AM  Vol.5 No.16 , September 2014
Discrete Differential Geometry of n-Simplices and Protein Structure Analysis
Author(s) Naoto Morikawa*
Affiliation(s)
Genocript, Zama, Japan.
ABSTRACT

This paper proposes a novel discrete differential geometry of n-simplices. It was originally developed for protein structure analysis. Unlike previous works, we consider connection between space-filling n-simplices. Using cones of an integer lattice, we introduce tangent bundle-like structure on a collection of n-simplices naturally. We have applied the mathematical framework to analysis of protein structures. In this paper, we propose a simple encoding method which translates the conformation of a protein backbone into a 16-valued sequence.


Cite this paper
Morikawa, N. (2014) Discrete Differential Geometry of n-Simplices and Protein Structure Analysis. Applied Mathematics, 5, 2458-2463. doi: 10.4236/am.2014.516237.
References
[1]   Morikawa, N. (2007) Discrete Differential Geometry of Tetrahedrons and Encoding of Local Protein Structure. arXiv: math.CO/0710.4596.

[2]   Morikawa, N. (2011) A Novel Method for Identification of Local Conformational Changes in Proteins. arXiv: q-bio. BM/1110.6250.

[3]   Bobenko, A.I. and Suris, Yu.B. (2008) Discrete Differential Geometry. Integrable Structure. Graduate Studies in Mathematics, 98, 404 p. arXiv:math/0504358.

[4]   Meyer, M., Desbrun, M., Schroder, P. and Barr, A.H. (2003) Discrete Differential-Geometry Operators for Triangulated 2-Manifolds. In: Hege, H.-C. and Polthier, K., Eds., Visualization and Mathematics III, Springer-Verlag, Berlin, 35-58.
http://dx.doi.org/10.1007/978-3-662-05105-4_2

[5]   Sillitoe, I., et al. (2013) New Functional Families (FunFams) in CATH to Improve the Mapping of Conserved Functional Sites to 3D Structures. Nucleic Acids Research, 41, D490-D498.
http://dx.doi.org/10.1093/nar/gks1211

[6]   Murzin, A.G., Brenner, S.E., Hubbard, T. and Chothia, C. (1995) SCOP: A Structural Classification of Proteins Database for the Investigation of Sequences and Structures. Journal of Molecular Biology, 247, 536-540.
http://dx.doi.org/10.1016/S0022-2836(05)80134-2

[7]   Rackovsky, S. and Scheraga, H.A. (1978) Differential Geometry and Polymer Conformation. 1. Comparison of Protein Conformations. Macromolecules, 11, 1168-1174.
http://dx.doi.org/10.1021/ma60066a020

[8]   Louie, A.H. and Somorjai, R.L. (1982) Differential Geometry of Proteins: A Structural and Dynamical Representation of Patterns. Journal of Theoretical Biology, 98, 189-209.
http://dx.doi.org/10.1016/0022-5193(82)90258-2

[9]   Montalvao, R.W., Smith, R.E., Lovell, S.C. and Blundell, T.L. (2005) CHORAL: A Differential Geometry Approach to the Prediction of the Cores of Protein Structures. Bioinformatics, 21, 3719-3725.
http://dx.doi.org/10.1093/bioinformatics/bti595

[10]   Gorielyn, A., Hausrath, A. and Neukirch, S. (2008) The Differential Geometry of Proteins and Its Applications to Structure Determination. Biophysical Reviews and Letters, 3, 77-101.
http://dx.doi.org/10.1142/S1793048008000629

[11]   Hu, S., Lundgren, M. and Niemi, A.J. (2011) The Discrete Frenet Frame, Inflection Point Solitons and Curve Visualization with Applications to Folded Proteins. arXiv:q-bio.BM/1102.5658.

[12]   Mavridis, L., et al. (2010) SHREC’10 Track: Protein Model Classification. Proceedings of Eurographics Workshop on 3D Object Retrieval, 117-124.

[13]   Boujenfa, K. and Limam, M. (2012) Consensus Decision for Protein Structure Classification. Journal of Intelligent Learning Systems and Applications, 4, 216-222.
http://dx.doi.org/10.4236/jilsa.2012.43022

 
 
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