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
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
Voronoi Diagram Research Center,
Hanyang Univ., Korea 7
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, Hanyang Univ., Korea 8
Shape vs. Function
Enzyme (1GMH)
Globular
Antibody (1NBV)
Chain
Muscle contraction (1RFQ)
Structural protein (1CGD)
Hormone (1ZNI)
Regulatory protein (1CMB)
Transport (1MBC)
Irregular Long stick
Voronoi Diagram Research Center ISVD 2011 Voronoi Diagram Research Center, Hanyang Univ., Korea 9
Protein
Peptide
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 Voronoi Diagram Research Center, Hanyang Univ., Korea 10
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
11
Protein Data Bank (PDB) http://www.rcsb.org/pdb
Public repository of protein structures
45,213 structures (Aug. 20, 2007)
Voronoi Diagram Research Center, Hanyang Univ., Korea 12
VD(P) vs. Delaunay Trian. (DT)
DT is a simplicial complex!
Voronoi Diagram Research Center, Hanyang Univ., Korea 13
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 set 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 14
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.
Voronoi Diagram Research Center, Hanyang Univ., Korea 16
Anomaly in QT
2 tetrahedra share 3 faces.
• same volumes • one positive vol. and one negative vol.
Voronoi Diagram Research Center, Hanyang Univ., Korea 17
2 tetrahedra share 2 faces.
• same volumes • one positive vol. and one negative vol.
Anomaly in QT
Voronoi Diagram Research Center, Hanyang Univ., Korea 18
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
20
Quasi-triangulation Voronoi diagram
b-ballb-ball
b-ballb-ball
b-ballb-ballb-ball
b-ballb-ballb-ballb-ball
b-ballb-ballb-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.
Voronoi Diagram Research Center, Hanyang Univ., Korea 21
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
22
[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
23
[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 Voronoi Diagram Research Center, Hanyang Univ., Korea 24
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
25
[PDB code: 1xbb]
Voronoi Diagram Research Center ISVD 2011 Voronoi Diagram Research Center, Hanyang Univ., Korea 27
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
occupation
28
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
0
10
20
30
40
50
60
0 1000 2000 3000 4000 5000# of atoms
Tim
e(m
illi sec.)
Vol(β) Area(β)
Vol(vdw) Area(vdw)
Voronoi Diagram Research Center, Hanyang Univ., Korea 31
Representative proteins
1lf1
(0.716)
2i49
(0.672)
1t4q
(0.646)
2gpo
(0.630)
2op6
(0.593)
2yz1
(0.563)
1zrs
(0.550)
1iz9
(0.525)
2h2r
(0.511)
1eai
(0.467)
2ol7
(0.402)
Voronoi Diagram Research Center ISVD 2011
Blind hold with
one opening
Beta-shape
33
[PDB code: 1xbb]
Voronoi Diagram Research Center ISVD 2011
Tunnel
Hole penetrating a
molecule
Beta-complex
& Quasi-triangulation
34
[PDB code: 1jd0]
Voronoi Diagram Research Center, Hanyang Univ., Korea
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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
0
2
0
1012
612
dihedralnV
anglebondK
lengthbondrrK
bondHr
D
r
C
ticelectrostar
Waalsdervanr
B
r
A
dihedrals
n
angles
bonds
r
ij
ij
ij
ij
icelecrostat ij
ji
vdW ij
ij
ij
ij
bondsH
Nonbonded
interactions
Bonded
interactions
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 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
36
36
Placement of ligand into pocket Superposition using SVD
Voronoi Diagram Research Center ISVD 2011 Voronoi Diagram Research Center, Hanyang Univ., Korea 37
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 of Topology Structure
On-the-fly
Pre-processed
39
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
41
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
Voronoi Diagram Research Center, Hanyang Univ., Korea 43
Voronoi Diagram Research Center, Hanyang Univ., Korea 45
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 Voronoi Diagram Research Center, Hanyang Univ., Korea 46
http://voronoi.hanyang.ac.kr
THANK YOU 감사합니다