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

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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

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Shape vs. Function

Enzyme (1GMH) Glob lobula lar Antibody (1NBV) Ch Chain Muscle contraction

(1RFQ)

Structural protein

(1CGD)

Hormone (1ZNI) Regulatory protein

(1CMB)

Transport (1MBC) Irr rregula lar Lo Long sti stick

Voronoi Diagram Research Center ISVD 2011

Voronoi Diagram Research Center, Hanyang Univ., Korea

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Protein

Peptide tide bond bond Carboxyl group

H O

Amino group H

H2O Cysteine(CYS, C)

R-group Amino group

C O H C N H H H H C H S

R-group Carboxyl group

H O C O H C N H H H H C Alanine(ALA, A)

one or more chains of amino acids

Voronoi Diagram Research Center ISVD 2011

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Protein Structure

 Secondary structure  Tertiary structure  Quaternary structure

LYS LYS GLY GLY LEU VAL ALA HIS

α - helix β - pleated sheet

 Primary structure

Voronoi Diagram Research Center, Hanyang Univ., Korea

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Protein Data Bank (PDB)

http://www.rcsb.org/pdb

 Public repository of protein structures  45,213 structures (Aug. 20, 2007)

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VD(P) vs. Delaunay Trian. (DT)

DT is a simplicial complex!

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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.

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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. Donguk Kim and Deok-Soo Kim, Region-Expansion for the Voronoi

Voronoi Diagram Research Center, Hanyang Univ., Korea

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.

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Anomaly in QT

2 tetrahedra share 3 faces.

• same volumes
• one positive vol. and one negative vol.

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2 tetrahedra share 2 faces.

• same volumes
• one positive vol. and one negative vol.

Anomaly in QT

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2 tetrahedra share 4 faces.

• same volumes
• one positive vol. and one negative vol.

Anomaly in QT

Disconnected Edge-graph in VD

Big-world and Small-world

Voronoi Diagram Research Center, Hanyang Univ., Korea

Voronoi Diagram Research Center ISVD 2011

Geometric Constructs for Molecule

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Quasi-triangulation Voronoi diagram

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

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. 1179-1191, 2006.

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Applications - Molecular surface(1ous)

(Pentium IV, 3.2 GHz, 1 GB RAM) (Realtime computation and visualization)

Voronoi Diagram Research Center ISVD 2011

Lee-Richards (Accessible) model

 Offset model of a van der Waals model  Beta-shape

 β-vertex  face  β-edge  edge  β-face  vertex

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[PDB code: 1xbb]

Voronoi Diagram Research Center ISVD 2011

Connolly model

 Blended model of a van der Waals model  Beta-shape

 β-vertex  contact patch  β-edge  rolling patch  β-face  link patch

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[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.

Voronoi Diagram Research Center ISVD 2011

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RasMol vs. BetaMol

Molecular surface from RasMol (v2.7.3) Molecular surface from BetaMol

Voronoi Diagram Research Center ISVD 2011

Interior Void

 Empty space in a molecule  Beta-shape

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[PDB code: 1xbb]

Voronoi Diagram Research Center, Hanyang Univ., Korea

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Voronoi Diagram Research Center ISVD 2011

Voronoi Diagram Research Center, Hanyang Univ., Korea

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Union of N spheres

 (N, m) m-fold intersections  2N – 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!

Voronoi Diagram Research Center ISVD 2011

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

• ccupation

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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.

Voronoi Diagram Research Center, Hanyang Univ., Korea 29

Computation times of volume and area

10 10 20 20 30 30 40 40 50 50 60 60 1000 1000 2000 2000 3000 3000 4000 4000 5000 5000 # o # of a atoms T ime(milli se sec.) V

• l(β)

Area(β) V

• l(v

(vdw) Area(v (vdw)

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Sphericity

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Representative proteins

1lf1 1lf1

(0. 0.716)

2i49 2i49

(0. 0.672)

1t4 1t4q

(0. 0.646)

2g 2gpo

(0. 0.630)

2o 2op6

(0. 0.593)

2y 2yz1

(0. 0.563)

1zr 1zrs

(0. 0.550)

1iz9 1iz9

(0. 0.525)

2h2r 2h2r

(0. 0.511)

1e 1eai

(0. 0.467)

2o 2ol7 l7

(0. 0.402)

Voronoi Diagram Research Center, Hanyang Univ., Korea

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Pocket Extraction Concept

Voronoi Diagram Research Center ISVD 2011

Pocket

 Blind hold with

• ne opening

 Beta-shape

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[PDB code: 1xbb]

Voronoi Diagram Research Center ISVD 2011

Tunnel

 Hole penetrating a molecule  Beta-complex & Quasi-triangulation

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[PDB code: 1jd0]

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Protein Docking

• Lennard-Jones potential model
• This example uses the
• conjugate gradient method only.
• Energy terms in Docking Computation

) ( )) cos( 1 ( 2 ) ( ) ( ) ( ) ( )

• (

) ( ) ( E

2 2 10 12 6 12

dihedral n V angle bond K length bond r r K bond H r D r C tic electrosta r q q Waals der van r B r A

dihedrals n angles bonds r ij ij ij ij ic elecrostat ij j i vdW ij ij ij ij

     

                                

    







Nonbonded interactions Bonded interactions

Chong-Min Kim, et al., Protein-Ligand Docking Based

• n Beta-shape, Transactions on Computational Science,

LNCS 6290, pp. 123-138, 2010. Deok-Soo Kim, et al., BetaDock: Shape-priority Docking Method Based on Beta-complex, Journal of Biomolecular Structure & Dynamics, Vol. 29, Issue 1, pp. 219-242, , 2011.

Voronoi Diagram Research Center, Hanyang Univ., Korea

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Placement of ligand into pocket

 Superposition using SVD

Voronoi Diagram Research Center ISVD 2011

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BetaDock vs. AutoDock (1/2)

BetaDock AutoDock 1kzk 1oyt 1t46 RMSD = 0.16 Å RMSD = 0.61 Å RMSD = 0.04 Å RMSD = 0.73 Å RMSD = 10.81 Å RMSD = 12.19 Å

Voronoi Diagram Research Center ISVD 2011

Computation Time

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Voronoi Diagram Research Center ISVD 2011

Computation of Topology Structure

 On-the-fly  Pre-processed

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n: the number of atoms m: the number of simplexes in the quasi-triangulation k: the number of simplexes in the beta-complex

Voronoi Diagram Research Center, Hanyang Univ., Korea 40

Designed by K. Drexler (http://www.nanorex.com)

• Position & relationship of each atom
• Detect collision and interference among

atoms

• Combinatorial Explosion
• Need S/W based on systemic methodology

NanoCAD

MarkIII(k) Planetary Gear Drexler-Merkle Differential Gear

Nano Machinery

Voronoi Diagram Research Center, Hanyang Univ., Korea

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Conclusions

 Geometric biology is …

– Geometry-priority study of biology problems – Quick and good initial solution to biology problems

 Biogeometry is … – Geometric theory suitable for biology problems  Beta-complex is the kernel !!!

– Consice abstraction – Precise representation

 Unified framework for ALL problems…  Softwares are available at

– http://voronoi.hanyang.ac.kr

Beta-complex, 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

Where is it going?

 X-ray  Quantum theory  DNA  Computational mathematics

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The Sc Scream (Sk (Skrik), Edvard Munch (18 (1893)

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Collaborations Wanted

 Graphics and geometric modeling – Rendering – Meshing  Applications to biology problems

– Better application algorithms

 Applications to other problems

Voronoi Diagram Research Center ISVD 2011

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dskim@hanyang.ac.kr http://voronoi.hanyang.ac.kr

THANK YOU 감사합니다