Abstract
A molecular linkage consists of a set of rigid bodies pairwise connected by revolute hinges where all hinge lines of each body are concurrent. It is an important problem in biochemistry, as well as in robotics, to efficiently analyze the motions of such linkages. The theory of generic rigidity of body-bar frameworks addresses this problem via fast combinatorial algorithms. However, recent work has shown that symmetry (a common feature of many molecular and mechanical structures) can lead to additional motions. These motions typically maintain the original symmetry of the structure throughout the path, and they can often be detected via simple combinatorial counts. In this paper, we outline how these symmetry-based mathematical counts and methods can be used to efficiently predict the motions of symmetric molecular linkages, and we numerically analyze configuration spaces supporting the symmetric Molecular Conjectures formulated herein.
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References
Asimov, L., Roth, B.: The rigidity of graphs. AMS 245, 279–289 (1978)
Atkins, P., Child, M., Phillips, C.: Tables for Group Theory. Oxford University Press, Oxford (1970)
Daniel, R.M., Dunn, R.V., Finney, J.L., Smith, J.C.: The role of dynamics in enzyme activity. Annu. Rev. Biophys. Biomol. Struct. 32, 69–92 (2003)
Goodsell, D., Olson, A.: Structural symmetry and protein function. Annu. Rev. Biophys. Biomol. Struct. 29, 105–153 (2000)
Henderson, M.E.: Multiparameter parallel search branch switching. Int. J. Bifurcat. Chaos Appl. Sci. Eng. 15(3), 967–974 (2005)
Henderson, M.E.: Multiple parameter continuation: computing implicitly defined \(k\)-manifolds. Int. J. Bifurcat. Chaos 12(3), 451–476 (2002)
Jacobs, D., Rader, A., Kuhn, L., Thorpe, M.: Protein flexibility predictions using graph theory. Proteins Struct. Funct. Genet. 44, 150–165 (2001)
Jagodzinski, F., Hardy, J., Streinu, I.: Using rigidity analysis to probe mutation-induced structural changes in proteins. J. Bioinform. Comput. Biol. 10(10), (2012)
Katoh, N., Tanigawa, S.: A proof of the molecular conjecture. Discrete Comput. Geom. 45(4), 647–700 (2011)
Lee, A., Streinu, I.: Pebble game algorihms and \((k, l)\)-sparse graphs. In: European Conference on Combinatorics, Graph Theory and Applications, Berlin pp. 181–186 (2005)
Porta, J.M., Ros, L., Bohigas, O., Manubens, M., Rosales, C., Jaillet, L.: The CUIK suite: motion analysis of closed-chain multibody systems (Submitted)
Porta, J.M., Ros, L., Thomas, F., Corcho, F., Cantó, J., Pérez, J.J.: Complete maps of molecular-loop conformational spaces. J. Comput. Chem. 28, 2170–2189 (2007)
Schulze, B., Sljoka, A., Whiteley, W.: How does symmetry impact the rigidity of proteins? Phil. Trans. R. Soc. A (2013, in press)
Schulze, B.: Symmetry as a sufficient condition for a finite flex. SIAM J. Discrete Math. 24(4), 1291–1312 (2010)
Schulze, B., Whiteley, W.: The orbit rigidity matrix of a symmetric framework. Discrete Comput. Geom. 46(3), 561–598 (2011)
Sljoka, A.: Algorithms in rigidity theory with applications to protein flexibility and mechanical linkages. Ph.D. Thesis, York University http://www.math.yorku.ca/~adnanslj/adnanthesis.pdf (2012)
Tanigawa, S.: Matroids of gain graphs in applied discrete geometry. arXiv:1207.3601 (2012)
Tay, T.S.: Rigidity of multi-graphs, linking rigid bodies in \(n\)-space. J. Comb. Theory, B 36, 95–112 (1984)
Tay, T.S., Whiteley, W.: Recent advances in generic rigidity of structures. Struct. Topol. 9, 31–38 (1985)
Whiteley, W.: Some matroids from discrete applied geometry. Contemp. Math. 197, 171–311 (1996)
Whiteley, W.: Counting out to the flexibility of molecules. Phys. Biol. 2, 1–11 (2005)
Acknowledgments
This work has been partially supported by the Spanish Ministry of Economy and Competitiveness under project DPI2010-18449.
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Porta, J.M., Ros, L., Schulze, B., Sljoka, A., Whiteley, W. (2014). On the Symmetric Molecular Conjectures. In: Thomas, F., Perez Gracia, A. (eds) Computational Kinematics. Mechanisms and Machine Science, vol 15. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-7214-4_20
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DOI: https://doi.org/10.1007/978-94-007-7214-4_20
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