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).