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					      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              T



  double                      single strands
stranded

     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     l


                                                   2d
                            substrate
       3d




At the wetting transition   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
1               directly by optical means

                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               Linearized Plasmid pNT1
Bizzaro et al, Mat. Res. Soc. Proc. 489, 73 (1998)        3.83 Kbp
T   G A                   Nucleotides:   A , T ,C , G
          C C A


  C T                           A – T ~ 320 K
A     G G T
                                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
    Bockelmann et al (1997)




   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)
 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     l



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


loop

                       l
                   s
           ( l )  c
                   l
               C=d/2




                  R l
       
                   V l    d /2

           R
Results: nature of the transition depends on c

    •         c 1        no transition

    •     1 c  2        continuous transition

    •         c2         first order transition


                           2c
        For 1  c  2                c=d/2
                           c 1
Loop-size distribution

                l /
            e
 P (l )            c
                l
                    1
                       1
                               1 c  2
      (TM  T )         c 1


       1
                                c2
     TM  T
 Outline of the derivation of the partition sum
                                         l2                 l4
typical configuration
                                l1                l3               l5
                             w l1                                 (2l4 )


             wl1 (2l2 ) wl3 ( 2l4 ) wl5 ...


                        sl           G ( L)   ...
        E
 we             (l )  c                       k     l1    l2     l2 k 1
                        l            l1  ...  l2 k 1  L
           Grand partition sum (GPS)
          
( z )   G ( L ) z l                                    ln ( z )
                              z - fugacity            L
         l 1                                               ln z

                                        1
                        ( z ) 
                                 1  V ( z )U ( z )

          
                                                                  E
V ( z)   w z  l   l
                               GPS of a segment          we
         l 1
           
              sl l
U ( z)   c z                 GPS of a loop
         l 1 l
                 1                     ln ( z )
( z )                            L
         1  V ( z )U ( z )              ln z


    L                V ( z )U ( z )  1


                                                          E
  Thermodynamic potential              z(w)         we
                                       ln z
  Order parameter                  q
                                       ln w
Non-interacting, self avoiding loops
(Fisher, 1966)

Loop entropy:
• Random self avoiding loop
• no loop-loop interaction
Degeneracy of a self avoiding loop

            l                    n = 3/4 for d=2
           s
    (l )  c    c  dn          n = 0.588 for d=3
           l
n      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             L/2
                                       l         L/2
 in a chain
                       Total length:       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                      l6
                         l4                 7
Example:
                  l3
                                           l
                                           i 1
                                                  i   L
             l1          l5           l7


                              L  G 1
             (l1 ,..., l7 )  s L

       G depends only on the topology!
                                                   
                  L  G 1
(l1 ,..., l7 )  s L              G  1  dnlo   nk k
                                                   k 1



     l0 no. of loops                nk no. of k-vertices

                        l2                   l6
                                     l4
   for example:
                             l3
                        l1          l5
                                              l7



        l0  1 n3  2 n4  1 n1  4
                1
d=2         k  (2  k )(9k  2)
                64


                                     
       
                                          2
d=4-       k         k (2  k )            k (k  2)(8k  21)
                  16                  512
                                 l

                 L/2
                                 l          L/2             G
           Total length:             L+l   l/L << 1


                                                 L 2 l  G 1
s   L  2l             G 1
               ( L  2l )        g (l / L)  s        L          g ( l / L)
                                 l

                 L/2
                                 l              L/2           G
           Total length:             L+l       l/L << 1


                                                   L 2 l  G 1
s   L  2l             G 1
               ( L  2l )        g (l / L)  s            L        g ( l / L)


For l/L<<1          ( L)  s L L 1                                           

                                       G 
   hence           g ( x)  x                     for x<<1
                L  1               G 
           s L  s (2l )2l



 hence          c   G

         1  2 1
with
        G  1  dn  2 3  2 1

                c  dn  2 3
For the configuration


                 c  dn  2 3
                                13
           d 2           c 2
                                32
                                 
           d  4       c 2
                                 8
           d 3          c  2.11

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

                                               l
                                             s
               Loop degeneracy:      ( l )  c
                                             l

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


c  d /2                  c  dn                     c  dn  2 3

  3/2                       1.76                          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.


                                          e l /            1
    loop-size distribution:        P(l )  c            
                                           l              TM  T

             l  - finite at TM        l 2  - diverges at TM
  “Rest of the chain”

                           line


                        Loop-line
                        structure

extreme case: macroscopic loop
                        c  dn   4

                    11
d 2          c 2
                    16
                        
d  4       c2
                      4
d 3          c  2.22
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 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)

                     l              r
P(r , l )           dn
                                 f( n)                    r
                 l                 l
                        l / 
                     e
P(r )   dl                 c
                                  r   d 1
                                             P(r , l )
            0
                         l
 where at criticality

             1
    P( r )   ,   1  (c - 2)n
            r
  In the bound phase (off criticality):

                   l       r
 P(r , l )        dn
                        f( n)
               l          l
                             1
           
f ( x)  x exp(  Dx        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   T-A A-T C-G …
                   A-T   A-T C-G G-C …


                10 energy parameters altogether



    Cooperativity parameter       0
     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 - fluorophore
                      F
                                              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           weak disorder is irrelevant
           c  3/ 2           weak disorder is relevant
         Strong disorder
         Y. Kafri, D. Mukamel, cond-mat/0211473

                consider a model with a bond energy distribution:



                               
                                   1              p
                        i                                   v  1
                                   v              1 p


             Phase diagram:


                                      bound                   denaturated




                         TG                   TM          T
Griffiths singularity
                 
                     1   p
         i                             v 
                     v   1 p


 free energy of a homogeneous segment of length N


                N 
                
     f N (t )  
                                0
                                 n
                                 t
                                       t0
                                        t0
                                                   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
                 c
                 R          k  1,2...
                     N

              To leading order
     wR                 i          i      1
           Z N  (t      )( t  ) t 2  2
                        N        N        N
                                     1 
            f N (T )  kT ln  t 2  2 
                                    N 



            t  TG  T  wR  wR
                               c
   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            2



    using             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     p(l )  1 / l c
                                                 1
Inter-strand distance distribution      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).

				
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