A Combinatorial Method for the Number of Components of DNA and Polypeptide Cages

Abstract

A single circular DNA or polypeptide strand can be used to construct nanostructures based on the mathematical concept of strong traces. It is known that if a special face graph of a 2-connected plane graph G is a tree, then G has a thickened graph $F\left(G\right)$ with only one boundary component. However, to ensure that this thickened graph contains only a single boundary component, it is not necessary for the special face graph of G to be a tree. In this paper, we summarize a combinatorial method, called the flat-based operations, for obtaining a matrix $M_0$ where $\det \left(M_0\right)$ determines whether $F\left(G\right)$ associated with G contains just one boundary component. By applying this method, we can easily determine if the number of boundary components of some $F\left(G\right)$ is one. Further, we characterize a type of 2-connected plane graph whose thickened graph has more than one boundary component and then has no antiparallel strong trace.

1. Introduction

DNA and polypeptides are two types of biomolecules that can self-assemble into complex three-dimensional shapes. One common shape is the polyhedron, which is a closed spatial graph composed of several plane polygons. DNA polyhedra have important applications in biology and biochemistry, such as drug delivery and disease treatment. To achieve these applications, researchers have developed and synthesized various polyhedral structures, such as DNA tetrahedra, cubes, hexahedra, dodecahedra, etc. [1,2,3,4,5,6,7,8,9,10]. These structures can be described by mathematical language, namely DNA polyhedral cages. To construct a specific DNA polyhedral structure, a specific arrangement of junctions is required at the vertices of the polyhedral graph, which can be realized by biological engineering methods.

By using orthogonal segments that form coiled-coils, a method is proposed to design self-assembled polypeptide nanostructures that have polyhedral shapes. A tetrahedral cage, TET12, can be folded from a single polypeptide chain with 12 coiled-coil segments [11], which is shown in Figure 1. Some other polyhedral shapes are also designed by using this method, such as triangular prisms and trigonal bipyramids [12,13]. Fijavž described these polypeptides [14] by using strong traces. How polypeptide nanostructures are modeled by antiparallel strong traces were studied by Kla$\check{\mathrm{v}}$zara and Rus, respectively [14,15].

Based on biologically stable configurations, strong trace is a good mathematical model for designing polypeptide and DNA nanostructures. Fijavž et al. [14] provided a necessary and sufficient condition for a graph G to have an antiparallel or parallel strong trace. Jonoska and Saito found that the quantity and shape of DNA structures are influenced by the boundary components of a thickened graph. Specifically, the number of circular DNA strands is linked to the boundary curves of the thickened graph [16]. Jonoska and Twarock designed nucleic acid dodecahedral molecules, and drew blueprints of DNA dodecahedra with only two strands by using 3-arms as building blocks [17]. Using only one or two strands, Cheng and Diao designed DNA structures by 3-arms, and figured out the smallest number and position of these 3-arms [18]. Subsequently, they also created more complex DNA cages with a single branch on a 2-connected planar graph, using junctions of odd-crossing curves and 0-crossing curves, where the construction of a special face graph is a tree [19]. They obtained a necessary and sufficient condition for the special face graph of a 2-connected plane graph G to be a tree. They also noted that, if the special face graph is not a tree, the number of components of the constructed DNA or polypeptide cage may be one or greater than one.

The components of DNA or polypeptide cages are of significant importance for the design and adaptation of increasingly complex cage-like structures and functions, which play an important role in catalysis, materials science, and medicine. The number of these components can alter the morphology, stability, and assembly/disassembly characteristics of protein cages. Furthermore, these components of protein cages can be modified through methods such as supercharging, cyclic permutation, direct evolution, and genetic fusion of biologically orthogonal functional groups, thereby enhancing their efficiency in drug encapsulation and release [20,21,22]. Notably, recent research highlights the relevance of this component-related exploration in the context of broader studies on DNA and polypeptide nanostructures. In 2022, Edwardson highlighted that the number of components within protein cages from a biological perspective, which is related to the boundary components we study, directly impacts drug encapsulation efficiency. He emphasized that understanding these components is essential for optimizing the functions of cages in biological applications [20]. In 2023, Majsterkiewicz et al. mathematically pointed out that topological features like boundary components play a decisive role in determining the connectability and the functional modifiability of cages [21].

In this paper, we continue the work of Diao et al., and provide a new combinatorial method called the flat-based operations for obtaining the target matrix denoted by $M_0$, where the determinant $\det \left(M_0\right)$ only equals to 0 or 1. If $\det \left(M_0\right)=0$, then the number of components of the constructed DNA or polypeptide cage is greater than one. If $\det \left(M_0\right)=1$, then the number of components of the constructed DNA or polypeptide cage is equal to one. Further, we also characterize a type of 2-connected plane graph whose thickened graph has more than one boundary component and then has no antiparallel strong trace. Our research provides new perspectives and tools for understanding and designing new polypeptide nanostructures.

2. Preliminaries

2.1. Double Trace and Its Properties

All graphs considered in this paper are finite and contain no loops or multiple edges. Some basic terminologies in graph theory can be found in [23], while some terminologies in algebraic topology can be found in [24]. Let G be a graph with $V\left(G\right)$ and $E\left(G\right)$ as its vertex set and edge set, respectively. Let $\left|V\right(G\left)\right|$ and $\left|E\right(G\left)\right|$ denote the number of vertices and edges of G, respectively. Let W be a closed walk that traverses every edge of G twice. Then W is called a double trace of G. If W traverses an edge e of G in opposite (or the same) direction, then e is called an antiparallel (or parallel) edge with respect to W. A double trace W is called an antiparallel (or parallel) trace if all the edges of G are traversed in the opposite (or the same) directions.

Proposition 1

([14]). Any connected graph G contains a double trace.

Let $W=v_0{e}_1{v}_1\dots v_{l-1}{e}_l{v}_0$ be a double trace of G of length l. Let v be a vertex of G and N be a subset of $N\left(v\right)$, where $N\left(v\right)$ denotes the set of all vertices adjacent to v. We say that W has an N-repetition at v if the following holds: For every $i\in \{0,1,2,\dots ,l-1\}$, if $v=v_i$, then $v_{i+1}\in N$ if and only if $v_{i-1}\in N$, where i is under modulo l. Intuitively, whenever we enter v from a vertex in N, we also exit to a vertex in N from v. If $\left|N\right|=k$, the repetition is also called a k-repetition. Figure 2 gives an example of 3- and 4-repetition at v where $d_G\left(v\right)=7$. The N-repetition is called trivial if N is either ∅ or $N\left(v\right)$. A double trace is called a strong trace if it has only trivial repetitions. Then we have the following theorem.

Theorem 1

([14]). Any connected graph G contains a strong trace.

A cellularly embedded graph G means that G is embedded in a closed surface $\Sigma$ such that every connected component of the complement of the union of the points and arcs associated with the vertices and edges of G, denoted by $\Sigma -G$, is homeomorphic to an open disk, called a face. Furthermore, if the number of faces is k, we say that G has a k-face embedding. A thickened graph $F\left(G\right)$ (of G) is defined as a compact orientable surface such that G is topologically embedded (as a 1-complex) in $F\left(G\right)$ as a deformation retract (see [24] for examples of deformation retract). By capping each boundary component of the thickened graph $F\left(G\right)$ with a topological disk, we obtain a closed orientable surface $S\left(G\right)$ without boundary. It implies that G is cellularly embedded in $S\left(G\right)$. Then we have the following theorem.

Theorem 2

([14]). A graph G can be cellularly embedded in an orientable surface with a single face if and only if it admits an antiparallel strong trace.

2.2. Some Basic Results in Knot Theory

The boundary of the thickened graph $F\left(G\right)$ can also be seen as a link in ${\mathbb{R}}^3$. Then we outline some basic terminologies and results in knot theory below. Some basic terminologies in knot theory can be found in [25,26].

Theorem 3

([25]). Let L be an arbitrary oriented knot (or link) in ${\mathbb{R}}^3$. Then there is an orientable connected surface F in ${\mathbb{R}}^3$ with L as its boundary.

The surface F is called a Seifert surface of L. Let D be a regular oriented diagram of L. A method called Seifert decomposition can be used to construct the Seifert surface and it is dependent on the choice of D. The knot book [25] explains how to construct the Seifert surface by Seifert decomposition where all types of crossings in D are smoothed out as illustrated in Figure 3.

Then we obtain a collection of circles that do not intersect each other, which are called Seifert circles. Each Seifert circle can be spanned by a topological disk. By attaching positive or negative bands as shown in Figure 4 among these Seifert circles according to the types of crossings of D, we can obtain a connected and orientable surface F. This surface is known as the Seifert surface and is constructed by Seifert decomposition. The boundary of F is the original L. Figure 5 gives an example of the Seifert decomposition of the Hopf link with its Seifert graph. Moreover, if we contrast each disk to a point, and narrow each band to an edge, then we obtain the Seifert graph of D. Denote the Seifert graph by $\mathcal{G}\left(D\right)$. Note that $\mathcal{G}\left(D\right)$ is bipartite and plane.

There are two types of crossings, positive and negative crossings, as shown in Figure 4. When we come across a crossing c of D as shown in Figure 4i, we assign $sign\left(c\right)$ a value of $+1$, while in the case shown in Figure 4ii, we assign $sign\left(c\right)$ a value of $-1$. Then we consider a 2-component link L, consisting of components $K_1$ and $K_2$, and the link is represented by an oriented regular diagram D. The crossings between the projections of $K_1$ and $K_2$ are labeled as $c_1$, $c_2,\dots ,c_m$. In this case, we are only considering the intersections between the two components, not the self-intersections of each component. The linking number of $K_1$ and $K_2$, denoted by $lk(K_1,K_2)$, is defined as half the sum of the signs of these crossings. In mathematical terms,

$$lk(K_1,K_2)={\displaystyle \frac{1}{2}}\{sign\left(c_1\right)+sign\left(c_2\right)+\dots +sign\left(c_m\right)\}.$$

This linking number provides valuable information about the topological relationship between $K_1$ and $K_2$ in the link L.

A matrix called Seifert matrix created from the Seifert surface F can be used to obtain a link invariant called the Alexander polynomial. There are several closed curves on F corresponding to the bounded face boundaries in $\mathcal{G}\left(D\right)$. The orientation of these closed curves are arbitrarily given. Let $\alpha$ be a closed curve on F corresponding to a bounded face boundary of $\mathcal{G}\left(D\right)$. Since F is orientable, we can thicken (or lift) F slightly by the right-hand rule, in other words, construct $F\times [0,1]$. Then we denote $\alpha =\alpha \times \left\{0\right\}$, ${\alpha }^{\#}=\alpha \times \left\{1\right\}$. Suppose $\beta$ is another closed curves on F corresponding to a bounded face boundary of $\mathcal{G}\left(D\right)$. Then the element of the Seifert matrix corresponding to $(\alpha ,\beta )$ is defined as $lk(\alpha ,{\beta }^{\#})$, where $lk(\alpha ,{\beta }^{\#})$ is the linking number of $\alpha$ and ${\beta }^{\#}$. Then the number of closed cuvers corresponding to face boundaries in the Seifert graph $\mathcal{G}\left(D\right)$ (excluding the unbounded face) is equal to the number of bounded face boundaries in $\mathcal{G}\left(D\right)$, which is the order of the constructed Seifert matrix. For more information about Seifert matrix, refer to [25]. Then we have the following theorem.

Theorem 4

([25]). Suppose $M_1$ and $M_2$ are two Seifert matrices of a knot (or link) L. Let r and s be the orders of $M_1$ and $M_2$, respectively. Then we have

$$t^{{\displaystyle \frac{r}{2}}}\det (M_1-t{M}_1^T)=t^{{\displaystyle \frac{s}{2}}}\det (M_2-t{M}_2^T).$$

Thus, if we suppose M is a Seifert matrix of L and k is the order of M, ${\Delta }_L\left(t\right)=t^{{\displaystyle \frac{k}{2}}}\det (M-t{M}^T)$ is an invariant, called the Alexander polynomial.

Then we have the following proposition.

Proposition 2

([25]). The link L is a knot (or a link with at least two components) if and only if ${\Delta }_L\left(1\right)=1$ (or ${\Delta }_L\left(1\right)=0$).

3. π-Junction, the Special Face Graph, and the Flat-Based Operations

In this section, we will use $\pi$-junction and $\tau$-junction to construct $F\left(G\right)$, as the thickened graph of a 2-connected plane graph G, and establish a combinatorial method for obtaining the matrix $M_0=M-M^T$ from G, where M is the Seifert matrix derived from the boundary of the thickened graph $F\left(G\right)$, denoted by $\partial F\left(G\right)$, of G. Then by Proposition 2, we have access to determining whether $F\left(G\right)$ has more than one boundary component by computing $det\left(M_0\right)$.

3.1. π-Junction and Special Face Graph

Suppose G is a plane graph (without loops) and v is a vertex of G which is not an isolated vertex. Let $E\left(v\right)=\{e_i:i=1,2,\dots ,k\}$ be the set of the edges incident with v as shown in Figure 6.

Then the $\pi$ (or $\tau$)-junction at v is defined as a cyclic permutation of $E\left(v\right)$ in the clock-wise ordering around v, denoted by ${\pi }_v$ (or ${\tau }_v$), which satisfies ${\pi }_v\left(e_i\right)=e_{i+2}$ (or ${\tau }_v\left(e_i\right)=e_{i+1}$), where i is under modulo k.

Note that the degree of v, denoted by $d_G\left(v\right)$, must be odd if a $\pi$-junction is assigned to v, since ${\pi }_v$ is a cyclic permutation. The thickened graph $F\left(G\right)$ that we want to construct can now be described through the two junctions. As for $\pi$-junction at a vertex v, we rearrange the edges incident with v as shown in Figure 7a, where the $\pi$-junction ${\pi }_v=(e_1,e_3,e_5,e_7,e_2,e_4,e_6)$. The local thickened graph is constructed from Figure 7a where each vertex is replaced with a vertex-disc and each edge is replaced with an edge-ribbon, as shown in Figure 7b. Figure 7c shows the crossing form realized by the $\pi$-junction, which is also the boundary curves shown in Figure 7b up to homeomorphism. We shall assign the orientation to the boundary of $F\left(G\right)$ as shown in Figure 7b. Similarly, Figure 7e shows the local thickened graph of $F\left(G\right)$ with $\tau$-junction. This type of junction is also called the 0-crossing junction.

Note that how to arrange the crossing types as shown Figure 7c is not essential, since we consider the boundary of the thickened graph, as shown in Figure 7b, up to homeomorphism. The vertex v is also called a $\pi$-vertex (or $\tau$-vertex) if a $\pi$-junction (or $\tau$-junction) is assigned to it. By connecting these bands along the edges of G in a natural way, we obtain the thickened graph denoted by $F\left(G\right)$. An example is shown in Figure 8b where the vertices in red are $\pi$-vertices and the vertices in black are $\tau$-vertices.

We can also regard the boundary of the thickened graph $F\left(G\right)$ as a link. Let D denote the regular oriented diagram of $\partial F\left(G\right)$. By Seifert decomposition and smoothing all the crossings in D, the Seifert surface F and the Seifert graph $\mathcal{G}\left(D\right)$ are constructed. Nevertheless, the Seifert graph can also be obtained by constructing the graph which is induced from G as defined below. This alternative way to construct the Seifert graph is more convenient once the $\pi$-vertices are given.

Definition 1.

Let G be a 2-connected plane graph where some vertices are π-vertices. If we place a vertex called face vertex in each face of G (including the unbounded face), and connect each of the π-vertices and its corresponding adjacent face vertices by single edges. Then the graph constructed by the π-vertices and face vertices with the new added edges is called the special face graph of G, denoted by $\mathcal{F}\left(G\right)$.

Note that $\mathcal{F}\left(G\right)$ is a simple and bipartite plane graph. Figure 8d gives an example of the construction of the special face graph where $E\left(\mathcal{F}\right(G\left)\right)$ consists of those in red. It is not difficult to verify that the construction of the Seifert graph of D can be replaced by $\mathcal{F}\left(G\right)$, we recommend readers refer to [19].

3.2. The Flat-Based Operations

By Seifert decomposition, as shown in Figure 8c, only one disk corresponding to the unbounded face of G in the Seifert surface F is not on the same level (plane) as the other disks of F. If all disks of a Seifert surface stay on the same level, we say that it is a flat. Then the surface as shown in Figure 8c is not flat.

Although F is not a flat, we can still obtain a flat Seifert surface from F. Without loss of generality, we suppose the disk corresponding to the unbounded face is at the bottom. The strategy is to use Reidemeister moves (refer to [25]) to replace the bottom disk with a narrow disk, as shown in Figure 9. Then by Seifert decomposition, we obtain the flat Seifert surface, denoted by $F^{\prime }$. Let $D^{\prime }$ be the link diagram of $\partial F^{\prime }$ as shown in Figure 9 and $\mathcal{G}\left(D^{\prime }\right)$ be the corresponding Seifert graph of $D^{\prime }$. Since $F^{\prime }$ is flat, we can construct $F^{\prime }$ directly from $\mathcal{G}\left(D^{\prime }\right)$ by thickening the vertices into disks and widening the edges into bands corresponding to the crossings in $D^{\prime }$ (see also Figure 9).

Then we have the following proposition.

Proposition 3

([25]). Let $M^{\prime }$ denote the Seifert matrix of the flat Seifert surfaces $F^{\prime }$. Then $M^{\prime }$ is ${\Lambda }_1$-equivalent to M, where ${\Lambda }_1:M\to PM{P}^T$, and P represents an invertible integer matrix with a determinant of $\pm 1$.

As Proposition 3 indicates,

$$\det (M^{\prime }-{\left(M^{\prime }\right)}^T)=\det (M-M^T)=\det \left(M_0\right).$$

If we want to calculate $det\left(M_0\right)$, we just need to calculate the linking numbers directly from $\mathcal{G}\left(D^{\prime }\right)$ by using the bounded face boundaries in $\mathcal{G}\left(D^{\prime }\right)$ and without considering the spacial structure of $F^{\prime }$ since $F^{\prime }$ is flat. In addition, the pendent edges are not essential. Then in the following discussion, we only consider the graphs induced from the special face graph $\mathcal{F}\left(G\right)$ and $\mathcal{G}\left(D^{\prime }\right)$ by deleting all 1-degree vertices, denoted by $R\left(\mathcal{F}\right(G\left)\right)$ and $R\left(\mathcal{G}\left(D^{\prime }\right)\right)$, respectively. $R\left(\mathcal{F}\right(G\left)\right)$ is also called the reduced special face graph (briefly RSF graph) of G.

Let $\alpha$, $\beta$ be the two different closed curves on $F^{\prime }$ corresponding to two bounded face boundaries A and B of $R\left(\mathcal{G}\left(D^{\prime }\right)\right)$, respectively. Let $lk(\alpha ,{\beta }^{\#})$, $lk(\beta ,{\alpha }^{\#})$ (or $lk(A,B^{\#}),lk(B,A^{\#})$), respectively, denote the pair of the linking numbers of $\alpha$ and $\beta$. If a and b have no common edges, then the linking numbers $lk(\alpha ,{\beta }^{\#})$, $lk(\beta ,{\alpha }^{\#})$ are both zeros, where ${\alpha }^{\#}$, ${\beta }^{\#}$ are the lifts of $\alpha$, $\beta$ respectively. If A and B have edges in common, then we establish a combinatorial method to compute the linking numbers $lk(\alpha ,{\beta }^{\#})$, $lk(\beta ,{\alpha }^{\#})$.

Further, $R\left(\mathcal{G}\left(D^{\prime }\right)\right)$ and $R\left(\mathcal{F}\right(G\left)\right)$ are isomorphic according to the construction. Then we can calculate the Seifert matrix $M^{\prime }$ of $F^{\prime }$ directly from $R\left(\mathcal{F}\right(G\left)\right)$. It does not change the number of components of the regular diagram D of $\partial F\left(G\right)$ when we exchange the overcrossing and undercrossing relationship of a crossing point. Thus, we suppose D is alternating, which refers to a specific arrangement where the crossings alternate between overcrossings and undercrossings. Besides, all the crossings in D are supposed to be positive (see Figure 4i).

When we lift $F^{\prime }$ from $F^{\prime }\times \left\{0\right\}$ to $F^{\prime }\times \left\{1\right\}$, accordingly, we also lift $\mathcal{F}\left(G\right)$ from $\mathcal{F}\left(G\right)\times \left\{0\right\}$ to $\mathcal{F}\left(G\right)\times \left\{1\right\}$. Then the face boundaries in $R\left(\mathcal{F}\right(G\left)\right)$ corresponding to the closed curves in F should also be lifted. To calculate the linking numbers, we assume that the orientations of all face boundaries in the RSF graph $R\left(\mathcal{F}\right(G\left)\right)$ are anti-clockwise.

Let $\alpha$ be a closed curve that travels along the band of the Seifert surface $F^{\prime }$ as shown in Figure 10i. When we lift $F^{\prime }$ (including the band), then there is a point in $\alpha$ such that the lift direction of that point is as shown in Figure 10i marked in blue. The edges in the RSF graph $R\left(\mathcal{F}\right(G\left)\right)$ correspond to positive bands of $F^{\prime }$ as shown in Figure 10ii where the point marked in red is a $\pi$-vertex, and the other is a face vertex. Accordingly, we attach a small arrow to each edge of $R\left(\mathcal{F}\right(G\left)\right)$ as shown in Figure 10iii, which is vertical with the edge according to Figure 10ii. The direction of the arrow represents the direction of the lift of the local point of the edge.

Let $C_1$ and $C_2$ be two bounded face boundaries of $R\left(\mathcal{F}\right(G\left)\right)$ as shown in Figure 11. Without loss of generality, suppose $C_1\cap C_2$ is a path of length k, denoted by $v_0{e}_1{v}_1\dots e_k{v}_k$. If $v_0$ is a $\pi$-vertex, when we lift $C_2$, $C_2^{\#}$ can be illustrated as shown in Figure 11i, then the linking number of $C_1$ and $C_2^{\#}$ is

$$lk(C_1,C_2^{\#})=\lfloor {\displaystyle \frac{k+1}{2}}\rfloor .$$

If $v_0$ is a face vertex, $C_2^{\#}$ can be illustrated as shown in Figure 11ii, then the linking number of $C_1$ and $C_2^{\#}$ is

$$lk(C_1,C_2^{\#})=\lceil {\displaystyle \frac{k-1}{2}}\rceil .$$

In general, $lk(C_1,C_2^{\#})$ equals to the number of the arrows (marked in blue) on $C_1\cap C_2$ which are not contained in the exterior of $C_1$. These arrows are also called edge-arrows.

Thus we can calculate the linking number by counting the number of the arrows. For instance, in Figure 10iii, $lk(D_1,D_3^{\#})=1$ since the number of the arrows on $D_1\cap D_3$ that are not contained in the exterior of $D_1$ is one. Similarly, $lk(D_3,D_1^{\#})=1$. This RSF graph $R\left(\mathcal{F}\right(G\left)\right)$ as illustrated in Figure 10 is also called the reduced special face graph with edge-arrows, denoted by $\overrightarrow{\mathcal{R}}\left(G\right)$. We have the following definition for constructing $M_0$.

Definition 2.

Let $\overrightarrow{\mathcal{R}}\left(G\right)$ be an RSF graph with edge-arrows of a 2-connected plane graph G. Let C be a face boundary of $\overrightarrow{\mathcal{R}}\left(G\right)$ and let S be any subset of $E\left(\overrightarrow{\mathcal{R}}\left(G\right)\right)$. We define $C\left(S\right)$ as the number of the arrows on $S\cap E\left(C\right)$ that are not contained in the exterior of C. Let $C\left[S\right]=2C\left(S\right)-|S\cap E(C\left)\right|$. Then $C\left[S\right]$ is called the arrow-value of C restricted by S.

Note that the arrow-value of C is the difference between the number of the arrows on $S\cap E\left(C\right)$ that are not contained in the exterior of C and the number of the arrows on $S\cap E\left(C\right)$ that are contained in the exterior of C, since $C\left[S\right]=C\left(S\right)-\left(\right|S\cap E\left(C\right)|-C(S\left)\right)$. If $S=E\left(C\right)$ or $S\cap E\left(C\right)=\varnothing$, then $C\left[S\right]=0$.

Since $M_0=M-M^T$, where M denote the corresponding Seifert matrix of $F^{\prime }$ by Seifert decomposition. Let $C_1$ and $C_2$ be two bounded face boundaries of $\overrightarrow{\mathcal{R}}\left(G\right)$. Then we have

$$lk(C_1,C_2^{\#})-lk(C_2,C_1^{\#})=C_1\left[E\left(C_2\right)\right],$$

which is the element of $M_0$ corresponding to the row of $C_1$ and column of $C_2$.

Note that $M_0$ is a skew-symmetric matrix, all diagonal elements of the matrix are zeros. The order of the matrix $M_0$ is the number of the bounded face boundaries in $\overrightarrow{\mathcal{R}}\left(G\right)$. For example, as shown in Figure 10, we consider the element corresponding to $D_1$ and $D_2$. Then $D_1\left[E\left(D_2\right)\right]=1-0=1$ and $D_2\left[E\left(D_1\right)\right]=0-1=-1$. Similarly, $D_3\left(E\left(D_1\right)\right)=0=D_1\left(E\left(D_3\right)\right)$. We have the matrix $M_0$ is ${\Lambda }_1$-equivalent to

$$\left(\begin{array}{cccc}0& 1& 0& 0\\ -1& 0& 1& -1\\ 0& -1& 0& 1\\ 0& 1& -1& 0\end{array}\right)$$

Proposition 2 implies that $\partial F\left(G\right)$, the boundary of the thickened graph $F\left(G\right)$, has only one component as shown in Figure 8b, since $\det \left(M_0\right)=1$. It also tells us, if we choose the $\pi$-vertices as Figure 10iii shows, an antiparallel strong trace can be obtained according to Theorem 2.

This method for obtaining the matrix $M_0$ contains several operations on $\mathcal{F}\left(G\right)$. Since the concept of flat plays an important role in the above discussion, we name this method the flat-based operations.

4. A Theorem on 2-Connected and Plane Graphs

As a Seifert graph of $\partial F\left(G\right)$, the construction of $\mathcal{F}\left(G\right)$ is more convenient. If $\partial F\left(G\right)$ has at least two boundary components, then the determinant of the target matrix of $\mathcal{F}\left(G\right)$ is zero. This implies that $\mathcal{F}\left(G\right)$ can not be a tree and then contains a face boundary. As we know, it is also possible that $\partial F\left(G\right)$ has one component even when the $\mathcal{F}\left(G\right)$ is not a tree. In this section, we characterize a type of 2-connected plane graph whose special face graph contains at least one face boundary, which is not a tree.

Theorem 5 provides a method for finding a graph G that is 2-connected and plane such that its thickened graph contains at least two boundary components. It implies that G admits no antiparallel strong trace according to Theorem 2. In the proof of Theorem 5, we use the flat-based operations to determine the number of boundary components of the corresponding thickened graph.

Lemma 1.

Let G be a 2-connected plane graph and $\overrightarrow{\mathcal{R}}\left(G\right)$ be an RSF graph with edge-arrows of G. Let S be the edge set of a subgraph of $\overrightarrow{\mathcal{R}}\left(G\right)$ and $C_0$ be a face boundary of $\overrightarrow{\mathcal{R}}\left(G\right)$. Then $C_0[S\Delta E\left(C_0\right)]=-C_0\left[S\right]$, where $S\Delta E\left(C_0\right)$ denotes the symmetric difference between S and $E\left(C_0\right)$.

Proof. 

We have

$$C_0[S\Delta E\left(C_0\right)]=C_0[S\setminus E\left(C_0\right)]+C_0[E\left(C_0\right)\setminus S],$$

since $S\setminus E\left(C_0\right)$ and $E\left(C_0\right)\setminus S$ contain all edges in $S\Delta E\left(C_0\right)$ and have no common edges. Similarly,

$$C_0\left[S\right]=C_0[S\setminus E\left(C_0\right)]+C_0[E\left(C_0\right)\cap S].$$

Since $S\setminus E\left(C_0\right)$ and $C_0$ have no common edges, then $C_0[S\setminus E\left(C_0\right)]=0$. Thus,

$$C_0[S\Delta E\left(C_0\right)]+C_0\left[S\right]=C_0[E\left(C_0\right)\setminus S]+C_0[E\left(C_0\right)\cap S]=C_0\left[E\left(C_0\right)\right]=0.$$

Then $C_0[S\Delta E\left(C_0\right)]=-C_0\left[S\right]$. □

Theorem 5.

Let G be a 2-connected plane graph and $\overrightarrow{\mathcal{R}}\left(G\right)$ be an RSF graph with edge-arrows of G. Suppose m bounded face boundaries of $\overrightarrow{\mathcal{R}}\left(G\right)$, denoted by $C_1,C_2,\dots ,C_m$ for some integer m, and let $D={\Delta }_{i=1}^{m}E\left(C_i\right)$ be the symmetric difference among $E\left(C_1\right),E\left(C_2\right),\dots ,E\left(C_m\right)$. For any arbitrary bounded face boundary C of $\overrightarrow{\mathcal{R}}\left(G\right)$, if $C\left[D\right]=0$, then the corresponding thickened graph $F\left(G\right)$ has at least two boundary components.

Proof. 

We claim that $C\left[D\right]={\sum }_{i=1}^{m}C\left[E\left(C_i\right)\right]$. If $m=1$, then the result holds. We assume that the result holds if $m=k$ for some $k>1$. Then we prove by induction when $m=k+1$. For all $1\le i\le k$, if $E(C_i\cap C_{k+1})=\varnothing$, then we have

$$C\left[D\right]=C[D\setminus E\left(C_{k+1}\right)]+C\left[E\left(C_{k+1}\right)\right],$$

since any of the arrows attached in $C_{k+1}$ is not attached in $D\setminus E\left(C_{k+1}\right)$. By the assumption, we have

$$C[D\setminus E\left(C_{k+1}\right)]=C\left[{\Delta }_{i=1}^{k}E\left(C_i\right)\right]=\sum _{i=1}^{k}C\left[E\left(C_i\right)\right],$$

thus $C\left[D\right]={\sum }_{i=1}^{k+1}C\left[E\left(C_i\right)\right]$.

If $E(C_j\cap C_{k+1})=\{e_1,e_2,\dots ,e_{k_j}\}\ne \varnothing$ for some $1\le j\le k$. Then we discuss two cases.

Case 1.

$E\left(C\right)$ contains an edge $e_t$ for some $1\le t\le k_j$. Since $\overrightarrow{\mathcal{R}}\left(G\right)$ is simple and $e_t$ is only contained in two face boundaries, then $E\left(C\right)$ is identical to either $E\left(C_j\right)$ or $E\left(C_{k+1}\right)$. Without loss of generality, suppose $E\left(C\right)=E\left(C_j\right)$. Let $D^{\prime }={\Delta }_{1\le i\le k+1,i\ne j}E\left(C_i\right)$, then according to Lemma 1, we have

$$C\left[D\right]=-C\left[D^{\prime }\right].$$

Then by the assumption, we have

$$C\left[D^{\prime }\right]=\sum _{1\le i\le k+1,i\ne j}C\left[E\left(C_i\right)\right]=0.$$

Since $C\left[E\left(C_j\right)\right]=0$, then

$$C\left[D\right]=0=\sum _{1\le i\le k+1}C\left[E\left(C_i\right)\right].$$

Case 2.

For all $1\le i\le k$, $E\left(C\right)$ contains no edges in $C_i\cap C_{k+1}$. Then

$$\begin{array}{cc}\hfill C\left[D\right]& =C[D\setminus E\left(C_{k+1}\right)]+C[E\left(C_{k+1}\right)\cap D]\hfill \\ \hfill & =C\left[{\Delta }_{i=1}^{k}E\left(C_i\right)\right]+C[E\left(C_{k+1)}\right]\hfill \\ \hfill & =\sum _{i=1}^{k+1}C\left[E\left(C_i\right)\right].\hfill \end{array}$$

Thus, $C\left[D\right]={\sum }_{i=1}^{m}C\left[E\left(C_i\right)\right]=0$.

Let $M_0$ be the target matrix of $\overrightarrow{\mathcal{R}}\left(G\right)$, and let $\overrightarrow{R_i}$ be the row vector of $M_0$ corresponding to $C_i$$(1\le i\le m)$. Since the element of $M_0$ corresponding to $(C,C_i)$ equals to $C\left[E\left(C_i\right)\right]$, then ${\sum }_{i=1}^{m}C\left[E\left(C_i\right)\right]=0$$=-{\sum }_{i=1}^m{C}_i\left[E\left(C\right)\right])$. Then we have

$$\sum _{i=1}^m\overrightarrow{R_i}=\overrightarrow{0}.$$

Thus $\overrightarrow{R_1},\overrightarrow{R_2},\dots ,\overrightarrow{R_m}$ are linearly dependent, and the determinant of the target matrix $M_0$ is equal to 0. By Proposition 2, the thickened $F\left(G\right)$ of G has at least two boundary components. □

Example 1.

As an application of Theorem 5, we have the following example as shown in Figure 12.

Figure 12i shows the 4-prism with four $\pi$-vertices assigned, and Figure 12ii shows the RSF graph with edge-arrows of G. Let $D=E\left(C_1\right)$. As we can see, $C_2\left[D\right]=1-1=0$. Similarly, $C_3\left[D\right]=C_1\left[D\right]=0$. Then by Theorem 5, the number of the boundary components of $F\left(G\right)$ is more than one. According to Theorem 2, it also implies that the graph G as shown in Figure 12 contains no antiparallel strong trace by these $\pi$-vertices.

Example 2.

The authors Cheng et al. pointed out that their method for obtaining the thickened graph with one boundary component is by no means the only approach in [19] and they gave an example, as shown in Figure 13, where the special face graph marked in red is not a tree.

By the discussion of Section 3, we can use the flat-based operation to obtained the target matrix $M_0$ corresponding to this special face graph. Obviously, $M_0$ is ${\Lambda }_1$-equivalent to

$$\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right).$$

Then $\det \left(M_0\right)=1$. The graph as shown in Figure 13 (marked in black) must admit a thickened graph with one boundary component although its special face graph is not a tree. Then this graph admits an antiparallel strong trace. By Theorem 5 in Section 4, we know that, if it has a thickened graph with only one boundary component, then we can not find some bounded face boundaries in $\overrightarrow{\mathcal{R}}\left(G\right)$ to construct D and any bounded face boundary C of $\overrightarrow{\mathcal{R}}\left(G\right)$ such that $C\left[D\right]=0$. Figure 13 shows only two bounded face boundaries $C_1$ and $C_2$, then D can only be $E\left(C_1\right)$ or $E\left(C_2\right)$ or $E\left(C_1\right)\Delta E\left(C_2\right)$. Then any $C_1\left[D\right]$ is not equal to 0.

5. Conclusions

In this study, we extend the research framework of DNA and polypeptide cage construction based on Diao et al., focusing on developing combinatorial methods to analyze the topological properties of thickened graphs $F\left(G\right)$ associated with 2-connected plane graphs G. The core objective is to determine the number of boundary components of $F\left(G\right)$, which directly influences nanostructure design.

The main contribution is the flat-based operations, a method for obtaining the target matrix $M_0=M-M^T$. Its determinant classifies $F\left(G\right)$ topology: $\det \left(M_0\right)=1$ indicates a single boundary component (enabling single-strand cages via antiparallel strong traces), while $\det \left(M_0\right)=0$ indicates multiple components. This overcomes previous limitations that required the special face graph $\mathcal{F}\left(G\right)$ to be a tree. For example, $F\left(G\right)$ can have a single component with a non-tree $\mathcal{F}\left(G\right)$ (e.g., Figure 13).

We also characterize the thickened graph $F\left(G\right)$ with multiple components using Theorem 5: if the bounded face boundaries in $\overrightarrow{\mathcal{R}}\left(G\right)$ yield $C\left[D\right]=0$ for some symmetric difference D, then $\det \left(M_0\right)=0$. The 4-prism example (Figure 12) confirms this, with 4 $\pi$-vertices leading to 4 face boundaries and $\det \left(M_0\right)=0$.

Topologically, single-component $F\left(G\right)$ (e.g., 3-prism, Figure 8) supports single-strand cages, while multi-component structures (e.g., 4-prism) require multi-strand designs. This links algebraic topology to biomolecular engineering, allowing strand requirements to be predicted via graph theory. Functionally, single-component cages (e.g., DNA tetrahedrons) are suitable for uniform applications like drug delivery, while multi-component cages offer flexibility for complex interactions. Manipulating $\pi /\tau$-junctions in the 2-connected plane graph G enables control over $M_0$ and nanostructure architecture.

In summary, our work not only enriches the mathematical modeling of DNA and polypeptide nanostructures, but also lays a theoretical foundation for their design and synthesis. Future research may explore the potential of the flat-based operations in broader classes of graphs and their applications in biophysical contexts.