Advertisement

A β-Shape from the Voronoi Diagram of Atoms for Protein Structure Analysis

  • Jeongyeon Seo
  • Donguk Kim
  • Cheol-Hyung Cho
  • Deok-Soo Kim
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3980)

Abstract

In this paper, we present a β-shape and a β-complex for a set of atoms with arbitrary sizes for a faster response to the topological queries among atoms. These concepts are the generalizations of the well-known α-shape and α-complex (and their weighted counterparts as well). To compute a β-shape, we first compute the Voronoi diagram of atoms and then transform the Voronoi diagram to a quasi-triangulation which is the topological dual of the Voronoi diagram. Then, we compute a β-complex from the quasi-triangulation by analyzing the valid intervals for each simplex in the quasi-triangulation. It is shown that a β-complex can be computed in O(m) time in the worst case from the Voronoi diagram of atoms, where m is the number of simplices in the quasi-triangulation. Then, a β-shape for a particular β consisting of k simplices can be located in O(log m + k) time in the worst case from the simplicies in the β-complex sorted according to the interval values.

Keywords

Voronoi diagram of spheres α-shape α-complex β-shape β-complex 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Lee, B., Richards, F.M.: The interpretation of protein structures: Estimation of static accessibility. Journal of Molecular Biology 55, 379–400 (1971)CrossRefGoogle Scholar
  2. 2.
    Richards, F.M.: Areas, volumes, packing, and protein structure. Annual Review of Biophysics and Bioengineering 6, 151–176 (1977)CrossRefGoogle Scholar
  3. 3.
    Connolly, M.L.: Analytical molecular surface calculation. Journal of Applied Crystallography 16, 548–558 (1983)CrossRefGoogle Scholar
  4. 4.
    Connolly, M.L.: Solvent-accessible surfaces of proteins and nucleic acids. Science 221, 709–713 (1983)CrossRefGoogle Scholar
  5. 5.
    Edelsbrunner, H., Mücke, E.P.: Three-dimensional alpha shapes. ACM Transactions on Graphics 13(1), 43–72 (1994)zbMATHCrossRefGoogle Scholar
  6. 6.
    Kim, D.S., Kim, D., Sugihara, K.: Voronoi diagram of a circle set from Voronoi diagram of a point set: I. topology. Computer Aided Geometric Design 18, 541–562 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Kim, D.S., Kim, D., Sugihara, K.: Voronoi diagram of a circle set from Voronoi diagram of a point set: II. geometry. Computer Aided Geometric Design 18, 563–585 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Edelsbrunner, H.: Weighted alpha shapes. Technical Report UIUCDCS-R-92-1760, Department of Computer Science, University of Illinois at Urbana-Champaign, Urbana, IL (1992)Google Scholar
  9. 9.
    Edelsbrunner, H., Facello, M., Liang, J.: On the definition and the construction of pockets in macromolecules. Discrete Applied Mathematics 88, 83–102 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Liang, J., Edelsbrunner, H., Woodward, C.: Anatomy of protein pockets and cavities: Measurement of binding site geometry and implications for ligand design. Protein Science 7, 1884–1897 (1998)CrossRefGoogle Scholar
  11. 11.
    Liang, J., Edelsbrunner, H., Fu, P., Sudhakar, P.V., Subramaniam, S.: Analytical shape computation of macromolecules: I. molecular area and volume through alpha shape. PROTEINS: Structure, Function, and Genetics 33, 1–17 (1998)CrossRefGoogle Scholar
  12. 12.
    Liang, J., Edelsbrunner, H., Fu, P., Sudhakar, P.V., Subramaniam, S.: Analytical shape computation of macromolecules: II. inaccessible cavities in proteins. PROTEINS: Structure, Function, and Genetics 33, 18–29 (1998)CrossRefGoogle Scholar
  13. 13.
    (RCSB Protein Data Bank Homepage), http://www.rcsb.org/pdb/
  14. 14.
    Kim, D.S., Cho, Y., Kim, D.: Euclidean Voronoi diagram of 3D balls and its computation via tracing edges. Computer-Aided Design 37(13), 1412–1424 (2005)CrossRefzbMATHGoogle Scholar
  15. 15.
    Kim, D.S., Cho, Y., Kim, D., Cho, C.H.: Protein sructure analysis using Euclidean Voronoi diagram of atoms. In: Proceedings of the International Workshop on Biometric Technologies (BT 2004), pp. 125–129 (2004)Google Scholar
  16. 16.
    Kim, D.S., Cho, Y., Kim, D., Kim, S., Bhak, J., Lee, S.H.: Euclidean Voronoi diagrams of 3D spheres and applications to protein structure analysis. In: Proceedings of the 1st International Symposium on Voronoi Diagrams in Science and Engineering (VD 2004), pp. 137–144 (2004)Google Scholar
  17. 17.
    Cho, Y., Kim, D., Kim, D.S.: Topology representation for the Voronoi diagram of 3D spheres. International Journal of CAD/CAM 5(3) (2005) (in press)Google Scholar
  18. 18.
    Kim, D.-S., Kim, D., Cho, Y., Sugihara, K.: Quasi-triangulation and interworld data struction in three dimensions. Computer-Aided Design (2005) (submitted)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Jeongyeon Seo
    • 1
  • Donguk Kim
    • 2
  • Cheol-Hyung Cho
    • 2
  • Deok-Soo Kim
    • 1
    • 2
  1. 1.Department of Industrial EngineeringHanyang UniversitySeoulSouth Korea
  2. 2.Voronoi Diagram Research CenterHanyang UniversitySeoulSouth Korea

Personalised recommendations