Topological shape analysis

The three-dimensional shape of this fuzzy body of the electronic distribution has many important features not revealed by the simple, skeletal ball and stick model. One of the most important tasks of topological shape analysis of molecules is the precise analysis and concise description of the three-dimensional electronic charge distributions, such as that illustrated by the selected MIDCO s of allyl alcohol in Figure 1.2. Various methods and computational techniques of such topological shape analyses are discussed in detail in this book. 

Molecular topology [155-158,190-199] presents a systematic framework for general shape analysis methods applicable, in principle, to all molecules. The same framework is also the basis for special shape analysis methods designed to exploit the typical features of some special, distinguished molecular families, such as the folding properties of polypeptides, proteins, and other chain biomolecules. Molecular topology and the associated topological shape analysis approaches form the basis of the present book. 

The stereochemical shape concept covers a wide range of possible resolutions, from the details of electron density distributions between pairs of nuclei in relatively small molecules to the structural organization of the tertiary structure of proteins [201-203], the architecture of supramolecular assemblies [204-230], the problems of shape selectivity in reactions of large molecules [231-233], and the intriguing shape features of self-replicating chemical systems [234-239]. In the following chapters we shall discuss various topological shape analysis techniques, suitable for the relevant level of resolution. 

In this book we shall be concerned only with a very limited. selection of the elements of topology, relevant to the basics of topological shape analysis of molecules. All the tools we shall use will be described in sufficient detail in the book. However, for readers interested in more details of the fundamentals, some introductory and advanced texts are listed among the references [113-122]. 

The above topological shape analysis techniques can replace visual shape comparisons of molecular models on the computer screen with precise, reliable, and reproducible numerical comparisons of topological shape codes. These comparisons and the similarity or complementarity rankings of molecular sequences can be performed by the computer automatically. This eliminates the subjective element of visual shape comparisons, a particularly important concern if large sequences (e.g. several thousands) of molecules are to be compared. In the data banks of most drug companies there is information stored on literally hundreds of thousands of molecules, and their detailed shape analysis by visual comparison on a computer screen is clearly not feasible. By contrast, automatic, numerical, topological shape analysis by computer is a viable alternative. 

The input data for the shape analysis methods are provided by well-established quantum chemical or empirical computational methods for the calculation of electronic charge distributions, electrostatic potentials, fused spheres Van der Waals surfaces, or protein backbones. The subsequent topological shape analysis is equally applicable to any existing molecule or to molecules which have not yet been synthesized. This is precisely where the predictive power of such shape analysis lies based on a detailed shape analysis, a prediction can be made on the expected activity of all molecules in the sequence and these methods can select the most promising candidates from a sequence of thousands of possible molecules. The actual expensive and time-consuming synthetic work and various chemical and biochemical tests of... 

Mezey, P.G. (1993d). Topological Shape Analysis of Chain Molecules An Application of the GSTE Principle. J.Math.Chem., 12, 365-374. 

In general, formal molecular fragments that are larger than the conventional functional groups can also be represented by fuzzy moieties of electron densities, dominated by several nuclei. The shapes of molecular fragments with density domains indicating separate identity have important chemical consequences, and these shapes can be characterized by topological shape analysis methods [40]. 

Hence while the topological shape analysis and shape similarity measures do not truly represent a Non-Quantum Similarity approach, nevertheless, in practical terms, they offer a powerful alternative. 

Several additional developments had very positive impact on the advances and applications of the topological shape analysis and similarity methods. Among these are the already mentioned establishment of the holographic property of the electron density clouds of real, boundary-less molecules [4,5] and the extension of many aspects of small-molecule quantum chemistry to macromolecules, such as proteins, by the linear-scaling MEDLA and ADMA methods [6-11]. 

Mezey, P.G. (1995) Methods of molecular shape-similarity analysis and topological shape design. 

Mezey, P.G.,"Methods of Molecular Shape-Similarity Analysis and Topological Shape Design". In Dean, P.M., ed., Molecular Similarity in Drug Design (Chapman Hall - Blackie Publishers, Glasgow, U.K., 1995). 

Figure 2.2 Selected families (DDj(aj)) of density domains of the water molecule, as calculated with the GAUSSIAN 90 ab initio program [253] and the GSHAPE 90 molecular shape analysis program [254], using a 6-3IG basis set. There are only two topologically different types of families of density domains either a single density domain, or a family of three density domains. The sequence of topologically distinct cases provides a topological description of chemical bonding.

Molecular bodies of quantum mechanical electron distributions or some other molecular functions such as electrostatic potentials can be represented on various levels of approximation. These representations have two main components the physical property or model used to define a formal molecular body, and the geometrical or topological method used to describe and analyze the model. If a representation of the molecular body is selected, then the boundaries of these approximate molecular bodies can be regarded as formal molecular surfaces. Hence, the molecular shape analysis problem can be formulated as the shape analysis problem of formal molecular surfaces. 

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]. 

We shall distinguish two types of methods for dynamic shape analysis. The methods of the first type are used to determine which nuclear arrangements are associated with a given topological shape. The methods of the second type determine the available topological shapes compatible with some external conditions, for example, with an energy bound. 

Within the simplest formulation of a dynamic shape analysis method of the first type, the invariance of topological descriptors within domains of the dynamic shape... 

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