Abstract
A major obstacle in applying distance geometry techniques is the analytical complexity of the Cayley-Menger determinants that are used to characterize euclidian spaces in terms of distances between points. In this paper we show that, with the aid of a theorem of Jacobi, the complex Cayley-Menger determinants can be replaced by simpler determinants, and we derive the concept of Cayley-Menger coordinates, a coordinate system in terms of which each point of En is characterized by n + 1 distances to n + 1 points of a reference. We also show that this coordinate system provides a natural norm for the incomplete embedding problem. This paper provides the tools to treat the problem of filling out an incomplete distance matrix so that our previous procedure can then be used to embed the corresponding structure in a three-dimensional space.
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Selected References
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- 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]
- Sippl M. J., Scheraga H. A. Solution of the embedding problem and decomposition of symmetric matrices. Proc Natl Acad Sci U S A. 1985 Apr;82(8):2197–2201. doi: 10.1073/pnas.82.8.2197. [DOI] [PMC free article] [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]