Katritch V, Olson W K, Pieranski P, Dubochet J, Stasiak A
Department of Chemistry, Rutgers the State University of New Jersey, New Brunswick 08903, USA.
Nature. 1997 Jul 10;388(6638):148-51. doi: 10.1038/40582.
The shortest tube of constant diameter that can form a given knot represents the 'ideal' form of the knot. 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 solution. 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.
能够形成给定纽结的直径恒定的最短管子代表了该纽结的“理想”形式。理想纽结提供了纽结的不可约表示，并且它们具有一些有趣的数学和物理特征，包括与溶液中纽结DNA分子的时间平均形状直接对应。在这里，我们描述了复合纽结的理想形式的性质——通过在同一根绳子上依次系两个或更多独立纽结（称为因子纽结）而得到的纽结。我们发现，复合纽结的扭曲（与交叉点的手性有关）是因子纽结理想形式的扭曲之和。通过将理想复合纽结与纽结的、热波动的DNA的模拟构型进行比较，我们得出结论，扭曲的可加性也适用于复合纽结及其相应因子纽结的随机扭曲构型。我们表明，具有多个因子纽结的复合纽结可能拥有不同的结构异构体，这些异构体只能通过解开纽结来相互转换。
