Operations preserving the global rigidity of graphs and frameworks in the plane

AbstractA straight-line realization of (or a bar-and-joint framework on) graph G in Rd is said to be globally rigid if it is congruent to every other realization of G with the same edge lengths. A graph G is called globally rigid in Rd if every generic realization of G is globally rigid. We give an algorithm for constructing a globally rigid realization of globally rigid graphs in R2. If G is triangle-reducible, which is a subfamily of globally rigid graphs that includes Cauchy graphs as well as Grünbaum graphs, the constructed realization will also be infinitesimally rigid.Our algorithm relies on the inductive construction of globally rigid graphs which uses edge additions and one of the Henneberg operations. We also show that vertex splitting, which is another well-known operation in combinatorial rigidity, preserves global rigidity in R2

Analyzing the simplicial decomposition of spatial protein structures

<p>Abstract</p> <p>Background</p> <p>The fast growing Protein Data Bank contains the three-dimensional description of more than 45000 protein- and nucleic-acid structures today. The large majority of the data in the PDB are measured by X-ray crystallography by thousands of researchers in millions of work-hours. Unfortunately, lots of structural errors, bad labels, missing atoms, falsely identified chains and groups make dificult the automated processing of this treasury of structural biological data.</p> <p>Results</p> <p>After we performed a rigorous re-structuring of the whole PDB on graph-theoretical basis, we created the RS-PDB (Rich-Structure PDB) database. Using this cleaned and repaired database, we defined simplicial complexes on the heavy-atoms of the PDB, and analyzed the tetrahedra for geometric properties.</p> <p>Conclusion</p> <p>We have found surprisingly characteristic differences between simplices with atomic vertices of different types, and between the atomic neighborhoods – described also by simplices – of different ligand atoms in proteins.</p

Being a binding site: Characterizing residue composition of binding sites on proteins

The Protein Data Bank contains the description of more than 45,000 three-dimensional protein and nucleic-acid structures today. Started to exist as the computer-readable depository of crystallographic data complementing printed articles, the proper interpretation of the content of the individual files in the PDB still frequently needs the detailed information found in the citing publication. This fact implies that the fully automatic processing of the whole PDB is a very hard task. We first cleaned and re-structured the PDB data, then analyzed the residue composition of the binding sites in the whole PDB for frequency and for hidden association rules. Main results of the paper: (i) the cleaning and repairing algorithm (ii) redundancy elimination from the data (iii) application of association rule mining to the cleaned non-redundant data set. We have found numerous significant relations of the residue-composition of the ligand binding sites on protein surfaces, summarized in two figures. One of the classical data-mining methods for exploring implication-rules, the association-rule mining, is capable to find previously unknown residue-set preferences of bind ligands on protein surfaces. Since protein-ligand binding is a key step in enzymatic mechanisms and in drug discovery, these uncovered preferences in the study of more than 19,500 binding sites may help in identifying new binding protein-ligand pairs

DECOMP: A PDB decomposition tool on the web

The protein databank (PDB) contains high quality structural data for computational structural biology investigations. We have earlier described a fast tool (the decomp_pdb tool) for identifying and marking missing atoms and residues in PDB files. The tool also automatically decomposes PDB entries into separate files describing ligands and polypeptide chains. Here, we describe a web interface named DECOMP for the tool. Our program correctly identifies multi­monomer ligands, and the server also offers the preprocessed ligand­protein decomposition of the complete PDB for downloading (up to size: 5GB

Gráfok és algoritmusok = Graphs and algorithms

A kutatás az elvárt eredménnyel zárult: tekintélyes nemzetközi konferenciákon és pubikációkban hoztuk nyilvánosságra az eredményéket, ideértve a STOC, SIAM és IEEE kiadványokat is, valamint egy könyvet is. A publikációk száma a matematikában elég magas (74). Ez nemzetközi összehasonlításban is kiemelkedő mutató a támogatás összegére vetítve. A projektben megmutattuk, hogy a gráfelmelet és a diszkrét matematika eszköztára számos helyen jól alkalmazható, ilyen terület a nagysebességű kommunikációs hálózatok tervezése, ezekben igen gyors routerek létrehozása. Egy másik terület a biológiai nagymolekulákon definiált gráfok és geometriai struktúrák. | The research concluded with the awaited results: in good international conferences and journals we published 74 works, including STOC conference, SIAM conferences and journals and one of the best IEEE journal. This number is high above average in mathematics research. We showed in the project that the tools of graph theory and discrete mathematics can be well applied in the high-speed communication network design, where we proposed fast and secure routing solutions. Additionally we also found applications in biological macromolecules