Knot Energy, Complexity, and Mobility of Knotted Polymers

The Newtonian energy EN of an object is defined by the energy required to charge a conductive object of arbitrary shape and scales inversely to the self-capacity C, a basic measure of object size and shape. It is known that C is minimized for a sphere for all objects having the same volume, and that c increases as the symmetry of an object is reduced at fixed volume. Mathematically similar energy functionals have been related to the average knot crossing number (m), a natural measure of knot complexity and correspondingly we find EN to be directly related to both (m) and the Stokes friction coefficient of knotted DNA. To establish this relation, we employ molecular dynamics simulations to generate knotted polymeric configurations having different length and stiffness, and minimum knot crossing number values m. We then compute EN for all these knotted polymers using the path-integration program ZENO and find that the average Newtonian knot energy (EN) is directly proportional to (m). The ratio of the hydrodynamic radius, radius of gyration, and the intrinsic viscosity of semi-flexible knotted polymers in comparison to the linear polymeric chains since these ratios should be useful in characterizing knotted polymers experimentally. Finally, the solution properties of other branched polymers, such as star polymers, and these results are compared with those of knotted ring polymers. The effect of chain stiffness on these properties is also discussed