Local shape codes

Local Shape Codes and Local Similarity Measures... 

In many chemical problems the comparisons of local molecular regions are more important than global comparisons. The presence of functional groups or other molecular moieties with specified shape properties often imply similar chemical behavior even if the molecules compared have very different global shapes. For this reason, local molecular shape descriptors and local shape codes are of major importance. 

The very same procedure that has been used to construct the global C(Mi) codes for the global shape matrices s(a,b,M]) along the parameter map (a,b) can also be applied to the set of local shape matrices Ib(a,b,Mi) along the parameter map (a,b), resulting in a local shape code vector... 

The global and local shape codes can be used for measuring global and local shape compexity, respectively. Let w(s(a,b,M)) and w(lli)(a,b,M)) denote the number of different entries of the n-dimensional global shape matrix s(a,b,M) and a k-dimensional local shape matrix lb(a,b,M), respectively. Simple global and local shape complexity measures of molecule M are defined as the following ratios ... 

For most practical applications, CIMM is used within the framework of local measures. These measures are based on local shape matrices or on the shape groups of local moieties, defined either by the density domain approach mentioned earlier, or by alternative conditions, such as the simple truncation condition replacing the "remainder" of the molecule by a generic domain [192], For proper complementarity, identity or close similarity of the patterns of the matched domains is an advantage, hence the parts Cl HM)) and Cl KM2) of the corresponding local shape codes are compared directly. For shape complementarity only the specified density range [bq - Aa, Bq + Aa] and a specified curvature range of the (a,b) parameter maps is considered. A local shape complementarity measure, denoted by... 

Shape codes [43,109,196,351,408]. The simplest topological shape codes derived from the shape group approach are the (a,b) parameter maps, where a is the isodensity contour value and b is a reference curvature against which the molecular contour surface is compared. Alternative shape codes and local shape codes are derived from shape matrices and the Density Domain Approach to functional groups [262], as well as from Shape Globe Invariance Maps (SGIM). 

The nonvisual shape similarity measures of molecules as well as molecular fragments, using the numerical shape code method, provide the basis for a shape complementarity measure. A simple transformation of the local shape codes generates a representation that is suitable for a direct evaluation of local shape complementarity. 

The nondifferentiability of these surfaces at the seams of interpenetrating spheres as well as the local nondifferentiability of solvent accessible surfaces or union surfaces, are a technical disadvantage. Local nondifferentiability limits the application of the shape group methods in their original form that requires second derivatives for curvature analysis. For example, at every point r of a VDWS where two or more atomic spheres interpenetrate one another, the surface is not smooth and is not differentiable. For such nondifferentiable molecular surfaces, alternative shape descriptors and shape codes have been introduced. 

Note that if the element s(a,b)i,i of the shape matrix is different from zero, then the information on the dimension n of the matrix can be deduced from the second element C2(s(a,b)) of the shape code vector c(s(a,b)). The special case of s(a,b)i 1 = 0 seldom occurs, since this implies that the largest domain is a locally concave Dq domain relative to the curvature parameter b. Nevertheless, in order to avoid ambiguity in such cases, the dimension n is specified as the first component ci(s(a,b)) of the shape code vector c(s(a,b)). 

The general methods applied for shape codes based on shape matrices are esf)ecially suitable for developing local shap>e codes. 

In such a case, the electron density fragment additivity principle provides a simple approach for analyzing and evaluating local shape similarity of molecules. The fragment electron densities F are well defined within any LCAO-based quantum chemical electron density, and it is simple to consider the family of MIDCOs, their shape groups, as well as their shape codes in a manner entirely analogous to the... 

Various shape complementarity measures can also be determined based on shape codes. This is an important problem since molecular recognition usually depends on the complementarity of local regions of molecules, where complementarity may refer to electron distributions, polarizability properties, electrostatic potentials, or simply geometric considerations. The powerful topological techniques are suitable for the quantification of the degree of molecular complementarity and can be used as tools for the study of molecular recognition. 

But there is one case in which this does not work well When you want to create an object, you must say which class you want it to belong to. However, there are a number of patterns, such as Factory, that help localize the dependencies, so that adding a new Shape to the drawing editor (for example) causes only one or two alterations to be necessary to the existing code. 

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. 

Studies on the effect of hydrodynamics on localized corrosion and electrochemical etching processes have been reviewed by West et al. Much of the work has been performed by Alkire and co-workers." They have used FIDAP, a commercial FEM code, to investigate the influence of fluid flow on geometries relevant to etching and to pitting corrosion. In most cases, Stokes flow was considered. The Stokes flow approximation is frequently valid inside the cavity because its characteristic dimension is small. However, the flow outside the cavity may not be in the Stokes flow regime. Since it is the external fluid motion that induces flow inside the cavity, under many (especially unsteady) situations, the use of the Stokes flow approximation may be problematic. Some of the work of Alkire and co-workers has been extended hy Shin and Economou, " who simulated the shape evolution of corrosion pits. Natural convection was also considered in their study. 

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