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Research Papers

Kinematics Meets Crystallography: The Concept of a Motion Space1

[+] Author and Article Information
Gregory S. Chirikjian

Robot and Protein Kinematics Laboratory,
Department of Mechanical Engineering,
Johns Hopkins University,
Baltimore, MD 21218
e-mail: gregc@jhu.edu

In reality such crystals exist only in three-dimensional Euclidean space, but for the purpose of generality in this introduction, the dimension n will be allowed to be general, with realistic examples having n = 2 or 3.

In the symmorphic case Pu=Fns=Pns.

1This paper was originally presented at the ASME 2014 International Design Engineering Technical Conferences as Paper No. DETC2014-34243.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTING AND INFORMATION SCIENCE IN ENGINEERING. Manuscript received August 23, 2014; final manuscript received August 28, 2014; published online February 2, 2015. Editor: Bahram Ravani.

J. Comput. Inf. Sci. Eng 15(1), 011012 (Mar 01, 2015) (7 pages) Paper No: JCISE-14-1259; doi: 10.1115/1.4028922 History: Received August 23, 2014; Revised August 28, 2014; Online February 02, 2015

In this paper, it is shown how rigid-body kinematics can be used to assist in determining the atomic structure of proteins and nucleic acids when using x-ray crystallography, which is a powerful method for structure determination. The importance of determining molecular structures for understanding biological processes and for the design of new drugs is well known. Phasing is a necessary step in determining the three-dimensional structure of molecules from x-ray diffraction patterns. A computational approach called molecular replacement (MR) is a well-established method for phasing of x-ray diffraction patterns for crystals composed of biological macromolecules. In MR, a search is performed over positions and orientations of a known biomolecular structure within a model of the crystallographic asymmetric unit, or, equivalently, multiple symmetry-related molecules in the crystallographic unit cell. Unlike the discrete space groups known to crystallographers and the continuous rigid-body motions known to kinematicians, the set of motions over which MR searches are performed does not form a group. Rather, it is a coset space of the group of continuous rigid-body motions, SE(3), with respect to the crystallographic space group of the crystal, which is a discrete subgroup of SE(3). Properties of these “motion spaces” (which are compact manifolds) are investigated here.

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References

Bottema, O., and Roth, B., 1990, Theoretical Kinematics, Dover, Mineola, NY.
Angeles, J., 1988, Rational Kinematics, Springer-Verlag, New York.
McCarthy, J. M., 1990, Introduction to Theoretical Kinematics, MIT Press, Boston, MA.
Park, F. C., 1991, “The Optimal Kinematic Design of Mechanisms,” Ph.D. thesis, Division of Engineering and Applied Sciences, Harvard University, Cambridge, MA.
Murray, R. M., Li, Z., and Sastry, S. S., 1994, A Mathematical Introduction to Robotic Manipulation, CRC, Boca Raton, FL.
Selig, J. M., 1996, Geometrical Methods in Robotics, Springer, New York.
Karger, A., and Novák, J., 1985, Space Kinematics and Lie Groups, Gordon and Breach, New York.
Park, F. C., and Ravani, B., 1995, “Bézier Curves on Riemannian Manifolds and Lie Groups With Kinematics Applications,” ASME J. Mech. Des., 117(1), pp. 36–40. [CrossRef]
Kazerounian, K., and Rastegar, J., 1992, “Object Norms: A Class of Coordinate and Metric Independent Norms for Displacement,” Flexible Mechanisms, Dynamics, and Analysis, ASME, Vol. 47, pp. 271–275.
Bradley, C., and Cracknell, A., 1972/2010, The Mathematical Theory of Symmetry in Solids, Oxford University, Oxford, UK.
Burns, G., and Glazer, A. M., 1990, Space Groups for Solid State Scientists, 2nd ed., Academic, Boston, MA.
Janssen, T., 1973, Crystallographic Groups, North Holland/Elsevier, New York.
Ball, R. S., 1900, A Treatise on the Theory of Screws, Cambridge University Press, Cambridge, UK.
Schoenflies, A., 1891, Theory of Crystal Structure, Teubner, Leipzig, Germany.
Fedorov, E. S., 1971, Symmetry of Crystals, American Crystallographic Association, Paper Number 7, (English translation of 1885 book in Russian).
Barlow, W., 1894, “Über die Geometrischen Eigenschaften homogener starrer Strukturen und ihre Anwendung auf Krystalle,” Z. Krystallogr. Mineral., 23, p. 1-63.
Hilton, H., 1963, Mathematical Crystallography and the Theory of Groups of Movements, Dover, Mineola, NY.
H. M.Berman, T.Battistuz, T. N.Bhat, W. F.Bluhm, P. E.Bourne, K.Burkhardt, Z.Feng, G. L.Gilliland, L.Iype, S.Jain, P.Fagan, J.Marvin, D.Padilla, V.Ravichandran, B.Schneider, N.Thanki, H.Weissig, J. D.Westbrook, and C.Zardecki, 2002, “The Protein Data Bank,” Acta Crystallogr., Sect. D: Biol. Crystallogr., 58(6), pp. 899–907. [CrossRef]
Rossmann, M. G., and Blow, D. M., 1962, “The Detection of Sub-Units Within the Crystallographic Asymmetric Unit,” Acta Crystallogr., 15(1), pp. 24–31. [CrossRef]
Rossmann, M. G., 2001, “Molecular Replacement - Historical Background,” Acta Crystallogr., Sect. D: Biol. Crystallogr., 57(10), pp. 1360–1366. [CrossRef]
Chirikjian, G. S., and Kyatkin, A. B., 2001, Engineering Applications of Noncommutative Harmonic Analysis, CRC Press, Boca Raton, FL.
Chirikjian, G. S., 2009/2012, Stochastic Models, Information Theory, and Lie Groups: Volumes I + II, Birkhäuser, Boston, MA.
Rupp, B., 2010, Biomolecular Crystallography: Principles, Practice, and Application to Structural Biology, Garland Science, Taylor and Francis Group, New York.
Senechal, M., 1980, “A Simple Characterization of the Subgroups of Space Groups,” Acta Crystallogr., Sect. A: Found. Crystallogr., 36(6), pp. 845–850. [CrossRef]
Chirikjian, G. S., 2011, “Mathematical Aspects of Molecular Replacement: I. Algebraic Properties of Motion Spaces,” Acta. Crystallogr., Sect. A: Found. Crystallogr., 67(5), pp. 435–446. [CrossRef]
Chirikjian, G. S., and Yan, Y., 2012, “Mathematical Aspects of Molecular Replacement: II. Geometry of Motion Spaces,” Acta. Crystallogr., Sect. A: Found. Crystallogr., 68(2), pp. 208–221. [CrossRef]
Yan, Y., and Chirikjian, G. S., 2011, “Molecular Replacement for Multi-Domain Structures Using Packing Models,” ASME Paper No. DETC2011-48583. [CrossRef]
Yan, Y., and Chirikjian, G. S., 2012, “Almost-Uniform Sampling of Rotations for Conformational Searches in Robotics and Structural Biology,” ICRA 2012, Minneapolis, MN, May 14–18, pp. 4254–4259.
Yan, Y., and Chirikjian, G. S., 2013, “Voronoi Cells in Lie Groups and Coset Decompositions: Implications for Optimization, Integration, and Fourier Analysis,” 52nd IEEE Conference on Decision and Control, Firenze, Italy, Dec. 10–13.
Aroyo, M., Perez-Mato, J. M., Capillas, C., Kroumova, E., Ivantchev, S., Madariaga, G., Kirov, A., and Wondratschek, H., 2006, “Bilbao Crystallographic Server: I. Databases and Crystallographic Computing Programs,” Zeitschrift für Kristallographie, 221(1), pp. 15–27. [CrossRef]

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