Knotting occurs in polymers and affects polymer properties. Physical understanding of polymer knots is limited due to the complex conformational space of knotted structures. The knotting problem can be handled by the tube model, which assumes that knotted polymer segments are confined in a virtual tube. Recently, we quantified this virtual tube using a computational algorithm. The algorithm was limited to the simplest knot: 31 knot. It remains unclear how the tube model and computational algorithm are applied to more complex knots. In this work, we apply the tube model to 41, 51, and 52 knots, resulting in several findings. First, the computational algorithm developed for 31 knot cannot be directly applied to 41 knot. After modifying the algorithm, we quantify the tubes for 41 knot. Second, we find that, for all four knot types, the knotcore region have less average bending energy density than unknotted regions when the chain bending stiffness is small. This counterintuitive result is explained by the tube model. Third, for all four knot types, polymer segments at the boundaries of knot cores adopt nearly straight conformations (almost zero bending) and exhibit lower local bending compared to other knot-core regions and unknotting regions. This local behavior is also consistent with prediction from the tube model. This counterintuitive result is also explained by the tube model. Fourth, for all four knot types, when a polymer has non-uniform bending stiffness, a knot prefers certain chain positions such that the knot boundary locates at one stiff segment. Overall, our work paves the way for applying the tube model to complex polymer knots and obtains many common results for different knot types, which can be useful in understanding many knotting systems, such as DNA knots in vivo.
Read more at Chinese Journal of Polymer Science:
https://link.springer.com/article/10.1007/s10118-025-3405-8#citeas
Photo caption:
Mirror reflection of knot conformation. (a) Classification of 41 knot conformations into 8 types based on the position and direction of the first bead (marked in red). The first conformation is labelled as L↑, where L indicates that the first bead is located at the left side of the last bead, and ↑ indicates that the direction from the second bead to the first bead points outward of the plane of paper. R indicates right and ↓ indicates the direction points inward of the plane of paper. ML↑, ML↓, MR↑, and MR↓ are mirror reflections of L↑, L↓, R↑, and R↓, respectively. (b) The new method, including mirror reflection, can transform any knot conformation to the target type. The traditional method cannot do it. Note that before mirror reflections, each conformation is rotated to make the first and second principal axes aligning with x- and y- directions, respectively.
10 Sep 2025
