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
…AATCGGTTTCCCC…
…TTAGCCAAAGGGG…
T
double
stranded
T
single strands
Helix to Coil transition
Single strand conformations: RNA folding
conformation changes in RNA
Schultes, Bartel (2000)
Unzipping of DNA by an external force
Bockelmann et al PRL 79, 4489 (1997)
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.
Wetting transition
interface
gas
liquid
2d
3d
At the wetting transition
substrate
l 
l
One is interested in features like




Loop size distribution P(l )
Order of the denaturation transition
Inter-strand distance distribution P (r )
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
DNA denaturation
fluctuating DNA
Persistence length lp
double strands lp ~ 100-200 bp
Single strands lp ~ 10 bp
Schematic melting curve
q = fraction of bound pairs
q
Melting curve is measured
directly by optical means
1
absorption of uv line
268nm
T
Linearized
Plasmid pNT1
3.83 Kbp
O. Gotoh, Adv. Biophys. 16, 1 (1983)
Melting curve of yeast DNA 12 Mbp long
Bizzaro et al, Mat. Res. Soc. Proc. 489, 73 (1998)
Linearized Plasmid pNT1
3.83 Kbp
T
G A
C C A
C T
G G T
A
Nucleotides:
A , T ,C , G
A – T ~ 320 K
C – G ~ 360 K
High concentration of C-G
High concentration of A-T
T
T
.
Experiments:
steps are steep
each step represents the melting
of a finite region, hence smoothened
by finite size effect.
Sharp (first order) melting transition
Recent approaches using single molecule experiments
yield more detailed microscopic information on the
statistics and dynamics of DNA configurations

unzipping by external force

fluorescence correlation spectroscopy (FCS)
Bockelmann et al (1997)
time scales of loop dynamics, and loop size distribution
Libchaber et al (1998, 2002)
Theoretical Approach
fluctuating microscopic configurations
Basic Model (Poland & Scheraga, 1966)
homopolymers
Bound segment:
• Energy –E per bond (complementary bp)
Loops:
l
s
• Degeneracy (l )  c
l
s - geometrical factor
c=d/2 in d dimensions
chain
(l ) - no. of configurations
(l )  s


S=4 for d=2
S=6 for d=3
l
loop
l
s
( l )  c
l
C=d/2


R l
V l
R
d /2
Results: nature of the transition depends on c
•
•
•
c 1
1 c  2
c2
For 1  c  2
no transition
continuous transition
first order transition
2c

c 1
c=d/2
Loop-size distribution
P (l ) 

e
l /
l
c
1
(TM  T )
1

TM  T
1
c 1
1 c  2
c2
Outline of the derivation of the partition sum
l4
l2
typical configuration
l1
l5
l3
(2l4 )
w l1
wl1 (2l2 ) wl3 (2l4 ) wl5 ...
we
E
sl
(l )  c
l
G ( L)   ... 
k
l1
l2
l1  ...  l2 k 1  L
l2 k 1
Grand partition sum (GPS)

( z )   G ( L ) z l
z - fugacity
l 1
 ln ( z )
L
 ln z
1
( z ) 
1  V ( z )U ( z )

V ( z)   w z
l
l 1

l
sl l
U ( z)   c z
l 1 l
GPS of a segment
GPS of a loop
we
E
1
( z ) 
1  V ( z )U ( z )
L 
 ln ( z )
L
 ln z
V ( z )U ( z )  1
Thermodynamic potential
Order parameter
z(w)
 ln z
q
 ln w
we
E
Non-interacting, self avoiding loops
(Fisher, 1966)
Loop entropy:
• Random self avoiding loop
• no loop-loop interaction
Degeneracy of a self avoiding loop
l
s
(l )  c
l
n
c  dn
n = 3/4 for d=2
n = 0.588 for d=3
Correlation length exponent
Thus for the self avoiding loop model one has c=1.76
and the transition is continuous.
The order-parameter critical exponent satisfies
2c

c 1
d=3:  =1 (PS)
  0.25 (Fisher)
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:
Interacting loops (Kafri, Mukamel, Peliti, 2000)
l
Loop embedded
in a chain
L/2
Total length:
l
L/2
L+l
l/L << 1
• Mutually self-avoiding configurations of a loop
and the rest of the chain
• Neglect the internal structure of the rest of the chain
Polymer network with arbitrary topology
(B. Duplantier, 1986)
l2
Example:
l1
l6
l4
l3
7
l
i 1
l5
l7
L  G 1
(l1 ,..., l7 )  s L
 G depends only on the topology!
i
L

 G  1  dnlo   nk k
L  G 1
(l1 ,..., l7 )  s L
k 1
nk no. of k-vertices
l0 no. of loops
l2
for example:
l1
l4
l3
l5
l0  1 n3  2 n4  1 n1  4
l6
l7
1
 k  (2  k )(9k  2)
64
d=2
d=4-

k 

16
k (2  k ) 

2
512
k (k  2)(8k  21)
l
L/2
l
Total length:
s
L  2l
 G 1
( L  2l )
G
L/2
L+l
l/L << 1
g (l / L)  s
L 2 l  G 1
L
g ( l / L)
l
L/2
l
Total length:
s
L  2l
 G 1
( L  2l )
For l/L<<1
hence
G
L/2
L+l
l/L << 1
g (l / L)  s
L 2 l  G 1
L

( L)  s L L 1
g ( x)  x
 G 
g ( l / L)
for x<<1
L  1
 G 
  s L  s (2l )
hence
with
2l
c   G
  1  2 1
 G  1  dn  2 3  2 1
c  dn  2 3
For the configuration
c  dn  2 3
d 2
d  4 
d 3
13
c 2
32
c 2

8
c  2.11
C>2 in d=2 and above. First order transition.
In summary
l
Loop degeneracy:
Random chain
c  d /2
3/2
s
( l )  c
l
Self-avoiding (SA) loop
c  dn
1.76
SA loop embedded in a chain
c  dn  2 3
2.1
Results: for a loop embedded in a chain

l
s
(l )  c
l
c  dn - 2 3
c=2.11
sharp, first order transition.

loop-size distribution:
 l  - finite at TM
e l / 
P(l )  c
l
1

TM  T
 l 2  - diverges at TM
“Rest of the chain”
line
Loop-line
structure
extreme case: macroscopic loop
c  dn   4
11
c 2
16
d 2
d  4 
d 3
C>2
c 2

4
c  2.22
(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 distribution
c = 2.10(2)
length distribution of the end segment
p(l )  1 / l
c'
c'  ( 1   3 )
c'  0.092
in d  3
Inter-strand distance distribution:
Baiesi, Carlon,Kafri, Mukamel, Orlandini, Stella (2002)
P(r , l ) 

l
l
P(r )   dl
0
r
f( n)
l
dn
e
l / 
l
c
r
d 1
P(r , l )
where at criticality
1
P( r )   ,   1  (c - 2)n
r
r
In the bound phase (off criticality):
P(r , l ) 
l
l
dn
r
f( n)
l

f ( x)  x exp(  Dx
1
1n
)
averaging over the loop-size distribution
exp(  r )
P(r ) 
s
r
n
  (TM  T )
More realistic modeling of DNA melting

Stacking energy:
A-T
A-T
T-A A-T C-G …
A-T C-G G-C …
10 energy parameters altogether


0
Cooperativity parameter
Weight of initiation of a loop in the chain
Loop entropy parameter
c
Blake et al, Bioinformatics,
15, 370 (1999)
G  H  TS
MELTSIM simulations
Blake et al Bioinformatics 15, 370 (1999).
4662 bp long molecule
C=1.7



 0  1.26 x 105
Small cooperativity parameter is
needed to make a continuous
transition look sharp.
It is thus expected that
taking c=2.1 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)
Recent single molecule experiments
fluorescence correlation spectroscopy (FCS)
G. Bonnet, A. Libchaber and O. Krichevsky (preprint)
Q
F
F - fluorophore
Q - quencher
18 base-pair long A-T chain
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.
2c  3

c 1
c  3/ 2
c  3/ 2
weak disorder is irrelevant
weak disorder is relevant
Strong disorder
Y. Kafri, D. Mukamel, cond-mat/0211473
consider a model with a bond energy distribution:
i 

1
p
v
v  1
1 p
Phase diagram:
bound
TG
Griffiths singularity
denaturated
TM
T
i 

1
p
v
1 p
v 
free energy of a homogeneous segment of length N
N 
f N (t ) 

0
t0
n
t0
t
n

1 /(c  1)
1
1 c  2
c2
t  (TG  T ) / TG
TG
- transition temperature of the homogeneous chain with
  1
the free energy of the heterogeneous chain
F (t )  (1  p) 2  p N f N (t )
N
f N (t ) is analytic for any finite N . It becomes singular at
t  0 (namely at T  TG ) in the limit N  .
the weight of f N (t ) decays exponentia lly to
zero in the large N limit.
This is a typical situation where Griffiths singularities in
the free energy F could develop.
Lee-Yang analysis of the partition sum
N
Z N ( w)   ( w  wi )
w  e  e 
i 1
N
f N ( w)  kT  ln( w  wi )
i 1
For c>2
wI
k
wi  w  i
N
c
R
k  1,2...
To leading order
wR
i
i
1
)( t  ) t 2  2
N
N
N
1 

f N (T )  kT ln  t 2  2 
N 

Z N  (t 
t  TG  T  wR  wRc
If, for simplicity, one considers only the closest zero to evaluate the
free energy, one has (for, say, c>2)
f N (t )  ln( t  1 / N )
2
using
2
F (t )  (1  p) 2  p N f N (t )
N

F  kT  e x ln( x 2t 2  1)dx
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
(l )  sl / l c
c  2.1
The interacting loops model yields first order melting transition.
Broad loop-size distribution at the melting point
Inter-strand distance distribution
p (l )  1 / l c
1
P( r )   ,   1  (c - 2)n
r
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).