Charged Knots

Charged Knots

"Ideal Knots," and "Knot Energies:" Can the knot type be determined from the `ideal' shape of the curve that minimizes a particular (two-body) potential? (ram video)

Jonathan Simon in Ideal Knots (1998):

"Suppose you have a knotted loop on a string, and you spread an electric charge along the string and then let go; what will happen?"
This question has been a common "cocktail party" topic among knot theorists for many years, ...

Consider a Charged Trefoil

Prime charged knots ... are tight

Composite charged knots ... are factored

Model & Simulations:

Results obtained from off-lattice Monte Carlo simulations of N monomers,
- each a sphere of diameter b;
- connected by tethers such that

Energy & length-scales:

Coulomb interaction, with screening length :

Effect of chaning the Bjerrum length

Screened Interactions:

In most physically relevant situations, Coulomb interactions are highly screened;
~1nm for DNA in vivo, but larger values can be achieved in vitro.

Do tight knots disappear for finite screening lengths?

Electrostatic (Odijk-Skolnick-Fixman) persistence length:

The tight knot is the global energy minimum as long as is comparable to size of the open chain.
The tight knot is a local energy minimum as long as is larger than the bare persistence length.

Dynamics:

Tight knots (metastable) slow down the relaxational dynamics:

The higher the charge density, the slower is the relaxation to the final state.
A knot in an open chain is untied by diffusion