Knots in open Polymers
Disclaimer: A
knot is not mathematically well defined in an open chain,
and its "size" is ambiguous. Results
somewhat depend on operational definitions.
Pulling
knots: Knots can be tied by pulling on
the ends of a chain.
 E.g., in experiments on single DNA
molecules:
We
used simulations of pulled knots to estimate the size:
 The
presence of the knot introduces a new length scale into the
problem,

Can
we use the lack of scaling to deduce a knot size?

 Interestingly,
the correction to scaling exponent of 0.56 in d=3
is consistent with t=0.4 0.1.
 However,
a recent study
claims t=0.75 0.14.
 B. Marcone, E. Orlandini,
A.L. Stella, and F. Zonta, PRE 75,
041105 (2007)
Reducing
complexity: To find and localize knots in long open polymers
 Close the chain by connecting its end through a
number of procedures, and look for knots.

A simplification (reduction) step is essential for calculation
of Alexander polynomials
 K. Koniaris and M. Muthukumar, Phys.
Rev. Lett. 66, 2211 (1991)
 W.R. Taylor, Nature 406,
916919 (2004)
Model
polyethylene: Monte Carlo
(MC) simulations of a coarsegrained model for polyethylene
 Beadspring chain (LJ+FENE): 1 bead ≅ 3 CH2
 Equilibrium configurations generated with standard MC techniques
(pivot, reptation, local moves)
Qualitative
results for polymers in coil (swollen), globule (compact), and confined sates.
 Knots are rare in the swollen
phase (1% for 3000 CH2)
 Knots are tight in the swollen
phase
Biopolymers:
 Knots are rare for DNA
in good solvent (0.5%4% for 10000 base pairs)
 Knots are also rare in globular
proteins (~1%  273/32,853 in PDB structures,
1/3/2007)
MIT
webserver for detection of knots: http://knots.mit.edu