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 CH2

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)

Simplification

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

Coil / Globule

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

Reduce chain, connect ends, calculate Alexander polynomial

Coil / Globule

At which chain length do knots occur?

unknot

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At which chain length do knots occur?

unknot

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At which chain length do knots occur?

Knots are rare in the swollen phase (1% for3000 CH2)

unknot

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At which chain length do knots occur?

Knots are common in a dense phase (80% for3000 CH2)

unknot

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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 knots localized ?

dilute rare (1% for 3000 CH2) yes

dense frequent (80%) no

Summary I

frequency of knots localized ?

dilute rare (1% for 3000 CH2) yes

dense frequent (80%) no

• Probabilities: Open polymers Loops ?

Summary I

frequency of knots localized ?

dilute rare (1% for 3000 CH2) yes

dense frequent (80%) no

• Probabilities: Open polymers Loops ?

• Excluded volume ?

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 ?

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

Polymers vs. Random Walks

Loops vs. Chains

unknot

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Knots are frequent

Loops vs. Chains

unknot

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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 knots localized ?

dilute rare (1% for 3000 CH2) yes

dense frequent (80%) no

RW very frequent extremely

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 10000 base pairs

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

The Protein Data Bank

www.pdb.org 02/2005 (24937)

The Protein Data Bank

www.pdb.org

Problems: 1. Missing atoms

2. Multiple Chains

3. Microheterogeneity

4. Same 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)

Knots in proteins

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

Final Summary

virnau@mit.edu

Early knot scientists …

Phrygia, 333 BC

The Alexander polynomial