Mathematical Challenges from Theoretical/Computational Chemistry (1995) / Chapter Skim
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Examples of Constructive Cross-Fertilization Between the Mathematical Sciences and Chemistry
Examples of Constructive Cross-Fertilization Between the Mathematical Sciences and Chemistry
Pages 19-42
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From page 19... ...
Not only are these compounds commercial successes for the companies that developed them, but they benefit mankind by aiding in the treatment of disease or increasing the food supply. This viewpoint has been so successful that recently a company, Arris, was founded to incorporate the direct involvement of mathematicians in the development of proprietary drug design software.
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:: '~; -~.'~ ' ;.~: ~ . ' .'BO:~..3.l Ra£ional Drug Design: .
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Chapman, 1994, Compass: Predicting biological activities from molecular surface properties; Performance comparisons on a steroid benchmark, Journal of Medicinal Chemistry 34:2315-2327. Kubinyi, H., ea., 1993, 3D QSAR in Drug Design: Theory, Methods, and Applications, ESCOM, Leiden.
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. Many computational chemistry codes have been adapted to work efficiently on parallel computers by a variety of techniques.
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In essence, distance geometry is a method to work in spaces with greater than three dimensions, allowing distance constraints to be satisfied that could not be satisf~ed in three dimensions. Distance geometry helps one move from collections of distances between points to possible coordinates for these points.
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Distance geometry ~ an i~or1~ 1ec~i~e in co~u1adonal cbemkiry. The focus of 1be original work was 10 predict protein structure Mom amino acid sequence (see Box 3.~)
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References Blaney, J.M., and J.S. Dixon, 1994, Distance geometry in molecular modeling, in Reviews in Computational Chemistry, vol.
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Venkataraghaven, 1986, The ensemble approach to distance geometry: Application to the nictotinic pharmacophore, J
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. Consequently, any striking deviation from conventional expectation about what interatomic or intermolecular forces can yield as solid-state ordering automatically concerns chemical researchers.
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has intervened to change the topological state (Wasserman and Cozzarelli, 1986~. Combinatorics, Graph Theory, and Chemical Isomer Enumeration In the nineteenth century, Arthur Cayley produced a body of work involving the enumeration of certain types of trees (connected acyclic I-complexes)
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, as such spectra sometimes reveal hidden structure associated with short-time molecular dynamics that can easily be interpreted. Recently there has been interest in using the methods of graph theory, in particular Cayley trees, to study complex spectra (Davis, 1993~.
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Group theory has also been applied to quantum chemical problems such as symmetry-adapted functions for molecular orbital theory, ligand field theory, and molecular vibrations (Cotton, 1971; National Research Council, in preparation)
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Recent striking advances in observation techniques such as fluorescence microscopy for single DNA molecules (Bustamante et al., 1990) make possible the verification and fine tuning of some of these mathematical theories of molecular conformation and dynamic molecular properties.
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1976. Chemical Applications of Graph Theory, Academic Press, London.
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Delbruck, M., 1962, Knotting problems in biology, in Mathematical Problems in the Biological Sciences (Proceedings of Symposia in Applied Mathematics, vol.14) , American Mathematical Society, Providence, R.I., pp.
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Graph Theory Application of Graph Theory to Organizing Chemical Literature To organize the chemistry literature for research or patent purposes, it is essential that scientists be able to search this literature by the chemical features of interest as well as with traditional text queries. Accordingly, a body of experience has been developed for using computers to recognize either total chemical structures or parts of them from an input structural diagram and to quickly identify in databases of millions of compounds those that match the search criteria (Ash et al., 19851.
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A key element to the success of such chemical information systems was the recognition that a twodimensional chemical structure diagram can be treated as a labeled graph (Sussenguth, 1965~. Many algorithms and concepts from graph theory (Harary, 1976)
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Rather, the chemical information specialists have followed the graph theory literature, and when a particular concept or algorithm seemed the appropriate solution for some problem, they would attempt to implement it. Sometimes this meant waiting until a graduate student with the particular skills was available.
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Since that time, 1986, the UlIman subgraph isomorphism algorithm has replaced the previous algorithms in all commercial chemical information systems (Willett, 1987~. Application of Graph Theory to Representation of Chem~c~1 Reactions The synthesis of organic chemicals is an art that takes many years of training and experience to master.
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The unchanged parts represent the chemical context in which the reaction occurs. In graph theory terms.
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Willett, 1980, Use of a maximal common subgraph algorithm in the automatic identification of the ostensible bond changes occurring in chemical reactions, J Chemical Information and Computer Sciences 21: 137-140.
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"It was recognized almost from the beginning that the diffraction pattern, the directions and intensities of the X-rays scattered by a crystal, is uniquely determined by the crystal structure; which is to say that if one knew the crystal structure the arrangement of the atoms in the crystal then one could calculate the diffraction pattern completely. It turns out that, conversely, diffraction patterns determine, in general, unique crystal and molecular structures, although this fact was not known until many years later.
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With this assumption the intensities and phases of the scattered X-rays, as functions of the atomic position vectors, are also random variables, and one can use the methods of modern mathematical probability theory to calculate the joint probability distribution of any collection of intensities and phases. A suitably chosen joint probability distribution leads directly to the conditional probability distribution of a specified structure invariant, assuming again an appropriately chosen set of diffraction intensities.
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Remark The committee can only add its belief that the last quoted sentence from Hauptman's account has wide applicability to problems of chemical interest, some of which are described in the next chapter. References Harker, D., and J.S.
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