Diffusion mechanisms of localised knots along a polymer

We consider the diffusive motion of a localised knot along a linear polymer chain. In particular, we derive the mean diffusion time of the knot before it escapes from the chain once it gets close to one of the chain ends. Self-reptation of the entire chain between either end and the knot position, during which the knot is provided with free volume, leads to an L-3 scaling of diffusion time; for sufficiently long chains, subdiffusion will enhance this time even more. Conversely, we propose local breathing, i.e., local conformational rearrangement inside the knot region (KR) and its immediate neighbourhood, as additional mechanism. The contribution of KR-breathing to the diffusion time scales only quadratically, similar to L-2, speeding up the knot escape considerably and guaranteeing finite knot mobility even for very long chains.