Universal Knot Spectra for Confined Polymers

Knotting is a prevalent phenomenon which occurs at the macroscale (e.g., in headphone cords) and at the microscale (e.g., in DNA and proteins). For a confined polymer, the knotting probability can rapidly approach 100% as the degree of confinement increases, while the mechanism of knot formation in a confined space is unclear. In this work, we use computer simulation to generate equilibrium conformations of a polymer confined in a sphere or a tube and then calculate the knotting probability, pknot, and the knot complexity that is quantified by the minimal crossing number, Ncross. Surprisingly, we find a universal correlation between pknot and ⟨Ncross⟩. Further analysis reveals that the universal correlation is caused by the fact that the distribution of knot types, i.e., the knot spectrum, of a confined polymer follows a universal behavior, only depending on the total knotting probability, regardless of the polymer length, bending stiffness, and degree of confinement. Such universal behavior reveals a possible mechanism of knot formation in a confined space via random threading of segments among other segments. The universal behaviors agree with prior experimental and simulation results of DNA knots and can be practically useful to infer ⟨Ncross⟩ from pknot, or vice versa, in the case that either one is difficult to be measured