Peter%20Virnau,%20Mehran%20Kardar,%20Yacov%20Kantor

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Capturing knots in (bio-) polymers
Peter Virnau, Mehran Kardar, Yacov Kantor
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History of knot science
Lord Kelvin (1867) Vortex atoms
P.G. Tait Knot tables
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Classification of knots
J.W. Alexander (1923) First algorithm which can
distinguish between knots ( somewhat)
2005 still no complete invariant
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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
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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
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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
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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
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Motivation Biology
Knots Why? Structure lt-gt Function Role of
entanglements?
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Motivation Biology
Knots How? Reference system Single homopolymer
in stretched and compact state
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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?
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Model
Polymer Coarse-grained model for polyethylene
Bead-spring chain (LJFENE) 1 bead _at_
3 CH2
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Model
Polymer Coarse-grained model for polyethylene
Bead-spring chain (LJFENE) 1 bead _at_
3 CH2 Equilibrium configurations are generated
with standard Monte Carlo techniques (pivot,
reptation, local moves)
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Simplification
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Coil / Globule
Polymer Coarse-grained model for polyethylene
Bead-spring chain (LJFENE) 1 bead _at_
3 CH2
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Coil / Globule
Polymer Coarse-grained model for polyethylene
Bead-spring chain (LJFENE) 1 bead _at_
3 CH2
Reduce chain, connect ends, calculate Alexander
polynomial
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At which chain length do knots occur?
unknot 31 41
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At which chain length do knots occur?
unknot 31 41
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At which chain length do knots occur?
unknot 31 41
Knots are rare in the swollen phase (1 for 3000
CH2)
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At which chain length do knots occur?
unknot 31 41
Knots are common in a dense phase (80 for 3000
CH2)
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Are knots localized or spread?
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Are knots localized or spread?
Knots are localized in the swollen phase
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Are knots localized or spread?
Knots are delocalized in a dense phase
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Summary I
frequency of knots localized
? dilute rare (1 for 3000 CH2)
yes dense frequent (80) no
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Summary I
frequency of knots localized
? dilute rare (1 for 3000 CH2)
yes dense frequent (80) no

• Probabilities Open polymers lt-gt Loops ?

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Summary I
frequency of knots localized
? dilute rare (1 for 3000 CH2)
yes dense frequent (80) no

• Probabilities Open polymers lt-gt Loops ?
• Excluded volume ?

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Summary I
frequency of knots localized
? dilute rare (1 for 3000 CH2)
yes dense frequent (80) no

• Probabilities Open polymers lt-gt Loops ?
• Excluded volume ?
• Distribution of sizes and location ?

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Summary I
frequency of knots localized
? dilute rare (1 for 3000 CH2)
yes dense frequent (80) no

• Probabilities Open polymers lt-gt Loops ?
• Excluded volume ?
• Distribution of sizes and location ?
• -gt simpler (faster) model Random walk

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Polymers vs. Random Walks
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Loops vs. Chains
unknot 31 41
Knots are frequent
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Loops vs. Chains
unknot 31 41
Loops and chains have similar knotting
probabilities
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Distribution of knot sizes
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Distribution of knot sizes
Knots are localized in random walks
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Distribution of knot sizes
Most likely knot size only 6 segments
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Distribution of knot sizes
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Distribution of knot sizes
Power-law tail in knot size distribution
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Where are knots located?
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Where are knots located?
Knots are equally distributed over the entire
polymer, but
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Where are knots located?
larger in the middle
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Where are knots located?
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Summary II
frequency of knots localized
? dilute rare (1 for 3000 CH2)
yes dense frequent (80) no RW very
frequent extremely DNA ???
??? Proteins ??? ???
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Knots in DNA?
Human DNA is wrapped around histone proteins

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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)
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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)
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The Protein Data Bank
www.pdb.org 02/2005 (24937)
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The Protein Data Bank
www.pdb.org Problems 1. Missing atoms 2.
Multiple Chains 3. Microheterogeneity 4.
Same Proteins
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Knots in proteins
Knots are very rare 230 / 24937
(1) Source mostly bacteria and viruses, but
also mouse, cow, human and spinach Depth
gt5 gt10 gt15 gt20 gt25 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)
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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_at_mit.edu
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Early knot scientists
Phrygia, 333 BC
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The Alexander polynomial