Unbinding of biopolymers:statistical physics of interacting loops

David Mukamel

unbinding phenomena

• DNA denaturation (melting)

• RNA melting

• Conformational changes in RNA

• DNA unzipping by external force

• Unpinning of vortex lines in type II superconductors

• Wetting phenomena

DNA denaturation

T T

double stranded

single strands

Helix to Coil transition

…AATCGGTTTCCCC……TTAGCCAAAGGGG…

Single strand conformations: RNA folding

conformation changes in RNASchultes, Bartel (2000)

Unzipping of DNA by an external force

Bockelmann et al PRL 79, 4489 (1997)

Unpinning of vortex lines from columnar defectsIn type II superconductors

Defects are produced by irradiation with heavy ions with high energyto produce tracks of damaged material.

Wetting transition

substrate

interface

3d

2d

gas

liquid l

At the wetting transition l

One is interested in features like

Loop size distribution )(lP

Order of the denaturation transition Inter-strand distance distribution )(rP

Effect of heterogeneity of the chain

outline

• Review of experimental results for DNA denaturation

• Modeling: loop entropy in a self avoiding molecule

• Loop size distribution

• Denaturation transition

• Distance distribution

• Heterogeneous chains

Persistence length lpdouble strands lp ~ 100-200 bpSingle strands lp ~ 10 bp

fluctuating DNA

DNA denaturation

Schematic melting curve = fraction of bound pairs

Melting curve is measureddirectly by optical means

absorption of uv line 268nm

T

1

O. Gotoh, Adv. Biophys. 16, 1 (1983)

LinearizedPlasmid pNT1

3.83 Kbp

Melting curve of yeast DNA 12 Mbp longBizzaro et al, Mat. Res. Soc. Proc. 489, 73 (1998)

Linearized Plasmid pNT1 3.83 Kbp

G AT C C A

AC T G G T

Nucleotides: A , T ,C , G

A – T ~ 320 KC – G ~ 360 K

High concentration of C-G

High concentration of A-T

T

T

Experiments:

steps are steep

each step represents the meltingof a finite region, hence smoothenedby finite size effect.

.

Sharp (first order) melting transition

Recent approaches using single molecule experimentsyield more detailed microscopic information on thestatistics and dynamics of DNA configurations

Bockelmann et al (1997) unzipping by external force

fluorescence correlation spectroscopy (FCS)time scales of loop dynamics, and loop size distribution Libchaber et al (1998, 2002)

Theoretical Approach

fluctuating microscopic configurations

Basic Model (Poland & Scheraga, 1966)

• Energy –E per bond (complementary bp)

Bound segment:

homopolymers

Loops:

• Degeneracyc

l

l

sl )(

s - geometrical factorc=d/2 in d dimensions

S=4 for d=2S=6 for d=3

lsl )(

chain

)(l - no. of configurations

c

l

l

sl )(

loop

C=d/2

2/dlV

lR

R

Results: nature of the transition depends on c

• no transition

• continuous transition

• first order transition

1c

21 c

2c

1

2 21

c

ccFor c=d/2

2c 1

2c1

)(

1

)(

1

1

/

TT

TT

l

elP

M

cM

c

l

Loop-size distribution

Outline of the derivation of the partition sum

Eew c

l

l

sl )(

l1 l3 l5

l2

1lw )2( 4l

l4

... )2( )2( 53142

lll wlwlw

Lll

LG

k

k l l l k

121 ...

...)(1 2 12

typical configuration

Grand partition sum (GPS)

1

)()(l

lzLGz z - fugacityz

zL

ln

)(ln

l

l

l zwzV

1

)( GPS of a segment

l

lc

l

zl

szU

1

)( GPS of a loop

Eew

)()(1

1)(

zUzVz

z

zL

ln

)(ln

)()(1

1)(

zUzVz

L 1)()( zUzV

Thermodynamic potential z(w)

Order parameter w

z

ln

ln

Eew

Non-interacting, self avoiding loops (Fisher, 1966)

Loop entropy:• Random self avoiding loop• no loop-loop interaction

Degeneracy of a self avoiding loop

Correlation length exponent

= 3/4 for d=2

= 0.588 for d=3c

l

l

sl )( dc

0.25 (Fisher)d=3:

=1 (PS)

1

2

c

c

Thus for the self avoiding loop model one has c=1.76and the transition is continuous.

The order-parameter critical exponent satisfies

In these approaches the interaction (repulsive, self avoiding)between loops is ignored.

Question: what is the entropy of a loop embedded in a line composed of a sequence of loops?

What is the entropy of a loop embedded in a chain?(ignore the loop-structure of the chain)

rather than:

L/2L/2

l

l

Total length: L+l l/L << 1

Interacting loops (Kafri, Mukamel, Peliti, 2000)

• Mutually self-avoiding configurations of a loop and the rest of the chain • Neglect the internal structure of the rest of the chain

Loop embedded in a chain

depends only on the topology!G

171 ),...,( GLsll L

Polymer network with arbitrary topology(B. Duplantier, 1986)

1l

2l

3l

4l

5l

6l

7l

Lli

i

7

1

Example:

kk

koG nld

1

1

no. of k-verticeskn

10 l 23 n 14 n 41 n

1l

2l

3l

4l

5l

6l

7l

171 ),...,( GLsll L

0l no. of loops

for example:

d=2 )29)(2(64

1 kkk

)218)(2(512

)2(16

2

kkkkkk

d=4-

)/()2( 12 LlglLs GlL )/( 12 LlgLs GlL

L/2L/2

l

l

Total length: L+l l/L << 1

G

)/()2( 12 LlglLs GlL )/( 12 LlgLs GlL

L/2L/2

l

l

Total length: L+l l/L << 1

G

For l/L<<1 1)( LsL L

Gxxg )( for x<<1hence

GlsLs lL )2(21

Gc

32 dc

13

1

221

21

dG

hence

with

For the configuration

32 dc

11.2c 38

2c 4

32

132c 2

d

d

d

C>2 in d=2 and above. First order transition.

Random chain Self-avoiding (SA) loop SA loop embedded in a chain

2/dc dc 32 dc

3/2 1.76 2.1

In summary

c

l

l

sl )(Loop degeneracy:

Results: for a loop embedded in a chain

c=2.11

sharp, first order transition.

32-dc )( c

l

l

sl

loop-size distribution:

TTl

elP

Mc

l

1 )(

/

M2

M Tat diverges - Tat finite - ll

line

Loop-linestructure

“Rest of the chain”

extreme case: macroscopic loop

22.2c 34

2c 4

16

112c 2

d

d

d

4 dc

C>2 (larger than the case )

Numerical simulations:

Causo, Coluzzi, Grassberger, PRE 63, 3958 (2000)(first order melting)

Carlon, Orlandini, Stella, PRL 88, 198101 (2002)loop size distributionc = 2.10(2)

length distribution of the end segment

'/1)( cllp

3d in 092.0'

)(' 31

c

c

Inter-strand distance distribution:Baiesi, Carlon,Kafri, Mukamel, Orlandini, Stella (2002)

r)(),( l

rf

l

llrP

d

),( )( 1

0

/

lrPrl

edlrP d

c

l

where at criticality

2)-(c1 , 1

)( r

rP

)(),( l

rf

l

llrP

d

In the bound phase (off criticality):

)exp()( 1

1

Dxxxf

averaging over the loop-size distribution

)(

) exp()(

TT

r

rrP

M

s

More realistic modeling of DNA melting

Stacking energy:

A-T T-A A-T C-G …A-T A-T C-G G-C …

10 energy parameters altogether

Cooperativity parameterWeight of initiation of a loop in the chain

Loop entropy parameter c

0

STHG

Blake et al, Bioinformatics,15, 370 (1999)

MELTSIM simulationsBlake et al Bioinformatics 15, 370 (1999).

4662 bp long molecule

C=1.75

0 10x 26.1

Small cooperativity parameter isneeded to make a continuoustransition look sharp.

It is thus expected thattaking c=2.1 should result in alarger cooperativity parameter

Indeed it was found that thecooperativity parameter should belarger by an order of magnitudeBlossey and Carlon, PRE 68, 061911(2003)

F

Q

Recent single molecule experimentsfluorescence correlation spectroscopy (FCS)G. Bonnet, A. Libchaber and O. Krichevsky (preprint)

F - fluorophoreQ - quencher

18 base-pair long A-T chain

Heteropolymers

Question: what is the nature of the unbinding transition in longdisordered chains?

Weak disorder

Harris criterion: the nature of the transition remainsunchanged if the specific heat exponent is negative.

1

32

c

c

relevant isdisorder weak 2/3

irrelevant isdisorder weak 2/3

c

c

Strong disorderY. Kafri, D. Mukamel, cond-mat/0211473

consider a model with a bond energy distribution:

i p

p

1 v

1

1v

Phase diagram:

MTGT T

denaturatedbound

Griffiths singularity

vi p

p

1 v

1

NN tf )( 0

0 0

tt

t

free energy of a homogeneous segment of length N

GG TTTt /)(

- transition temperature of the homogeneous chain withGT 1

2 1

21 )1/(1

c

cc

N

NN tfpptF )()1()( 2

the free energy of the heterogeneous chain

.limit in the )at (namely 0

at singular becomesIt . finiteany for analytic is )(

NTTt

Ntf

G

N

limit. large in the zero

lly toexponentia decays )( of weight the

N

tfN

This is a typical situation where Griffiths singularities inthe free energy F could develop.

Lee-Yang analysis of the partition sum

)()(1

N

iiN wwwZ eew

)ln()(1

N

iiN wwkTwf

Rw

IwFor c>2

...2,1 kN

kiww c

Ri

To leading order

22

22

1ln)(

1))((

NtkTTf

Nt

N

it

N

itZ

N

N

cRRG wwTTt

If, for simplicity, one considers only the closest zero to evaluate thefree energy, one has (for, say, c>2)

)/1ln()( 22 NttfN

N

NN tfpptF )()1()( 2

using

dxtxekTF x )1ln( 22

0

Singular at t=0 with finite derivatives to all orders. Griffiths type singularity.

Summary

Scaling approach may be applied to account for loop-loop interaction.

For a loop embedded in a chain 1.2ccl lsl /)(

The interacting loops model yields first order melting transition.

Broad loop-size distribution at the melting pointcllp /1)(

Inter-strand distance distribution 2)-(c1 , 1

)( r

rP

Larger cooperativity parameter

Future directions: dynamics of loops, RNA melting etc.

selected references

Reviews of earlier work:

O. Gotoh, Adv. Biophys. 16, 1 (1983).R. M. Wartell, A. S. Benight, Phys. Rep. 126, 67 (1985).D. Poland, H. A. Scheraga (eds.) Biopolymers (Academic, NY, 1970).

Poland & Scheraga model:

D. Poland, Scheraga, J. Chem. Phys. 45, 1456, 1464 (1966);M. E. Fisher, J. Chem. Phys. 45, 1469 (1966)Y. Kafri, D. Mukamel, L. Peliti PRL, 85, 4988, 2000; EPJ B 27, 135, (2002); Physica A 306, 39 (2002).M. S.Causo, B. Coluzzi, P. Grassberger, PRE 62, 3958 (2000).E. Carlon, E. Orlandini, A. L. Stella, PRL 88, 198101 (2002).M. Baiesi, E. Carlon, A. L. Stella, PRE 66, 021804 (2002).

Directed polymer approach:

M. Peyrard, A. R. Bishop, PRL 62, 2755 (1989)

Simulations of real sequences:

R.D. Blake et al, Bioinformatics, 15, 370 (1999).R. Blossey and E. Carlon, PRE 68, 061911 (2003).

Analysis of heteropolymer melting:

L. H. Tang, H. Chate, PRL 86, 830 (2001).Y. Kafri, D. Mukamel, PRL 91, 055502 (2003).

Interband distance distribution:

M. baiesi, E. carlon, Y. kafri, D. Mukamel, E. Orlandini, A. L. Stella,PRE 67, 021911 (2003).