Molecular Contour Surfaces

One approach for the generation of an appropriate superposition of contour surfaces is based on the distance geometry approach of Crippen. Most of the software packages used in drug design (see, for example, refs. 83-93) provide a variety of tools for the related computations, and have been applied by medicinal chemists (see, for example, refs. 94-96). 

The optimum selection of criteria for the construction of approximate molecular surfaces is dependent on the chemical problem the molecular surface is supposed to represent. Whereas fused spheres VDW surfaces adequately represent the space requirements of formal, rigid molecular conformations, isodensity contours and MEP contour surfaces are also of importance for the study of molecular interactions. 

As an example, one may consider an isodensity contour Gi = Gi( 2i) and a MEP contour Gz = G2( t2) of the same molecule, where the contour threshold values are a and az, respectively. Whether an interpenetration occurs at all depends on the choice of the threshold values ai and az-If interpenetration occurs, then it defines one or several closed loops on both surfaces. The points along these loops belong to both contour surfaces. Hence on each surface the loops define the boundaries of subsets characterized by the function value of the other physical property the value of the property is either greater than the threshold for all points within a subset, or it is lower than the threshold for all points in the subset. In fact, the interpenetration pattern can be generated simultaneously for a series of threshold values. For example, by considering k different MEP threshold values, az, az, 22, . .. J2 the pattern of interpenetration may be used to generate ranges of the electrostatic potential 

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In this chapter we shall combine some of the ideas described in Chapters 3 and 4 the applications of topological concepts and methods for the study of various representations of molecular shapes. Among the shape representations molecular contour surfaces have a prominent role, but we shall also consider alternatives, primarily for the purposes of characterizing the large scale shape features of biological macromolecules. 

For the characterization of the shapes of molecular contour surfaces, such as MIDCO s and MEPCO s, one may subdivide the surface into domains fulfilling some local shape criteria. One can distinguish two types of criteria, relative, and absolute, leading to a relative shape domain or an absolute shape domain subdivision of the molecular contour surface. 

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. 

Several topological methods of shape analysis of molecular contour surfaces have been designed to take advantage of such relative and absolute shape domain subdivisions of the contours, according to. some physical or geometrical conditions [155-158,199]. 

Figure 5.3 The shape domains of relative local convexity of a MIDCO surface G(a) of Figure 5.1, relative to a tangent sphere T of curvature b (radius 1/b) are shown. A geometrical interpretation of the classitication of points r of G(a) into locally concave Dq, locally saddle-type D, and locally convex D2 domains relative to b is given when comparing local neighborhoods of the surface to the tangent sphere T. The classification depends on whether at point r the surface G(a) is curved more in all directions, or more in some and less in some other directions, or less in all directions, than the test sphere T of radius 1/b. In the corresponding three types of domains Do(b). Dm,), and D2(b), or in short Dp, D, and D2, the molecular contour surface G(a) is locally concave, of the saddle-type, and convex, respectively, relative to curvature b.

In the above discussion we have assumed that the molecular contour surface G(a) is twice differentiable. This condition is required for gradients and local Hessian matrices of the local elevation function at all points along the surface, and for the local canonical curvatures of G(a) at each point r of G(a), needed for their classification into shape domains. 

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... 

Consider a family of shape domains defined on a molecular contour surface G(a), and the truncated contour surface G(a,n) obtained from G(a) by excising a selected subfamily of shape domains. The shape groups of the contour surface G(a), with respect to the given family of shape domains, are the homology groups of the truncated contour surfaces G(a,p). 

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]. 

Fused sphere surfaces, such as fused sphere Van der Waals surfaces (VDWS ) are simple approximations to molecular contour surfaces. By specifying the locations of the centers and the radii of formal atomic spheres in a molecule, the fused sphere surface is fully defined as the envelope surface of the fused spheres and can be easily generated by a computer. Although fused sphere VDW surfaces are not capable of representing the fine details of molecular shape, such surfaces are very useful for an approximate shape representation. 

Consider a given molecular contour surface G(a). If the size s of the cubes is chosen small enough, then any finite polycube P can fit within G(a). As in the two-dimensional case, we do not consider orientation constraints and we assume that the contour surface G(a) and polycube P may be translated and rotated with respect to one another the relative orientation of G(a) and the cubic grid is not fixed. In this model, the identity of a polycube is independent of its orientation. Two polycubes P and F are regarded identical if and only if they can be superimposed on one another by translation and rotation in 3D space. Note, however, that the polycube method of shape analysis and determination of resolution based similarity measures can be augmented with orientation constraints, suitable for the study of molecular recognition and shape problems in external fields or within enzyme cavities [240,243]. 

The three-dimensional RBSM method relies on the shape properties of interior filling polycubes Pj(G(a),n) inscribed in molecular contour surfaces G(a) when assessing the similarity of the G(a) contours and the formal molecular bodies B(a) enclosed by them. In order to define levels of resolution scaled relative to the molecular size, the absolute size parameter s is not used directly. One obtains more comparable shape characterizations of both small and large objects when using the same number of cubes. Consequently, each level of resolution is defined by the number n of cubes of interior filling polycubes Pi(G(a),n), which depends on the relative size of the object G(a) as compared to the cube size s. 

The family of all interior filling polycubes P (G,n) of the molecular contour surface G at level n is denoted by F(G,n),... 

This set F(G,n) provides an absolute shape characterization of G and the body B enclosed by it. By analogy with the two-dimensional case, we may use these F(G,n) sets to introduce a relative measure for shape similarity of two molecular contour surfaces Gi and G2. These surfaces may belong to two different molecules, or... 

The similarity index io(Gi,G2) of two molecular contour surfaces Gj and G2 is the smallest n value at and above which all interior filling polycubes of contour surfaces Gj and G2 are different,... 

If the two molecular contour surfaces G1 and G2 have identical shapes then their degree of similarity s(G],G2) = 1, otherwise it is a smaller positive number. 

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). 

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.

Figure M-10. Solvent-accessible molecular surface defined by the centres of spheres rolled along the molecular contour surface.

Mezey, P.G. (1988c). Shape Group Studies of Molecular Similarity Shape Groups and Shape Graphs of Molecular Contour Surfaces. J.Math.Chem.,2,299. 

Mezey, P.G. (1988c) Shape group studies of molecular similarity shape groups and shape graphs of molecular contour surfaces. /. Math. Chem., 2, 299. 

In closing, we should also include the molecular graphs introduced by Bader and co-workers.These graphs are derived from the entire electron density function p(r, K) and not from one molecular contour surface. In this sense, these are 2D descriptors of molecular 3D models. 

Figure 5 A cross section of an approximate solvent accessible surface of the molecule hown in Figure 1. The solvent accessible surface is defined by the centers of spheres rolled along the molecular contour surface, where the radius of the sphere is chosen according to the size of the solvent molecule.

By using an appropriate neighbor relation to describe the mutual arrangements of the D, domains along the molecular contour surface G(a), a shape matrix s a,b) and the corresponding shape graph g, a,b) can be defined. 

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