Reference curvature

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. 

Complementarity of the curvature types for truncation is not sufficient for a direct comparison of the two (a.b)-maps, since one must also take into account the required complementarity of density thresholds a and reference curvatures b. This... 

The local canonical curvatures can be compared to a reference curvature parameter b [156,199]. For each point r of the molecular surface G(a) a number X = x(r,b) is defined as the number of local canonical curvatures [the number of eigenvalues of the local Hessian matrix H(r) that are less than this reference value b. The special case of b=0 allows one to relate this cla.ssification of points to the concept of ordinary convexity. If b=0, then p is the number of negative eigenvalues, also called the index of critical point r. As mentioned previously, in this special case the values 0, 1, or 2 for p(r,0) indicate that at the point r the molecular surface G(a) is locally concave, saddle-type, or convex, respectively [199]. 

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. 

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. 

The numerical value of the reference curvature b can be specified in absolute units or in units scaled relative to the size of the object G(a). If absolute units are used, then a relative convexity characterization of G(a) involves size information if an object G(a) is scaled twofold, then its shape remains the same, but with respect to a fixed, nonzero b value a different relative convexity characterization is obtained. That is, the pattern of relative shape domains Do(b)> D (b), and D2(b) defined with respect to some fixed, nonzero reference curvature value b (b K)) is size-dependent. On the other hand, if the reference curvature b is specified with respect to units proportional to the size of G(a), then a simple. scaling of the object does not alter the pattern of relative shape domains with respect to the scaled reference curvature b. In this case, the shape characterization is size-invariant, that is, a "pure" shape characterization is obtained. 

The relative curvature domains Do(bG)- Dl(bG)> D2(bG). specified in terms of the scaled reference curvature be provide a size-independent shape characterization of the object G(a) for all curvatures. 

In an alternative approach, the reference curvature b is scaled by the diameter d(K) of the 3D nuclear configuration K. If r(K) = 0.5 d(K) is the radius of the smallest sphere that encloses all the nuclei of the given nuclear configuration K, then the scaled relative curvature parameter bK is defined as... 

The resulting curvature domains Do(bK)> D (bK). and D2(bK) are not invariant with respect to the size of the G(a) objects (this size is dependent on the contour parameter a), nevertheless, the scaling is specific for the size of the nuclear arrangement K, hence these shape domains provide a valid shape comparison of MIDCO s or other molecular surfaces of molecules of different sizes. This approach is simpler than the fully size-invariant approach using the reference curvature be, where a new scaling factor r(G(a)) is required for each new MIDCO G(a). 

For the special case of reference curvature b = 0 (i.e., for the tangent plane T of ordinary convexity), the pattern of the original curvature domains Do(0), D](0), and D2(0) is already size-invariant. 

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. 

Figure 5.6 Three of the topologically different shape domain patterns of the G(O.Ol) MIDCO of the equilibrium nuclear configuration of the allyl alcohol molecule are shown, corresponding to reference curvature parameter values b=0, b=0.005, and b=-0.008, respectively.

For the study of most intermolecular interactions, valence shell properties, and for practical applications in drug design, the lower density MIDCO s are more important than those at high threshold values. Consequently, the [0.001,0.1] density interval for the a values usually provides sufficient information for shape comparisons. Furthermore, a finite grid on the (a,b) map appears satisfactory for shape characterization. In some recent applications [263], a grid of 41 x 21 = 861 points have been used, taking 41 values from the above density interval and 21 values from the [-l,+ l] interval for reference curvature value b. 

If a different reference curvature value b is chosen, then the shape matrix may be different, although the matrix is invariant within small enough intervals of the b values. As examples, the three different shape matrices s(0.0l,0), s(0.01,0.005), and s(0.01,-0.008) of the three shape domain partitionings of the ally I alcohol MIDCO G(O.OI) shown in Figure 5.6 are given below. The index... 

Both the shape matrix s(a,b) and the shape graph g(a,b) give a detailed shape characterization of the MIDCO surface G(a), with respect to the selected reference curvature b. 

Figure 5.9 The construction of Shape Globe Invariance Maps (SGIM s), of MIDCO relative convexity shape domain patterns for two reference curvature values, b = 0 and b < 0.

A practical implementation of the above approach is the following a global shape property of the molecule is assigned to each point of the sphere S, followed by the determination of those domains of S where this shape property is invariant. A pair of examples is shown in Figure 5.9, where the shape globe invariance domains of a MIDCO surface for two relative convexity shape domain partitionings (P) with respect to two reference curvatures, b = 0, and b < 0, are given. As... 

As it has been pointed out in Section 5.2, it is natural to formulate dynamic shape analysis aproaches in terms of the dynamic shape space D described earlier [158]. The reader may recall that the dynamic shape space D is a composition of the nuclear configuration space M, and the space of the parameters involved in the shape representation, for example, the two-dimensional parameter space defined by the possible values of the density threshold a, and the reference curvature parameter b of a given MIDCO surface. 

For example, consider a given electron density threshold and reference curvature parameter pair (a,b), and assume that the associated truncated contour surface falls into seven pieces. Furthermore, assume that a shape group analysis gives the sequence... 

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. 

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 local curvature properties of the surface G(m) in each point r of the surface are given by the eigenv ues of the local Hessian matrix. Moreover, for a defined reference curvature b, the number p,(r, b) is defined as the number of local canonical curvatures (Hessian matrix eigenvalues) that are less than b. Usually b is chosen equal to zero and therefore the number p(r, 0) can take values 0,1, or 2 indicating that at the point r the molecular surface is locally concave, saddle-type, or convex, respectively. The three disjoint subsets Ao, Ai, and A2 are the collections of the surface points at which the molecular surface is locally concave, saddle-type, or convex, respectively the maximum connected components of these subsets Ao, Aj, and A2 are the surface domains denoted by Do,, Diand D2, where the index k refers to an ordering of these domains, usually according to decreasing surface size. 

Considering a finite number of threshold values m, a set of contour surfaces G(m) is studied for each molecule combined with a set of reference curvature values b. Therefore, for each pair (m, b) of parameters, the curvature domains Do(m, b), b) and T>2 m, b) are computed and the truncation of contour surfaces G m) is performed by removing all curvature domains of specified type p (in most applications p = 2) from the contour surface, thus obtaining a truncated surface G(m, p) for each m, b) pair. For most small changes of the parameter values, the truncated surfaces remain topologically equivalent, and only a finite number of equivalence classes is obtained for the entire range of m and b values. 

The curvature-based shape analysis of each MIDCO surface G(K,a) can be repeated for a whole range of reference curvature values b, providing a detailed shape characterization of G(K,a). It is important to point out that for the complete range of chemically relevant reference curvature values b, there exist only a finite number of topologically different truncated MIDCOs G(K,a,p) obtained from G(K,a). When these truncated surfaces are characterized by their topological invariants, then a numerical shape characterization is obtained. 

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