Knot Energy, Complexity, and Mobility of Knotted Polymers
Luis F. Vargas Lara, Ahmed M. Hassan, Marc A. Mansfield, Jack F. Douglas
The Newtonian capacity C of an object is defined by the equilibrium energy, EN =1/C, of a charge distribution on a conductive object and provides a classical measure of object shape. It is well known, for example, that C is minimized for a sphere for all objects having the same volume, and that C tends to increase as the symmetry of an object is reduced at fixed volume. Recent calculations have shown that this "energy functional" is related to the translational friction coefficient ft of Brownian particles having general shape. Mathematically, similar energy functionals are known to be related to the average crossing number of knotted polymers, a natural measure of knot complexity. EN vanishes for smooth" curves in 3 dimensions, but this quantity is finite for Brownian and self-avoiding paths because of their fractal shape. In the present paper, we find EN to be a measure of knot complexity that is directly related to the mobility of knotted DNA. To establish these relations, we employ molecular dynamics simulations to generate polymeric knot configurations having different length and stiffness, and minimum knot crossing number values m in the range, 0 <= m <= 9. As a reference point for comparison to knotted semi-flexible polymer properties, we charge the polymer segments to create ideal knotted chain configurations or "canonical forms" having apparently unique shapes that serve to classify a family of knots. We then compute EN for all these knotted polymers using the numerical path-integration program ZENO. We find that the average Newtonian energy, , of knotted polymers normalized by E0, the energy of a canonical unknotted ring, is directly proportional to the average crossing number , to a high approximation. Moreover, the universal scaling is nearly independent of chain rigidity. Our observations potentially provide a rationale for understanding the observed dependence of the electrophoretic mobilities of knotted DNA on .