On the Bond Graphs in the Delaunay-Tetrahedra of the Simplicial Decomposition of Spatial Protein Structures

  • Rafael ördög
  • Vince Grolmusz
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5518)


The examination of straightforwardly definable discrete structures in nucleic acids and proteins turned out to be perhaps the most important development in our present knowledge and understanding the their form and function. These discrete structures are sequences of nucleotides and amino acid residues, respectively. Bioinformatics was born as the science of analyzing these sequences. The discretization of the biological information into easy-to-handle sequences of 4 or 20 symbols made possible the application of deep mathematical, combinatorial and statistical tools with enormous success. The tools, resulting from this process, changed our perception of genetics, molecular biology, and life itself.

Straightforward discrete structures can also be defined in the spatial descriptions of proteins and nucleic acids. The definition and examination of discrete objects, using the spatial structure of proteins instead of amino acid sequences would intercept spatial characteristics, that are more conservative evolutionary than the polypeptide sequences.

In the present work we analyze the Delaunay tessellations of more than 5700 protein structures from the Protein Data Bank. The Delaunay tessellations of the heavy atoms of these protein structures give certainly a more complex structure than the polymer sequences themselves, but these tessellations are still easily manageable mathematically and statistically, and they also well describe the topological simplicial complex of the protein.

Our main result is Table 1, describing the relation between van der Waals and covalent bonds in the edges of the Delaunay tessellation. Among other findings, we show that there is only a single one Delaunay tetrahedron in the analyzed 5757 PDB entries with more than 81 million tetrahedra, where all six edges of the tetrahedron correspond to atom-pairs in van der Waals distance, but none of them to atom-pairs in covalent distance.


Heavy Atom Degree Sequence Discrete Structure Bond Graph Covalent Distance 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Berman, H., Westbrook, J., Feng, Z., Gilliland, G., Bhat, T., Weissig, H., Shindyalov, I., Bourne, P.: The Protein Data Bank. Nucleic Acids Research 28, 235–242 (2000)CrossRefGoogle Scholar
  2. 2.
    Ördög, R., Szabadka, Z., Grolmusz, V.: Analyzing the simplicial decomposition of spatial protein structures. BMC Bioinformatics 9(S11) (2008)Google Scholar
  3. 3.
    Barber, C.B., Dobkin, D.P., Huhdanpaa, H.: The quickhull algorithm for convex hulls. ACM Transactions on Mathematical Software 22(4), 469–483 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Singh, R.K., Tropsha, A., Vaisman, I.I.: Delaunay tessellation of proteins: Four body nearest-neighbor propensities of amino acid residues. Journal of Computational Biology 3(2), 213–222 (1996)CrossRefGoogle Scholar
  5. 5.
    Masso, M., Hijazi, K., Parvez, N., Vaisman, I.I.: Computational mutagenesis of E. coli lac repressor: Insight into structure-function relationships and accurate prediction of mutant activity. In: Măndoiu, I., Sunderraman, R., Zelikovsky, A. (eds.) ISBRA 2008. LNCS (LNBI), vol. 4983, pp. 390–401. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  6. 6.
    Ördög, R.: Pydet, a pymol plug-in for visualizing geometric concepts around proteins. Bioinformation 2(8), 346–347 (2008)CrossRefGoogle Scholar
  7. 7.
    Szabadka, Z., Grolmusz, V.: Building a structured PDB: The RS-PDB database. In: Proceedings of the 28th IEEE EMBS Annual International Conference, New York, NY, August 30-September 3, 2006, pp. 5755–5758 (2006)Google Scholar
  8. 8.
    Rovner, S.L.: Chemical ’naming’ method unveiled. Chem. & Eng. News 83, 39–40 (2005)CrossRefGoogle Scholar
  9. 9.
    Adam, D.: Chemists synthesize a single naming system. Nature 417(369) (2002)Google Scholar
  10. 10.
    Lovász, L., Plummer, M.D.: Matching theory. North-Holland Mathematics Studies, vol. 121. North-Holland Publishing Co., Amsterdam (1986); annals of Discrete Mathematics 29zbMATHGoogle Scholar
  11. 11.
    Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to algorithms, 2nd edn. MIT Press, Cambridge (2001)zbMATHGoogle Scholar
  12. 12.
    Bentley, J.L.: Multidimensional binary search trees used for associative searching. Communications of the ACM 18(9), 509–517 (1975)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Rafael ördög
    • 1
    • 2
  • Vince Grolmusz
    • 1
    • 2
  1. 1.Protein Information Technology Group Department of Computer ScienceEötvös UniversityBudapestHungary
  2. 2.Uratim Ltd., H-4400NyíregyházaHungary

Personalised recommendations