Abstract
A solution of the problem of calculating cartesian coordinates from a matrix of interpoint distances (the embedding problem) is reported. An efficient and numerically stable algorithm for the transformation of distances to coordinates is then obtained. It is shown that the embedding problem is intimately related to the theory of symmetric matrices, since every symmetric matrix is related to a general distance matrix by a one-to-one transformation. Embedding of a distance matrix yields a decomposition of the associated symmetric matrix in the form of a sum over outer products of a linear independent system of coordinate vectors. It is shown that such a decomposition exists for every symmetric matrix and that it is numerically stable. From this decomposition, the rank and the numbers of positive, negative, and zero eigenvalues of the symmetric matrix are obtained directly.
Full text
PDF




Selected References
These references are in PubMed. This may not be the complete list of references from this article.
- Bernstein F. C., Koetzle T. F., Williams G. J., Meyer E. F., Jr, Brice M. D., Rodgers J. R., Kennard O., Shimanouchi T., Tasumi M. The Protein Data Bank: a computer-based archival file for macromolecular structures. J Mol Biol. 1977 May 25;112(3):535–542. doi: 10.1016/s0022-2836(77)80200-3. [DOI] [PubMed] [Google Scholar]
- Braun W., Bösch C., Brown L. R., Go N., Wüthrich K. Combined use of proton-proton Overhauser enhancements and a distance geometry algorithm for determination of polypeptide conformations. Application to micelle-bound glucagon. Biochim Biophys Acta. 1981 Feb 27;667(2):377–396. doi: 10.1016/0005-2795(81)90205-1. [DOI] [PubMed] [Google Scholar]
- Wüthrich K., Wider G., Wagner G., Braun W. Sequential resonance assignments as a basis for determination of spatial protein structures by high resolution proton nuclear magnetic resonance. J Mol Biol. 1982 Mar 5;155(3):311–319. doi: 10.1016/0022-2836(82)90007-9. [DOI] [PubMed] [Google Scholar]