Unbinding of biopolymers: statistical physics of interacting loops

1. Unbinding of biopolymers: statistical physics of interacting loops David Mukamel

2. 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

3. T T single strands double stranded Helix to Coil transition DNAdenaturation …AATCGGTTTCCCC… …TTAGCCAAAGGGG…

4. Single strand conformations: RNA folding

5. conformation changes in RNA Schultes, Bartel (2000)

6. Unzipping of DNA by an external force Bockelmann et al PRL 79, 4489 (1997)

7. 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.

8. Wetting transition interface gas liquid 2d substrate 3d At the wetting transition

9. Loop size distribution Order of the denaturation transition Inter-strand distance distribution Effect of heterogeneity of the chain One is interested in features like

10. 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

11. DNA denaturation fluctuating DNA Persistence length lp double strands lp ~ 100-200 bp Single strands lp ~ 10 bp

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

13. Linearized Plasmid pNT1 3.83 Kbp O. Gotoh, Adv. Biophys. 16, 1 (1983)

14. Melting curve of yeast DNA 12 Mbp long Bizzaro et al, Mat. Res. Soc. Proc.489, 73 (1998) Linearized Plasmid pNT1 3.83 Kbp

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

16. T T

17. . steps are steep each step represents the melting of a finite region, hence smoothened by finite size effect. Experiments: Sharp (first order)melting transition

18. fluorescence correlation spectroscopy (FCS) 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)

19. Theoretical Approach fluctuating microscopic configurations

20. Degeneracy Basic Model (Poland & Scheraga, 1966) homopolymers Bound segment: • Energy –E per bond (complementary bp) Loops: s - geometrical factor c=d/2 in d dimensions

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

22. R loop C=d/2

23. Results: nature of the transition depends on c • no transition • continuoustransition • first order transition c=d/2

24. Loop-size distribution

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

26. z - fugacity GPS of asegment GPS of aloop Grand partition sum (GPS)

27. Thermodynamic potentialz(w) Order parameter

28. n Correlation length exponent n = 3/4 for d=2 n = 0.588 for d=3 Non-interacting, self avoiding loops (Fisher, 1966) • Loop entropy: • Random self avoiding loop • no loop-loop interaction Degeneracy of a self avoiding loop

29. =1 (PS) d=3: 0.25 (Fisher) Thus for the self avoiding loop model one has c=1.76 and the transition is continuous. The order-parameter critical exponent satisfies

30. In these approaches the interaction (repulsive, self avoiding) betweenloops is ignored. Question: what is the entropy of a loop embedded in a line composed of a sequence of loops?

31. What is the entropy of a loop embedded in a chain? (ignore the loop-structure of the chain) rather than:

32. Interacting loops(Kafri, Mukamel, Peliti, 2000) l Loop embedded in a chain L/2 l L/2 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

33. depends only on the topology! Polymer network with arbitrary topology (B. Duplantier, 1986) Example:

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

35. d=2 d=4-

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

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

38. hence with

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

40. Loop degeneracy: Random chain Self-avoiding (SA) loop SA loop embedded in a chain 2.1 1.76 3/2 In summary

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

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

43. (larger than the case ) C>2

44. 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)

45. length distribution of the end segment

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

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

50. 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