Peter Virnau, Mehran Kardar, Yacov Kantor

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

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 includedin thestandard 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. Motivation: Biology 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?

11. Model Polymer: Coarse-grained model for polyethylene Bead-spring chain(LJ+FENE): 1 bead @ 3 CH2

12. Model Polymer: Coarse-grained model for polyethylene Bead-spring chain(LJ+FENE): 1 bead @ 3 CH2 Equilibrium configurations are generated with standard Monte Carlo techniques (pivot, reptation, local moves)

13. Simplification

15. Coil / Globule Polymer: Coarse-grained model for polyethylene Bead-spring chain(LJ+FENE): 1 bead @ 3 CH2

16. Coil / Globule Polymer: Coarse-grained model for polyethylene Bead-spring chain(LJ+FENE): 1 bead @ 3 CH2 Reduce chain, connect ends, calculate Alexander polynomial

17. At which chain length do knots occur? unknot 31 41

18. At which chain length do knots occur? unknot 31 41

19. At which chain length do knots occur? unknot 31 41 Knots are rare in the swollen phase (1% for3000 CH2)

20. At which chain length do knots occur? unknot 31 41 Knots are common in a dense phase (80% for3000 CH2)

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 knots localized ? dilute rare (1% for 3000 CH2) yes dense frequent (80%) no

25. Summary I frequency of knots localized ? dilute rare (1% for 3000 CH2) yes dense frequent (80%) no • Probabilities:Open polymers <-> Loops ?

26. Summary I frequency of knots localized ? dilute rare (1% for 3000 CH2) yes dense frequent (80%) no • Probabilities: Open polymers <-> Loops? • Excluded volume ?

27. Summary I frequency of knots localized ? dilute rare (1% for 3000 CH2) yes dense frequent (80%) no • Probabilities: Open polymers <-> Loops? • Excluded volume ? • Distribution of sizes and location ?

28. Summary I frequency of knots localized ? dilute rare (1% for 3000 CH2) yes dense frequent (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 31 41 Knots are frequent

31. Loops vs. Chains unknot 31 41 Loops and chains have similar knotting probabilities

32. Distribution of knot sizes

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

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

35. Distribution of knot sizes

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

37. Where are knots located?

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

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

40. Where are knots located?

41. Summary II frequency of knots localized ? dilute rare (1% for 3000 CH2) yes dense frequent (80%) no RW very frequent extremely DNA ??? ??? Proteins ??? ???

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

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

44. Knots in DNA? Human DNA is wrapped around histone proteins 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 in proteins 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, 52 Functions: mostly enzymes (13 transferases)

50. Final Summary frequency of knots localized ? dilute rare (1% for 3000 CH2) yes dense frequent (80%) no RW very frequent extremely DNA in vivo: probably few in vivo: - Proteins very few not enough statistics virnau@mit.edu