1. Peter Virnau, Mehran Kardar, Yacov Kantor Capturing knots in (bio-) polymers …

2. History of knot science Lord Kelvin (1867): “Vortex atoms” P.G. Tait: Knot tables

3. Classification of knots J.W. Alexander (1923): First algorithm which can distinguish between knots (… somewhat) 2005: still no complete invariant

4. Motivation: Polymers Knots are topological invariants (self-avoiding) ring polymers A sufficiently long polymer will have knots (Frisch & Wassermann (1961), Delbrück (1962)) Knots are not included in the standard theories Knots modify dynamics of polymers; e.g. relaxation or electrophoresis

5. Motivation: Polymers Knots are topological invariants (self-avoiding) ring polymers A sufficiently long polymer will have knots (Frisch & Wassermann (1961), Delbrück (1962)) Knots are not included in the standard theories Knots modify dynamics of polymers; e.g. relaxation or electrophoresis

6. Motivation: Polymers Knots are topological invariants (self-avoiding) ring polymers A sufficiently long polymer will have knots: (Frisch & Wassermann (1961), Delbrück (1962)) Knots are not included in the standard theories Knots modify dynamics of polymers; e.g. relaxation or electrophoresis

7. Motivation: Polymers Knots are topological invariants (self-avoiding) ring polymers A sufficiently long polymer will have knots (Frisch & Wassermann (1961), Delbrück (1962)) Knots are not included in the standard theories Knots modify dynamics of polymers; e.g. relaxation or electrophoresis

8. Motivation: Biology Knots: Why? Structure  Function Role of entanglements?

9. Motivation: Biology Knots: How? Reference system: Single homopolymer in stretched and compact state

10. Knots: How? Reference system: Single homopolymer in stretched and compact state 1. At which chain length do knots occur? 2. Are knots localized or spread? Motivation: Biology

11. Model Polymer: Coarse-grained model for polyethylene Bead-spring chain (LJ+FENE): 1 bead  3 CH 2

12. Model Polymer: Coarse-grained model for polyethylene Bead-spring chain (LJ+FENE): 1 bead  3 CH 2 Equilibrium configurations are generated with standard Monte Carlo techniques (pivot, reptation, local moves)

13. Simplification

15. Polymer: Coarse-grained model for polyethylene Bead-spring chain (LJ+FENE): 1 bead  3 CH 2 Coil / Globule

16. Polymer: Coarse-grained model for polyethylene Bead-spring chain (LJ+FENE): 1 bead  3 CH 2 Reduce chain, connect ends, calculate Alexander polynomial Coil / Globule

17. At which chain length do knots occur? unknot 3 1 4 1

18. At which chain length do knots occur? unknot 3 1 4 1

19. At which chain length do knots occur? Knots are rare in the swollen phase (1% for  3000 CH 2 ) unknot 3 1 4 1

20. At which chain length do knots occur? Knots are common in a dense phase (80% for  3000 CH 2 ) unknot 3 1 4 1

21. Are knots localized or spread?

22. Are knots localized or spread? Knots are localized in the swollen phase

23. Are knots localized or spread? Knots are delocalized in a dense phase

24. Summary I frequency of knotslocalized ? diluterare (1% for 3000 CH 2 ) yes densefrequent (80%) no

25. Summary I frequency of knotslocalized ? diluterare (1% for 3000 CH 2 ) yes densefrequent (80%) no Probabilities: Open polymers  Loops ?

26. Summary I frequency of knotslocalized ? diluterare (1% for 3000 CH 2 ) yes densefrequent (80%) no Probabilities: Open polymers  Loops ? Excluded volume ?

27. Summary I frequency of knotslocalized ? diluterare (1% for 3000 CH 2 ) yes densefrequent (80%) no Probabilities: Open polymers  Loops ? Excluded volume ? Distribution of sizes and location ?

28. Summary I frequency of knotslocalized ? diluterare (1% for 3000 CH 2 ) yes densefrequent (80%) no Probabilities: Open polymers  Loops ? Excluded volume ? Distribution of sizes and location ?  simpler (faster) model: Random walk

29. Polymers vs. Random Walks

30. Loops vs. Chains unknot 3 1 4 1 Knots are frequent

31. Loops vs. Chains unknot 3 1 4 1 Loops and chains have similar knotting probabilities

32. Distribution of knot sizes

33. Knots are localized in random walks Distribution of knot sizes

34. Most likely knot size: only 6 segments Distribution of knot sizes

35. Distribution of knot sizes

36. Power-law tail in knot size distribution Distribution of knot sizes

37. Where are knots located?

38. Knots are equally distributed over the entire polymer, but… Where are knots located?

39. … larger in the middle Where are knots located?

40. Where are knots located?

41. Summary II frequency of knotslocalized ? diluterare (1% for 3000 CH 2 ) yes densefrequent (80%) no RWvery frequentextremely DNA??? ??? Proteins??? ???

42. Human DNA is wrapped around histone proteins Knots in DNA?

43. Human DNA is wrapped around histone proteins Knots in DNA? DNA coiled in phage capsid, but some indication of knotting inside Arsuaga et al., PNAS 99, 5373 (2002)

44. Human DNA is wrapped around histone proteins Knots in DNA? DNA coiled in phage capsid, but some indication of knotting inside Arsuaga et al., PNAS 99, 5373 (2002) DNA in good solvent: 0.5%-4% for 10000 base pairs Rybenkov et al., PNAS 90, 5307 (1991)

45. The Protein Data Bank www.pdb.org 02/2005 (24937)

46. The Protein Data Bank www.pdb.org Problems: 1. Missing atoms 2. Multiple Chains 3. Microheterogeneity 4. Same Proteins

47. Knots are very rare:230 / 24937 (1%) Source:mostly bacteria and viruses, but also mouse, cow, human and spinach Depth >5>10>15>20>25 # structures 35 33 28 28 25 (0.1%) # proteins 26 (9) 24 20 20 17 Size: 43% of protein, but variations from 17% to 82% Complexity: 23 trefoils, 2 figure-eights, 5 2 Functions: mostly enzymes (13 transferases) Knots in proteins

50. frequency of knotslocalized ? diluterare (1% for 3000 CH 2 ) yes densefrequent (80%) no RWvery frequentextremely DNAin vivo: probably fewin vivo: - Proteinsvery few not enough statistics Final Summary virnau@mit.edu

52. Early knot scientists … Phrygia, 333 BC

53. The Alexander polynomial