Sedimentation of Knotted Polymers

We investigate the sedimentation of knotted polymers by means of stochastic rotation dynamics, a molecular dynamics algorithm that takes hydrodynamics fully into account. We show that the sedimentation coefficient s, related to the terminal velocity of the knotted polymers, increases linearly with the average crossing number n_c of the corresponding ideal knot. To the best of our knowledge, this provides the first direct computational confirmation of this relation, postulated on the basis of experiments in "The effect of ionic conditions on the conformations of supercoiled DNA. I. sedimentation analysis" by Rybenkov et al., for the case of sedimentation. Such a relation was previously shown to hold with simulations for knot electrophoresis. We also show that there is an accurate linear dependence of s on the inverse of the radius of gyration R_g^-1, more specifically with the inverse of the R_g component that is perpendicular to the direction along which the polymer sediments. When the polymer sediments in a slab, the walls affect the results appreciably. However, R_g^-1 remains to a good precision linearly dependent on n_c. Therefore, R_g^-1 is a good measure of a knot's complexity.