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Properties of ideal composite knots

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Abstract

The shortest tube of constant diameter that can form a given knot represents the ‘ideal’ form of the knot1,2. Ideal knots provide an irreducible representation of the knot, and they have some intriguing mathematical and physical features, including a direct correspondence with the time-averaged shapes of knotted DNA molecules in solution1,2. Here we describe the properties of ideal forms of composite knots — knots obtained by the sequential tying of two or more independent knots (called factor knots) on the same string. We find that the writhe (related to the handedness of crossing points) of composite knots is the sum of that of the ideal forms of the factor knots. By comparing ideal composite knots with simulated configurations of knotted, thermally fluctuating DNA, we conclude that the additivity of writhe applies also to randomly distorted configurations of composite knots and their corresponding factor knots. We show that composite knots with several factor knots may possess distinct structural isomers that can be interconverted only by loosening the knot.

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Figure 1: Ideal geometrical representations of several kinds of composite knots as viewed down their longest principal axes of inertia.
Figure 2: Additivity of the writhe and subadditivity of the relative lengths of ideal representations of composite knots.
Figure 3

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Acknowledgements

This work was supported by the US Public Health Service, the Polish Committee of Scientific Research, the Swiss National Foundation and by Foundation Herbette, University of Lausanne. We thank R. Scharein for making available his KnotPlot program.

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Authors and Affiliations

  1. *Department of Chemistry, Rutgers the State University of New Jersey, New Brunswick, 08903, New Jersey, USA

    Vsevolod Katritch & Wilma K. Olson

  2. †Institute of Physics, Poznan University of Technology, 60-965 Poznan and Institute of Molecular Physics, 60-159, Poznan, Poland

    Piotr Pieranski

  3. ‡Laboratoire d'Analyse Ultrastructurale, Bâtiment de Biologie, Université de Lausanne, CH-1015, Lausanne-Dorigny, Switzerland

    Jacques Dubochet & Andrzej Stasiak

Authors
  1. Vsevolod Katritch
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  2. Wilma K. Olson
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  3. Piotr Pieranski
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  4. Jacques Dubochet
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  5. Andrzej Stasiak
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Correspondence to Andrzej Stasiak.

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Katritch, V., Olson, W., Pieranski, P. et al. Properties of ideal composite knots. Nature 388, 148–151 (1997). https://doi.org/10.1038/40582

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