Abstract
The Coulomb energy EC is defined by the energy required to charge a conductive object 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 EC to be directly related to 〈m〉 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 for a wide class of knot types relevant to the real DNA. We then compute EC for all these knotted polymers using the program ZENO and find that the average Coulomb energy 〈EC〉 is directly proportional to 〈m〉. Finally, we calculate estimates of 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.
Highlights
Experiments have shown a remarkable correlation between the migration speed of knotted DNA in gel electrophoresis and average knot crossing number, the number of places where a knotted polymer crosses itself when projected onto a surface[1,2]
Minimum surface area directly corresponds to a minimizing energy, e.g., the interfacial energy of a droplet defines an “energy functional” and fluid droplets of ordinary fluids are spherical in order to minimize their interfacial energy and their surface area
We explore the validity of this theoretically motivated approximation through a consideration of knotted polymer chains generated by molecular dynamics simulations and we use ZENO to determine 〈EC〉
Summary
The Coulomb energy functional in Eq (2) is a natural functional to define the complexity of knotted DNA since DNA is a highly charged macromolecule. These images correspond to representative knot configurations for the polymers interacting with a bending energy amplitude kbend = 10 ε (blue triangles). We compute the average Coulomb energy 〈EC〉, m, and 〈m〉 for the polymeric knots configurations generated using the coarse–grained model described in the previous section for a selected family of knot types relevant to the characterization of real DNA. The average Coulomb energy 〈EC〉 as a function of the minimum crossing number m (upper panel) and average crossing number 〈m〉 (lower panel) for polymers having the same length L = 352.8 nm and different chain rigidities.
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