Mechanics Inspired Bioinformatics: Predicting the Function of Eukaryotic

Scaffold/Matrix Attachment Region (SMAR) by Single Molecule

DNA Mechanics

International Workshop on Continuum Modeling of Biomolecules Sept. 14-16, 2009 in Beijing, China

Zhong-can Ou-YangInstitute of Theoretical Physics Chinese Academy of SciencesBeijing 100080, oy@itp.ac.cn

Outline:

I. Stretching single molecule DNA

II. Mechanics-inspired Bioinformatics :An example S/MARs on Eukaryotic Chromosome, predicting the location and function

In the past decadePhysical techniques such as hydrodynamic drag [4], magnetic beads [5], optical tweezers [6], glass needles [7] and AFM [8,9] offer the opportunity to study DNA/RNA and p

rotein mechanics with single molecules.

[4] J. T. Perkins, D. E. Smith, R. G. Larson, S. Chu, Science 268 (1995) 83-87

[5] S. B. Smith, L. Finzi, C. Bustamantl, Science 258 (1992) 1122-1126

[6] S. B. Smith, Y. Cui, C. Bustmantl, Science 271 (1996) 795-799

[7] P. Cluzel et al., Science 271 (1996) 792-794

[8] M . Rief, H. C.-Schauman, H. E. Gaub, Nat. Struct. Biol. 6 (1999) 346-349

[9] David J. Brockwell et al., Nat. struct. Biol. 10 (2003) 731

I. Stretching single molecule DNA

Stretching double-stranded DNA can be treated as a uniform polymer

Zhou, Zhang, Ou-Yang, PRL, 82, 4560(1999)

* Two classical models:

** Freely jointed chain (FJC)

, Kuhn segment 2

coth 1

3 for low force

1 for high force

p

B

B

B

L Nb b l

z L fb K T x x x

bf K T

K T fb

L ff L

** Worm-like chain (WLC)

2

0

2

1

2

1 41

4 1

2 3 for low force

L

B p

B

p

p B

E K Tl k ds

K T zf

l Lz L

z L l K T f

As a Hookian spring with Hooke's constant

5

3 2

10 pN nm 10 m

DNA B pK K T l

L

Zhou, Zhang, Ou-Yang, PRL, 82, 4560(1999)

* Introduction of a new structural parameter, the folding angle .

* Without consideration of force-induced melting and nick.

s

• Our Model

2Rb s

LJU

0r

2R

• Mathematically

* Two backbones ,

* Central axis ,

1r s 2 1 2r s r Rb s

1 21 2r s r r

t dr ds

11

22

cos sin

cos sin

drt t n

dsdr

t t nds

n b t

• Bending energy2 2 2 2

41 220 0

sin2

L L

b

dt dt dt dE ds ds

ds ds ds ds R

* decomposed into the bending energy of central axis plus folding energy

* Introduction of a new structural parameter, the folding angle .

* Without consideration of force-induced melting and nick.

s

• Our Model

2Rb s

LJU

0r

2R

16 base-pair stacking potentials

* The asymmetric Lennard-Jones potential ensures relaxed DNA in B-form, a right-handed double-helix.

*

* is harmonic in low force/extension regimes. (FJC and WLC)

* strongly prevent right-handed overtwist, weakly so for left-handed one and allow a torque-induced B-to-Z-form transition in dsDNA

thermal energy stable dsDNABK T

Biophys. J. 78(2000) 1979-1987

12 60 0

12 6

0 0

0

cos 2cos , 0

cos cos 2 cos cos , 0

62

U

• The Potential of External Force F

0

0

0

cos

cos

L

L

f

F fz

R t ds

E F R f t z ds

• The Energy of External Torque

0

00

2

sin

1

t

L

E Lk

z t dt dsds

R z t

T

T

0

,

b LJ f t

L

e

E E E E E

r s ds r t

，

2

2e

m dr drA r V r

ds ds

• Total Elasticity Energy

f

,t

i

Particle moves in a field A and a potential V if look s as time t

• Polymer Dynamics and Path Integral Method

* Partition function:

, ; ,0

1 , Green function is

, ; , exp ,

f f f i i i i

B

r s

er s

Z L dr r G r L r r dr

K T

G r s r s r ds r s

D

* “Schrödinger equation”

2 2

, , ; , ,2 2 2

r r rA A Ar s dr G r s r s s s V

s m m m m

* , Bs t K T i

* z F f

lnBF K T Z* Free energy:

• Extension/Force of 0T

2 24

0* 2 2 2

cossin

4 4P

P P B B

f lt z

s l t l K T K T R

is Hermitian, it is real Schrödinger Eq.

Extension:

00

2

0 0

0

cos

cos

normalized ground-state eigengfunction

Lz t z ds

L t z dtd

* * ,P B P Bl K T l K T

Extension of Torsionally Relaxed DNA

Zhou, Zhang, Ou-Yang, PRL, 82, 4560(1999)

Distribution of Folding of Backbones

2

0 ,P t dt

Above calculationss are interesting for pure theoretical physicists but not for biologists and IT scientists. Both they are interested in the information and function hided in their sequence (AGCT….). The Bioinformatics is based on pure statistic mathematics, our propose is a Mechanic

s-Inspired Bioinformatics.

4 types of nucleotides: Adenine, Guanine, Thymine, CytosineWatson-Crick base pair: A-T, G-CIntrinsic right-handed helix (torsional state)B-DNA: uniform, sequence-independent

4-letter text:…ATTTTAATGTCATGATAAAGTTACTTCCTTTTTTTTTAAGTTACTTCTATAATATATGTAAATTACTTTTAATCTCTACTGAAATTACTTTTATATATCTAAGAAGTATTTAGTGAAATCTAAAAGTAATTTAGATATAATATAAAAGTAATTTGTATTTTTTTCATCAAAATATAATCATGTGAGACCTTGTTATAAAGATTTAA…

II. Mechanics-inspired Bioinformatics :An example S/MARs on Eukaryotic Chromosome, predicting the location and function

DNA: ~ centimeters (human cell 2meters)

DNA in lily cell 30 meters. Nucleus: ~ microns compaction ratio: ~1/8000 DNA must undergo

significant mechanical force in the nucleus

The elastic response is vital for DNA

Elasticity Plays the Key Role… !

compaction ratio: ~ 1/8000 considerable force exerted

on DNA (stretching,

bending and twisting) S/MARs:

topologically independent

domains

basement of chromatin loops S/MAR(Scaffold/Matrix Attachment Region)

Chromosome AssemblyChromatin Loop Model

Chirality Variable bubble

cruciform

H-Bond Broken

Structure Heterogeneity Induced by Mechanical Structure Heterogeneity Induced by Mechanical Force:Force:

Secondary StructuresSecondary Structures

How to predict SMAR location and function ? it’s difficult in the framework of conventional

bioinformatics methods because there is very little similarity among SMAR sequences, thus

sequence comparison cannot work well.

S/MARs have been observed to adopt noncanonical DNA structures, bubble configuration (stress-induced unwound elements * )

* Bode J., et al., Science, 1992, 255: 195-197

Standard B-form DNA

Local bubble

The unwinding stress can induce the formation of local bubbles

Lk=Wr+Tw, writhing number—axis self-linking number, Tw—inter-winding number of two strands

topological parameters for ds-DNALk : linking number, number of helical turns when DNA is

imposed in planar conformation

Lk0 : linking number of relaxed ds-DNA. Lk0= N/10.5Tw : twisting number, number of helical turns

Wr : writhing number, coiling times of the central axis

(supercoiling). for planar conformation, Wr = 0

σ: superhelical density, defined as (Lk – Lk0)/ Lk0

σ< 0, negative supercoiling ;σ> 0, positive supercoiling

For eukaryotes, σ~ - 0.06σ* Lk0 = Lk – Lk0 = △Tw (r, r’) + △Wr (r)

Can we make the prediction on bubbles (S/MARs) by taking account of the unwinding stress, i.e., the energy correspond

ing toσ (~ -0.06 ) ?

Bubble Formation is Sequence Dependent Benham Model

Bauer WR, Benham CJ., J Mol Biol. 1993, 234(4):1184-96.

2N configurations{…10111111100…}

local bubble

a : initiation energy of bubble formation

jn = 0 … base paried

jn = 1 … base unparied

j : rewinding angle of the denatured region

ATb GCb : base unparing energy

A : 10.5 bp per helical turn of B-DNA

: superhelical densityσ

N

jjnn

1

N

j

jjn

1 2

A

nTw

total change in twisting turns upon bubble formation

Benham Model

twisting energy of DNA

interwinding energy of the two strands in bubble regions

unpairing energy in bubble (sequence dependent )

initiation energy of bubble formation from the intact helix,The boundary energy between bubble and B-form, interface energy

4321 HHHHH

21

1( )

2 rH K Lk

j

jjnc

H 22 2

j

jjbnH 3

j

jj nnaarH )1( 14

total energy

0 0, :r Lk Lk cLk L onsk Tw Tw t

1

2

2

n

K

CnLk

N

jjjsbarnC

nnLkKE

1

2

2

2

1

22

1

N

jjj ssr

111=å j

j

n s

Base-stackingEnergy form:

dE/dt=0

jj

jj psn

Stress-induced melting profile

H （ n ） , Hj （ n ） calculated by transfer matrix method (e.g., circular DNA)

N

n

nj

N

jiij xnHMMTr

0

1'

N

n

nN

jj xnHMTr

0

1

0

Constrains on specific sites can be realized as following ：

(sk= 0)

sj=0 sj=1

8.10a 1molkcal

58.3c 21 radmolkcal

2350K NRT

255.0ATb 1molkcal

301.1GCb 1molkcal

5.10A

Different unpairing e

nergy

The following calculation is indeed insensitive to the parameters except the difference bet

ween bAT and bGC

Unpairing Probability ProfileBenham Model

M. Li, Z.C. Ou-Yang, Thin Solid Film, 499:207-212 (2006)

{ }

{ }

( 1)Hj

sj H

s

e n

pe

Unpairing Probability for any base pair

M.Li, Z.C. Ou-Yang, J. Phys:Condens. Matter 17 S2853-S2860

(2005)Nucleosome:

Core of 8 histone molecules:2(H3—H4—H2A—H2B)—link H1

Drosophila melanogaster: Real DNA Sequence: Histone Gen

e Cluster

Drosophila melanogaster: Real DNA Sequence: Histone Gene Cluster

5- —H3—H4—H2A—H2B—H1— -3MAR MAR

Arrow: transcriptional direction

Experimentally find a SMAR for the cluster of the above five geneswith known DNA sequence (X14215, NCBI), calculation with 2 repeats

The position of the two distinct peaks coincide with the identified S/MARt DB (SM0000037)

S/MAR identified between H1 and H3

The two SMARs define a single structure unit

Result shows nicely: Where Are They ?

Flanking SMARs as barriers to retain the unwinding stress Possible LRAE: SMARs fixation onto the matrix induces unpairing events elsewhere Function Unit:the new unpairing events may play a role in transcriptional termination between H4-(weaker SMAR ?)

5—H3—H4—H2A—H2B—H1—3

Take out Flanking SMARs, find new bubbles: Why They Are There? Long Range Allosteric Effect (LRAE) play the role…

Unwinding stress induces strong bubbles (SMARs) (strong) SMARs may inversely function in gene regulation by protecting the unwinding stress on the chromatin loop chromatin loop as both structure and function unit Mechanics analysis is hopefully a new approach complementary to sequence analysis, especially on the study of DNA function

Summary

Thanks for your

attention !

topological parameters for ds-DNA

Lk : linking number, number of helical turns when DNA is imposed in planar conformation Lk0 : linking number of relaxed ds-DNA. Lk0= N/10.5 Tw : twisting number, number of helical turns Wr : writhing number, coiling times of the central axis (supercoiling). for planar conformation, Wr = 0 σ: superhelical density, defined as (Lk – Lk0)/ Lk0

σ< 0, negative supercoiling ;σ> 0, positive supercoiling

For eukaryotes, σ~ - 0.06 σ* Lk0 = Lk – Lk0 = △Tw (r, r’) + △Wr (r)

DNA Topology : Ribbon Model

3

1 ( ) ( ) [ ( ) ( )]

4 ( ) ( )

dr s dr s r s r sLk

r s r s

12

( )( )

1]

2

de se s

dsTw ds

3

1 ( ) ( ) [ ( ) ( )]

4 ( ) ( )

dr s dr s r s r sWr

r s r s

Circular dsDNA: topological invariant Lk (r, r’) = Tw (r, r’) + Wr (r)

Central axis of dsDNA

one strand

local frame

Ribbon (r, r’) : central axis + one strand

Adapted from: Wang, J.C. 1991. DNA topoisomerases: why so many? Journal of Biological Chemistry 266:6659-6662.

Some geometrical parameters to characterize ds-DNA

2Rb s

LJU

0r

2R

The double-helical DNA taken as a flexible ladder with rigid rungs of fixed length 2R. Central axis R0 (s) , its arc length denoted as s. The tangent vector of R0 (s) denoted as t The two strands R1(s), R2 (s). The tangent vector of R1(s), R2(s) denoted as t1 , t2 . The distance between nearest rungs: along R1(s) or R2(s): r0 , fixed and along R0(s): U , variable The folding angle between t and t1 (or t2): . ~ 57o for standard B-DNA

3

the coiling number of the central axis.

1

4s s

Wr

r s r s r s r sWr dsds

r s r s

0 linking number of natural (standard) B-DNA

interwinding times of the two strands

link number of twisted DNA

Lk

Lk

1 0 0

the times of one strand coiling around the central axis

1 1 sin( , )

2 2

L L

Tw

dbTw r r t b ds ds

ds R

3

1 ( ) ( ) [ ( ) ( )]

4 ( ) ( )

dr s dr s r s r sLk

r s r s

a word about twist: given the link shown below, the twist tells us basically which component ‘wraps  around’ which.                                           

 We need three vectors to parameterize a surface:- Correspondence vector: pointing from one curve to the other and tracing out the surface between the two curves).- T: unit tangent vector at x- V: unit vector perpendicular to T but lies on the surface defined by correspondence vector.

Now we can define twist more rigorously:

Definition:

                                                     

                                         

( )r s

the number of Complete Revolutions of one DNA strand about the other

the total number of turns of the DNA duplex itself

total number of turns about the superhelical axis itself

Central axis of dsDNA

one strand

local frame

Central axis of dsDNA

one strand

local frame

compaction ratio: ~ 1/8000 considerable force exerted

on DNA (stretching,

bending and twisting) S/MARs:

topologically independent

domains

basement of chromatin loops S/MAR(Scaffold/Matrix Attachment Region)

Chromosome AssemblyChromatin Loop Model