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

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.