Convex and concave, Escher (1955) Voronoi Diagram Research Center, - PowerPoint PPT Presentation

Convex and concave, Escher (1955)

Voronoi Diagram Research Center, Hanyang Univ., Korea 2

Voronoi Diagram, Biogeometry, and Geometric Modeling for Molecular Biology Deok-Soo Kim Hanyang University Dept. of Industrial Engineering Voronoi Diagram Research Center 2011. 9.7. UK-Korea Geometric Modeling Workshop British Embassy, Seoul, Korea

Human Cell/Chromosome

Gene vs. DNA

Decoding The Book of Life (June 26, 2000) Bill Clinton, J Craig Venter (Celera Genomics) and Dr. Francis Collins (National Institute of Health) East Room of the White House

Protein • Organic compounds in living bodies • Essential parts in biological processes • Functions – Antigen, antibody – Enzyme – Muscle Tyrosine Kinase(1XBA) + Gleevec TK: key protein of leukemia Voronoi Diagram Research Center, 7 Hanyang Univ., Korea

Shape vs. Function Glob lobula lar Irr rregula lar Ch Chain Long sti Lo stick Hormone (1ZNI) Structural protein (1CGD) Transport (1MBC) Regulatory protein Muscle contraction Enzyme (1GMH) Antibody (1NBV) (1CMB) (1RFQ) Voronoi Diagram Research Center, Hanyang Univ., Korea 8

Protein  one or more chains of amino acids Peptide tide bond bond H H H H O O Amino Carboxyl Amino Carboxyl group group group group H N C H H N C H C O C O H C H C H H H R-group S H R-group H 2 O Alanine(ALA, A) Cysteine(CYS, C) ISVD 2011 9 Voronoi Diagram Research Center Voronoi Diagram Research Center, Hanyang Univ., Korea

Protein Structure  Quaternary structure  Primary structure  Tertiary structure  Secondary structure LYS LYS GLY GLY LEU VAL ALA HIS α - helix β - pleated sheet ISVD 2011 10 Voronoi Diagram Research Center Voronoi Diagram Research Center, Hanyang Univ., Korea

Protein Data Bank (PDB) http://www.rcsb.org/pdb Public repository of protein structures  45,213 structures (Aug. 20, 2007)  11 Voronoi Diagram Research Center, Hanyang Univ., Korea

VD(P ) vs. Delaunay Trian. (DT) DT is a simplicial complex! Voronoi Diagram Research Center, Hanyang Univ., Korea 12

Voronoi Diagram of Disks (2D) Deok-Soo Kim, Donguk Kim, and Kokichi Sugihara, Voronoi Diagram of a circle set from Voronoi Diagram of a point set: II. Geometry , Computer Aided Geometric Design , Vol. 18, pp. 563-585, 2001. Deok-Soo Kim, Donguk Kim, and Kokichi Sugihara, Voronoi Diagram of a circle from Voronoi Diagram of a point set: I. Topology , Computer Aided Geometric Design , Vol. 18, pp. 541-562, 2001. Voronoi Diagram Research Center, Hanyang Univ., Korea 13

Voronoi Diagram of Balls (3D) Deok-Soo Kim, Youngsong Cho, and Donguk Kim, Euclidean Voronoi diagram of 3D balls and its computation via tracing edges , Computer-Aided Design , Vol. 37, No. 13, pp. 1412-1424, 2005. Voronoi Diagram Research Center, Hanyang Univ., Korea 14 Donguk Kim and Deok-Soo Kim, Region-Expansion for the Voronoi

Quasi-triangulation (QT)  Dual structure of Voronoi diagram of spheres [ Kim, Sugihara et al. (2006) CAD ] Deok-Soo Kim, Donguk Kim, Youngsong Cho, and Kokichi Sugihara, Quasi-Triangulation and Interworld Data Structure in Three Dimensions , Computer-Aided Design , Vol. 38, No. 7, pp. 808-819, 2006. Deok-Soo Kim, Youngsong Cho, and Kokichi Sugihara, Quasi-worlds and Quasi-operators on Quasi-triangulations , Computer-Aided Design , Vol. 42, No. 10, pp. 874-888, , 2010. Voronoi Diagram Research Center, Hanyang Univ., Korea

Anomaly in QT 2 tetrahedra share 3 faces. • same volumes • one positive vol. and one negative vol. Voronoi Diagram Research Center, Hanyang Univ., Korea 16

Anomaly in QT 2 tetrahedra share 2 faces. • same volumes • one positive vol. and one negative vol. Voronoi Diagram Research Center, Hanyang Univ., Korea 17

Anomaly in QT 2 tetrahedra share 4 faces. • same volumes • one positive vol. and one negative vol. Voronoi Diagram Research Center, Hanyang Univ., Korea 18

Disconnected Edge-graph in VD Big-world and Small-world Voronoi Diagram Research Center, Hanyang Univ., Korea

Geometric Constructs for Molecule b -ball b -ball b -ball b -ball b -ball b -ball b -ball b -ball b -ball b -ball b -ball b -ball b -ball b -ball Voronoi diagram Quasi-triangulation Beta-complex Beta-shape Deok-Soo Kim, Youngsong Cho, Kokichi Sugihara, Joonghyun Ryu, and Donguk Kim, Three-dimensional Beta-shapes and Beta-complexes via Quasi-triangulation , Computer-Aided Design , Vol. 42, Issue 10, pp. 911-929, , 2010. Deok-Soo Kim, Jeongyeon Seo, Donguk Kim, Joonghyun Ryu, and Cheol-Hyung Cho, Three-Dimensional Beta Shapes , Computer-Aided Design , Vol. 38, No. 11, pp. 20 1179-1191, 2006. ISVD 2011 Voronoi Diagram Research Center

Applications - Molecular surface(1ous) (Pentium IV, 3.2 GHz, 1 GB RAM) (Realtime computation and visualization) Voronoi Diagram Research Center, Hanyang Univ., Korea 21

Lee-Richards (Accessible) model  Offset model of a van der Waals model  Beta-shape  β -vertex  face  β -edge  edge  β -face  vertex [PDB code: 1xbb] ISVD 2011 22 Voronoi Diagram Research Center

Connolly model  Blended model of a van der Waals model  Beta-shape  β -vertex  contact patch  β -edge  rolling patch  β -face  link patch [PDB code: 1orx] Joonghyun Ryu, Rhohun Park, and Deok-Soo Kim, Molecular Surfaces on Proteins via Beta Shapes , Computer-Aided Design , Vol. 39, No. 12, pp. 1042-1057, 2007. Joonghyun Ryu, Youngsong Cho, and Deok-Soo Kim, Triangulation of molecular surfaces , Computer-Aided Design , Vol. 41, No. 6, pp. 463-478, 2009. ISVD 2011 23 Voronoi Diagram Research Center

RasMol vs. BetaMol Molecular surface from RasMol (v2.7.3) Molecular surface from BetaMol ISVD 2011 24 Voronoi Diagram Research Center Voronoi Diagram Research Center, Hanyang Univ., Korea

Interior Void  Empty space in a molecule  Beta-shape [PDB code: 1xbb] ISVD 2011 25 Voronoi Diagram Research Center

26 Voronoi Diagram Research Center, Hanyang Univ., Korea

Union of N spheres  (N, m) m-fold intersections  2 N – 1 terms in the full expansion  Exponential # Terms  Experimental report – A 31-mer chain • The highest order term: 17-fold intersect • Over 80,000 terms of 8-fold intersect – C200 chain • Impossible to compute – Dodd and Theodorou, Analytical treatment of the volume and surface area of molecules formed by an arbitrary collection of unequal spheres intersec, Molecular Physics 72(6) (1991) 1313-1345  b -complex: Exponential  Linear # Terms! ISVD 2011 27 Voronoi Diagram Research Center Voronoi Diagram Research Center, Hanyang Univ., Korea

van der Waals Volume (2D)  Volume of the union of atoms  Beta-complex  β -face  interior triangle  β -edge  apex-exposed triangle  β -vertex  exposed atomic occupation Deok-Soo Kim, Joonghyun Ryu, Hayong Shin, and Youngsong Cho, Beta-decomposition for the Volume and Area of the Union of Three-dimensional Balls , Journal of Computational Chemistry , (under revision), 2011. ISVD 2011 28 Voronoi Diagram Research Center

Computation times of volume and area 60 60 V ol(β) Area(β) 50 50 V ol(v (vdw) Area(v (vdw) sec.) 40 40 ime(milli se 30 30 20 20 T 10 10 0 0 1000 1000 2000 2000 3000 3000 4000 4000 5000 5000 # of a # o atoms Voronoi Diagram Research Center, Hanyang Univ., Korea 29

Sphericity Voronoi Diagram Research Center, Hanyang Univ., Korea 30

Representative proteins 1lf1 1lf1 2i49 2i49 1t4 1t4q 2gpo 2g 2op6 2o 2yz1 2y 1zrs 1zr 1iz9 1iz9 2h2r 2h2r 1e 1eai 2o 2ol7 l7 (0. 0.716) (0. 0.672) (0. 0.646) (0. 0.630) (0. 0.593) (0. 0.563) (0. 0.550) (0. 0.525) (0. 0.511) (0. 0.467) (0. 0.402) Voronoi Diagram Research Center, Hanyang Univ., Korea 31

Pocket Extraction Concept 32 Voronoi Diagram Research Center, Hanyang Univ., Korea

Pocket  Blind hold with one opening  Beta-shape [PDB code: 1xbb] ISVD 2011 33 Voronoi Diagram Research Center

Tunnel  Hole penetrating a molecule  Beta-complex & Quasi-triangulation [PDB code: 1jd0] ISVD 2011 34 Voronoi Diagram Research Center

Protein Docking  • Energy terms in Docking Computation     A B    ij ij   E ( van der Waals )  12 6  r r   vdW ij ij q q  Nonbonded  i j ( electrosta tic )   interactions r elecrostat ic ij     C D    ij ij   ( H - bond )   12 10  r r    H bonds ij ij    2 K ( r r ) ( bond length ) r 0 bonds      Bonded 2 K bond angle ( ) ( )  0 interactions angles V       n ( 1 cos( n )) ( dihedral ) 2 dihedrals Chong-Min Kim, et al., Protein-Ligand Docking Based on Beta-shape , Transactions on Computational Science , LNCS 6290, pp. 123-138, 2010. Deok-Soo Kim, et al., BetaDock: Shape-priority • Lennard-Jones potential model Docking Method Based on Beta-complex , Journal of • This example uses the Biomolecular Structure & Dynamics , Vol. 29, Issue 1, pp. • conjugate gradient method only. 219-242, , 2011. 35 Voronoi Diagram Research Center, Hanyang Univ., Korea

Placement of ligand into pocket  Superposition using SVD 36 Voronoi Diagram Research Center, Hanyang Univ., Korea 36