Shape local

Mezey, P.G. (1996) Local shape analysis of macromolecular electron densities. In Computational Chemistry Reviews and Current Trends, Leszczynski, J. (Ed.), World Scientific Publ., Singapore. 

However, the local shape of a PES is given by ZN — 6 locally independent coordinates. From the argument above, we must make different choices for local coordinates in different locales. How this can be accomplished is detailed elsewhere.52 Very briefly, the local changes in the Zn relative to changes in the Cartesian coordinates Xi, are given by the matrix B (a variant of the Wilson B matrix53) ... 

One can analyze the Jeffery-Hamel flow using a nondimensional velocity scaled by the maximum velocity at a radial location [429]. This approach permits determination of the local shape of the velocity profiles but still requires an integral mass-flow constraint to determine the local maximum velocity and hence the specific velocity profiles (i.e., in m/s). Nevertheless, using this approach and the limit of small angle, but large Reynolds number, permits the determination of the separation point as a function of the combined parameter Rea2 alone. 

Refinements, parameterization, and the complete COSMO-RS 113 7.1.4 Local shape index... 

Local Shape Changes Induced by Molecular Environment... 

Shape Similarity Measures of Functional Groups Based on Local Shapes... 

Local Shape Complementarity Measures for Functional Groups... 

An important concern is the efficient detection of local shape changes introduced by chemical changes in remote locations of a molecule. One simple approach [20] applied a truncation method, compatible with the truncation process already used within the shape group methods for molecular shape analysis [41-44]. 

One of the alternative computational methods for diagnosing local shape variations of the electron density was suggested by Walker (P.D. Walker (1992), see also [66]), by introducing a pseudo-density matrix Pk of a formal molecular fragment for a subset k of the nuclei of a molecule. The pseudo-density matrix Pk is defined by... 

Walker s pseudo-densities and their complements are designed for enhanced detection of local shape variations and are valuable for local shape analysis. Some more advanced variants of non-additive pseudo-densities are described in ref. [66]. 

The additive fuzzy electron density fragmentation scheme of Mezey is the basis of the Molecular Electron Density Lego Assembler (MEDLA) method [67,70-72], reviewed in section 4. of this report, where additional details and applications in local shape analysis are discussed. The MEDLA method was used for the generation of the first ab initio quality electron densities for macromolecules such as proteins [71,72] and other natural products such as taxol [66],... 

The shapes of these density domains are characteristic to the set of nuclei enclosed by them, to the nuclear geometry, and also to the location of these density domains within the molecule, collectively represented by the configuration variable K, as well as to the actual density threshold a. The sequence of density domains as a function of density threshold a, augmented with the results of a local shape analysis of these density domains [2], provides a detailed description of chemical bonding within the methane molecule. 

The reactivities of functional groups are highly dependent on the molecular surroundings, and the effects of the global molecular environment on the local shape variations can be significant. 

In the following sections the actual electron density variations will be used for local shape analysis. 

The fundamental principle we shall follow in the local shape analysis of functional groups and local molecular moieties is a strict analogy with the shape analysis of complete molecules. Accordingly, instead of molecular isodensity contour (MIDCO) surfaces, the main tool of analysis will be the fragment isodensity contour (FIDCO) surfaces. Some of the ideas and concepts described in this section are illustrated in Figure 1. 

Two choices for the representation of a local molecular moiety will be discussed. For the first choice, describing the local shapes of non-interacting functional groups within a molecule, we define a FIDCO for a fragment A in a molecule AB as follows ... 

The local shape analysis can no longer be carried out on an "isolated" FIDCO GA(a) if the interactions of various molecular fragments in a molecule AB are fully taken into account, beyond simply using these interactions for a truncation of the isolated FIDCO GA(a). If a detailed description of the interactions is required, then a new contour calculation is needed for the interactive FIDCO GA(B)(a) in molecule AB, where GA(a) is defined as... 

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

In most interactions between two reactants, local shape complementarity of functional groups is of importance. A local shape complementarity of molecular electron densities represented by FIDCOs implies complementary curvatures for complementary values of the charge density threshold parameters a. For various curvature domains of a FIDCO, we shall use the notations originally proposed for complete molecues [2], For example, the symbol D2(b),i(a, Fj) stands for the i-th locally convex domain of a FIDCO G(a) of functional group Fj, where local convexity, denoted by subscript 2(b), is interpreted relative to a reference curvature b. For locally saddle type and locally concave domains relative to curvature b, the analogous subscripts 1(b) and 0(b) are used, respectively. 

In general, a locally convex domain D2(b),j(a> Fj) of a functional group F, relative to a reference curvature b, shows local shape complementarity with a locally concave domain Do( b),j(a, F2) of a complementing functional group F2, relative to a reference curvature of -b. The threshold values a and a are also likely to complement each other the shape complementarity between the higher electron density contours of one functional group and the lower electron density contours of the other functional group is relevant. 

If a contact density threshold ao can be chosen for a given interaction between two functional groups, then the local shape complementarity between G(ao, Fj) and G(a<), F2) is clearly of importance. However, complementarity should also manifest itself within a whole range of density thresholds. One may consider the local shape complementarity of FIDCOs G(ao-a, F[) and G(ao+a, F2) in a density interval containing the contact density threshold ao,... 

Note that the complementarity of the local shapes of those FIDCOs are important where the thresholds deviate from the contact density value ao in the opposite sense. 

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