Shape Analysis of Molecular Surfaces

Molecular shape affects many chemical properties, for example, it has an important role in structure-activity relations. For most purposes, molecular surfaces can represent the shape of molecules. A molecular surface is fully specified if sufficient information is given that allows one to reconstruct the surface. For example, if the surface can be described by some equation of closed form, then each of its points can be calculated, and the shape of the molecular surface can be analyzed relatively easily. Expressions of closed forms 

The same approach is applicable for the comparison of molecular surfaces belonging to two different molecules. If the neighbor relations among the corresponding domains on the two molecular surfaces are the same, then the shapes of the two contours are similar within the above context.  

As an illustration, here we shall outline only one of the simplest of the nonvisual, topological methods for shape characterization, applicable for smooth (differentiable) molecular surfaces. This method is based on the classification of the points of a molecular surface into convex, concave, and saddle-type domains using local curvature properties, and on the representation of the mutual arrangements of these domains by a matrix or by an equivalent graph. One of the advantages of the method is the fact that the generation 

The local curvature properties can be compared to a reference curvature parameter For each point r of the molecular surface G(a) a number p, 

This number p.(r,fe) is the tool used for a classification of points r of the contour G a) into various domains. Each point r of the contour surface G a) belongs to one of three disjoint subsets of G(a), denoted by Ao, At, or Az, depending on whether at point r none, one, or both, respectively, of the local canonical curvatures hi and hz are smaller than the reference value b. Evidently, the union of the three sets Ao, At, and Az generates the entire contour surface, G a) = Ao U i4i U Az, where U is the union symbol. 

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Duncan BS, Olson A (1993) Shape analysis of molecular surfaces. Biopolymers 33 231-238... 

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]. 

Different physical properties and molecular models have been used to define the molecular surface the most common are reported below together with the descriptors proposed as measures of surface areas and molecular volume (- volume descriptors). Molecular surface area and volume are parameters of molecules that are very important in understanding their structure and chemical behaviour such as their ability to bind ligands and other molecules. An analysis of molecular surface shape is also an important tool in QSAR and - drug design-, in particular, both - molecular shape analysis and - Mezey 3D shape analysis were developed to search for similarities among molecules, based on their molecular shape. 

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. 

In some instances, the interactions of various molecular regions in a macromolecule RM or in a composite "supramolecule" of a host-guest system are of interest. In these cases, a local shape analysis of the "isolated" RIDCO surfaces GR(a) is no longer sufficient, and the study of the interactions requires... 

At another level molecular shape is linked to the external surface of a molecule. Although it is generally recognized that quantum-mechanically molecules do not have clearly defined surfaces, new definitions of molecular shape and surface appear in the literature on a regular basis. Variables such as molecular surface area and volume are useful in the analysis of molecular recognition and other surface-dependent properties that assume a clearly defined surface. 

The segmentation of molecular surfaces can be carried out on the basis of different surface qualities.Here we shall focus on a segmentation with respect to topographical criteria. The shape analysis is based on the surface topography index (STI) described previously. Following Heiden and Brickmann, a six class linguistic variable... 

This observation of Parr and Berk provides the basis for a simple approach to molecular shape analysis and molecular similarity analysis, described below. Although the molecular shapes, as defined by the electronic density, differ somewhat from the shapes of the nuclear potentials, their similarity can be exploited the nuclear potential contour surfaces provide a simple approximation of the shape of molecules. We shall refer to the isopotential surfaces of the nuclear potential contours as NUPCO surfaces. These surfaces have a major advantage the computation of NUPCO s is a trivially simple task as compared to the calculation of electronic densities. Furthermore, nuclear potential is a useful molecular property in its own right, without any reference to electronic density a comparison of NUPCO s of various molecules can provide a valid tool for evaluating molecular similarity. The superposition of potentials of different sets of nuclei can result in similar composite potentials, consequently, the comparison of NUPCO s is better... 

Shape Analysis of Fused Sphere Van der Waals Surfaces and Other Locally Nondifferentiable Molecular Surfaces... 

A similarity assessment of the shapes encoded by these vectors can be carried out on various levels. For two molecules (or conformers) Mj and M2, the number of matches along their C(M ) and C(M2) vectors, divided by the dimension (3 x 861), provides a simple, numerical similarity measure [262]. The comparison of the first 2 x 861 components [i.e., the comparison of vectors C( )(Mi) and C(2)(Mi) to C( )(M2) and C(2)(M2), respectively] appears the most important for a crude shape analysis, since these vectors store the information on the number and type of shape domains occurring for various a and b parameter values. By contrast, a direct comparison of vectors C<3)(M ) and C(3)(M2) gives information on the similarities of the patterns of different shape domains on the two families of molecular surfaces. 

Several shape descriptors are defined within more general approaches to - molecular descriptors. This is the case of - Kier shape descriptors, -> shape profiles, -> shadow indices, -> WHIM shape descriptors, - Sterimol shape parameters L/Bj and B1/B5, molecular - periphery codes, and -> centric indices. Other approaches to the study of molecular surface and shape are Mezey 3D shape analysis and Hopfinger - molecular shape analysis. -> Triangular descriptors have also been used to characterize molecular shape to search for similarities among molecules. 

We assume that conditions can be controlled to minimize additional relaxation effects such as magnetic dipole-dipole interactions. As the number of relaxation mechanisms decreases, the information necessary for a line shape analysis of the spectra also decreases. Thus, a poorer signal-to-noise ratio can be tolerated, and the signal can be smoothed by curve fitting techniques. Since little is known at the molecular level about two-dimensional transport coefficients, such as the surface viscosity, large uncertainties can be tolerated. In this sense, we believe that much can be learned from monolayer experiments using spin label surfactants. 

For completeness, we mention that there are several other methodologies for the characterization of the shape of molecular surfaces from local geometric features. The use of Fourier shape descriptors is an interesting alternative, adapted recently for the analysis of macromolecular surfaces. 

The purpose of this review is to provide the reader with a brief introduction to the background and to some of the more recently developed applications of molecular surfaces. Some of these developments are related to the concept of molecular shape, which is of fundamental importance in the analysis of similarities, regularities, and correlations among molecules and in the modeling of chemical and biochemical processes. The study of molecular surfaces has predictive value in various fields of chemistry.In particular, pharmaceutical drug design, toxicity analysis, and the development of new... 

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