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 • c2 first order transition 2c 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 c2 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 we (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 we 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) we 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 2c 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 c2 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 1n ) 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 TS MELTSIM simulations Blake et al Bioinformatics 15, 370 (1999). 4662 bp long molecule C=1.7 0 1.26 x 105 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 t0 t0 n 1 /(c 1) 1 1 c 2 c2 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).