Unbinding of biopolymers:

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Unbinding of biopolymers statistical physics of
interacting loops
David Mukamel
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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

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DNA denaturation
AATCGGTTTCCCC TTAGCCAAAGGGG
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Single strand conformations RNA folding
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conformation changes in RNA
Schultes, Bartel (2000)
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Unzipping of DNA by an external force
Bockelmann et al PRL 79, 4489 (1997)
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Unpinning of vortex lines from columnar
defects In type II superconductors
Defects are produced by irradiation with heavy
ions with high energy to produce tracks of
damaged material.
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Wetting transition
interface
gas
liquid
2d
substrate
3d
At the wetting transition
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One is interested in features like
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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

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DNA denaturation
fluctuating DNA
Persistence length lp double strands lp
100-200 bp Single strands lp 10 bp
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Schematic melting curve
q
fraction of bound pairs
Melting curve is measured directly by optical
means absorption of uv line 268nm
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Linearized Plasmid pNT1 3.83 Kbp
O. Gotoh, Adv. Biophys. 16, 1 (1983)
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Melting curve of yeast DNA 12 Mbp long Bizzaro et
al, Mat. Res. Soc. Proc. 489, 73 (1998)
Linearized Plasmid pNT1 3.83 Kbp
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Nucleotides A , T ,C , G
A T 320 K C G 360 K
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Experiments
Sharp (first order) melting transition
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Recent approaches using single molecule
experiments yield more detailed microscopic
information on the statistics and dynamics of DNA
configurations
unzipping by external force
Bockelmann et al (1997)
time scales of loop dynamics, and loop size
distribution Libchaber et al (1998, 2002)
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Theoretical Approach
fluctuating microscopic configurations
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Basic Model (Poland Scheraga, 1966)
homopolymers
Bound segment

• Energy E per bond (complementary bp)

Loops
s - geometrical factor cd/2 in d dimensions
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chain
- no. of configurations
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loop
Cd/2
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Results nature of the transition depends on c

• no transition
• continuous
  transition
• first order
  transition

cd/2
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Loop-size distribution
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Outline of the derivation of the partition sum
typical configuration
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Grand partition sum (GPS)
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Non-interacting, self avoiding loops (Fisher,
1966)

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

Degeneracy of a self avoiding loop
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Thus for the self avoiding loop model one has
c1.76 and the transition is continuous.
The order-parameter critical exponent satisfies

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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?
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What is the entropy of a loop embedded in a chain?
(ignore the loop-structure of the chain)
rather than
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Interacting loops (Kafri, Mukamel, Peliti, 2000)
l
Loop embedded in a chain
L/2
l
L/2
Total length Ll l/L ltlt 1

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

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Polymer network with arbitrary topology (B.
Duplantier, 1986)
Example
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for example
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d2
d4-
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For l/Lltlt1
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hence
with
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For the configuration
Cgt2 in d2 and above. First order transition.
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In summary
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Results for a loop embedded in a chain
sharp, first order transition.
loop-size distribution
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Rest of the chain
line
Loop-line structure
extreme case macroscopic loop
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(larger than the case )

Cgt2
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Numerical simulations Causo, Coluzzi,
Grassberger, PRE 63, 3958 (2000) (first order
melting) Carlon, Orlandini, Stella, PRL 88,
198101 (2002) loop size distribution c 2.10(2)

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length distribution of the end segment
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Inter-strand distance distribution Baiesi,
Carlon,Kafri, Mukamel, Orlandini, Stella (2002)
where at criticality
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In the bound phase (off criticality)
averaging over the loop-size distribution
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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 parameter Weight of initiation of a
loop in the chain
Loop entropy parameter c
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Blake et al, Bioinformatics, 15, 370 (1999)
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MELTSIM simulations Blake et al Bioinformatics
15, 370 (1999).
4662 bp long molecule
Small cooperativity parameter is needed to make a
continuous transition look sharp.
It is thus expected that taking c2.1 should
result in a larger cooperativity parameter
Indeed it was found that the cooperativity
parameter should be larger by an order of
magnitude Blossey and Carlon, PRE 68,
061911 (2003)
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Recent single molecule experiments fluorescence
correlation spectroscopy (FCS) G. Bonnet, A.
Libchaber and O. Krichevsky (preprint)
F - fluorophore Q - quencher
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18 base-pair long A-T chain
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Heteropolymers
Question what is the nature of the unbinding
transition in long disordered chains?
Weak disorder
Harris criterion the nature of the transition
remains unchanged if the specific heat exponent
is negative.
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Strong disorder Y. Kafri, D. Mukamel,
cond-mat/0211473
consider a model with a bond energy distribution
Phase diagram
denaturated
bound
Griffiths singularity
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free energy of a homogeneous segment of length N
- transition temperature of the homogeneous
chain with
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the free energy of the heterogeneous chain
This is a typical situation where Griffiths
singularities in the free energy F could develop.

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Lee-Yang analysis of the partition sum
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For cgt2
To leading order
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If, for simplicity, one considers only the
closest zero to evaluate the free energy, one has
(for, say, cgt2)
Singular at t0 with finite derivatives to all
orders. Griffiths type singularity.
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Summary
Scaling approach may be applied to account for
loop-loop interaction.
For a loop embedded in a chain
The interacting loops model yields first order
melting transition.
Broad loop-size distribution at the melting point
Inter-strand distance distribution
Larger cooperativity parameter
Future directions dynamics of loops, RNA melting
etc.
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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)
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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).