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)