Cyclic Oritatami Systems Cannot Fold Infinite Fractal Curves

TL;DR

This paper proves that cyclic Oritatami Systems cannot generate certain infinite fractal curves like Koch and Minkowski, highlighting limitations in modeling infinite self-similar structures through this computational framework.

Contribution

It establishes a negative result showing the impossibility of folding specific infinite fractals with cyclic OS and provides conditions under which cyclic OS cannot produce infinite aperiodic curves.

Findings

Cyclic OS cannot generate Koch or Minkowski fractals.

Sufficient conditions are identified where cyclic OS cannot fold infinite aperiodic curves.

The study highlights fundamental limitations of cyclic OS in modeling infinite fractal structures.

Abstract

RNA cotranscriptional folding is the phenomenon in which an RNA transcript folds upon itself while being synthesized out of a gene. The oritatami system (OS) is a computation model of this phenomenon, which lets its sequence (transcript) of beads (abstract molecules) fold cotranscriptionally by the interactions between beads according to the binding ruleset. The OS is an useful computational model for predicting and simulating an RNA folding as well as constructing a biological structure. A fractal is an infinite pattern that is self-similar across different scales, and is an important structure in nature. Therefore, the fractal construction using self-assembly is one of the most important problems. We focus on the problem of generating an infinite fractal instead of a partial finite fractal, which is much more challenging. We use a cyclic OS, which has an infinite periodic transcript, to generate an infinite structure. We prove a negative result that it is impossible to make a Koch curve or a Minkowski curve, both of which are fractals, using a cyclic OS. We then establish sufficient conditions of infinite aperiodic curves that a cyclic OS cannot fold.

Equations

$$C_{i{+}1}\in\mathop{\rm argmin}\limits_{\begin{subarray}{c}C\in\mathcal{C}_{\leq\alpha}\ \mbox{s.t.}\\ C_{i}\stackrel{{\scriptstyle\mathcal{H}}}{{\to}}_{w[i{+}1]}C\end{subarray}}\min\left\{U(C^{\prime})\Bigm{|}C\stackrel{{\scriptstyle\mathcal{H}}}{{\to}}_{w[i{+}2:i{+}k]}^{*}C^{\prime},k\leq\delta,C^{\prime}\in\mathcal{C}_{\leq\alpha}\right\}.$$

Full text

1 Introduction

Self-assembly is the process where smaller components—usually molecules—autonomously assemble and form a larger complex structure. Self-assembly plays an important role in constructing biological structures and high polymers [16]. One well-known mathematical model of the self-assembly phenomenon is the abstract tile assembly model (aTAM) [17]. Recently, Geary et al. [2] proposed a new computation model called the oritatami system (OS) that simulates the cotranscriptional self-assembly based on the experimental RNA transcription called RNA origami [3]. In general, the OS assumes that a sequence of molecules is transcribed linearly, and predicts the geometric shape of the autonomous folding of the sequence based on the reaction rate of the folding. The OS consists of a sequence of beads (which is the transcript) and a set of rules for possible intermolecular reactions between beads. For each bead in the sequence, the system takes a lookahead of a few upcoming beads and determines the best location of the bead that maximizes the number of possible interactions from the lookahead. The lookahead represents the reaction rate of the cotranscriptional folding and the number of interactions represents the energy level (See Figure 2 for the analogy between RNA origami and oritatami system).

The OS is a computation model based on geometric aspects, and, thus, it is important to design and analyze the OS in both computational and geometric perspectives. From the computational point of view, the OS is proved to be Turing complete using a cyclic tag system simulator [2]. Designs for other computational problems such as binary counting [1] and Boolean formula simulation [5] are also established. Researchers also proved a few decision properties and proposed various optimization methods for the OS design [4, 6, 11]. From the geometric point of view, Rogers and Seki [12] proved the decidability of geometric structure constructions based on the delay. Recently, Masuda et al. [10] proposed how to construct a finite Heighway dragon using a cyclic OS, which has a periodic transcript.

A fractal is an infinite pattern that is self-similar across different scales, and is an important structure in nature. The construction of fractals is one of the most important topics in both geometric computation models [7, 8, 9] and experiments [14, 15]. For constructing fractals by the OS, we need a few assumptions.

Since the OS transcribes an RNA single strand, it is natural to consider fractal curves, which are infinite sequences of segments (and points). 2. 2.

We only consider a deterministic OS (that only folds into a unique conformation). Note that it is trivial to make a nondeterministic OS that may fold into several different structures including a target structure. 3. 3.

We consider an infinite fractal construction. Masuda et al. [10] proposed how to construct a finite fractal by implementing a counter and an automaton periodically inside the transcript. This approach can be used to construct an arbitrary long fractal but not an infinite one, since the counter is finite. 4. 4.

Since we need an infinite transcript with a formal finite representation, it is a natural choice to use a cyclic OS that has an infinitely repeated transcript. Researchers already used cyclic OSs to construct conformations that can grow infinitely [1, 2].

Although fractals have self-similarity, they do not have repeated structures with any fixed period. In aTAM, it is known that we can assemble a certain type of fractals infinitely even with limited tile types [9]. In OS, we claim that under some reasonable assumptions, it is impossible to fold some infinite fractals with a cyclic OS. We first define the folding of the curve by an OS as mapping segments and points of the curve to sets of points on the OS triangular lattice.

Before presenting our main contributions on the impossibility of OS folding, we start with two well-known fractal curves (Koch and Minkowski curves) with respect to the OS folding and provide a brief idea on the impossibility results. We, in particular, prove that regardless of the delay and the period, it is impossible to fold Koch and Minkowski curves in Sections 3 and 4, respectively. Then we generalize this impossibility to infinite aperiodic curves and establish sufficient conditions together with main contributions in Section 5.

We then examine two well-known fractal curves (Koch and Minkowski curves), and prove that regardless of the delay and the period, it is impossible to fold such curves under our assumptions. We expand the result to infinite aperiodic curves and establish sufficient conditions to prove impossibility of the folding.

2 Preliminaries

Let $w=a_{1}a_{2}\cdots a_{n}$ be a string over $\Sigma$ for some integer $n$ and bead types $a_{1},\ldots,a_{n}$

$\in\Sigma$. The length $|w|$ of $w$ is $n$. For two indices $i,j$ with $1\leq i\leq j\leq n$, we let $w[i,j]$ be the substring $a_{i}a_{i{+}1}\cdots a_{j{-}1}a_{j}$; we use $w[i]$ to denote $w[i,i]$. We use $w^{n}$ to denote the concatenation of $n$ copies of $w$.

Oritatami systems operate on the triangular lattice $\Lambda_{o}$ with the vertex set $V$ and the edge set $E$. A configuration is a triple $(P,w,H)$ of a directed path $P$ in $\Lambda_{o}$, $w\in\Sigma^{*}\cup\Sigma^{\omega}$, and a set H\subseteq\left\{(i,j)\bigm{|}1\leq i,i+2\leq j,\{P[i],P[j]\}\in E\right\} of interactions. This is to be interpreted as the sequence $w$ being folded while its $i$-th bead $w[i]$ is placed on the $i$-th point $P[i]$ along the path and there is an interaction between the $i$-th and $j$-th beads if and only if $(i,j)\in H$. Configurations $(P_{1},w_{1},H_{1})$ and $(P_{2},w_{2},H_{2})$ are congruent provided $w_{1}=w_{2}$, $H_{1}=H_{2}$, and $P_{1}$ can be transformed into $P_{2}$ by a combination of a translation, a reflection, and rotations by $60$ degrees. The set of all configurations congruent to a configuration $(P,w,H)$ is called the conformation of the configuration and denoted by $C=[(P,w,H)]$. We call $w$ a primary structure of $C$.

A ruleset $\mathcal{H}\subseteq\Sigma\times\Sigma$ is a symmetric relation specifying between which bead types can form an interaction. An interaction $(i,j)\in H$ is valid with respect to $\mathcal{H}$, or simply $\mathcal{H}$-valid, if $(w[i],w[j])\in\mathcal{H}$. We say that a conformation $C$ is $\mathcal{H}$-valid if all of its interactions are $\mathcal{H}$-valid. For an integer $\alpha\geq 1$, $C$ is of arity $\alpha$ if the maximum number of interactions per bead is $\alpha$, that is, if for any $k\geq 1$, \bigl{|}\{i\mid(i,k)\in H\}\bigr{|}+\bigl{|}\{j\mid(k,j)\in H\}\bigr{|}\leq\alpha and this inequality holds as an equation for some $k$. By $\mathcal{C}_{\leq\alpha}$, we denote the set of all conformations of arity at most $\alpha$.

Oritatami systems grow conformations by elongating them under their own ruleset. For a finite conformation $C_{1}$, we say that a finite conformation $C_{2}$ is an elongation of $C_{1}$ by a bead $b\in\Sigma$ under a ruleset $\mathcal{H}$, written as $C_{1}\stackrel{{\scriptstyle\mathcal{H}}}{{\to}}_{b}C_{2}$, if there exists a configuration $(P,w,H)$ of $C_{1}$ such that $C_{2}$ includes a configuration $(P\cdot p,w\cdot b,H\cup H^{\prime})$, where $p\in V$ is a point not in $P$ and H^{\prime}\subseteq\left\{(i,|P|{+}1)\bigm{|}1\leq i\leq|P|-1,\{P[i],p\}\in E,(w[i],b)\in\mathcal{H}\right\}. This operation is recursively extended to the elongation by a finite sequence of beads as follows: For any conformation $C$, $C\stackrel{{\scriptstyle\mathcal{H}}}{{\to}}_{\lambda}^{*}C$; and for a finite sequence of beads $w$ and a bead $b$, a conformation $C_{1}$ is elongated to a conformation $C_{2}$ by $w\cdot b$, written as $C_{1}\stackrel{{\scriptstyle\mathcal{H}}}{{\to}}_{w\cdot b}^{*}C_{2}$, if there is a conformation $C^{\prime}$ that satisfies $C_{1}\stackrel{{\scriptstyle\mathcal{H}}}{{\to}}_{w}^{*}C^{\prime}$ and $C^{\prime}\stackrel{{\scriptstyle\mathcal{H}}}{{\to}}_{b}C_{2}$.

An oritatami system (OS) is a $6$-tuple $\Xi=(\Sigma,w,\mathcal{H},\delta,\alpha,C_{\sigma}=[(P_{\sigma},w_{\sigma},H_{\sigma})])$, where $\mathcal{H}$ is a ruleset, $\delta\geq 1$ is a delay, and $C_{\sigma}$ is an $\mathcal{H}$-valid initial seed conformation of arity at most $\alpha$, upon which its transcript $w\in\Sigma^{*}\cup\Sigma^{\omega}$ is to be folded by stabilizing beads of $w$ one at a time and minimize energy collaboratively with the succeeding $\delta-1$ nascent beads. The energy of a conformation $C=[(P,w,H)]$ is $U(C)=-|H|$; namely, the more interactions a conformation has, the more stable it becomes. The set $\mathcal{F}(\Xi)$ of conformations foldable by this system is recursively defined as follows: The seed $C_{\sigma}$ is in $\mathcal{F}(\Xi)$; and provided that an elongation $C_{i}$ of $C_{\sigma}$ by the prefix $w[1:i]$ be foldable (i.e., $C_{0}=C_{\sigma}$), its further elongation $C_{i{+}1}$ by the next bead $w[i{+}1]$ is foldable if

[TABLE]

Once we have $C_{i{+}1}$, we say that the bead $w[i{+}1]$ and its interactions are stabilized according to $C_{i{+}1}$. A conformation foldable by $\Xi$ is terminal if none of its elongations is foldable by $\Xi$. An OS is deterministic if, for all $i$, there exists at most one $C_{i{+}1}$ that satisfies (1). Namely, a deterministic OS folds into a unique terminal conformation. An OS is cyclic if its transcript $w=w_{o}^{\omega}$ is repetition of a string $w_{o}$. We say that the OS has a period $|w_{o}|$.

Figure 3 illustrates an example of an OS with delay $3$, arity $4$, ruleset $\{(a,\overline{a})\}$ and transcript $w=\overline{a}\overline{a}\overline{a}aaa\overline{a}\overline{a}\overline{a}$; in (a), the system tries to stabilize the first bead $\overline{a}$ of the transcript, and the elongation $P_{1}$ gives $2$ interactions, while the elongation $P_{2}$ gives $4$ interactions, which is the most stable one. Thus, the first bead $\overline{a}$ is stabilized according to the location in $P_{2}$. In (b) and (c), $P_{2}$ is the most stable elongation and $\overline{a}$’s are stabilized according to $P_{2}$. As a result, the terminal conformation is given as in (d). Note that the system grows the terminal conformation straight without external interactions, and we can use $w=(\overline{a}\overline{a}\overline{a}aaa)^{\omega}$ to fold an infinite periodic conformation. This example is called a glider [1].

The bead stabilization in OS is a local optimization of finding the best position of the bead using the next $\delta$ beads. Thus, the stabilization of a bead $w[i]$ in a delay-$\delta$ OS is not affected by any bead whose distance from $w[i{-}1]$ is greater than $\delta+1$. On the triangular lattice, we can draw a hexagonal border of radius $\delta+1$ from $w[i{-}1]$ to identify the set of points that may affect the stabilization of $w[i]$. While stabilizing a bead $w[i]$, we define the event horizon of $w[i]$ to be a partial conformation within this hexagon. Namely, the event horizon is the context used to stabilize $w[i]$. Thus, if two beads $w[i]$ and $w[j]$ have the same event horizon, then $w[i]$ and $w[j]$ are stabilized congruently (See Figure 4). We define an event horizon for a partial transcript $w[i,j]$ to be the union of event horizons of beads in $w[i,j]$ while stabilizing $w[i]$.

An L-system is a parallel rewriting system and its recursive nature makes the system easy to describe fractal-like structures [13]. An L-system is defined as $G=(V,C,\omega,P)$, where

• $\bullet$

$V$ is the set of variables that can be replaced by production rules,

• $\bullet$

$C$ is the set of constants that do not get replaced,

• $\bullet$

$\omega\in(V\cup C)^{*}$ is the axiom, the initial string, and

• $\bullet$

$P\subseteq V\times(V\cup C)^{*}$ is the set of production rules defining rewriting of variables.

The system starts with $\omega$, and as many rules as possible are applied simultaneously for each iteration. With graphical semantics on variables and constants, the L-system is often used to represent self-similar fractals. In this paper, we assume that curves are represented by strings, whose characters represent turns and unit segments. Then, an infinite curve is periodic if there exists a periodic string representation with a fixed finite period, and is aperiodic otherwise. Note that all fractal infinite curves are aperiodic.

3 Impossibility of the Infinite Koch Curve

We start with one example of infinite fractal curves—the Koch curve. The Koch curve can be constructed by starting with a segment, then recursively altering each line segment as follows:

Divide the line segment into three segments of equal length. 2. 2.

Draw an equilateral triangle that has the middle segment from step 1 as its base and points outward. 3. 3.

Remove the line segment that is the base of the triangle from step 2.

Using the L-system, the Koch curve can be encoded as follows:

• $\bullet$

Variable: $F$

• $\bullet$

Constants: $+,-$

• $\bullet$

Axiom: $F$

• $\bullet$

Production Rule: $F\to F+F-F+F$,

where $F$ denotes a segment, $-$ denotes $120^{\circ}$ right turn and $+$ denotes $60^{\circ}$ left turn. Figure 5 illustrates the Koch curve after three iterations.

For this specific curve, we first make a few assumptions about the curve representation by a cyclic OS and the OS itself. We prove that it is impossible to draw an infinite Koch curve using a cyclic OS under our assumptions.

Let a shape be a set of points on the triangular lattice $\Lambda_{o}$, whose grid graph is connected. A curve can be represented as a sequence of alternating points and segments. We say that a sequence of shapes represents a curve if there exists one-to-one correspondence between shapes and alternating points and segments, and a point and a segment should be adjacent on the curve if and only if two shapes corresponding to them are adjacent. We formally define the drawing of the curve by a deterministic OS as follows:

Definition 1**.**

Given a (possibly infinite) curve on the plane and the (possibly infinite) sequence $(S_{k})$ of shapes that represents the curve, we say that a deterministic OS draws the curve if the following condition holds: There exists a (possibly infinite) sequence $(i_{k})$ of indices that corresponds to the sequence of shapes, where, for all $k$’s, there exists a partial configuration for $w[i_{k{-}1}{+}1:i_{k}]$ that folds within $S_{k}$. We say that the OS covers $S_{k}$ with the partial transcript $w[i_{k{-}1}{+}1:i_{k}]$.

Figure 6 shows two examples of curve drawing by an OS. Here, the target curve is represented by three shapes $(S_{1},S_{2},S_{3})$. By Definition 1, an OS draws the curve in Figure 6 (a) but does not in Figure 6 (b) since the partial configuration for $w[3:11]$ is not within $S_{2}$. Note that shapes limit paths of conformations, and it is not necessary to fill all points in the shape with the conformation.

From the design perspective, it is crucial to assume this locality of partial configuration. OS designs are usually modular [1, 2, 5, 10]—a partial transcript is folded locally under a controlled context, and we connect these partial transcripts to perform complex computations. Especially, for an infinite transcript, it becomes almost impossible to remove unintended interferences without this locality. Previous cyclic OSs such as a binary counter [1] or a cyclic tag system [2] follow this assumption. Remark that the drawing in Definition 1 is general in the sense that it does not restrict the OS to be infinite or cyclic, and the curve to be infinite.

We assume the followings to draw the Koch curve:

The Koch curve consists of an infinite sequence of alternating points and segments on the triangular lattice $\Lambda_{c}$, which is with vertical rows with unit triangles pointing left and right. We use a hexagon $S_{p}$ of side length $d$ to represent a point on the curve. For a segment on the curve, we use a shape $S_{l}$ of $d$ points ($l+1$ rows in total) and $d+1$ points ($l$ rows in total) in alternative positions which are orthogonal to the direction of the segment (See Figure 7 (a)). The OS starts with covering the first $S_{p}$, and denote the $i$th $S_{p}$ ($S_{l}$) by $S_{p}[i]$ ($S_{l}[i]$). 2. 2.

We use constant numbers of beads for segments or points—$p_{p}$ beads in $S_{p}$, and $p_{l}$ beads in $S_{l}$. This assumption is reasonable in the modular design of the OS. 3. 3.

The period $p$ of the OS is $p=p_{p}+p_{l}$.

Figure 7 shows an example of shapes that can be used to draw the Koch curve, and a part of an OS that draws the curve, following the above assumptions. In Figure 7 (a), $S_{p}$’s are drawn in red, and $S_{l}$’s are drawn in blue, where $d=2$ and $l=3$. In Figure 7 (b), from assumption (2), the number of beads for segments or points are constant, even if there are different paths in different $S_{p}$’s or $S_{l}$’s. For example, paths in $S_{p}$ use $11$ beads and paths in $S_{l}$ use $9$ beads.

Under these assumptions, we claim the following theorem.

Theorem 1**.**

There is no deterministic OS that can draw the Koch curve.

Proof.

Assume that there exists a deterministic cyclic OS $\Xi$ with delay $\delta$ that draws the Koch curve. First, we assume that $\delta<3d+3l+2$, which is one less than the distance between two $S_{p}$’s on $\Lambda_{o}$—apart by two unit distances on $\Lambda_{c}$. If the delay $\delta$ has an upper bound, we denote the event horizon for the maximum delay as the “maximum” event horizon, and omit the term maximum if the context is clear. For convenience, we use $S_{pl}[i]$ to denote the shape resulted from connecting $S_{p}[i]$ and $S_{l}[i]$.

We consider bead stabilization in $S_{pl}[i]$. We have an event horizon $E(i)$ that is used to fold the conformation in $S_{pl}[i]$. Due to the delay upper bound, all $S_{l}$’s that overlap with $E(i)$ are at most three unit distances apart from $S_{p}[i]$, and all $S_{p}$’s that overlap with $E(i)$ are at most two unit distances apart from $S_{p}[i]$ (See Figure 8).

Since the Koch curve does not touch itself, if a point in $\Lambda_{c}$ is adjacent to two segments of the curve, there is no segment other than two that are adjacent to the point. Moreover, due to the self-similarity of the curve, the same statement holds for any scale of the power of $3$—for a point $q_{0}$ in $\Lambda_{c}$, if there exist two points $q_{1}$ and $q_{2}$ on the curve that are $3^{n}$ unit distances straight away from $q_{0}$, then there is no segment within $3^{n}$ unit distances from $q_{0}$, other than segments on the curve from $q_{1}$ and $q_{2}$ (See Figure 9).

We have nine different cases for the beads within $S_{p}$’s and $S_{l}$’s that overlap with $E(i)$ as shown in Figure 10. In each case, we first transcribe a sequence of blue shapes (if they exist), and then a sequence of red shapes. Shapes $S_{p}[i]$ and $S_{l}[i]$ are distinguished by points within the shape. The thick hexagons represent $q_{0}$’s of distance $3$. We can observe that in all cases, all $S_{p}$’s and $S_{l}$’s are at most three unit distances apart from one of $q_{0}$’s, and they are empty except the red and blue shapes. Moreover, in all cases, all beads within $E(i)$ are from $S_{p}[i{-}4]$ to $S_{l}[i{-}1]$, a consecutive sequence of shapes before $S_{l}[i]$. In other words, a partial conformation in $S_{pl}[i]$ is dependent to beads in the sequence of shapes from $S_{p}[i{-}4]$ to $S_{l}[i{-}1]$.

Since the period of the OS is $p_{1}+p_{2}$, we have exactly the same partial transcripts that fold within all $S_{pl}$’s, and, therefore, having the same path yields the same partial conformation. The upper bound for the number of possible paths within $S_{p}$ ($S_{l}$) is $5^{p_{1}}$ ($5^{p_{2}}$). We have beads from $S_{p}[i{-}4]$ to $S_{l}[i{-}1]$ that determine the partial conformation in $S_{pl}[i]$. Thus, we have $i$ and $j$ such that $1\leq i<j\leq 1+5^{4p_{l}+4p_{p}}$, and $S_{pl}[i]$ and $S_{pl}[j]$ have exactly the same conformation. Moreover, since beads from $S_{p}[i{-}4]$ to $S_{l}[i{-}1]$ are consecutively transcribed, $S_{pl}[i]$ and $S_{pl}[j]$ result in a periodic sequence of segment turns of length $j-i$ (See Figure 11). Since the Koch curve is aperiodic, we know that the first assumption $\delta<3l+3d+2$ is wrong.

Now we assume that $3l+3d+2\leq\delta<12l+12d+11$, which is the distance between two $S_{p}$’s apart by $8$ unit distances in the lattice for the Koch curve. Suppose that we want to stabilize beads in $S_{pl}[i]$. The event horizon for beads in $S_{pl}[i]$ covers only $S_{p}$’s and $S_{l}$’s which are within $8$ unit distances from $S_{p}[i]$ in the lattice for the Koch curve. Similar to the previous case, we can find at most two $q_{i}$’s of distance $9$, and all $S_{p}$’s and $S_{i}$’s are empty except those who represent the curve from $q_{i}$’s. Due to the length constraint, there can be at most $32$ $S_{p}$’s and $S_{i}$’s that overlap with the event horizon. Thus, among $S_{pl}[1]$ to $S_{pl}[5^{32p_{l}+32p_{p}}+1]$, there should exist $S_{pl}[i]$ and $S_{pl}[j>i]$ that have exactly the same event horizon for points within, and we know that the second assumption is also wrong. Similarly, we can expand the proof for arbitrarily large $\delta$. Therefore, we know that for any given $\delta$, there is no delay-$\delta$ deterministic OS that can draw the Koch curve.

Note that we can remove the third assumption about the OS period. If a period $p$ is not $p_{p}+p_{l}$, beads in $S_{pl}[i]$ and $S_{pl}[i{+}pj]$ are exactly the same. Thus, for instance, if we assume that $\delta<3l+3d+2$, among $S_{pl}[1]$ to $S_{pl}[p\cdot 5^{4p_{l}+4p_{p}}{+}1]$, there should exist $S_{pl}[i]$ and $S_{pl}[j>i]$ that have exactly the same event horizon and produce the same conformation. ∎

4 Impossibility of the Infinite Minkowski Curve

We study another example of infinite fractal curves—the Minkowski curve. The Minkowski curve starts from a segment, then recursively alternates each line segment as follows:

Divide the line segment into four segments (we call these segment 1 to 4 from the start) of equal length. 2. 2.

Draw a square with segment 2 as a side to the left of the original segment, and the other square with segment 3 as a side to the right. 3. 3.

Remove segments 2 and 3.

Using the L-system, the Minkowski curve can be encoded as follows:

• $\bullet$

Variable: $F$

• $\bullet$

Constants: $+,-$

• $\bullet$

Axiom: $F$

• $\bullet$

Production Rule: $F\to F+F-F-FF+F+F-F$,

where $F$ denotes a segment, $-$ denotes $90^{\circ}$ right turn and $+$ denotes $90^{\circ}$ left turn. Figure 12 illustrates the Minkowski curve after three iterations.

For the Minkowski curve case, since the OS works on the triangular lattice, we assume that the Minkowski curve is slanted to fit into the triangular lattice—a square in the square lattice is mapped into a rhombus in the triangular lattice. We call the lattice the rhombus lattice. We assume the followings for drawing the Minkowski curve:

The (tilted) Minkowski curve consists of an infinite sequence of alternating points and segments on the rhombus lattice $\Lambda_{r}$. We use a parallelogram $S_{l}$ of width $d$ and length $l$ to represent a segment on the curve, and a rhombus $S_{p}$ of side length $d$ to represents a point on the curve (See Figure 13). The OS starts with covering the first $S_{p}$, and denote the $i$th $S_{p}$ ($S_{l}$) by $S_{p}[i]$ ($S_{l}[i]$).

For convenience, we set up a coordinate $(x,y)$ for the points of the rhombus lattice based on an arbitrary origin, where the unit $x$ vector heads right and the unit $y$ vector heads upper right. Based on the coordinate, we use $S_{l}(x,y,x^{\prime},y^{\prime})$ to represent an $S_{l}$ that starts from the point $(x,y)$ and the direction is given by the vector $(x^{\prime},y^{\prime})$. We also use $S_{p}(x,y)$ to represent an $S_{p}$ at the point $(x,y)$ (See Figure 13).

We use constant numbers of beads for a connecting line and a point—$p_{p}$ beads for $S_{p}$, and $p_{l}$ beads for $S_{l}$. 3. 3.

The OS period $p$ is $p=p_{p}+p_{l}$.

Under these assumptions, we claim the following theorem.

Theorem 2**.**

There is no deterministic OS that can draw the tilted Minkowski curve.

Proof.

Similar to the proof for Theorem 1, we fist assume that $\delta<3l+3d+5$. Suppose we want to transcribe beads in $S_{p}[i]=S_{p}(x,y)$ and $S_{l}[i]=S_{l}(x,y,0,1)$, which represent the segment going up from the point $(x,y)$. We have three cases for the previous segment:

$S_{l}[i{-}1]=S_{l}(x,y,-1,0)$, where the previous segment came from the left. 2. 2.

$S_{l}[i{-}1]=S_{l}(x,y,1,0)$, where the previous segment came from the right. 3. 3.

$S_{l}[i{-}1]=S_{l}(x,y,0,-1)$, where the previous segment came from the low.

Considering all three cases, we can draw the event horizon $E(i)$ for all beads in $S_{pl}[i]$ as in Figure 14:

We observe the property of the curve based on self-similarity. For a given $n$, let $q_{1}$ be the starting point of a periodic substructure of length $8^{n}$, and $q_{2}$ be the ending point. Points $q_{1}$ and $q_{2}$ should be $4^{n}$ distance away. Now, suppose we draw the square of size $2\cdot 4^{n}$ with the center $q_{2}$. Then, the point $q_{3}$ that appears first in the square (including the boundary) is at most $7\cdot 8^{n}$ segments away from $q_{1}$ (See Figure 15 (a)). Moreover, the point $q_{4}$ that appears last in the square is at most $8\cdot 8^{n}$ segments away from $q_{1}$ (See Figure 15 (b)).

We have eleven different cases for the beads within $S_{p}$’s and $S_{l}$’s that overlap with $E(i)$ (See Figure 16). In each case, shapes $S_{p}[i]$ and $S_{l}[i]$ are distinguished by points within the shape. The thick rhombi represent $q_{2}$’s for $n=1$. We can observe that in all cases, the union of event horizons is covered by at most two squares with different $q_{2}$’s, whose shapes range from $S_{p}[i{-}5]$ to $S_{p}[i{+}6]$. According to the observation, we can say that a partial conformation in $S_{pl}[i]$ is dependent to beads in the sequence of shapes from $S_{p}[i{-}61]$ to $S_{l}[i{-}1]$. Thus, we have $i$ and $j$ such that $1\leq i<j\leq 1+5^{61p_{l}+61p_{p}}$ and $S_{pl}[i]$ and $S_{pl}[j]$ have exactly the same conformation. Since this results in a periodic sequence of segment turns, our assumption $\delta<3l+3d+5$ is wrong.

Now we assume that $3l+3d+5\leq\delta<15l+15d+29$. Suppose that we want to stabilize beads in $S_{pl}[i]$. The event horizon covers only $S_{p}$’s and $S_{l}$’s that are within the square of size $32$ unit distances with the center $S_{p}[i]$. Similar to the previous case, we can find at most two $q_{2}$’s with $n=2$ where the corresponding squares completely cover the event horizon. The earliest $S_{p}$ that can be such $q_{2}$ is $S_{p}[i{-}64]$, and we can say that a partial conformation in $S_{pl}[i]$ is dependent to beads in the sequence of shapes from $S_{p}[i{-}512]$ to $S_{l}[i{-}1]$. Thus, we have $i$ and $j$ such that $1\leq i<j\leq 1+5^{512p_{l}+512p_{p}}$ and $S_{pl}[i]$ and $S_{pl}[j]$ have exactly the same conformation. In a similar way, we can extend the proof for arbitrarily large $\delta$. Similar to the Koch curve case, we can also remove the assumption about the OS period.

∎

5 Impossibility of Infinite Aperiodic Curves

We expand the results from Sections 3 and 4 to impossibility of infinite aperiodic curves, which includes fractals. Construction of infinite periodic curves using a cyclic OS seems to be reasonable if we can design a partial OS that folds one period of the curve, and we have one running example—the glider in Section 2. On the other hand, for infinite aperiodic curves, we propose sufficient conditions that makes curves impossible to fold. We make the following assumptions:

• $\bullet$

The curve is on an arbitrary lattice $\Lambda$, and each point (segment) in $\Lambda$ is mapped to a shape $S_{p}$ ($S_{l}$) in the triangular lattice $\Lambda_{o}$.

• $\bullet$

The OS uses $p_{p}$ ($p_{l}$) beads for $S_{p}$ ($S_{l}$). We say $p_{pl}=p_{p}+p_{l}$.

• $\bullet$

The OS has the period of $p_{o}$.

• $\bullet$

The curve is represented by an infinite alternation of $S_{p}$ and $S_{l}$, starting from $S_{p}[1]$. For convenience, we refer to the union of $S_{p}[i]$ and $S_{l}[i]$ as $S_{pl}[i]$.

Let $\delta(n)$ be an upper bound of the delay $\delta$, dependent to a given integer $n$. We propose the condition that curves are not foldable when $\delta\leq\delta(n)$, and expand the result to all possible delays. Suppose we want to stabilize beads in $S_{pl}[i]$. Then, we have the maximum event horizon $E(i,n)$ for beads in $S_{pl}[i]$, which is the union of all event horizons of radius $\delta(n)+1$ whose centers are points in $S_{pl}[i]$ and points neighboring $S_{p}[i]$ in $S_{l}$’s adjacent to $S_{p}[i]$. Now, for each $i$, we have $S_{p}[r_{i,n}]$ that appears first in $E(i,n)$. Let $D_{i,n}=\max_{1\leq j\leq i}(j-r_{j,n})$ to be the maximum difference between $j$ and $r_{j,n}$ for all $j\leq i$. Then, it takes $1+gcd(p_{o},p_{pl})\cdot 5^{D_{i,n}p_{pl}}$ to have exactly the same conformation for the previous $D_{i,n}$ beads, which results in the same maximum event horizon and the same conformation for two shapes $S_{pl}[j]$ and $S_{pl}[k>j]$. After $S_{pl}[j]$ and $S_{pl}[k]$, it is assured that previous $D_{i,n}$ beads have exactly the same conformation for the consecutively following shapes, and these shapes fold exactly the same. Since $D_{i,n}$ is dependent on $i$, we have the following theorem.

Theorem 3**.**

If there exists $i$ such that $1+gcd(p_{o},p_{pl})\cdot 5^{D_{i,n}p_{pl}}\leq i$, then it is impossible to draw a given infinite aperiodic curve with a cyclic OS whose delay is less than or equal to $\delta(n)$ and period is $p_{o}$.

In practice, the delay of the OS is bounded by the transcript length. If we consider a cyclic OS that has an infinite transcript, the delay can be arbitrarily large. We extend Theorem 3 for arbitrarily large delays and obtain the following statement.

Theorem 4**.**

Suppose there exists a function $\delta(n)$, where for all $n\geq 1$, there exists $i$ such that $1+gcd(p_{o},p_{pl})\cdot 5^{D_{i,n}p_{pl}}\leq i$. Then, it is impossible to draw a given aperiodic infinite curve with a cyclic OS whose period is $p_{o}$.

If $D_{i,n}$ is dependent only on $n$ and not $i$, then we may use an arbitrarily large $i$ to satisfy the conditions for all $n$’s regardless of $gcd(p_{o},p_{pl})$. When we have such a case, the following statement holds.

Theorem 5**.**

If there exists a function $\delta(n)$ such that, for all $n$, $D_{i,n}$ is independent of $i$, then it is impossible to draw a given infinite aperiodic curve with a cyclic OS regardless of the delay and the period of the OS.

Note that both in the proofs for Theorems 1 and 2, we have $\delta(n)$ that makes $D_{i,n}$ independent of $i$, which is the Theorem 5 case.

6 Conclusions

The oritatami system (OS) is a computational model inspired by RNA cotranscriptional folding, where an RNA transcript folds upon itself while synthesized out of a gene. Since the OS is a geometric computation model, it is natural to consider the problem of constructing fractal curves using the OS. We have formally defined the drawing of the curve by an OS. Then we have proved that it is impossible to draw two infinite fractal curves (Koch curve and Minkowski curve) by a cyclic OS. Moreover, we have proposed sufficient conditions that make the folding of general infinite curves impossible. Our conjecture is that all fractal curves made by edge replacements are not foldable. However, we cannot directly apply the same approach to all curves, especially to curves that touch themselves such as Heighway dragons. Thus it is open to develop a new approach for these self-touching curves.

Acknowledgements

Han was supported by the Basic Science Research Program through NRF funded by MEST (2015R1D1A1A01060097). This work is partially supported by NIH R01GM109459, and by NSF’s CCF-1526485 and DMS-1800443. This research was also partially supported by the Southeast Center for Mathematics and Biology, an NSF-Simons Research Center for Mathematics of Complex Biological Systems, under National Science Foundation Grant No. DMS-1764406 and Simons Foundation Grant No. 594594.