Shape comparisons

Trend data can be used in the following ways (1) to compare with specific reference values, (2) mode-shape comparisons, and (3) cross-machine comparisons. 

Dynamic motion of the alkyl stationary phases can also be obtained from NMR studies through an analysis of line shapes, comparisons between single-pulse (SP) and CP-MAS spectra, and various relaxation time constants. Zeigler and Maciel... 

Grant, J.A., Gallardo, M.A., and PiGKUP, B.T. A fast method of molecular shape comparison a simple application of a Gaussian description of molecular shape./. Comb. Chem. 1996, 37, 1653-1666. 

An application of the ROCS program has been reported recently (82). New scaffolds for small molecule inhibitors of the ZipA-FtsZ protein-protein interaction have been found. The shape comparisons are made relative to the bioactive conformation of a HTS hit, determined by X-ray crystallography. A followup X-ray crystallographic analysis also showed that ROCS accurately predicted the binding mode of the inhibitor. This result offers the first experimental evidence that validates the use of ROCS for scaffold hopping purposes. 

B. (1993) Molecular shape comparison of angiotensin II receptor antagonists. / Med Chem 36, 1230-1238. 

Figure 19.7 Monovalent and trivalent ASAXS signal comparison. This figure illustrates differences in shape between the ASAXS signals for monovalent Rb around DNA and trivalent cobalt hexammine around DNA. The curves have been scaled to match at the lowest q to enable shape comparison. The increase in high-angle scatter associated with the trivalent relative to monovalent ion signal, suggests that the trivalent ions are more tightly localized to the DNA.

The shape analysis and shape comparisons of electron densities of molecular regions provide information relevant to their interactions. In the next section a brief review of a shape analysis method is given. 

Shape Comparison of Isoelectronic Cation and Molecule Pairs... 

In the work of Zachmann et al. new approaches to the quantification of surface flexibility have been suggested. The basis data for these approaches are supplied by molecular dynamics (MD) simulations. The methods have been applied to two proteins (PTI and ubiquitin). The calculation and visualization of the local flexibility of molecular surfaces is based on the notion of the solvent accessible surface (SAS), which was introduced by Connolly. For every point on this surface a probability distribution p(r) is calculated in the direction of the surface normal, i.e., the rigid surface is replaced by a soft surface. These probability distributions are well suited for the interactive treatment of molecular entities because the former can be visualized as color coded on the molecular surface although they cannot be directly used for quantitative shape comparisons. In Section IV we show that the p values can form the basis for a fuzzy definition of vaguely defined surfaces and their quantitative comparison. 

Molecular surfaces representing different physical properties are often markedly different. These differences, as interrelations among various molecular surfaces of the same molecule, can be easily represented by the pattern of interpenetration of two or several such surfaces. The same general technique of interpenetrating surfaces can be applied for two molecular surfaces of the same physical property of two different molecules. In this latter case, the interpenetrating surfaces provide a tool for direct shape comparisons. 

The resulting curvature domains Do(bK)> D (bK). and D2(bK) are not invariant with respect to the size of the G(a) objects (this size is dependent on the contour parameter a), nevertheless, the scaling is specific for the size of the nuclear arrangement K, hence these shape domains provide a valid shape comparison of MIDCO s or other molecular surfaces of molecules of different sizes. This approach is simpler than the fully size-invariant approach using the reference curvature be, where a new scaling factor r(G(a)) is required for each new MIDCO G(a). 

In a further generalization of the concept of convexity, the ellipsoid T may be replaced by any other differentiable surface, for example, by a contour surface of another molecule [199], The resulting shape domains can be used for a direct shape comparison and a direct similarity test for these molecules. 

For the study of most intermolecular interactions, valence shell properties, and for practical applications in drug design, the lower density MIDCO s are more important than those at high threshold values. Consequently, the [0.001,0.1] density interval for the a values usually provides sufficient information for shape comparisons. Furthermore, a finite grid on the (a,b) map appears satisfactory for shape characterization. In some recent applications [263], a grid of 41 x 21 = 861 points have been used, taking 41 values from the above density interval and 21 values from the [-l,+ l] interval for reference curvature value b. 

One technique which is applicable for surfaces that are not everywhere differentiable is also suitable for the shape characterization of dot representations of molecular surfaces such as the Connolly surfaces [87], which are not only nondifferentiable, but are not even continuous. The method of 1-hulls [351] is based on a generalization of the concept of convex hull. The convex hull of a set A is the smallest convex set that contains A. Consider a three-dimensional body T. The T-hull of a point set A is the intersection of all rotated and translated versions of T which contain A. The T-hull method is suitable for shape comparisons with a common reference shape, chosen as that of the body T. Alternatively, when the shapes of two molecules, T and A are compared, one molecular body can be chosen as T and the T-hull of the other molecular body A provides a direct shape comparison [351]. 

The first example of similarity measure we shall consider is a resolution based similarity measure (RBSM). This particular realization of a RBSM is conceptually simple, but it is not recommended for highly detailed shape comparisons since its practical applications are computationally feasible only for relatively low levels of resolution [240,243],... 

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 above topological shape analysis techniques can replace visual shape comparisons of molecular models on the computer screen with precise, reliable, and reproducible numerical comparisons of topological shape codes. These comparisons and the similarity or complementarity rankings of molecular sequences can be performed by the computer automatically. This eliminates the subjective element of visual shape comparisons, a particularly important concern if large sequences (e.g. several thousands) of molecules are to be compared. In the data banks of most drug companies there is information stored on literally hundreds of thousands of molecules, and their detailed shape analysis by visual comparison on a computer screen is clearly not feasible. By contrast, automatic, numerical, topological shape analysis by computer is a viable alternative. 

In multiple shape comparisons, efficient, algorithmic shape analysis methods are of particular importance. The shape group and shape code methods provide a framework for such analysis however, the input information they require, such as the 3D electron densities or electrostatic potentials often involve time consuming calculations. This is the case for large molecules or molecular systems, important in drug design and molecular engineering applications. Efficient calculation and representation of these molecular functions is of special importance in such cases. 

Grant, J.A., Gallardo, M.A. and Pickup, B.T. (1996). A Fast Method of Molecular Shape Comparison A Simple Application of a Gaussian Description of Molecular Shape. J.Com-putChem., 17,1653-1666. 

Holzgrabe, U. and Hopfinger, A.J. (1996). Conformational Analysis, Molecular Shape Comparison, and Pharmacophore Identification of Different Allosteric Modulators of Muscarinic Receptors. J.Chem.lnfComput.ScL, 36,1018-1024. 

The series-of-stirred-tanks model could not represent the RTD for the laminar-flow reactor shown in Fig. 6-7. However, the RTD data given in Example 6-2 can be simulated approximately. The dashed curve in Fig. 6-1 Oh is a plot of this RTD. While no integer value of n coincides with this curve for all 6/6, the curve for = 5 gives approximately the correct shape. Comparison of the fit in Figs. 6-9 and 6-lOh indicates that about the same... 

Holzgrabe, U., Hopfinger, A.J., 1996. Conformational analysis, molecular shape comparison and pharmacophore identification of different allosteric modulators of muscarinic receptors. J. Chem. Inf. Comp. Sci. 36, 1018-1024. 

Often, the shape comparisons of local regions of molecules are of interest, which, in many instances, may appear more important than the evaluation of the global similarities of molecules. 

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