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 TT double stranded single strands Helix to Coil transition …AATCGGTTTCCCC… …TTAGCCAAAGGGG…

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 substrate interface 3d 2d gas liquid At the wetting transition

One is interested in features like Loop size distribution Order of the denaturation transition Inter-strand distance distribution 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 lp double strands lp ~ bp Single strands lp ~ 10 bp fluctuating DNA DNA denaturation

Schematic melting curve  = fraction of bound pairs Melting curve is measured directly by optical means absorption of uv line 268nm  T 1

O. Gotoh, Adv. Biophys. 16, 1 (1983) Linearized Plasmid pNT Kbp

Melting curve of yeast DNA 12 Mbp long Bizzaro et al, Mat. Res. Soc. Proc. 489, 73 (1998) Linearized Plasmid pNT Kbp

GA T C C A A C T G G T Nucleotides: A, T,C, G A – T ~ 320 K C – G ~ 360 K High concentration of C-G High concentration of A-T

TT

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 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: Degeneracy s - geometrical factor c=d/2 in d dimensions

S=4 for d=2 S=6 for d=3 chain - no. of configurations

loop C=d/2 R

Results: nature of the transition depends on c no transition continuous transition first order transition c=d/2

Loop-size distribution

Outline of the derivation of the partition sum l1 l3 l5 l2 l4 typical configuration

Grand partition sum (GPS) z - fugacity GPS of a segment GPS of a loop

Thermodynamic potential z(w) Order parameter

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 = for d=3

0.25 (Fisher) d=3: =1 (PS) Thus for the self avoiding loop model one has c=1.76 and 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/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! Polymer network with arbitrary topology (B. Duplantier, 1986) Example:

no. of k-vertices no. of loops for example:

d=2 d=4-

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

L/2 l l Total length: L+l l/L << 1 For l/L<<1 for x<<1 hence

with

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

Random chainSelf-avoiding (SA) loop SA loop embedded in a chain 3/ In summary Loop degeneracy:

Results: for a loop embedded in a chain c =2.11 sharp, first order transition. loop-size distribution:

line Loop-line structure “Rest of the chain” extreme case: macroscopic loop

C>2 (larger than the case )

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

length distribution of the end segment

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

In the bound phase (off criticality): averaging over the loop-size distribution

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

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

MELTSIM simulations Blake et al Bioinformatics 15, 370 (1999) bp long molecule C=1.7 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, (2003)

F Q Recent single molecule experiments fluorescence correlation spectroscopy (FCS) G. Bonnet, A. Libchaber and O. Krichevsky (preprint) 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.

Strong disorder Y. Kafri, D. Mukamel, cond-mat/ consider a model with a bond energy distribution: Phase diagram: denaturatedbound Griffiths singularity

free energy of a homogeneous segment of length N - transition temperature of the homogeneous chain with

the free energy of the heterogeneous chain This is a typical situation where Griffiths singularities in the free energy F could develop.

Lee-Yang analysis of the partition sum

For c>2 To leading order

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

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, (2002). M. Baiesi, E. Carlon, A. L. Stella, PRE 66, (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, (2003). Analysis of heteropolymer melting: L. H. Tang, H. Chate, PRL 86, 830 (2001). Y. Kafri, D. Mukamel, PRL 91, (2003). Interband distance distribution: M. baiesi, E. carlon, Y. kafri, D. Mukamel, E. Orlandini, A. L. Stella, PRE 67, (2003).