Reference shape

The comparison to a reference shape, which is generally the convex bounding polygon provides another set of descriptors, at a mesoscopic level of details. The index of concavity in length or in surface measures globally how the object departs from a convex shape ... 

We define the continuous symmetry measure (CSM) as a quantifier of the minimum effort required to turn a given shape into a symmetric shape. This effort is measured by the sum of the square distances each point is moved from its location in the original shape to its location in the symmetric shape. Note that no a priori symmetric reference shape is assumed. 

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]. 

Tlie main assumption of this approach is that the shape of the molecule is closely related to the shape of the - binding site cavity and, as a consequence, to the biological activity. Therefore, a shape reference compound is chosen which represents the binding site cavity, and the similarity (or commonality) measured between the reference shape and the shape of other compounds is used to determine the biological activity of these compounds. As well as the shape similarity measures, other molecular descriptors such as those in - Hansch analysis can be used to evaluate the biological response. The MSA model is thus defined as ... 

The most active compound is usually assumed as the reference structure, but a set of overlapped structures can also be assumed to define a reference shape. 

According to their function in the description of engineering objects, relationships can be represented between model entities as simple relationships, parameters, references, shape modifications, common shape modifications of several parts, variants, combinations, and structures (Figure 2-19). A simple relationship relates parameters of entities. In the example of Figure 2-19a, dimensions... 

This parameter fully characterizes the shape of a vesicle with reduced volume v. Note that eq. (5) captures the physics of the two sources of preferred membrane curvature. Whereas Cq is only a property of the vesicle membrane, the result of changing Auq depends also on the reference shape via its integrated mean curvature, which is proportional to Aa. A large preferred differential area Auq greater than Aa tends to curve the membrane further outward, whereas a smaller value would lead to more invaginated shapes. Let us now discuss a few mechanisms for membrane curvature. 

These images show an idealised version of the pattern of flow. The real boundaries of flow channels often vary a little from time to time because they depend quite sensitively on small changes in the packing of particles (Arnold 1991). Further, the idealised pattern is shown with the silo completely full, but the pattern cannot develop until some solid has come out at the bottom (unless it is being continuously replenished). However, because the critical design condition is almost always when the silo is fuU, this is the idealised reference shape. 

To choose the reference shape for MSA, each available conformation is used in turn as a reference to calculate the pairwise molecular similarity to all other conformations of all other molecules. The conformation of each molecule that has the highest overlap volume with the current reference is used as the similarity measure for that reference. Thus, given M conformations in the database, there will be M MSA parameters that describe the shapes of the compounds. In a 1994 study, the overlapped structures of four molecules were merged to define a reference shape. 

MSA parameters appear in a regression equation as linear or quadratic terms. Positive correlations with V(, emphasize that the greater the similarity between a given molecule and the reference (the most potent compound), the higher the expected potency of the former. Alternatively, negative correlations with NV() emphasize that potency tends to decrease as the dissimilarity between the molecule and the reference shape increases. 

Equations that have only linear terms in MSA descriptors suggest that there are no compounds more potent than the reference, because by definition there can be no compound that has a V value, or forecast potency, higher than that of the reference." To minimize the nonoverlap steric volume NV, the best we can do is to use the reference shape for which NVq will be zero and forecast potency equal to that of the reference. These problems result from lumping multiple regions of 3D space into one shape descriptor. Even when the most potent compound in the training set is close to the ideal, other 3D-QSAR methodologies might reveal bad steric contacts that could be removed to produce an increase in potency. 

If the model suggests a parabolic relationship between bioactivity and Vo, and if the coefficient of the linear term is positive and the coefficient of the squared term is negative, the reference shape is too large for an optimal fit into the receptor cavity. An optimum value of V,) would correspond to a structure sized between the common parent structure and the reference compound. If... 

The disagreements between pore sizes are particularly expected when the true pore shape of the material studied is very different from a cylinder with circular cross section, which has been chosen as a reference shape for the development of the theories of the various analysis methods. This is certainly the case for sol-gel materials and more generally for materials with a structure of interconnected filaments and whose pores are under no circumstance cylinders with circular sections. 

With respect to a reference shape T, the T-hull of a molecular body M is computed as the intersection of all rotated and translated versions of T, which contains M. T-hulls provide shape similarity measures for molecules and for their solvent accessible surfaces. T-hulls are also the basis for a shape quantization . ... 

Thus, if a shape has the desired symmetry, 5(G) = 0. A shape s symmetry measure increases as the shape departs from G symmetry and it reaches a maximal value (not necessarily 100 - see Appendix D in Ref. 2b). Equations (1) and (2) are general and allow one to evaluate the symmetry measure of any shape relative to any symmetry group or element. No reference shape is assumed at the beginning of the analysis, though it is obtained as an end outcome. 

With the reference block method the distance law of a model reflector is established experimentally prior to each ultrasonic test. The reference reflectors, mostly bore holes, are drilled into the reference block at different distances, e.g. ASME block. Prior to the test, the reference reflectors are scanned, and their maximised echo amplitudes are marked on the screen of the flaw detector. Finally all amplitude points are connected by a curve. This Distance Amplitude Curve (DAC) serves as the registration level and exactly shows the amplitude-over-distance behaviour" of the reference reflector for the probe in use. Also the individual characteristics of the material are automatically considered. However, this curve may only be applied for defect evaluation, in case the reference block and the test object are made of the same material and have undergone the same heat treatment. As with the DGS-Method, the value of any defect evaluation does not consider the shape and orientation of the defect. The reference block method is safe and easy to apply, and the operator need not to have a deep understanding about the theory of distance laws. 

Equation V-64 is that of a parabola, and electrocapillary curves are indeed approximately parabolic in shape. Because E ax tmd 7 max very nearly the same for certain electrolytes, such as sodium sulfate and sodium carbonate, it is generally assumed that specific adsorption effects are absent, and Emax is taken as a constant (-0.480 V) characteristic of the mercury-water interface. For most other electrolytes there is a shift in the maximum voltage, and is then taken to be Emax 0.480. Some values for the quantities are given in Table V-5 [113]. Much information of this type is due to Gouy [125], although additional results are to be found in most of the other references cited in this section. 

The characteristic isotherm concept was elaborated by de Boer and coworkers [90]. By accepting a reference from a BET fit to a standard system and assuming a density for the adsorbed film, one may convert n/rim to film thickness t. The characteristic isotherm for a given adsorbate may then be plotted as t versus P/P. For any new system, one reads t from the standard r-curve and n from the new isotherm, for various P/P values. De Boer and co-work-ers t values are given in Table XVII-4. A plot of t versus n should be linear if the experimental isotherm has the same shape as the reference characteristic isotherm, and the slope gives E ... 

Another important application of perturbation theory is to molecules with anisotropic interactions. Examples are dipolar hard spheres, in which the anisotropy is due to the polarity of tlie molecule, and liquid crystals in which the anisotropy is due also to the shape of the molecules. The use of an anisotropic reference system is more natural in accounting for molecular shape, but presents difficulties. Hence, we will consider only... 

The label liquid crystal seems to be a contradiction in tenns since a crystal cannot be liquid. However, tire tenn refers to a phase fonned between a crystal and a liquid, witli a degree of order intennediate between tire molecular disorder of a liquid and tire regular stmcture of a crystal. Wlrat we mean by order here needs to be defined carefully. The most important property of liquid crystal phases is tliat tire molecules have long-range orientational order. For tliis to be possible tire molecules must be anisotropic, whetlier tliis results from a rodlike or disclike shape. 

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