Local shape complementarity

Local Shape Complementarity Measures for Functional Groups... 

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,... 

The catalytic activity of zeolites also involves shape complementarity. The cavities of zeolites are interconnected by various channels, consequently, in a strict sense, a cavity does not fully surround a molecule that enters the zeolite. As a result, for high density zeolite MIDCO s only local shape complementarity is possible. Nevertheless, the low density MIDCO s of zeolites contain closed internal contour surfaces corresponding to the cavities and for these parts of MIDCO s global shape complementarity is relevant. The shapes of these MIDCO s may approximately complement the shapes of the MIDCO s of a molecule inside the cavity. 

For the simpler case of molecule pair interactions, such as the interactions between two reactants, local shape complementarity is of importance. The basic principle of local shape similarity measures is also applicable for the construction of local shape complementarity measures. 

If in an approximate model of molecular interactions a contact density value ag can be chosen, then the local shape complementarity between G(ag, M]) and G(ag, M2) is of relevance. In a more general model, one considers the local shape complementarity of MIDCO s G(ag-a, Mi) and G(ag+a, M2) in a narrow density interval... 

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... 

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. 

Consider the conformations Ki and K2 of interacting molecules M, and M2, respectively, and their local shape complementarity, with reference to the MIDCOs G(Ki,aQ) and G(K2,a0) for the given interaction and a contact density value aQ. Naturally, shape complementarity is limited to MIDCOs of a single density threshold value aQ, and, generally, one should consider the local shape complementarities of all MIDCO pairs G(Ki,a0—a ) and G(K2,a0 + a ) in some narrow density interval surrounding the reference density threshold ... 

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. 

Shape complementarity of functional groups involves matches between locally concave and locally convex domains, and also matches between properly aligned saddle-type domains, that is, between curvature domain pairs of the following combinations ... 

It is now possible to analyze macromolecular electron densities at a resolution far exceeding the resolution of current x-ray diffraction and other experimental and macromolecular computational techniques. The MEDLA method presents a new perspective for the analysis of global and local shape, molecular similarity, and complementarity. 

In nearly all chemically important problems, shape complementarity refers to local shape properties. Most of the typical molecular interactions where shape complementarity is relevant involve only some local moieties of the molecules. Global shape complementarity is more difficult to achieve and seldom plays a role. 

Whereas the curvature types for truncation are complementary, the above two (a,b) maps cannot yet be compared directly, since in a direct comparison of these maps, identical, and not complementary, a and b values occur for the two molecules. However, the complementarity of density thresholds and curvatures can be taken into account by a simple transformation by inverting the (a,b) parameter map of molecule M2 centrally with respect to the point (ao,0), and by comparing the centrally inverted (a,b) map of M2 to the original (a,b) map of M ]. This transformation ensures that domain types, density thresholds, and curvature parameters are matched properly, as required by the pairing scheme (6.77) - (6.79). For example, the locally convex domains of MIDCO G(ao-a, M ) relative to the reference curvature b are tested for shape complementarity against the locally concave domains of MIDCO G(ao+a, M2) relative to a reference curvature - b. 

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. 

On the local level, shape complementarity implies matches between locally concave and locally convex domains, as well as matches between properly placed saddle-type domains, where a directional convex-concave match is important. Replacing the simple D (K,a) notation, the more elaborate notation D b),i(K,a) is used sometimes when studying the complementarity of local shape domains, where the notation includes the relative convexity specification fi(b). This quantity takes values... 

Evidently, the locally convex domains of the MIDCO G(Ki,a0—a ) of fragment F, relative to the reference curvature b are tested for shape complementarity against the locally concave domains of the MIDCO G(K2,a0 + a ) of fragment F2 relative to a reference curvature —b. This is precisely what is required for both curvature and density threshold. 

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