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

Diszkrét és folytonos: a gráfelmélet, algebra, analízis és geometria találkozási pontjai = Discrete and Continuous: interfaces between graph theory, algebra, analysis and geometry

Sok eredmény született a gráfok növekvő konvergens sorozataival és azok limesz-objektumaival, ill. az ezek vizsgálatára szolgáló gráf-algebrákkal kapcsolatban. Kidolgozásra kerültek a nagyon nagy sűrű gráfok (hálózatok) matematikai elméletének alapjai, és ezek alkalmazásai az extremális gráfelmélet területén. Aktív és eredményes kutatás folyt a diszkrét matematika más, klasszikus matematikai területekkel való kapcsolatával kapcsolatban: topológia (a topológiai módszer alkalmazása gráfok magjára, ill a csomók elmélete), geometriai szerkezetek merevsége (a Molekuláris Sejtés bizonyítása 2 dimenzióban), diszkrét geometriai (Bang sejtésének bizonyítása), véges geometriák (lefogási problémák, extremális problémák q-analogonjai), algebra (félcsoport varietások, gráfhatványok színezése), számelmélet (additív számelmélet, Heilbronn probléma), továbbá gráfalgoritmusok (stabilis párosítások, biológiai alkalmazások)) területén. | Several results were obtained in connection with convergent growing sequences of graphs and their limit objects, and with graph algebras facilitating their study. Basic concepts for the study of very large dense graphs were worked out, along with their applications to extremal graph theory. Active and successful research was conducted concerning the interaction of discrete mathematics with other, classical areas of mathematics: topology (applications of topology in the study of kernels of graphs, and the theory of knots), rigidity of geometric structures (proof of the Molecular Conjecture in 2 dimensions), discrete geometry (proof of the conjecture of Bang), finite geometries (blocking problems, q-analogues of extremal problems), algebra (semigroup varieties, coloring of graph powers), number theory (additive number theory, heilbronn problem), and graph algorithms (stable matchings, applications in biology)

Catalytic mechanism of alpha-phosphate attack in dUTPase is revealed by X-ray crystallographic snapshots of distinct intermediates, 31P-NMR spectroscopy and reaction path modelling.

Enzymatic synthesis and hydrolysis of nucleoside phosphate compounds play a key role in various biological pathways, like signal transduction, DNA synthesis and metabolism. Although these processes have been studied extensively, numerous key issues regarding the chemical pathway and atomic movements remain open for many enzymatic reactions. Here, using the Mason-Pfizer monkey retrovirus dUTPase, we study the dUTPase-catalyzed hydrolysis of dUTP, an incorrect DNA building block, to elaborate the mechanistic details at high resolution. Combining mass spectrometry analysis of the dUTPase-catalyzed reaction carried out in and quantum mechanics/molecular mechanics (QM/MM) simulation, we show that the nucleophilic attack occurs at the alpha-phosphate site. Phosphorus-31 NMR spectroscopy (31P-NMR) analysis confirms the site of attack and shows the capability of dUTPase to cleave the dUTP analogue alpha,beta-imido-dUTP, containing the imido linkage usually regarded to be non-hydrolyzable. We present numerous X-ray crystal structures of distinct dUTPase and nucleoside phosphate complexes, which report on the progress of the chemical reaction along the reaction coordinate. The presently used combination of diverse structural methods reveals details of the nucleophilic attack and identifies a novel enzyme-product complex structure

1

The Protein Data Bank (PDB) is the largest, most comprehensive, freely available depository of protein structural information, containing more than 36 500 deposited structures. On one hand, the form and the organization of the PDB seems to be perfectly adequate for gathering in-formation from specific protein structures, by using the bibliographic ref-erences and the informative remark fields. On the other hand, however, it seems to be impossible to automatically review remark fields and journal references for processing hundreds or thousands of PDB files. We present here a family of combinatorial algorithms to solve some of these problems. Our algorithms are capable to automatically analyze PDB structural information, identify missing atoms, repair chain ID information, and most importantly, the algorithms are capable of identifying ligands and binding sites, facilitating data mining studies of proven protein-ligand binding and testing virtual docking algorithms. The algorithms can also be used for predicting flexible sub-chains of the structure