convex and concave, escher (1955)doc.gold.ac.uk/uk-korea-geomod2011/slides/08-deok-soo-kim.pdf ·...

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

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


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


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)


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

DT is a simplicial complex!


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 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


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.


Anomaly in QT

2 tetrahedra share 3 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 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.


Applications - Molecular surface(1ous)

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


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]


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.


RasMol vs. BetaMol

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


Interior Void

Empty space in a

molecule

Beta-shape

25

[PDB code: 1xbb]


26


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!


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.


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)


Sphericity


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)


32

Pocket Extraction Concept


Pocket

Blind hold with

one opening

Beta-shape

33

[PDB code: 1xbb]


Tunnel

Hole penetrating a

molecule

Beta-complex

& Quasi-triangulation

34

[PDB code: 1jd0]


35

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

qq

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.


36

36

Placement of ligand into pocket Superposition using SVD


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


Computation Time

38


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


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


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


The Scream (Skrik), Edvard Munch (1893)


Collaborations Wanted

Graphics and geometric modeling

– Rendering

– Meshing

Applications to biology problems

– Better application algorithms

Applications to other problems


[email protected]

http://voronoi.hanyang.ac.kr

THANK YOU 감사합니다

convex and concave, escher (1955)doc.gold.ac.uk/uk-korea-geomod2011/slides/08-deok-soo-kim.pdf ·...

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