Local convexity

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. 

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 ... 

The molecular density range can be subdivided into subranges. At the highest threshold values within the molecular range, the DD has some local "neck" region, that is, there is at least one topological belt or some other multiply connected set on the surface of the density domain along which the surface is not locally convex. (Note that within any multiply connected set there are loops which cannot... 

In Figure 3.2 a formal molecular surface G and its subdivision into various cells are shown. The surface G is subdivided into five locally convex domains, denoted by A, B, C, D, and E, and the remaining sixth domain F, where the local... 

Convexity and curvature properties. In the above discussion and examples we have already used the concepts of convexity and locally convex domains in an intuitive manner. Whereas our goal is to provide a topological shape characterization for molecules, we shall often use geometrical tools at intermediate steps toward a topological description. These steps often involve the concepts of convexity, curvature, and a characterization of critical points of functions. 

An absolute shape characterization is obtained if a molecular contour surface is compared to some standard surface, such as a plane, or a sphere, or an ellipsoid, or any other clo.sed surface selected as standard. For example, if the contour surface is compared to a plane, then the plane can be moved along the contour as a tangent plane, and the local curvature properties of the molecular surface can be compared to the plane. This leads to a subdivision of the molecular contour surface into locally convex, locally concave, and locally saddle-type shape domains. These shape domains are absolute in the above sense, since they are compared to a selected standard, to the plane. A similar technique can be applied when using a different standard. By a topological analysis and characterization of these absolute shape domains, an absolute shape characterization of the molecular surface is obtained. 

Figure 5.1 The shape domains of local convexity of a MIDCO surface G(a) are shown. A geometrical interpretation of the classification of points r of G(a) into locally concave Dp. locally saddle-type D, and locally convex D2 domains is given when comparing local neighborhoods of the surface to a tangent plane T. Each point r of G(a) is classified into domains Dp. D, and D2 depending on whether at point r a local neighborhood of point r on the tangent plane (r not included) falls within the interior of the surface G(a), or it cuts into the surface G(a) within any small neighborhood of point r, or it falls on the outside of G(a).

As an example of absolute shape criteria, the local curvature properties of a MIDCO can be used for defining absolute shape domains on it [156], and for a subsequent global shape characterization. In Figure 5.1 a MIDCO G(a) is shown as an illustration of some of the concepts discussed. The simplest method [155] is based on comparisons to a reference of a tangent plane what leads to the identification of locally convex, concave, and saddle-type domains, as mentioned previously, although much finer characterizations are also possible [156,199]. 

Figure 5.2 A cellular subdivision based on the local convexity shape domains of the MIDCO surface of Figure 5.1 is shown.

By generalizing the idea of local convexity for any reference curvature value b [199], the number p(r,b) is the tool used for a classification of points r of the contour G(a) into various domains. For any fixed b, each point r of the contour surface G(a) belongs to one of three disjoint subsets of G(a), denoted by Aq, A, or A2, depending on whether at point r none, one, or both, respeetively, of the local canonical curvatures h and h2 are smaller than the reference value b [156]. The union of the three sets Aq, A, and A2 generates the entire contour surface, that is. 

The case of b=() corresponds to the shape domain subdivision of G(a) in terms of ordinary local convexity 155, 99], Geometrically, this case corresponds to comparing the local regions of the molecular contour surface to a test surface of zero curvature, that is, to a tangent plane. Local convexity and the corresponding classification of points r of G(a) into various domains, in the present case... 

The more general case of b >t0 corresponds to a generalization of the concept of convexity [156,199]. This concept of relative local convexity has a u.seful geometrical interpretation. Eor a fixed value of parameter b, relative local convexity classifies the points r of G(a) into domains D x(b), depending whether at point r the surface G(a) is curved more in all directions, more in. some and less in some other directions, or less in all directions, than a test sphere T of radius 1/b. The corresponding three types of domains are denoted by Do(b), D (b), and D2(b), where the molecular contour surface G(a) is locally concave, of the saddle-type, and convex, respectively, relative to curvature b. 

Within the general scheme of relative convexity, the conventional, ordinary local convexity is obtained as a special, degenerate case of relative local convexity, with a tangent sphere of infinite radius as reference, that is, with a tangent plane of reference curvature b = 0. 

Figure 5.4 Two sets of shape domains of oriented relative local convexity of the MIEKDO surface G(a) of Figure 5.1, relative to two orientations of a tangent ellipsoid T are shown.

For example, if the shape domains are defined in terms of local convexity, and if we select the locally convex domains, then the shape groups of G(a) are the homology groups of the truncated isodensity contour surface G(a,2), obtained from the molecular contour surface G(a) by eliminating all domains of index p = 2. This family of shape groups, obtained by cutting out all locally convex domains of G(a), has been studied in most detail for several molecules [192,262,263,342]. 

If the shape domains are defined by relative local convexity, then the notation HP j (a,b), p = 0, 1,2, is u.sed for the shape groups of MIDCO surfaces G(a), where besides the dimension p of the homology group, the truncation type p, the charge density contour parameter a, and the reference curvature parameter b are also specified. For the special case of ordinary local convexity, b=0, the second argument in the parentheses can be omitted and one may simply write HP (a). Usually, we are interested in the Betti numbers of the groups HP (a,b) and HP (a) for these numbers the bp x(a,b) and bpp(a) notations are used, respectively. 

A shape domain partitioning in terms of relative local convexity of parameter b=0.005 leads to a simpler pattern. We obtain... 

The third shape domain partitioning shown has been calculated for the relative local convexity parameter b= - 0.008. We obtain... 

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. 

Figure M-2. A subdivision of a molecular contour surface G(m), based on local curvature properties. The contour surface is subdivided into locally concave (Do, ), saddle-type (Di, ), and locally convex (D2, ) domains with respect to the local tangent plane in each point r. The second index k refers to an ordering of these domains according to decreasing surface area.

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