Summary
Several recent advances in both the theory and algorithms for the application of Distance Geometry are reported. These advances were developed with the application to Protein Folding and Drug Design in mind. However, several results are of general interest and are applicable in statistics and the social sciences where the corresponding problems are known as multidimensional scaling. The spectral gradient algorithm is proposed as an alternate method to the majorization algorithm for solving the metric stress problem. The ALS Algorithm that is used to determine molecular conformations is similar to the alternating least squares approach used in nonmetric multidimensional scaling. Several theorems relating geometric structure and the properties of distance matrices are explored. These include a geometrical characterization of the structures which comprise a face of the cone of Euclidean distance matrices. The structure of Euclidean matrices which arise from points which lie on the surface of a hypersphere are investigated.
This work was partially supported by NSF Grant CHE-9301120.
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References
BENAYADE, M. (1992): Distances of Spherical Type: A New Characterization, Distancia″92. International meeting on distance analysis. Rennes, France.
CRIPPEN, G. (1991): Chemical Distance Geometry: Current Realization and Future Projection, J. Math. Chem. 6, 307–324
CRIPPEN, G. and HAVEL, T. (1988): Distance Geometry and Molecular Conformation, Wiley, New York.
CRITCHLEY, F. (1988): On Certain Linear Mappings Between Inner-product and Squared-distance Matrices, Linear Algebra and Appl. 105, 91–107
CRITCHLEY, F. and FICHET, B. (1993): Spherical Distances and Their Relationship with Euclidean and City-Block Distances, Research Report, Dept. Statistics, Univ. of Warwick, Coventry
DE LEEUW, J. (1988): Convergence of the Majorization Method for Multidimensional Scaling, Journal of Classification, 5, 163–180
EASTHOPE, P. and HAVEL, T. (1989): Computational Experience with an Algorithm for Tetrangle Inequality Bound Smoothing, Bull. Math. Bio. 51, 173–194
GLUNT, W., HAYDEN, T., and LIU, W. (1991): The Embedding Problem for Predistance Matrices, Bull, of Math. Biol. 53, 769–796
GLUNT, W., HAYDEN, T., and RAYDAN, M. (1993): Molecular Conformations from Distance Matrices, J. Comp. Chem. 14, 114–120
GLUNT, W., HAYDEN, T., and RAYDAN, M. (1993a-submitted): Preconditioners for Distance Matrix Algorithms, Jour. Comp. Chem.
GLUNT, W., HAYDEN, T., SHELLING, J., WARD, D., and WELLS, C. (1993b-submitted): Weighting and Chirality Strategies for Distance Geometry Algorithms, Jour. Math. Chem.
GORDON, R., BREUER, E., PADILLA, F., SMEJKAL, R., and CHIANG, P. (1989): Distance Geometry of α-substituted 2,2-diphenylpropionate Antimuscarinics, Mol. Pharmacol. 36, 766–779
GOWER, J. (1985): Properties of Euclidean and Non-Euclidean Distance Matrices, Linear Algebra and Appl. 67, 81–97
HAVEL, T. (1990): The Sampling Properties of Some Distance Geometry Algorithms Applied to Unconstrained Polypeptide Chains: A Study of 1830 Independently Computed
HAVEL, T. and WUTHRICH, K. (1984): A Distance Geometry Program for Determining the Structures of Small Proteins and Other Macromolecules from Nuclear Magnetic Resonance Measurements of Intra-molecular H-l-H-1 Proximities in Solution, Bull. Math. Bio. 46, 699–744
HAVEL, T. (1991): An Evaluation of Computational Strategies for Use in the Determination of Protein Structure from Distance Constraints Obtained by Nuclear Magnetic Resonance, Prog. Biophys. Molec. Biol. 56, 43–78
HAYDEN, T. and TARAZAGA, P. (1993): Distance Matrices and Regualar Figures, Linear Algebra and Appl. 195, 9–16
HAYDEN, T., WELLS, J., LIU, W., and TARAZAGA, P. (1991): The Cone of Distance Matrices, Linear Algebra and Appl. 144, 153–169
RIPKA, W. (1987): Led. Heterocycl. Chern. 9, 95–102
TARAZAGA, P., HAYDEN, T., and WELLS, J. (1994-in press): Circum-Euclidean Distance Matrices and Faces, Linear Algebra and Appl
WAGNER, G., HYBERTS, S., and HAVEL, T. (1992): NMR Structure Determination in Solution: A Critique and Comparison with X-Ray Crystallography, Ann. Rev. Biophys. Biomol. Struct. 21, 167–198
WARD, D. (1991): Peptide Pharmaceuticals, Elsevier, New York, N. Y.
WUTHRICH, K. (1989): Protein Structure Determination in Solution by Nuclear Magnetic Resonance Spectroscopy, Science 243, 45–50
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Hayden, T.L. (1994). Applications of Distance Geometry to Molecular Conformation. In: Diday, E., Lechevallier, Y., Schader, M., Bertrand, P., Burtschy, B. (eds) New Approaches in Classification and Data Analysis. Studies in Classification, Data Analysis, and Knowledge Organization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-51175-2_41
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DOI: https://doi.org/10.1007/978-3-642-51175-2_41
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