Knots in open Polymers
knot is not mathematically well defined in an open chain,
and its "size" is ambiguous. Results
somewhat depend on operational definitions.
knots: Knots can be tied by pulling on
the ends of a chain.
- E.g., in experiments on single DNA
used simulations of pulled knots to estimate the size:
presence of the knot introduces a new length scale into the
we use the lack of scaling to deduce a knot size?
the correction to scaling exponent of 0.56 in d=3
is consistent with t=0.4 0.1.
a recent study
claims t=0.75 0.14.
- B. Marcone, E. Orlandini,
A.L. Stella, and F. Zonta, PRE 75,
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,
polyethylene: Monte Carlo
(MC) simulations of a coarse-grained model for polyethylene
- Bead-spring chain (LJ+FENE): 1 bead ≅ 3 CH2
- Equilibrium configurations generated with standard MC techniques
(pivot, reptation, local moves)
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
- 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,
web-server for detection of knots: http://knots.mit.edu