Quaternion
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... 221 -- 11.1 The Quaternions 221 -- 11.2 What Happened Next 226 -- 11.3 Valedictory 230 -- Contents -- ... ... /th Complex Roots 239 -- Appendix C How Do Quaternions Represent Rotations? ... ... 243 -- Appendix D Why Are the Quaternions So Special? 247 -- References 255 -- Index 257 ...
First published xi, 260 stran
... 4.9.2 On decomposition of tensors on manifolds 198 -- 4.9.3 F-traceless decomposition 203 -- 4.9.4 Quaternionic ...
1. elektronické vydání 1 online zdroj (676 stran)
There is a paramount need to develop new techniques and tools that will extract as much information as possible from the ever growing repository of protein 3D structures. We report here on the development of a software tool for the multiple superimposition of large sets of protein structural motifs. Our superimposition methodology performs a systematic search for the atom pairing that provides the best fit. During this search, the RMSD values for all chemically relevant pairings are calculated by quaternion algebra. The number of evaluated pairings is markedly decreased by using PDB annotations for atoms. This approach guarantees that the best fit will be found and can be applied even when sequence similarity is low or does not exist at all. We have implemented this methodology in the Web application SiteBinder, which is able to process up to thousands of protein structural motifs in a very short time, and which provides an intuitive and user-friendly interface. Our benchmarking analysis has shown the robustness, efficiency, and versatility of our methodology and its implementation by the successful superimposition of 1000 experimentally determined structures for each of 32 eukaryotic linear motifs. We also demonstrate the applicability of SiteBinder using three case studies. We first compared the structures of 61 PA-IIL sugar binding sites containing nine different sugars, and we found that the sugar binding sites of PA-IIL and its mutants have a conserved structure despite their binding different sugars. We then superimposed over 300 zinc finger central motifs and revealed that the molecular structure in the vicinity of the Zn atom is highly conserved. Finally, we superimposed 12 BH3 domains from pro-apoptotic proteins. Our findings come to support the hypothesis that there is a structural basis for the functional segregation of BH3-only proteins into activators and enablers.
