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Capturing knots in (bio-) polymers …. Peter Virnau, Mehran Kardar, Yacov Kantor. 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). - PowerPoint PPT Presentation
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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
31
41
At which chain length do knots occur?
unknot
31
41
At which chain length do knots occur?
Knots are rare in the swollen phase (1% for3000 CH2)
unknot
31
41
At which chain length do knots occur?
Knots are common in a dense phase (80% for3000 CH2)
unknot
31
41
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
31
41
Knots are frequent
Loops vs. Chains
unknot
31
41
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
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