Todd J.Taylor, Iosif I.Vaisman Abstract: A method of protein structural domain assignment using an Ising/Potts-like model on a lattice derived from the Delaunay tessellation of a protein structure is described. The method is very simple and agrees well with previously published methods. Protein Structural Domain Assignment with a Delaunay Tessellation Derived Lattice

Protein structures have been analyzed with a technique from computational geometry known as Delaunay tessellation (DT). Each amino acid is abstracted to a point and the points are then joined by edges to form a set of non-overlapping, irregular, space-filling tetrahedra each having the property that the sphere on the surface of which all four vertices reside does not contain a vertex from any other tetrahedron. The union of the surface faces of the tessellated protein forms the convex hull of the Cα point set. Surface irregularities are ‘paved over’ by long edges (20Å+) which form contacts between residue pairs that are too far apart to be ‘true’ neighbors. It is sometimes expedient therefore to impose an edge length cutoff in the DT analysis. Cα Delaunay tessellation of phosphoglycerate kinase (16pk) with no edge cutoff and with a 10Å cutoff

Structural domains: Wetlaufer (1973), Definition - continuous segment(s) of the main chain that form a compact, stable structure with a hydrophobic core and potentially could fold and function independently from the rest of the structure Delaunay-Potts: Sequence of domain labels is S={s 1,s 2, …, s N }, initialized to residue numbers. s i t+1 = s i t + U[∑ J(s i t,s j t ) ], i =1, …, N, where j varies over the Delaunay neighbors of i and U(x) = x/|x| Pick residue at random and immediately update (asynchronous updating). Iterate until shape of domain label profile meets ending 'stairstep' criteria. 1 if s j > s i and d ij ≤ r J(s i t,s j t ) = -1 if s j < s i and d ij ≤ r cutoff distance r, typically Å 0 if d ij > r Smooth in a window around residue i, replacing the label at i with the median in the window. Post-processing fine tunes assignment: no domains smaller than 40 residues, no domain boundary cuts a beta sheet. Protein domain assignment and DePot

domain 1 domain 2 Schematic of Delaunay-Potts (DePot) procedure

Example assignments and evolution of domain labels 2laodomain1domain2 Expert1-90, DALI1-89, CATH1-90, PDP1-90, DomainParser21-89, DEE1-89, DDBASE5-91, Islam1-88, SCOP1-238 DOMS1-90, DePot1-91, avhAdomain1domain2domain3domain4 Expert DALI3-86, CATH PDP3-140, DomainParser23-89, DEE DDBASE Islam SCOP3-320 DOMS DePot

same # overlapVIRand DOMS SCOP DePot Islam DDBASE DEE Domain Parser PDP CATH DALI Depot along with several other methods was tested on a set of 100 structures from three previously published domain assignment papers. The overlap score (used before in the literature) was used to measure similarity wrt expert assignments as well as two other scoring schemes, not applied to domain assignment before from the clustering literature. Performance on combined Jones, Taylor, and Veretnik test set wrt expert assignment

[1] Singh RK, Tropsha A, Vaisman II (1996) Delaunay tessellation of proteins: four body nearest-neighbor propensities of amino acid residues. J Comput Biol 3(2): [2] Taylor TJ, Vaisman II (2006) Protein structural domain assignment with a Delaunay tessellation derived lattice, Proceedings of the 3 rd International Symposium on Voronoi Diagrams in Science and Engineering. [3] Taylor WR (1999) Protein structural domain identification. Protein Eng 12: [4] Veretnik S, Bourne PE, Alexandrov NN, Shindyalov IN (2004) Toward consistent assignment of protein domains in proteins. J Mol Biol 339: [5] Holland TA, Veretnik S, Shindyalov IN, Bourne PE. (2006) Partitioning protein structures into domains: why is it so difficult? J Mol Biol. 361(3): [6] Jones S, Stewart M, Michie A, Swindells MB, Orengo C, Thornton JM (1998) Domain assignment for protein structures using a consensus approach: characterization and analysis. Protein Sci 7: [7] Okabe A (2000) Spatial tessellations : concepts and applications of Voronoi diagrams. Wiley Assignment server Acknowledgements W.R. Taylor for the DOMS method and code. Stella Veretnik for discussions regarding her work with domain assignment. NSF for funding. Selected references