DNA origami words, graphical structures and their rewriting systems

Abstract

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.

1 Introduction

Self-assembly is a process where smaller components (usually molecules) autonomously assemble to form a larger structure. This process is essential in building biomolecular structures and high order polymers (Whitesides and Boncheva 2002). Applications of self-assembly range from electric circuits at nano level (Bhuvana et al. 2009; Eichen et al. 1998) to smart drug delivery systems (Li et al. 2013; Verma and Hassan 2013). A well-known self-assembly variant is the DNA origami system introduced by Rothemund (2006). In DNA origami, a single-stranded DNA plasmid, called the scaffold, outlines a shape, while short DNA strands, called staples, connect different parts of the scaffold, fixing the terminal rigid structure. The top of Fig. 1 shows a segment of schematic DNA origami where the scaffold is depicted by a black line while staples are represented by colored lines with arrows. Experimental results of several DNA origami shapes from Rothemund’s original paper (Rothemund 2006) are shown at the bottom of Fig. 1.

Fig. 1

(Top) A schematic representation of a DNA origami structure. The scaffold is a black line and staples are colored lines with arrows. (Bottom) Various shapes made by DNA origami. Both figures are from Rothemund (2006). (Color figure online)

Theoretical approaches to analyze DNA origami have been mainly focused on efficient sequence design of staples as well as synthetic scaffolds that fold into the target shape (Rothemund 2005; Veneziano et al. 2016). However, the same outlined shape can be obtained by different scaffold and staple organizations. In this paper, we use graphical description to describe different scaffold/staple organization within the same origami shape. We identify unit building blocks (modules) for the graphical representations whose composition (one on top of another) through connecting the corresponding staple/scaffold strands builds up a larger structure. The unit blocks correspond to symbols in an alphabet, and concatenation of symbols correspond to composition of the modules. We observe that the unit structures within DNA origami resemble the diagram representation of the generators of the Jones monoid, a monoid variant of the well studied Temperley–Lieb algebras (Borisavljević et al. 2002; Jones 1983; Kauffman 2001). We assign symbols to the unit structures, and define rewriting rules that provide equivalence of words corresponding to their graphical representation equivalence. Winfree et al. (2001) proposed word representation of simple DNA tiles by modulization, but this is the first approach to describe general DNA origami structures by words. We propose two types of basic module structures and their corresponding rewriting rules. For each type, we provide the number of distinct equivalence classes of words, which corresponds to the possible DNA origami structures. We also compute the size of the shortest word within each class.

An extended abstract of this work had been published in the proceedings of the 18th International Conference on Unconventional Computation and Natural Computation (Garrett et al. 2019).

2 Preliminaries

An alphabet $\Sigma$ is a non-empty finite set of symbols. A word $w=w_1w_2\cdots w_n\in \Sigma ^n$ is a finite sequence of n symbols over $\Sigma$, and $|w|=n$ denotes the size of the word. We use $\epsilon$ to denote the empty word. A subword, or a factor, of a word $w=w_1w_2\cdots w_n$ is $w'=w_i\cdots w_j$ where $1\le i\le j\le n$. We use $\Sigma ^*$ to denote the set of all words over $\Sigma$. Concatenation of two words x and y is denoted by $x\cdot y$, or simply xy.

A word rewriting system $(\Sigma ,R)$ consists of an alphabet $\Sigma$ and a set $R\subseteq \Sigma ^*\times \Sigma ^*$ of rewriting rules. In this paper, both $\Sigma$ and R are finite. R generates an equivalence relation $\hat{R}$ on $\Sigma ^*$. An element (x, y) of R is called a rewriting rule, and is written as $x\leftrightarrow y$. In general, we rewrite uxv as uyv for $u,v\in \Sigma ^*$ if $(x,y)\in R$, and denote such rewriting by $uxv\leftrightarrow uyv$. For a sequence of words $u=x_1\leftrightarrow x_2\leftrightarrow \cdots \leftrightarrow x_n=v$ in a rewriting system $(\Sigma , R)$, we write $u\sim v$. We consider $\hat{R}$ and denote an equivalence class of a word w as [w]. A word $w_0\in [w]$ is irreducible if $|w_0|\le |w'|$ for all $w'\in [w]$. For an ordered alphabet $\Sigma$, we use the lexicographically first irreducible word $\hat{w}$ of [w] as the representative word of [w]. We consider the set of equivalence classes $\mathcal {O}$. The reader may refer to Book and Otto (1993) for more information about word rewriting systems.

The Temperley–Lieb algebra has been extensively studied in physics and knot theory (Kauffman 2001). A monoid version of Temperley–Lieb algebras, called the Jones monoid $\mathcal J_n$, has also been studied (Borisavljević et al. 2002; Jones 1983; Lau and FitzGerald 2006). The generators of $\mathcal J_n$ are $h_1,\dots , h_{n-1}$ and satisfy three classes of relations:

• 1. $h_ih_jh_i = h_i$ for $|i-j|=1$

• 2. $h_ih_i = h_i$

• 3. $h_ih_j = h_jh_i$ for $|i-j|\ge 2$

Fig. 2

Graphical representation of the Jones monoid $\mathcal {J}_5$. a The generator $h_3$. b The relation $h_1h_2h_1=h_1$. c The relation $h_1h_1=h_1$. d The relation $h_1h_3=h_3h_1$

The generators and relations can be represented graphically as in Fig. 2 (Lau and FitzGerald 2006). There are n endpoints at the top and the bottom of graphical representations of elements of $\mathcal J_n$. A generator $h_i$ connects the ith and $i{+}1$st top endpoints and the ith and $i{+}1$st bottom endpoints, while other endpoints are connected by vertical lines. The generator $h_3$ in $\mathcal J_5$ is presented in Fig. 2a, connecting the top 3rd and 4th and the bottom 3rd and 4th points, respectively. Multiplication of two elements corresponds to concatenation of diagrams, placing the diagram of the first element on top of the second, and removing closed loops. The relations 1, 2 and 3 can also be expressed graphically as in Fig. 2b, d, respectively. Two elements in the Jones monoid are equal if their graphical representations are equivalent, that is, they have the same set of top-bottom connecting segments after deleting internal loops. For any two words that have equivalent diagrams, one word can be rewritten to the other using the sequence of relations 1 to 3. In simplification of the DNA origami structure, we take a similar approach where we only take into account the endpoints of scaffolds and staples that are visible at the top and the bottom borderline of the whole structure. Thus, we use the Jones monoid as a base to construct DNA origami words and corresponding rewriting systems.

3 DNA origami words and rewriting systems

3.1 DNA origami words

We focus on rectangular DNA origami structures. They can be formed by a variety of scaffold-strand folds and connecting staples. We introduce an algebraic way to distinguish these different folds yielding the same overall shape. We use basic unit structures (modules) that build the shape and associate symbols (generators) to these basic modules. Based on graphical diagrams, and inspired by the Jones monoid diagrams, we define equivalence of two origami structures, and define corresponding rewriting rules that realize the equivalence in the graphical diagrams.

In this schematics of the DNA origami structure, we consider columns made of scaffolds, and staples that go along the scaffolds as follows: there are places where two adjacent scaffolds connect the two columns, and also places where two adjacent staples connect the two columns. In addition, because DNA is oriented, the scaffolds and staples have directions: adjacent scaffolds are anti-parallel, and a staple is anti-parallel to a scaffold it connects to. A graphical structure corresponding to DNA origami is thus presented with types of directed segments and the corresponding end-point connections. In addition, in order to define composition of structures when some parts of the structures are missing, we consider ‘virtual’ staples and scaffold. We use $s=i_t$ ($i_b$) to represent a point at the top (bottom) of the ith column. We assume that scaffolds at the ith column go upward if i is odd, and downward if i is even. The structure of width n is defined as follows: To the set $E_n=\{i_t,i_b\mid 1\le i\le n\}$ of points, we associate two partitions $(P_{c},Q_{c})$ and $(P_{p},Q_{p})$ where each set in the partitions consists of n points. The partition $(P_{c},Q_{c})$ is associated to the scaffold strands and $(P_{p},Q_{p})$ is associated to the staples. We define a bijection from $P_c$ to $Q_c$ which we describe as a set of ordered pairs $\{(s,t)\mid s,t,\in E_n\}$. Each pair (s, t) corresponds to a line segment that starts at point s and terminates at point t. This set of pairs is further partitioned to $({\mathcal {R}}_c, {\mathcal {V}}_c)$. We call the line segments ${\mathcal {R}}_c$ real scaffolds and $\mathcal {V}_c$ virtual scafolds. Similarly a bijection from $P_p$ to $Q_p$ represented by ordered pairs is partitioned to $(\mathcal {R}_p, \mathcal {Q}_p)$ where $\mathcal {R}_p$ is the set of real staples and $\mathcal {V}_p$ is the set of virtual staples. Then a graphical structure is defined as a tuple $(\mathcal {R}_{c},\mathcal {V}_{c},\mathcal {R}_{p},\mathcal {V}_{p})$ of four sets of pairs or four sets of line segments. Note that such virtual scaffolds and staples are merely used as auxiliary tools for concise definition of concatenation of graphical structures, where the resulting graphical structure is defined by a set of connections of different types of scaffolds and staples—otherwise, we may leave such virtual scaffolds and staples empty, which might look more natural, but such emptiness leads to introduction of complicated ‘extensions’ of scaffolds and staples aside from connections in the definition of concatenation of graphical structures, and we choose to use the former.

For given width n, we define basic modules and corresponding generators $\Sigma _n=\{\alpha _i,\beta _i\mid 1\le i\le n-1\}$ as an alphabet for DNA origami words with the order $\alpha _1<\cdots<\alpha _{n{-}1}<\beta _1<\cdots <\beta _{n{-}1}$. We say that $\alpha _i$ is complementary to $\beta _i$, and vice versa. For each generator $\alpha _i$, $\beta _i$, Table 1 shows the functions that describe their structures between the ith and the $i{+}1$st columns. The four maps that describe $\alpha _i$ (resp. $\beta _i$) are called units for $\alpha _i$ (resp. $\beta _i$). The units of the generators $\alpha _i$ and $\beta _i$ ($i=1,2$) are shown in Fig. 3.

Table 1 Units for generators of odd i’s (maps are inversed for even i’s)

Fig. 3

Graphical representation of units of $\alpha _i$ and $\beta _i$ ($i=1,2$). Scaffolds are represented by black lines and staples are represented by red dotted lines. For better visibility, staples are shifted right. (Color figure online)

In addition to the units for each generator, we must also define maps corresponding to the columns left of the ith and right of the $i{+}1$st column. Unlike the Jones monoid, the choice of structure for these surrounding columns is not trivial. We define three possible systems for this surrounding structure through choices of real or virtual scaffolds and staples. The structure of each generator $\gamma _i\in \Sigma _n$ has a context $\mathcal C(\gamma _i)$ which consists of pairs $(k_t, k_b)$ and their inverses where $k\not \in \{i,i+1\}$. The context $\mathcal C(\gamma _i)$ can have real or virtual pairs. Depending on the choice of virtual versus real context, there can be different structural descriptions. Table 2 defines three situations that can be used for three different descriptions of graphical structures $\mathcal G_{max(n)}$, $\mathcal G_{mid(n)}$, $\mathcal G_{min(n)}$, each representing a possible choice for the generator $\gamma _i$. We note that in $\mathcal {G}_{max(n)}$ the context $\mathcal C(\gamma _i)$ has both $\mathcal V_{c}$ and $\mathcal V_{p}$ empty, i.e. the whole context is real. In $\mathcal {G}_{mid(n)}$, the context $\mathcal C(\gamma _i)$ has $\mathcal V_{c}=\emptyset$, that is, the scaffold context is real but the staple context is virtual. In the case of $\mathcal {G}_{min(n)}$, the whole context $\mathcal C(\gamma _i)$ is virtual. The corresponding graphical structures of $\alpha _2$’s in different $\mathcal {G}$’s are shown in Fig. 4.

Table 2 Definition of three real and virtual scaffold and staple contexts for $\gamma _i$ when i is odd. The maps are inversed for even i’s

Concatenation of words and the corresponding graphical structure is defined similarly as in the Jones monoid diagrams. Graphical structures that correspond to words in $\Sigma _n^*$ are obtained by joining graphical structures of generators as explained below. The graphical structure corresponding to concatenation of two words is obtained by placing the graphical structure of the first word on top of the graphical structure of the second and connect the vertical lines that meet. In the case of virtual staples or scaffolds the connection follows the rule: If a real scaffold (staple) meets a virtual scaffold (staple), then the virtual scaffold (staple) becomes real. This process simulates the real structure extending through the empty space represented by the virtual structure.

Fig. 4

Different graphical structures of $\alpha _2$’s in $\mathcal {G}_{max(5)}$, $\mathcal {G}_{mid(5)}$ and $\mathcal {G}_{min(5)}$. Virtual scaffolds and staples are colored in gray

Fig. 5

Concatenation of $\alpha _1\beta _2$ and $\alpha _1$ under $\mathcal {G}_{min(3)}$

Figure 5 shows concatenation of $\alpha _1\beta _2$ and $\alpha _1$ under $\mathcal {G}_{min(3)}$. Formally, the graphical structure of a word $w=w_1w_2$ is defined as follows: Suppose $G(w_1)=(\mathcal {R}_{c1},\mathcal {V}_{c1},\mathcal {R}_{p1},\mathcal {V}_{p1})$ and $G(w_2)=(\mathcal {R}_{c2},\mathcal {V}_{c2},\mathcal {R}_{p2},\mathcal {V}_{p2})$ are graphical structures of $w_1$ and $w_2$, respectively. The graphical structure $G(w)=G(w_1w_2)=(\mathcal {R}_{c},\mathcal {V}_{c},\mathcal {R}_{p},\mathcal {V}_{p})$ is obtained with the following: The scaffold sets ($\mathcal {R}_{c}$ and $\mathcal {V}_{c}$) are obtained as follows (the staples follow an equivalent procedure):

• 1. For all pairs in $\mathcal {R}_{c1}\cup \mathcal {V}_{c1}$, replace the subscript b by m.

• 2. For all pairs in $\mathcal {R}_{c2}\cup \mathcal {V}_{c2}$, replace the subscript t by m.

• 3. Given set $\mathcal {R}_{c1}\cup \mathcal {V}_{c1}\cup \mathcal {R}_{c2}\cup \mathcal {V}_{c2}$, for each sequence $(q_0,q_1),(q_1,q_2),\ldots ,(q_{\nu {-}1},q_{\nu })$ of pairs where points $q_0$ and $q_\nu$ have subscripts t or b,

  • if any of these pairs is in $\mathcal {R}_{c1}\cup \mathcal {R}_{c2}$, add $(q_0,q_{\nu })$ to $\mathcal {R}_c$.

  • Otherwise, add $(q_0,q_{\nu })$ to $\mathcal {V}_c$.

We observe that the composition of graphical structures is associative, that is $G(w_1w_2w_3)=G((w_1w_2)w_3)=G(w_1(w_2w_3))$.

Fig. 6

Scaffold concatenation of $G(w_1)$ and $G(w_2)$. Real scaffolds are represented by thick lines for better visibility

Figure 6 describes the concatenation process of scaffolds. We replace the subscripts for the bottom points of $w_1$ and the top points of $w_2$ by m, which denotes the middle points in the concatenation. Then, connect pairs of scaffolds that meet in the middle. We regard the connected scaffold to be virtual only if both original scaffolds were virtual (step 4 (c)). Finally, we delete all pairs of scaffolds whose endpoints are at the middle, this includes all internal loops. Based on Tables 1 and 2, we define the set $\mathcal {G}\in \{\mathcal G_{max},\mathcal G_{mid},\mathcal G_{min}\}$ as the set of all graphical structures that can be constructed by concatenation of generators in the model $\mathcal G$. We denote the graphical structure of a word w under $\mathcal {G}_{max}$ as $G_{max}(w)$, and similarly define $G_{mid}(w)$ and $G_{min}(w)$.

3.2 DNA origami rewriting systems

It is straightforward that for any alphabet, given two words, the graphical structure of their concatenation is unique. Thus, if $G(w_1)=G(w_2)$ under a model $\mathcal {G}$, we say $w_1\sim w_2$ in $\mathcal {G}$ and define a rewriting rule $w_1\leftrightarrow w_2$ between two equivalent words. Due to difference of context structures, rewriting rules for $\mathcal {G}_{max(n)}$, $\mathcal {G}_{mid(n)}$ and $\mathcal {G}_{min(n)}$ differ from each other. For each structure, we find the set of basic rewriting rules that generate the equivalence and analyze the set of distinct equivalence classes.

3.2.1 $\mathcal {G}_{max(n)}$ case

We first observe that all staples and scaffolds in $\mathcal {G}_{max(n)}$ are real. We observe that except for added directions, their concatenation results in a bijection between scaffolds and staples without lasting conflict of the directions. Moreover, scaffolds in $\alpha _i$ (and staples in $\beta _i$) are straight and do not affect the structure of scaffolds (staples) when concatenated. For convenience, we use $\gamma$ and $\delta$ to represent an arbitrary generator, and ${\overline{\gamma }}$ to denote the complementary generator of $\gamma$. By concatenating generators we obtain the following rules that form the set $R_{max(n)}$ (Fig. 7 shows the inter-commutation rule):

• 1. (inter-commutation rule) $\gamma _i\overline{\gamma _j}\leftrightarrow \overline{\gamma _j}\gamma _i$

• 2. (idempotency rule) $\gamma _i\gamma _i\leftrightarrow \gamma _i$

• 3. (intra-commutation rule) $\gamma _i\gamma _j\leftrightarrow \gamma _j\gamma _i$ for $|i-j|\ge 2$

• 4. (TL relation rule) $\gamma _i\gamma _j\gamma _i\leftrightarrow \gamma _i$ for $|i-j|=1$

Fig. 7

Inter-commutation rewriting rule for $\mathcal {G}_{max(3)}$

We define the set $\mathcal {O}_{max(n)}$ of equivalence classes based on $\hat{R}_{max(n)}$. We say that a rule is non-increasing if the right-handed side word is not longer than the left-handed side word. A sequence of rewriting rules with only non-increasing rules is said to be a non-increasing rewriting.

We partition $\Sigma _n$ into $\Sigma _{(\alpha )n}=\{\alpha _1,\ldots ,\alpha _{n{-}1}\}$ and $\Sigma _{(\beta )n}=\{\beta _1,\ldots ,\beta _{n{-}1}\}$. Using the inter-commutation rule, we may rewrite any word w to $w_aw_b$, where $w_a\in \Sigma _{(\alpha )n}^*$ and $w_b\in \Sigma _{(\beta )n}^*$. We say that such a word $w_aw_b$ is in an inter-commutation-free form. Also, using intra-commutation rules, we may set additional conditions for $w_a=u_1u_2\cdots u_p$, where $u_i=(\alpha _{j_i}\alpha _{j_i{-}1}\cdots \alpha _{k_i})$, $j_p$ is the maximum subscript in $w_a$, $j_{i{+}1}>j_{i}$ and $k_{i{+}1}>k_{i}$ for $1\le i<p$, and a similar condition for $w_b$. Such $w_a$ and $w_b$ are unique (Jones 1983), and we call such $w_aw_b$ a commutation-free form of w.

We regard the graphical structure of a word as pairs of scaffolds and staples, which can be seen as two independent Jones monoid diagrams. Knowing that the relations 1 to 3 of the Jones monoid can sufficiently describe equivalence of diagrams, we have the following theorem:

Theorem 1

For all $w_1,w_2\in \Sigma _n^*$ , $G_{max}(w_1)=G_{max}(w_2)$ if and only if $w_1\sim w_2$ under $\hat{R}_{max(n)}$ . In other words, there exists a bijection between $\mathcal {G}_{max(n)}$ and $\mathcal {O}_{max(n)}$ .

Given n, the number of elements of $\mathcal {J}_n$ is equal to the Catalan number $\displaystyle C_n=\frac{1}{n+1}\left( {\begin{array}{c}2n\\ n\end{array}}\right)$ (Lau and FitzGerald 2006), and the maximum size of an element in $\mathcal J_n$ is $\displaystyle \left\lfloor \frac{n^2}{4}\right\rfloor$ (Dolinka and East 2017; Jones 1983). Thus, the following remark holds.

Remark 1

Given n, $|\mathcal {O}_{max(n)}|=\displaystyle \left( \frac{1}{n+1}\left( {\begin{array}{c}2n\\ n\end{array}}\right) \right) ^2$, and the maximum size of an irreducible word in $\mathcal {O}_{max(n)}$ is $\displaystyle 2\left\lfloor \frac{n^2}{4}\right\rfloor$.

Since the graphical structures in $\mathcal {G}_{max(n)}$ correspond to products of two Jones monoid diagrams, we use the following Lemma to obtain the proposed optimization algorithm.

Lemma 1

Given two elements  $w_1,w_2\in \mathcal {J}_n$ where $|w_2|\le |w_1|$ , there is a non-increasing rewriting $w_1\sim w_2$.

Proof

We recall the rewriting rules for $\mathcal {J}_n$.

• 1. $h_ih_i\leftrightarrow h_i$

• 2. $h_ih_j\leftrightarrow h_jh_i$ for $|i-j|\ge 2$

• 3. $h_ih_jh_i\leftrightarrow h_i$ for $|i-j|=1$

Suppose $w_1,w_2\in \mathcal {J}_n$, $|w_2|\le |w_1|$ and we can rewrite $w_1$ as $w_2$. Let $w'_1$ be the last word in the sequence of rewriting such that an increasing rule is applied to $w'_1$. If such $w'_1$ does not exist, then the whole rewriting is non-increasing. Otherwise, we claim that there exists another sequence of non-increasing rewriting from $w'_1$ to $w_2$. Note that rule 2 is the only rule that changes location of generators. Moreover, applying rules 1 or 3 on a word does not result in an additional pair of generators that rule 2 can be applied.

• If the first rule is rule 1 ($h_i\leftrightarrow h_ih_i$), the resulting two $h_i$’s should be involved in non-increasing rule 1 or 3 in the following sequence. The vertical sequence of Fig. 8a shows an example of such sequences where the left $h_i$ is used in rule 1 and the right $h_i$ is used in rule 3, where blue areas represent generators that $h_i$ can switch the location with using rule 2. For such sequences, we can always find another sequence without increasing rules as the right sequence of Fig. 8a.

• If the first rule is rule 3 ($h_i\leftrightarrow h_ih_jh_i$ for $|i-j|=1$), the resulting $h_j$ should be involved in non-increasing rule 1 or 3 in the following sequence. Without loss of generality, we assume that $j=i+1$. Since $h_{i{+}1}h_i\ne h_ih_{i{+}1}$, $h_j$ cannot be used in rule 1. The vertical sequence of Fig. 8b shows an example of such sequence where the middle $h_j=h_{i{+}1}$ is used in rule 3, and blue areas represent generators that $h_{i{+}2}$ can switch the location with. For such a sequence, we can always find another non-increasing sequence as the right sequence of Fig. 8b.

We may continuously find an increasing rewriting in the sequence and an alternate sequence of non-increasing rewriting to $w_2$. $\square$

Fig. 8

Examples of sequences of rewriting rules that leads to the same element, where one uses only non-increasing rules. a The first rule is rule 1. b The first rule is rule 3

Theorem 2

Given a word $w_0\in \Sigma ^*_n$ of size m , an irreducible word in $[w_0]$ can be obtained within $O(nm^2)$ time.

Proof

For a given $w_0$, we rewrite $w_0$ as w in the inter-commutation-free form, which takes $O(m^2)$ time. For each $i=1,\ldots ,n$, determine whether one of the following conditions in w holds and rewrite w accordingly:

• 1. If $w=v_1\gamma _i v_2\gamma _i v_3$ where $v_1,v_2,v_3\in \Sigma _n^*$ and $v_2$ does not have $\gamma _{i{+}1},\gamma _i$ and $\gamma _{i{-}1}$, then rewrite w as $v_1v_2\gamma _iv_3$. This is possible by the rule 3 and then 2.

• 2. If $w=v_1\gamma _i v_2\gamma _j v_3\gamma _i v_4$ where $v_1,v_2,v_3,v_4\in \Sigma _n^*$, $|i-j|=1$ and $v_2,v_3$ do not have $\gamma _{i{+}1},\gamma _i$ and $\gamma _{i{-}1}$, then rewrite w as $v_1v_2\gamma _iv_3v_4$. This is possible by the rule 3 and then 4.

It is straightforward that both rewritings are non-increasing. For each i, the iteration checking whether w satisfies the given form in the conditions 1 and 2 for $(i,j=i+1$ or $i-1)$’s is done in O(m) time. It takes O(nm) time to do one rewriting in 1 or 2, which decreases the size of the word by one or two symbols. Thus, it takes at most $O(nm^2)$ to finish the whole process. For the final word $w'$, conditions in 1 and 2 are no longer satisfied and there is no sequence of non-increasing rewriting that decreases the size of the word. Thus, $w'$ is irreducible by Lemma 1. $\square$

3.2.2 $\mathcal {G}_{mid(n)}$ case

Similar to the $\mathcal {G}_{max(n)}$ case, we have the following rewriting rules:

• 1. (Inter-commutation) $\gamma _i\overline{\gamma _j}\leftrightarrow \overline{\gamma _j}\gamma _i$

• 2. (Idempotency) $\gamma _i\gamma _i\leftrightarrow \gamma _i$

• 3. (Intra-commutation) $\gamma _i\gamma _j\leftrightarrow \gamma _j\gamma _i$ for $|i-j|\ge 2$

Fig. 9

Examples of equivalence in $\mathcal {G}_{mid(3)}$. a $\alpha _1\alpha _2\alpha _1\not \sim \alpha _1$ b $\alpha _1\alpha _2\alpha _1\beta _2\sim \alpha _1\beta _2$

Due to the lack of default real staples in generators, we cannot directly introduce the rewriting rule $\gamma _i\gamma _j\gamma _i\leftrightarrow \gamma _i$ for $|i-j|=1$. For example, we cannot rewrite $\alpha _1\alpha _2\alpha _1$ as $\alpha _1$, since $\alpha _1\alpha _2\alpha _1$ has a straight real staple at the third column while $\alpha _1$ does not (see Fig. 9a). We introduce the span of a word w as the set of columns $span(w)=\displaystyle \bigcup \nolimits _{\gamma _j \text{ in } w}\{j,j+1\}$.

Lemma 2

Under $\mathcal {G}_{mid(n)}$ , the span equals to the set of columns where real staples exist.

Proof

We can prove the statement by induction on the size of the word. It is straightforward that the statement holds for generators. Assume that the statement holds for all $|w|=m$. For a word $w'=w\gamma _i$, if i and $i+1$ are both in span(w), concatenation does not change the span and the statement holds. If i or $i+1$ are not in span(w), then the span has new columns, and the graphical structure of $w'$ has real staples for these columns, which makes the statement true. $\square$

Directly from Lemma 2, it follows that two equivalent words have the same span, and rewriting rules in the Jones monoid can be applied when both sides have the same span. For example, $\alpha _1\alpha _2\alpha _1\beta _2\leftrightarrow \alpha _1\beta _2$ holds as in Fig. 9b because $\beta _2$ adds 3 to $span(\alpha _1\beta _2)$ and $span(\alpha _1\alpha _2\alpha _1\beta _2)=span(\alpha _1\beta _2)$. In general, we have the following additional rewriting rules, where $\delta \in \{\alpha ,\beta \}$ and $v\in \Sigma _n^*$:

$4^{*}$.:

$\delta _jv\gamma _i\gamma _{i{-}1}\gamma _i\leftrightarrow \delta _jv\gamma _i$ if $j=i-1$ or $i-2$

$5^{*}$.:

$\delta _jv\gamma _i\gamma _{i{+}1}\gamma _i\leftrightarrow \delta _jv\gamma _i$ if $j=i+1$ or $i+2$

$6^{*}$.:

$\gamma _i\gamma _{i{-}1}\gamma _iv\delta _j\leftrightarrow \gamma _iv\delta _j$ if $j=i-1$ or $i-2$

$7^{*}$.:

$\gamma _i\gamma _{i{+}1}\gamma _iv\delta _j\leftrightarrow \gamma _iv\delta _j$ if $j=i+1$ or $i+2$

The symbol $\delta _j$ in the right hand side of rules 4$^*$ to 7$^*$ ensures that the span of the two words is the same. Note that $\{v\}=\Sigma _n^*$ in rules 4$^*$ to 7$^*$ is infinite, but we prove that $\{v\}$ can be reduced to a finite set of words while maintaining the equivalent rewriting system in Theorem 3. For better description of such a subset, we define a zig-zag word $w\in \Sigma _n^*$ to be a word where each pair of adjacent generators in w have adjacent indices. We call a maximal subword of increasing (decreasing) indices as zig (zag). For example, $w=\alpha _3\alpha _4\alpha _3\alpha _2\alpha _1\alpha _2$ is a zig-zag word with a zig-zag-zig sequence in the word: $\alpha _3\alpha _4$ being the zig, $\alpha _4\alpha _3\alpha _2\alpha _1$ being the zag, and $\alpha _1\alpha _2$ being the zig. Using rules 2 to 7$^*$, we can rewrite any zig-zag word that consists of single generators type either in $\Sigma _{(\alpha )n}$, or $\Sigma _{(\beta )n}$, as a zig-zag word with at most three zigs or zags, which we call the zig-zag normal form, or ZNF in short.

Theorem 3

Rules 1 to 3 and $4^*$ to $7^*$ generate the same equivalence relation on $\Sigma ^*$ as rules 1 to 7 below where $v=\epsilon$ , or $v\in \Sigma _{(\gamma )n}^*$ and $\gamma _jv\gamma _i$ in rules 4 and 5 ( $\gamma _iv\gamma _j$ in rules 6 and 7) is in ZNF.

• 4. $\gamma _jv\gamma _i\gamma _{i{-}1}\gamma _i\leftrightarrow \gamma _jv\gamma _i$ if $j=i-1$ or $i-2$

• 5. $\gamma _jv\gamma _i\gamma _{i{+}1}\gamma _i\leftrightarrow \gamma _jv\gamma _i$ if $j=i+1$ or $i+2$

• 6. $\gamma _i\gamma _{i{-}1}\gamma _iv\gamma _j\leftrightarrow \gamma _iv\gamma _j$ if $j=i-1$ or $i-2$

• 7. $\gamma _i\gamma _{i{+}1}\gamma _iv\gamma _j\leftrightarrow \gamma _iv\gamma _j$ if $j=i+1$ or $i+2$

Proof

Given rule $4^*$: $w_1=\delta _jv\gamma _i\gamma _{i{-}1}\gamma _i\leftrightarrow w_2=\delta _jv\gamma _i$ if $j=i-1$ or $i-2$, here we prove that rule $4^*$ can be replaced with rules 4 to 7 to generate the same equivalence relation. Similar simplification works for rules 5$^*$ to 7$^*$, and we focus on rule 4$^*$ in the proof. The proof consists of establishing three assumptions on $\delta$ and v.

First, we prove that we may assume $\delta =\gamma$. If $\delta ={\overline{\gamma }}$, we can rewrite v as a concatenation of a prefix $v_\gamma \in \Sigma _{(\gamma )n}^*$ and a suffix $v_\delta \in \Sigma _{(\delta )n}^*$. Then, $\delta _jv_\gamma v_\delta \gamma _i\gamma _{i{-}1}\gamma _i\leftrightarrow v_\gamma \delta _j\gamma _i\gamma _{i{-}1}\gamma _i v_\delta \leftrightarrow v_\gamma \delta _j\gamma _i v_\delta \leftrightarrow \delta _j v_\gamma v_\delta \gamma _i=\delta _j v \gamma _i$, and vice versa.

Second, we prove that we may assume $v\in \Sigma ^*_{(\gamma )n}$. If $v\notin \Sigma ^*_{(\gamma )n}$, we can rewrite v as a concatenation of a prefix $v_\gamma \in \Sigma _{(\gamma )n}^*$ and a suffix $v_{{\overline{\gamma }}}\in \Sigma _{({\overline{\gamma }})n}^*$. Then, $\gamma _jv_\gamma v_{{\overline{\gamma }}} \gamma _i\gamma _{i{-}1}\gamma _i\leftrightarrow v_{{\overline{\gamma }}} \gamma _j v_\gamma \gamma _i\gamma _{i{-}1}\gamma _i \leftrightarrow v_{{\overline{\gamma }}}\gamma _j v_\gamma \gamma _i\leftrightarrow$ $\gamma _j v_\gamma v_{{\overline{\gamma }}}\gamma _i=\gamma _j v \gamma _i$, and vice versa.

Third, we prove the given statement under the previously proved assumptions $\delta =\gamma$ and $v\in \Sigma _{(\gamma )n}^*$, by induction on |v|. It is straightforward that the statement holds for $v=\epsilon$. For $|v|=1$, we may assume that $v=\gamma _h$. According to h, we have the following cases (see Fig. 10):

• (1) If $h>j+1$ or $h<j-1$, we can switch $\gamma _h$ and $\gamma _j$.

• (2) If $h=j+1$, then $h=i$ or $i-1$.

  • (2-a) If $h=i$, $\gamma _h\gamma _i$ can be rewritten as $\gamma _i$.

  • (2-b) If $h=i-1$, $\gamma _j\gamma _h\gamma _i$ is in ZNF.

• (3) If $h=j$, we can rewrite $\gamma _j\gamma _h$ as $\gamma _j$.

• (4) If $h=j-1$, then $h=i-2$ or $i-3$.

  • (4-a) If $h=i-2$, we may regard $\gamma _h$ as new $\gamma _j$.

  • (4-b) If $h=i-3$, we can move $\gamma _h$ to the right of $\gamma _i\gamma _{i{-}1}\gamma _i$.

Fig. 10

Base case analysis when $|v|=1$

Fig. 11

Inductive step

Then, assume that the claim holds for all v where $|v|\le m$. For $|v|=m+1$, we will prove that we can rewrite $w_1$ to find a subword $w'_1=\gamma _jv'\gamma _i\gamma _{i{-}1}\gamma _i$ such that $|v'|\le m$, which leads to the induction hypothesis. Let $\gamma _jp$ be the maximal zig-zag prefix of $\gamma _jv\gamma _i$, and assume that $p\ne v$. Let $\gamma _t$ be the last generator of p, and $\gamma _h$ be the first generator after p, which is in v. Let $\max (\gamma _jp)$ ($\min (\gamma _jp)$) be the largest (smallest) index of generators in $\gamma _jp$. According to h, we have the following cases (see Fig. 11):

• (1) If $h>\max (\gamma _jp)+1$, we can move $\gamma _h$ to the left of $\gamma _j$, which reduces the size of v by 1.

• (2) If $h=\max (\gamma _jp)+1$, since $\gamma _jp$ is the maximal zig-zag prefix of v, $t\ne \gamma _{\max (\gamma _jp)}$. We consider two cases:

  • (2-a) If $j=\max (\gamma _jp)$, $h=j+1$.

    • (2-a-i) If $j=i-1$, then the first generator of p is $\gamma _{i{-}2}$, and we may regard it as a new $\gamma _j$.

    • (2-a-ii) If $j=i-2$, then $h=i-1$, and we may regard $\gamma _h$ as a new $\gamma _j$.

  • (2-b) If $j\ne \max (\gamma _jp)$, then there exists the rightmost subword $\hat{w}=\gamma _{\max (\gamma _jp){-}1}\gamma _{\max (\gamma _jp)}\gamma _{\max (\gamma _jp){-}1}$ in $\gamma _jp$. We can move $\gamma _h$ to the right of $\hat{w}$, and reduce $\hat{w}\gamma _h$ to $\gamma _{\max (\gamma _jp)}\gamma _h$, which reduces the size of v by 2.

• (3) If $\min (\gamma _jp)\le h\le \max (\gamma _jp)$, we consider four cases:

  • (3-a) If $\min (\gamma _jp)=\max (\gamma _jp)$, $p=\epsilon$, $h=j$ and we may reduce $\gamma _j\gamma _h$ as $\gamma _j$, which reduces the size of v by 1.

  • (3-b) If $\gamma _jp$ has more than three zigs or zags, it can be rewritten using rule 4 to 7, which reduces the size of v by at least 2.

  • (3-c) If $\gamma _jp$ has at most three zigs or zags,

    • (3-c-i) If $j=\max (\gamma _jp)$ or $j=\min (\gamma _jp)$ or $t=\max (\gamma _jp)$ or $t=\min (\gamma _jp)$, then $\gamma _jp$ can be rewritten using rule 4 to 7, which reduces the size of v by at least 2.

    • (3-c-ii) Otherwise, we can move $\gamma _h$ into $\gamma _jp$ so that $\gamma _jp$ has a subword $\gamma _h\gamma _s\gamma _h$ where $|s-h|=1$, since $\gamma _jp$ is the maximal zig-zag prefix of $\gamma _jv\gamma _i$ and $\gamma _h$ cannot ’extend’ the zig-suffix or the zag-suffix of $\gamma _jp$. Such subword can be rewritten as $\gamma _h$, which reduces the size of v by 2.

• (4) If $h=\min (\gamma _jp)-1$, the case is similar to $h=\max (\gamma _jp)+1$ case.

• (5) If $h<\min (\gamma _jp)-1$, we can move $\gamma _h$ to the left of $\gamma _j$, which reduces the size of v by 1.

Fig. 12

Inductive step when $p=v$

When $p=v$, $\gamma _h=\gamma _i$. The case analysis is similar to the above, but we cannot move $\gamma _h$ to the left of $\gamma _j$, which breaks the subword $\gamma _i\gamma _{i{-}1}\gamma _i$. Thus, we check the following two cases such that we moved $\gamma _h$ to the left of $\gamma _j$ (see Fig. 12):

• (1) If $h=i>\max (\gamma _jp)+1$, since $j=i-1$ or $i-2$, $\max (\gamma _jp)=i-2$.

  • (1-a) If $t=i-2$, we may regard $\gamma _t$ as a new $\gamma _j$.

  • (1-b) If $t\le i-3$, we may move $\gamma _t$ to the right of $\gamma _i\gamma _{i{-}1}\gamma _i$, which reduces the size of v by 1.

• (2) Since $j=i-1$ or $i-2$, $h=i$ cannot be less than $\min (\gamma _jp)-1$.

$\square$

Thus the rules 4$^*$ to 7$^*$ reduce to the following rewriting rules for $v\in \Sigma _{(\gamma )n}^*$ such that either $v=\epsilon$, or $\gamma _jv\gamma _i$ is in ZNF in rule 4 and 5 ( $\gamma _i v \gamma _j$ in rule 6 and 7):

• 4. $\gamma _jv\gamma _i\gamma _{i{-}1}\gamma _i\leftrightarrow \gamma _jv\gamma _i$ if $j=i-1$ or $i-2$

• 5. $\gamma _jv\gamma _i\gamma _{i{+}1}\gamma _i\leftrightarrow \gamma _jv\gamma _i$ if $j=i+1$ or $i+2$

• 6. $\gamma _i\gamma _{i{-}1}\gamma _iv\gamma _j\leftrightarrow \gamma _iv\gamma _j$ if $j=i-1$ or $i-2$

• 7. $\gamma _i\gamma _{i{+}1}\gamma _iv\gamma _j\leftrightarrow \gamma _iv\gamma _j$ if $j=i+1$ or $i+2$

Comparing the rewriting rules with $R_{max(n)}$, rules 4 to 7 are extensions of rule 4 of $R_{max(n)}$, but the ruleset stays to be finite. Note that if a zig-zag word that consists of single generators type has more than three zigs or zags, it can be successfully reduced to a ZNF word using rules 2 to 7.

For given n, let the set $R_{mid(n)}$ of rewriting rules consist of the above seven kinds of rules for $1\le i,j\le n$. As observed, since there are only finitely many words in ZNF, the set $R_{mid(n)}$ is finite. Then $\mathcal {O}_{mid(n)}=\Sigma ^*_n/ \hat{R}_{mid(n)}$.

Theorem 4

For all $w_1,w_2\in \Sigma _n^*$ , $G_{mid}(w_1)=G_{mid}(w_2)$ if and only if $w_1\sim w_2$ under $\hat{R}_{mid(n)}$ . In other words, there exists a bijection between $\mathcal {G}_{mid(n)}$ and $\mathcal {O}_{mid(n)}$.

Proof

Note that if $w_1\sim w_2$ under $\hat{R}_{mid(n)}$, then $G_{mid}(w_1)=G_{mid}(w_2)$ from definitions of rewriting rules. If $G_{max}(w_1)=G_{max}(w_2)$ and $span(w_1)=span(w_2)$, then $G_{mid}(w_1)=G_{mid}(w_2)$ from Lemma 2. Moreover, if $G_{max}(w_1)=G_{max}(w_2)$ and $span(w_1)\ne span(w_2)$, then $G_{mid}(w_1)\ne G_{mid}(w_2)$ from Lemma 2. Thus, $G_{mid}(w_1)=G_{mid}(w_2)$ if and only if $G_{max}(w_1)=G_{max}(w_2)$ and $span(w_1)=span(w_2)$. From Theorem 1, the set of the following (general) rules are sufficient to describe equivalence under $\mathcal {G}_{max(n)}$ when $v_1,v_2\in \Sigma _n^*$.

• (i) $v_1\gamma _i\overline{\gamma _j}v_2\leftrightarrow v_1\overline{\gamma _j}\gamma _iv_2$

• (ii) $v_1\gamma _i\gamma _iv_2\leftrightarrow v_1\gamma _iv_2$

• (iii) $v_1\gamma _i\gamma _jv_2 \leftrightarrow v_1\gamma _j\gamma _iv_2$ for $|i-j|\ge 2$

• (iv) $v_1\gamma _i\gamma _j\gamma _iv_2\leftrightarrow v_1\gamma _iv_2$ for $|i-j|=1$

In the proof, $R_{max(n)}$ denotes the set of these general rules. Now, there exists a set P of pairs of words $(w_1,w_2)$ where $G_{max}(w_1)=G_{max}(w_2)$ and $G_{mid}(w_1)=G_{mid}(w_2)$, a set H of pairs of words $(w_3,w_4)$ where $G_{max}(w_3)=G_{max}(w_4)$ and $G_{mid}(w_3)\ne G_{mid}(w_4)$, and another set N of pairs of words $(w_5,w_6)$ where $G_{max}(w_5)\ne G_{max}(w_6)$ and $G_{mid}(w_5)\ne G_{mid}(w_6)$. The sets P, H, N are disjoint and $P\cup H\cup N=\Sigma _n^*\times \Sigma _n^*$. Now, $R_{max(n)}$ can be partitioned into two sets $R_{diff}$ and $R_{same}$, where all rules in $R_{diff}$ have different span for two sides and all rules in $R_{same}$ have the same span for two sides. Then, the following statements hold:

• 1. For a pair $(w_1,w_2)\in P$, there exists a sequence of rewriting that rewrites $w_1$ as $w_2$ only using rules in $R_{same}$: For a pair $(w_1,w_2)\in P$, from Lemma 1, there exists a sequence of non-increasing rewriting that rewrite $w_1$ as $w_2$ given $|w_2|\le |w_1|$. We observe that the only rules that change the size of the word are rules (ii) and (iv). Rule (ii) does not change the span, and non-increasing rules in rule (iv) do not increase the size of the span. Since $w_1\sim w_2$ in $\mathcal {G}_{mid(n)}$, $span(w_1)=span(w_2)$. Thus, there exists a sequence of rewriting that rewrites $w_1$ as $w_2$ only using rules in $R_{same}$.

• 2. For a pair $(w_3,w_4)\in H$, all sequences of rewriting that rewrite $w_3$ as $w_4$ have a rule from $R_{diff}$. In other words, we cannot rewrite $w_3$ as $w_4$ only using rules in $R_{same}$.

• 3. For a pair $(w_5,w_6)\in N$, $w_5$ cannot be rewritten as $w_6$ using $R_{same}$, since $R_{same}\subseteq R_{max(n)}$.

Based on these statements, we can claim that a pair $(w_1,w_2)$ is in P if and only if $w_1\sim w_2$ under $\hat{R}_{same}$. Rules (i) to (iii) from $R_{max(n)}$ have the same span for the both sides, and they are in $R_{same}$. For rule (iv) from $R_{max(n)}$, the subset of the rules where both sides have the same span is rules 4 to 7 in $R_{mid(n)}$. Thus, $\hat{R}_{mid(n)}=\hat{R}_{same}$ and $G_{mid}(w_1)=G_{mid}(w_2)$ if and only if $w_1\sim w_2$ under $\hat{R}_{mid(n)}$. $\square$

We compute the number of equivalence classes of words in $\mathcal {O}_{mid(n)}$. We use a binary string of length n to represent the graphical structures in $\mathcal {G}_{mid(n)}$ such that the ith bit equals 1 if and only if the ith staple and the ith scaffold is a straight line. Each binary string is uniquely determined with a tuple $(a_1,b_1,\ldots ,a_k,b_k)$ where $a_i$ ($b_i$) represents the number of ith consecutive 0’s (1’s). For example, the 8-bit binary string 00111000 corresponds to a tuple (2, 3, 3, 0). In particular, the bit 0 corresponds to (1, 0) and the bit 1 to (0, 1). Let ${\mathbb {N}}^0={\mathbb {N}}\cup \{0\}$. The set of tuples corresponding to binary strings of length n is denoted $T_n=\{p=(a_1,b_1,\ldots ,a_k,b_k)\mid k\ge 1, \text{ for } \text{ all } i, a_i,b_i\in {\mathbb {N}}^0 \text { and }\displaystyle \sum _{i=1}^k(a_i+b_i)=n\}$. Note that $T_n$ is the set of partitions of n in 2k summands.

Theorem 5

Given $n\in {\mathbb {N}}^0$ , for each tuple  $p\in \displaystyle \bigcup \nolimits _{1\le i\le n} T_i$ , let $D(p)\in {\mathbb {N}}^0$ be recursively defined as follows:

• for $p\in T_0$, $D(0,0)=1$.

• for $p\in T_1$, $D(p)=1$ if $p=(0,1)$ and $D(p)=0$ if $p=(1,0)$.

• for $p=(a_1,b_1,\ldots ,a_k,b_k)\in T_n$, $(n>0)$ we have $D(p)=\displaystyle \prod \nolimits _{i=1}^k D(a_i,0)$.

• for $n>1$, we have $D(0,n)=1$ and

  $$\begin{aligned} D(n,0)=\left( \frac{1}{n+1}\left( {\begin{array}{c}2n\\ n\end{array}}\right) \right) ^2-\sum _{p\in T_{n}\setminus \{(n,0)\}}D(p). \end{aligned}$$

Then, $|\mathcal {O}_{mid(n)}|$ is given as

$$\begin{aligned} \displaystyle |\mathcal {O}_{mid(n)}|=d(n)=\sum _{p\in T_n}\left( D(p)\times x(p)\right) -n, \end{aligned}$$

where $x(a_1,b_1,\ldots ,a_k,b_k)=$

• $(b_1+1)$ if $k=1$,

• $(b_1+1)\cdot \displaystyle \prod \nolimits _{i=2}^{k-1}\left( \frac{b_i(b_i+1)}{2}+1\right) \cdot (b_k+1)$ if $k\ne 1,a_1=0$,

• $\displaystyle \prod \nolimits _{i=1}^{k-1}\left( \frac{b_i(b_i+1)}{2}+1\right) \cdot (b_k+1)$ if $k\ne 1,a_1>0$.

Fig. 13

The set of representative words in $\mathcal {O}_{mid(3)}$. Gray headers represent representative words corresponding to elements in $\mathcal {J}_3$, and the thick box represents the set of representative words in $\mathcal {O}_{max(3)}$

Proof

Figure 13 enumerates representative words in $\mathcal {O}_{mid(3)}$. We observe that the set of representative words in $\mathcal {O}_{mid(n)}$ is a superset of the set of representative words in $\mathcal {O}_{max(n)}$. Graphically, we observe that $\mathcal {G}_{mid(n)}$ is a superset of the set of a pair of diagrams of $\mathcal {J}_n$, where we regard one as scaffolds and the other as staples. In particular, when a consecutive set of columns adjacent to the span is occupied with both real straight scaffold and staple, there also exists a structure with virtual straight staples in these columns as in Fig. 14.

Fig. 14

Since the third and the fourth columns of $\alpha _1\alpha _2\alpha _3\alpha _2\alpha _1$ are occupied with both real straight scaffold and staple, we also have structures where the staple at the fourth column is virtual ($\alpha _1\alpha _2\alpha _1$) and staples at the third and the fourth columns are virtual ($\alpha _1$)

We can classify graphical structures in $\mathcal {G}_{max(n)}$ by using a binary b of length n, where the ith digit has 1 if the ith column has both straight scaffold and staple, and 0 otherwise. The set of binaries of length n has bijection with the set $T_n$ previously defined. Thus, let D(p) be the number of equivalence classes of words whose graphical structures correspond to $p\in T_n$. It is straightforward that $D(0,1)=1$ and $D(1,0)=0$. To calculate $D(a_1,b_1,\ldots ,a_k,b_k)$, columns that has 1’s in the binary has only one case (straight scaffold and staple), so we need to multiply all $D(a_i,0)$’s. For $D(a_1,0)$, once we know all D(p) where $p\in T_{a_1}$, we can calculate $D(a_1,0)$ using the fact that

$$\begin{aligned} |\mathcal {G}_{max(a_1)}|^2=\displaystyle \left( \frac{1}{a_1+1}\left( {\begin{array}{c}2a_1\\ a_1\end{array}}\right) \right) ^2=\displaystyle \sum _{p\in T_{a_1}}D(p). \end{aligned}$$

For each D(p), we calculate the number of distinct graphical structures in $\mathcal {G}_{mid(n)}$. Suppose we have j consecutive 1’s at the start or the end of the binary that corresponds to p. For such sequence, we can have $j+1$ distinct sets of virtual straight staples. When we have j consecutive 1’s between two consecutive 0’s, there are $\displaystyle \sum \nolimits _{i=1}^{j}i+1=\frac{j(j+1)}{2}+1$ distinct sets of virtual straight staples. The number of cases for each consecutive 1’s should be multiplied, which results in x(p) in the Theorem. The only exception for this calculation is D(0, n) case, where we have real straight scaffolds and staples for all columns. The only word that corresponds to the structure is the empty word $\epsilon$, and we need to subtract n from $D(0,n)\cdot x(0,n)=n+1$, which results in the formula in the theorem. Figure 15 shows how we count the number of cases for each D(p).

Fig. 15

A graphical structure corresponding to $p=(0,3,2,3,3,4)$, and calculation of D(p) and x(p). The binary that corresponds to the structure is stated on the structure. For consecutive 1’s in the binary, we may have arbitrary consecutive virtual staples within, which is counted by purple arrows to calculate x(p). (Color figure online)

To justify the counting of virtual staples cases, we claim the following statement: For a graphical structure in $\mathcal {G}_{mid(n)}$, let $t_i$ be 1 if the ith column has real straight scaffold and straight staple, and 0 otherwise. Then, a set of maximal consecutive columns with real straight scaffolds and staples should be adjacent to the ith column where $t_i=0$, as in Fig. 16a. In other words, there is no set of maximal consecutive columns with real straight scaffolds and staples where both ends are adjacent to straight scaffolds with straight virtual staples, as in Fig. 16b. We prove the statement by induction on the size of the word. It is straightforward that the statement holds for the generators. Assume that the statement holds for all $|w|=m$. For a word $w'=w\gamma _i$, we observe that a column in the graphical structure of $w'$ can have a virtual straight staple only if it had a virtual straight staple in the graphical structure of w. Thus, if i or $i+1$ is in span(w), the statement holds since the columns that have virtual straight staples do not change. If i and $i+1$ are not in span(w), we insert the unit of the generator in a set of consecutive columns with straight scaffolds and virtual staples. In that case, concatenation does not create columns that have straight scaffolds and real staples, and the statement holds. $\square$

Fig. 16

The set of maximal consecutive columns with real straight scaffolds and staples is represented by a blue box. a The set is adjacent to the ith column with $t_i=0$ on the left. b Both ends are adjacent to straight scaffolds with straight virtual staples, which is impossible. (Color figure online)

For each graphical structure in $\mathcal {G}_{max(n)}$, we may assign a unique binary number of size n, where the ith digit is 1 if the ith column has both straight vertical scaffolds and staples, and 0 otherwise. Then, D(p) is the number of graphical structures in $\mathcal {G}_{max(n)}$ whose assigned binary number corresponds to p. In particular, the sum of all D(p) for $p\in T_n$ is equal to the square of the nth Catalan number by Remark 1. For each p, the term x(p) gives the number of possible combinations of virtual straight staples within the segment of the graphical representation that consist of only vertical straight lines.

The sequence d(n) for $1\le n\le 10$ is 1, 4, 31, 253, 2247, 21817, 227326, 2499598, 28660639, 339816259. It is not listed in the OEIS (https://oeis.org/) list of sequences, and the non-recursive formula of d(n) is still open.

Theorem 6

An upper bound of the size u(n) of an irreducible word in $\mathcal {O}_{mid(n)}$ is given by $2^{n-1}$ for $n\ge 2$.

Proof

There exists a representative word $w_aw_b\in \mathcal {O}_{mid(n)}$ in an inter-commutation-free form where $|w_a|=|w_b|=\displaystyle \left\lfloor \frac{n^2}{4}\right\rfloor$, and $u(n)\ge \displaystyle \left\lfloor \frac{n^2}{4}\right\rfloor$. Aside from $w_aw_b$, we may have representative words in $\mathcal {O}_{mid(n)}$ that exploit rules 4 to 7, not satisfying the condition that $\gamma _i\gamma _j\gamma _i$ is reduced to $\gamma _i$ when $|i-j|=1$. Namely, we may rewrite $\gamma _i$ in a word w as $\gamma _i\gamma _j\gamma _i$ and have a distinct word if staples in the resulting word occupy at least one new column.

Given an irreducible word v of size l(v), let s(v) be the size of the span of v. Then, $\displaystyle \frac{s(v)}{2}\le l(v)\le u(s(v)-1)$ holds when $v\ne \epsilon$. Now, suppose for a word $v=v_\alpha v_\beta$ in an inter-commutation-free form, we want to continuously rewrite $\gamma _i$ as $\gamma _i\gamma _j\gamma _i$ as far as possible while making the resulting words distinct. Let $v_\alpha =\alpha _{i_1}\cdots \alpha _{i_p}$ and $v_\beta =\beta _{j_1}\cdots \beta _{j_q}$. If $v_\alpha =\epsilon$, there exists a longer word $\alpha _1 v_\beta$ which is distinct. Thus, without loss of generality, we may assume that $v_\alpha ,v_\beta \ne \epsilon$. Now, for the word v, $l(v)=l(v_\alpha )+l(v_\beta )$ and $\max {(s(v_\alpha ),s(v_\beta ))}\le s(v)\le s(v_\alpha )+s(v_\beta )$. Each rewriting of $\gamma _i$ to $\gamma _i\gamma _j\gamma _i$ increases s(v) by 1 and l by 2, and such rewriting becomes impossible once s(v) becomes n. Without loss of generality, we assume that $s(v_\alpha )\ge s(v_\beta )$. Then, we may have at most $n-s(v_\alpha )$ number of rewriting steps, which results in a word of size $l(v_\alpha )+l(v_\beta )+2(n-s(v_\alpha ))=2n-2s(v_\alpha )+l(v_\alpha )+l(v_\beta )\le 2n-2s(v_\alpha )+2u(s(v_\alpha )-1)$. Since $u(n)\ge \displaystyle \left\lfloor \frac{n^2}{4}\right\rfloor$, $2n-2s(v_\alpha )+2u(s(v_\alpha )-1)$ increases as $s(v_\alpha )$ increases. For $s(v_\alpha )=n$, $u(n)\le 2n-2n+2u(n-1)=2u(n-1)$. From $u(2)=2$, we have the upper bound $u(n)=2^{n-1}$. $\square$

The bound in Theorem 6 is not tight, and the exact size of a maximum irreducible word is open.

Theorem 7

Given a word $w_0\in [w_0]\in \mathcal {O}_{mid(n)}$ of size  m , we can find an irreducible word of $[w_0]$ within $O(nm^2)$ time.

The proofs of Lemma 1 and Theorem 2 work similarly for Theorem 7. We first rewrite $w_0$ as w in the inter-commutation-free form. Then we repeatedly find one of the following conditions in w if possible and rewrite w accordingly:

• 1. If $w=v_1\gamma _i v_2\gamma _i v_3$ where $v_1,v_2,v_3\in \Sigma _n^*$ and $v_2$ does not have $\gamma _{i{+}1}$, $\gamma _i$ and $\gamma _{i{-}1}$, then rewrite w as $v_1v_2\gamma _iv_3$.

• 2. If $w=v_1\gamma _i v_2\gamma _{i{+}1} v_3\gamma _i v_4$ where $v_1,v_2,v_3,v_4\in \Sigma _n^*$, $v_2,v_3$ do not have $\gamma _{i{+}1}$, $\gamma _i$ and $\gamma _{i{-}1}$, there exists $\delta _{i{+}1}$ in $v_1$ or $v_4$, or $\delta _{i{+}2}$ in $v_1,v_2,v_3$ or $v_4$, then rewrite w as $v_1v_2\gamma _iv_3v_4$.

• 3. If $w=v_1\gamma _i v_2\gamma _{i{-}1} v_3\gamma _i v_4$ where $v_1,v_2,v_3,v_4\in \Sigma _n^*$, $v_2,v_3$ do not have $\gamma _{i{+}1}$, $\gamma _i$ and $\gamma _{i{-}1}$, there exists $\delta _{i{-}1}$ in $v_1$ or $v_4$, or $\delta _{i{-}2}$ in $v_1,v_2,v_3$ or $v_4$, then rewrite w as $v_1v_2\gamma _iv_3v_4$.

3.3 Concluding remarks

We have proposed modules and corresponding generators for DNA origami structures, defined concatenation of words and rewriting rules, and analyzed equivalence classes based on graphical equivalence. One model that we have not discussed is $\mathcal {G}_{min(n)}$. For $\mathcal {G}_{min(n)}$, seven types of rewriting rules for $\mathcal {G}_{mid(n)}$ hold. Moreover, we may prove that Theorem 4 holds for $\mathcal {G}_{min(n)}$ using the similar proof. It turns out that there is bijection between $\mathcal {G}_{mid(n)}$ and $\mathcal {G}_{min(n)}$, and $\mathcal {O}_{mid(n)}=\mathcal {O}_{min(n)}$.

Note that the set of words in the first row of Fig. 13 is bijective with the set of distinct elements in the “modified” Jones monoid corresponding to the staple structure, assuming that there are two types (real and virtual) of lines in the graphical representation. However, $\mathcal {O}_{mid(3)}$ is not a bijection of the cross product of the original Jones monoid (for scaffolds) and such modified Jones monoid (for staples), since $\beta _1$ and $\beta _2$ also have virtual staples.

Graphical structures corresponding to generators $\alpha _i$’s and $\beta _i$’s in Fig. 3 describe crossing of scaffolds and staples in DNA origami well, while using only two types of generators. Here we explore possible further development of generators that are more plausible to DNA origami.

The first observation on the current generators is that they are vertically and horizontally symmetric (without directions), which causes the graphical structure to always have a cup-shaped fragment of a real scaffold at the top as in Fig. 17a. DNA origami does not have such fragments at the border of the structure, which leads us to revise generators to define such borders. Figure 17b proposes four different generators that are used to substitute $\alpha _1$. In these generators, we introduce asymmetric structures that can be used to construct borders of the structure. We may define generators for $\beta$ similarly. Under the assumption that we use the same concatenation procedure, for a graphical structure that corresponds to $\alpha _1$, we can make an arbitrary number of scaffolds and staples virtual by concatenation of four new generators as in Fig. 17c. Now, suppose we define the rewriting system based on equivalence under such generators. For each pair of diagrams of $\mathcal {J}_n$, we have 2n staples and scaffolds which can become virtual. From analysis similar to the proof of Theorem 5, the size of the set of equivalence classes becomes $\displaystyle \left( \left( \frac{1}{n+1}\left( {\begin{array}{c}2n\\ n\end{array}}\right) \right) ^2-1\right) \cdot 2^{2n}+1$. The set of rewriting rules that are sufficient to describe equivalence under such generators is open.

Fig. 17

a In a graphical structure generated from $\alpha _i$’s and $\beta _i$’s, there always exists a cup-shaped fragment of a real scaffold at the top (and cap-shaped fragment at the bottom). b Four revised generators that substitutes $\alpha _1$. Virtual scaffolds and staples are colored in gray. c We can make arbitrary staples and scaffolds in $\alpha _1$ virtual. (Color figure online)

The second observation on the current generators is that we do not consider which side of the scaffold the staple is on. In the DNA origami structure, staples can be on the left or the right of the scaffold, and these two cases are distinguished. Moreover, for two adjacent staples at the opposite side of the same scaffold, they either disconnect or connect by crossing the scaffold. To model this observation, we may introduce revised graphical structures for $\alpha$ and $\beta$ as in Fig. 18a. Staples are either at the left or the right of the scaffold, and some staple ends are extending which can be connected to other staples regardless of the side. We assume that two adjacent staples can be connected except when two are non-extending ends and at the opposite side. This additional condition for staple connection changes some of the commutation rewriting rules—for example, $\alpha _1\beta _1\nleftrightarrow \beta _1\alpha _1$ as in Fig. 18b. Algebraic analysis on relations based on such generators is done by Garrett et al. (2019). The set of rewriting rules that are sufficient to describe equivalence under such generators is still open.

Fig. 18

a Revised generators $\alpha _1$ and $\beta _1$. Two diagonal ends of staples in $\alpha _1$ represent extending staple-ends. b Adjacent staples can be connected except when two are non-extending ends and at the opposite side