains. (From Ref. l) (A) Profiles of typical sand grain trapped in a sieve mesh (B) length and width distributions of sand grains trapped in a sieve mesh and (C) elongation ratio (shape) distribution for sand grains trapped in a sieve mesh.

However, much more detailed shape description is obtained if the tangent planes are systematically replaced by some other objects. Typically, a MIDCO is compared to a series of tangent spheres of various radii r, but one may find advantageous in direction-dependent problems to use a series of oriented tangent ellipsoids T, especially if a characterization itself involves some reference directions. In the case of oriented tangent ellipsoids, we assume that they can be translated but not rotated as they are brought into tangential contact with the MIDCO surface G K,a). 

Bemis G, Kuntz ID. Fast and efficient method for 2D and 3D molecular shape description. J Comput-Aided Mol Des 1992 6 607-628. 

It is convenient to consider the shape description of all three functions within a common framework. For any molecular property P that is described by a 3D function P(r) which is continuous in r, such as the electronic density p(r), and the composite nuclearpotential Vn(r). or the Somoyai function 5(r,v) with a constant v parameter, the level sets F a) for any constant value a of function P(r) are defined as the following collection of points ... 

By definition, the shape groups of the property P(r) are the algebraic homology groups of the family of topological equivalence classes of the truncated surfaces. The family of all of these equivalence classes involves all property thresholds a as well as all reference curvatures b. The shape groups provide a detailed shape description of the entire 3D property P(r). 

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