We classify rectangular DNA origami structures according to their scaffold and staples organization by associating a graphical representation to each scaffold folding. Inspired by well studied Temperley–Lieb algebra, we identify basic modules that form the structures. The graphical description is obtained by ‘gluing’ basic modules one on top of the other. To each module we associate a symbol such that gluing of modules corresponds to concatenating the associated symbols. Every word corresponds to a graphical representation of a DNA origami structure. A set of rewriting rules defines equivalent words that correspond to the same graphical structure. We propose two different types of basic module structures and corresponding rewriting rules. For each type, we provide the number of all possible structures through the number of equivalence classes of words. We also give a polynomial time algorithm that computes the shortest word for each equivalence class.

我们通过将图形表示与每个支架折叠相关联，根据矩形DNA折纸结构根据其支架和钉书钉的组织进行分类。受深入研究的Temperley-Lieb代数的启发，我们确定了构成结构的基本模块。通过将基本模块一个接一个地“胶合”获得图形描述。对于每个模块，我们将一个符号关联起来，以便模块的粘合对应于关联关联的符号。每个单词都对应于DNA折纸结构的图形表示。一组重写规则定义了与相同图形结构相对应的等效词。我们提出两种不同类型的基本模块结构和相应的重写规则。对于每种类型，我们通过单词的等价类的数量来提供所有可能结构的数量。我们还给出了多项式时间算法，该算法计算每个等价类的最短单词。