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

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

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

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

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

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

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

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

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

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

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

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)

Simplification

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

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

At which chain length do knots occur? unknot

At which chain length do knots occur? unknot

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

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

Are knots localized or spread?

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

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

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

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

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

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 ?

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

Polymers vs. Random Walks

Loops vs. Chains unknot Knots are frequent

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

Distribution of knot sizes

Knots are localized in random walks Distribution of knot sizes

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

Distribution of knot sizes

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

Where are knots located?

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

… larger in the middle Where are knots located?

Where are knots located?

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

Human DNA is wrapped around histone proteins Knots in DNA?

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)

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 base pairs Rybenkov et al., PNAS 90, 5307 (1991)

The Protein Data Bank 02/2005 (24937)

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

Knots are very rare:230 / (1%) Source:mostly bacteria and viruses, but also mouse, cow, human and spinach Depth >5>10>15>20>25 # structures (0.1%) # proteins 26 (9) 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

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

Early knot scientists … Phrygia, 333 BC

The Alexander polynomial