PERSISTENT HOMOLOGY ON PHONOLOGICAL DATA: A PRELIMINARY STUDY
Persistent Homology for biomolecular systems...Persistent Homology for biomolecular systems Möbius...
Transcript of Persistent Homology for biomolecular systems...Persistent Homology for biomolecular systems Möbius...
Guowei Wei
Department of Mathematics
Center for Mathematical Molecular Biosciences
Michigan State University
http://www.math.msu.edu/~wei
SIAM Life Conference
Charlotte, August, 2014
Grant support:
NIH R01GM, NSF DMS (Focus Research Group), CCF, IIS Michigan State University
Persistent Homology for biomolecular systems
Möbius Strips (1858) Klein Bottle (1882)
Trefoil Knot
Classical topological objects
Torus Double Torus
Sphere
Topological invariants: Betti numbers
0 is the number of connected components.
1 is the number of tunnels or circles.
2 is the number of cavities or voids.
Circle TorusPoint Sphere
0
0
1
2
1
0
0
1
1
2
1
0
1
0
1
2
1
0
1
2
1
2
1
0
Topological invariants: Euler characteristic
c =V -E +F -C = -1( )k
åk
bk
Network topology
(Zhang et al., Nature, 452, 198 (2008))
Topology in nano-bio systems
Mug Doughnut
Opportunities, challenges and promises
Challenges with topological methods:
Geometric methods are often inundated with too
much structural detail.
Topological tools incur too much reduction of
original geometric information.
Topology is hardly used for quantitative prediction.
Opportunities from topological methods:
New approach for big data characterization and classification.
Dramatic reduction of dimensionality and data size.
Applicable to a variety of fields.
Promises from persistent homology:
Embeds geometric information in topological invariants.
Bridges the gap between geometry and topology.
Vietoris-Rips complexes of planar point sets
dSvvvvdSS R },,),(|{)(V 2121 σεσε
Simplicial complex:
0-simplex 1-simplex 2-simplex 3-simplex
kk
k
kk
kk
kk
H
BZ
H
B
Z
Rank
Im
Ker
1
ki
k
i
ik
k vvvv ,...,,...,,)1( 10
0
σ
k
i
i
ic
Simplicial complex:
0-simplex 1-simplex 2-simplex 3-simplex
Boundary operator:
Homology
Frosini and Nandi (1999),
Robins (1999),
Edelsbrunner, Letscher and
Zomorodian (2002),
Kaczynski, Mischaikow and Mrozek
(2004),
Zomorodian and Carlsson (2005),
Ghrist (2008),
……
k-chain:
Chain group: ),( 2ZKCk
Persistent homology is created through a
filtration process
r=1.0
r=4.0
r=2.8r=2.0
r=1.4
Distance filtration:
2
2
exp1
ij
ij
rM
Matrix filtration:KKKKK m 2100
Intrinsic
bar
(Xia, Feng, Tong & Wei, 2014)
0
1
2
'
22
2
1
/1
j jLN
A
AE
Persistent homology prediction of
fullerence heat formation energies
(Xia, Feng, Tong & Wei, 2014)
Persistent homology prediction of
fullerence isomer total strain energies
CC=0.96
C44
2/1 βLE
C40
CC=0.95
(Xia, Feng, Tong & Wei, 2014)
Topological fingerprints of an alpha helix
(Xia & Wei, IJNMBE, 2014)
Protein topological fingerprints
Beta sheet Alpha helix
All atom representation
vs
Coarse-grained model
(Xia & Wei, IJNMBE, 2014)
Topological fingerprints of beta barrel
(Xia & Wei, IJNMBE, 2014)
Topology-function relationship of
protein flexibility
j
jLA 11
2
2
exp1
ij
ij
rM
j
ij
i
rB
2
2
exp1
(Xia & Wei, IJNMBE, 2014)
Topology-function relationship of
protein unfolding
j
jLE 1/1 β
1I2T
(Xia & Wei, IJNMBE, 2014)
Microtubule analysis with cryo-EM data
EMD_1129Molecular structure
fitted with tubulin 1JFF
Helix backbone
(Xia & Wei, 2014)
Topological fingerprints at different noise levels
Original data Ten-iteration
denoising
Twenty-iteration
denoising
Forty-iteration
denoising
(Xia & Wei, 2014)
Persistent homology aided structure prediction
Processed original
data
Fitted with one-
type of tubulins
Fitted with two-
types of tubulins
(Xia & Wei, 2014)
Concluding remarks
Introduced molecular topological fingerprints. Introduced topology-function relationship. Introduced persistent-homology based quantitativepredictions of protein flexibility, protein folding andmicrotubule composition.
P Bates
(MSU)
G Wang
(VPI)
N Baker
(PNNL)
Y Tong
(MSU)
Y Zhou
Colorado S
S.N. Yu
Boston
Y Wang
(MSU)J Wang
(NSF)
Zhan ChenMinnesota
Duan ChenUNCCS Yang
City U KH
W Geng
SMUShan Zhao
Alabama
Y.H. Sun
Denver
X Ye
(UKLR)Z Burton
(MSU)