Tube Model for Complex Polymer Knots

Jian-Song Yang; Yong-Jian Zhu; Qi-Yuan Qiu; Wen-Xuan Xu; Liang Dai

doi:10.1007/s10118-025-3405-8

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Tube Model for Complex Polymer Knots

RESEARCH ARTICLE | Updated：2025-10-30

- Tube Model for Complex Polymer Knots
- Tube Model for Complex Polymer Knots
- 高分子科学（英文） 2025年43卷第11期页码：2138-2149
- Affiliations：
  
  a.Department of Physics, City University of Hong Kong, Hong Kong 999077, China
  b.Shenzhen Research Institute, City University of Hong Kong, Shenzhen 518057, China
- Author bio：
  
  liangdai@cityu.edu.hk
- Funds：
  
  This work was financially supported by the National Natural Science Foundation of China (No. 22273080) and Research Grants Council of Hong Kong (Nos. 11313322 and 11307224).;Electronic supplementary information (ESI) is available free of charge in the online version of this article at http://doi.org/10.1007/s10118-025-3405-8.
- DOI：10.1007/s10118-025-3405-8
  中图分类号：
- 收稿：2025-06-09，
  
  修回：2025-06-24，
  
  录用：2025-06-27，
  
  网络首发：2025-09-10，
  
  纸质出版：2025-11-05
- Accepted：
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Yang, J. S.; Zhu, Y. J.; Qiu, Q. Y.; Xu, W. X.; Dai, L. Tube model for complex polymer knots. Chinese J. Polym. Sci. 2025, 43, 2138–2149

Jian-Song Yang, Yong-Jian Zhu, Qi-Yuan Qiu, et al. Tube Model for Complex Polymer Knots[J]. Chinese Journal of Polymer Science, 2025, 43(11): 2138-2149.
Yang, J. S.; Zhu, Y. J.; Qiu, Q. Y.; Xu, W. X.; Dai, L. Tube model for complex polymer knots. Chinese J. Polym. Sci. 2025, 43, 2138–2149 DOI： 10.1007/s10118-025-3405-8.

Jian-Song Yang, Yong-Jian Zhu, Qi-Yuan Qiu, et al. Tube Model for Complex Polymer Knots[J]. Chinese Journal of Polymer Science, 2025, 43(11): 2138-2149. DOI： 10.1007/s10118-025-3405-8.

摘要

We extend the tube model to 4₁

5₁

and 5₂ knots

revealing universal behaviors across knot types. Knot boundaries tend to align with stiff segments

and bending energy is surprisingly lower in knot-core boundaries—findings that deepen our understanding of polymer knot physics.

Abstract

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: 3

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 4

and 5

knots

resulting in several findings. First

the computational algo

rithm developed for 3

knot cannot be directly applied to 4

knot. After modifying the algorithm

we quantify the tubes for 4

knot. Second

we find that

for all four knot types

the knot-core 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

关键词

Keywords

references

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