{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,2]],"date-time":"2026-05-02T10:01:15Z","timestamp":1777716075300,"version":"3.51.4"},"reference-count":53,"publisher":"SAGE Publications","issue":"2-3","license":[{"start":{"date-parts":[[2005,2,1]],"date-time":"2005-02-01T00:00:00Z","timestamp":1107216000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["The International Journal of Robotics Research"],"published-print":{"date-parts":[[2005,2]]},"abstract":"<jats:p>High-throughput nuclear magnetic resonance (NMR) structural biology and NMR structural genomics pose a fascinating set of geometric challenges. A key bottleneck in NMR structural biology is the resonance assignment problem. We seek to accelerate protein NMR resonance assignment and structure determination by exploiting a priori structural information. In particular, a method known as nuclear vector replacement (NVR) has been proposed as a method for solving the assignment problem given a priori structural information. Among several different types of input data, NVR uses a particular type of NMR data known as residual dipolar couplings (RDCs). The basic physics of RDCs tells us that the data should be explainable by a structural model and set of parameters contained within the \u201cSaupe alignment tensor\u201d.<\/jats:p>\n                  <jats:p>In the NVR algorithm, one estimates the Saupe alignment tensors and then proceeds to refine those estimates. We would like to quantify the accuracy of such estimates, where we compare the estimated Saupe matrix to the correct Saupe matrix. In this work, we propose a way to quantify this comparison. Given a correct Saupe matrix and an estimated Saupe matrix, we compute an upper bound on the probability that a randomly rotated Saupe tensor would have an error smaller than the estimated Saupe matrix. This has the advantage of being a quantified upper bound, which also has a clear interpretation in terms of geometry and probability. While the specific application of our rotation probability results is given to NVR, our novel methods can be used for any RDC-based algorithm to bound the accuracy of the estimated alignment tensors. Furthermore, they could also be used in X-ray crystallography or molecular docking to quantitate the accuracy of calculated rotations of proteins, protein domains, nucleic acids, or small molecules.<\/jats:p>","DOI":"10.1177\/0278364905050351","type":"journal-article","created":{"date-parts":[[2005,1,26]],"date-time":"2005-01-26T07:27:56Z","timestamp":1106724476000},"page":"165-182","source":"Crossref","is-referenced-by-count":3,"title":["A Probability-Based Similarity Measure for Saupe Alignment Tensors with                 Applications to Residual Dipolar Couplings in NMR Structural Biology"],"prefix":"10.1177","volume":"24","author":[{"given":"Anthony K.","family":"Yan","sequence":"first","affiliation":[{"name":"Dartmouth Computer Science Department, Hanover, NH 03755, USA"}]},{"given":"Christopher J.","family":"Langmead","sequence":"additional","affiliation":[{"name":"Dartmouth Computer Science Department, Hanover, NH 03755, USA"}]},{"given":"Bruce Randall","family":"Donald","sequence":"additional","affiliation":[{"name":"6211 Sudikoff Laboratory, Dartmouth Computer Science Department, Hanover,                         NH 03755, USA, Dartmouth Chemistry Department and Dartmouth Department of                         Biological Sciences, Hanover, NH 03755, USA,"}]}],"member":"179","published-online":{"date-parts":[[2005,2,1]]},"reference":[{"key":"atypb1","doi-asserted-by":"crossref","unstructured":"Al-Hashimi, H. M., and Patel, D. J. 2002. Residual dipolar couplings: synergy between NMR and structural genomics . Journal of Biomolecular NMR 22(1): 18 .","DOI":"10.1023\/A:1013801714041"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1016\/S0022-2836(02)00160-2"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1023\/A:1013334513610"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1107\/S0567740882004075"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1021\/ja005766d"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1089\/106652700750050934"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1021\/bi00052a006"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1006\/jmre.1998.1419"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1021\/ja9812610"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1021\/ja993603n"},{"key":"atypb11","doi-asserted-by":"crossref","unstructured":"Diaconis, P., and Shahshahani, M. 1999. The subgroup algorithm for generating uniform random variables . Problems in Engineering and Information Science 1: 15-32 .","DOI":"10.1017\/S0269964800000255"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1016\/0022-2836(74)90598-1"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1038\/nature00860"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1016\/S1074-5521(00)80022-8"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1006\/jmbi.2000.4199"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1021\/bi00181a032"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1023\/A:1021954812977"},{"key":"atypb18","doi-asserted-by":"crossref","unstructured":"Girard, E., Chantalat, L., Vicat, J., and Kahn, R. 2001. Gd-HPDO3A, a complex to obtain high-phasing-power heavy atom derivatives for SAD and MAD experiments: results with tetragonal hen egg-white lysozyme . Acta Crystallographica D 58: 1-9 .","DOI":"10.1107\/S0907444901016444"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1126\/science.1871600"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1006\/jmre.2002.2569"},{"key":"atypb21","unstructured":"Huttenlocher, D. P., and Kedem, K. 1992. Distance metrics for comparing shapes in the plane. Symbolic and Numerical Computation for Artificial Intelligence, Chapter 8, B. R. Donald, D. Kapur, and J. Mundy, editors. Academic Press, Harcourt Jovanovich, London , pp. 201-219."},{"key":"atypb22","doi-asserted-by":"crossref","unstructured":"Huttenlocher, D. P., and Kedem, K. 1990. Computing the minimum Hausdorff distance for point sets under translation . Proceedings of the 6th ACM Symposium on Computational Geometry, Berkeley, CA, pp. 340-349 .","DOI":"10.1145\/98524.98599"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1016\/S0969-2126(99)80123-3"},{"key":"atypb24","unstructured":"Kendall, M. G., and Moran, P. A. P. 1963 Geometrical Probability. Griffin, London , p. 93."},{"key":"atypb25","unstructured":"Kinsey, L. C. 1991. Miscellany. Topology of Surfaces, chapter 10. Springer-Verlag, Berlin , pp. 199-203."},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1107\/S0907444994009261"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1021\/ja9843730"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1023\/B:JNMR.0000019247.89110.e6"},{"key":"atypb29","doi-asserted-by":"crossref","unstructured":"Langmead, C. J., Yan, A. K., Wang, L., Lilien, R., and Donald, B. R. 2003. A polynomial time nuclear vector replacement algorithm for automated NMR resonance asignments . Proceedings of the 7th Annual International Conference on Computational Molecular Biology (RECOMB), Berlin, Germany, April 10-13, pp. 176-187 .","DOI":"10.1145\/640075.640098"},{"key":"atypb30","doi-asserted-by":"publisher","DOI":"10.1089\/1066527041410436"},{"key":"atypb31","doi-asserted-by":"publisher","DOI":"10.1107\/S0907444998002844"},{"key":"atypb32","doi-asserted-by":"publisher","DOI":"10.1006\/jmre.1999.1754"},{"key":"atypb33","doi-asserted-by":"publisher","DOI":"10.1093\/protein\/14.2.105"},{"key":"atypb34","unstructured":"Mitchell, J. C. 2004. The orthogonal image method for generating uniform distributions of rotation matrices. Proteins: Structure, Function and Genetics submitted for publication."},{"key":"atypb35","unstructured":"Munkres, J. R. 1984. Elements of Algebraic Topology. Addison-Wesley, Reading, MA ."},{"key":"atypb36","doi-asserted-by":"crossref","unstructured":"Oki, H., Matsuura, Y., Komatsu, H., and Chernov, A. A. 1999. Refined structure of orthorhombic lysozyme crystallized at high temperature: correlation between morphology and intermolecular contacts . Acta Crystallographica D 55: 114 .","DOI":"10.1107\/S0907444998008713"},{"key":"atypb37","doi-asserted-by":"publisher","DOI":"10.1016\/S0959-440X(97)80085-1"},{"key":"atypb38","doi-asserted-by":"publisher","DOI":"10.1042\/bj2990151"},{"key":"atypb39","doi-asserted-by":"publisher","DOI":"10.1021\/ja016880e"},{"key":"atypb40","doi-asserted-by":"publisher","DOI":"10.1107\/S0365110X62000067"},{"key":"atypb41","doi-asserted-by":"crossref","unstructured":"Rotman, J. J. 1988. Excision and applications. An Introduction to Algebraic Topology, chapter 6. Springer-Verlag, Berlin , p. 123.","DOI":"10.1007\/978-1-4612-4576-6_7"},{"key":"atypb42","unstructured":"Salamin, E. 1972. Quaternions. HAKMEM. Artificial Intelligence Memo No. 239, Massachusetts Institute of Technology, AI Laboratory , M. Beeleret al., editors, February 29. Item 107. http:\/\/www.inwap.com\/pdp10\/hbaker\/hakmem\/hakmem.html."},{"key":"atypb43","unstructured":"Salamin, E. 1979. Applications of quaternions to computation with rotations. Internal working paper, Stanford University, AI Laboratory ."},{"key":"atypb44","doi-asserted-by":"publisher","DOI":"10.1002\/anie.196800971"},{"key":"atypb45","doi-asserted-by":"publisher","DOI":"10.1110\/ps.43301"},{"key":"atypb46","doi-asserted-by":"publisher","DOI":"10.1126\/science.274.5292.1531"},{"key":"atypb47","doi-asserted-by":"publisher","DOI":"10.1021\/ja011806h"},{"key":"atypb48","doi-asserted-by":"publisher","DOI":"10.1126\/science.278.5340.1111"},{"key":"atypb49","doi-asserted-by":"crossref","unstructured":"Vaney, M. C., Maignan, S., Ries-Kautt, M., and Ducruix, A. 1996. High-resolution structure (1.33 angstrom) of a HEW lysozyme tetragonal crystal grown in the APCF apparatus. Data and structural comparison with a crystal grown under microgravity from SpaceHab-01 mission . Acta Crystallographica D 52: 505-517 .","DOI":"10.1107\/S090744499501674X"},{"key":"atypb50","doi-asserted-by":"publisher","DOI":"10.1016\/0022-2836(87)90679-6"},{"key":"atypb51","doi-asserted-by":"publisher","DOI":"10.1023\/B:JNMR.0000032552.69386.ea"},{"key":"atypb52","doi-asserted-by":"publisher","DOI":"10.1023\/A:1014206617752"},{"key":"atypb53","doi-asserted-by":"publisher","DOI":"10.1006\/jmbi.1997.1052"}],"container-title":["The International Journal of Robotics Research"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.1177\/0278364905050351","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.1177\/0278364905050351","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T10:17:16Z","timestamp":1777457836000},"score":1,"resource":{"primary":{"URL":"https:\/\/journals.sagepub.com\/doi\/10.1177\/0278364905050351"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2005,2]]},"references-count":53,"journal-issue":{"issue":"2-3","published-print":{"date-parts":[[2005,2]]}},"alternative-id":["10.1177\/0278364905050351"],"URL":"https:\/\/doi.org\/10.1177\/0278364905050351","relation":{},"ISSN":["0278-3649","1741-3176"],"issn-type":[{"value":"0278-3649","type":"print"},{"value":"1741-3176","type":"electronic"}],"subject":[],"published":{"date-parts":[[2005,2]]}}}