Liang C, Jiang Y
Institute of Theoretical Chemistry, Jilin University, Changchun, China.
J Theor Biol. 1992 Sep 21;158(2):231-43. doi: 10.1016/s0022-5193(05)80721-0.
The chirality of ground DNA knots and links is described and characterized in terms of color symmetry groups (CSG), i.e. color symmetry groups I and II, which correspond to topochirality (topological chirality) and topoachirality (topological achirality) which bear an uncanny resemblance to point groups I (proper) and point groups II (improper) used for testing geochirality (geometrical chirality) and geoachirality (geometrical achirality), respectively. By regarding these two crossing modes in mirror images as white and black vertices, DNA knots and links with minimal crossings can be mapped to vertex-bicolored graphs under a working hypothesis that DNA knots and links exist in ground states with minimal energy m0. The color symmetry group of a vertex-bicolored graph G is defined as the set of all permutations and permutation asymmetrizations of the vertices of G that preserve its topology (connectivity), where asymmetrization, denoted as (a), is the operation of changing vertices' colors, and a permutation followed by an (a) is a permutation asymmetrization. The color symmetry groups I contains only permutations, whereas color symmetry groups II comprise permutation asymmetrizations as well as permutations. Four DNA knots and links in nature are analyzed and tabulated consisely. In addition, the well-known figure-of-eight knot and Borromean rings are discussed in much the same way.
从颜色对称群（CSG）的角度描述并刻画了基态DNA纽结和链环的手性，即颜色对称群I和II，它们分别对应拓扑手性和拓扑非手性，这与用于测试地球手性（几何手性）和地球非手性（几何非手性）的点群I（真）和点群II（非真）有着惊人的相似之处。通过将镜像中的这两种交叉模式视为白色和黑色顶点，在DNA纽结和链环以最低能量m0处于基态的工作假设下，具有最小交叉数的DNA纽结和链环可以映射到顶点双色图。顶点双色图G的颜色对称群被定义为G的所有保持其拓扑结构（连通性）的顶点置换和置换不对称化的集合，其中不对称化，记为(a)，是改变顶点颜色的操作，一个置换后跟一个(a)就是一个置换不对称化。颜色对称群I只包含置换，而颜色对称群II既包括置换不对称化也包括置换。对自然界中的四个DNA纽结和链环进行了分析并简要列表。此外，以大致相同的方式讨论了著名的八字纽结和博罗梅安环。
